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The Review of Financial Studies
, Volume Advance Article – Apr 23, 2018

42 pages

/lp/ou_press/short-rate-expectations-and-unexpected-returns-in-treasury-bonds-Uza61X3Kzr

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
- ISSN
- 0893-9454
- eISSN
- 1465-7368
- D.O.I.
- 10.1093/rfs/hhy051
- Publisher site
- See Article on Publisher Site

Abstract I document large and persistent errors in investors’ expectations about the short-term interest rate over the business cycle. The largest errors arise in economic downturns and during Fed easings when investors overestimate future short rates and, thus, underestimate future bond returns. At a one-year horizon, errors about the path of the real rate (as opposed to inflation) account for 80% of short-rate forecast error variance, with more than half of that number attributed to the Fed easing more aggressively than the public expected. Short-rate forecast errors induce ex post predictability of excess returns on Treasury bonds that is not due to time-varying risk premium. Received June 10, 2016; editorial decision February 1, 2018 by Editor Robin Greenwood. The author has furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. Separating short-rate expectations from risk premiums is key for understanding the economic determinants of interest rates and, thus, for inferring investors’ expectations about the economy and risks they perceive. Its importance was summarized by Governor Donald L. Kohn, then-Governor of the Federal Reserve Board, on July 21, 2005: Investors’ expectations are reflected in asset prices, but so are risk premiums, and inferences about future economic conditions obtained from market prices are conditional on estimates of those premiums. Neglecting or grossly misestimating risk premiums will lead to misperceptions of the market’s outlook and thus potentially to market moves that we did not anticipate. (...) To what extent are long-term interest rates low because investors expect short-term rates to be low in the future (...), and to what extent do low long rates reflect narrow term premiums (...)? Clearly, the policy implications of these two alternative explanations are very different. In this paper, I highlight how errors in short-rate expectations can challenge the usual way we perform the decomposition of interest rates into risk premiums and short-rate expectations. A common approach to estimating risk premiums in Treasury bonds relies on the assumption of full-information rational expectations (FIRE). As such, one can use predictive regressions of bond excess returns on various conditioning variables as a way to measure the risk-premium variation, that is, the compensation that investors required to be willing to hold Treasuries at different moments in time. I argue that the frictionless view of short-rate expectations that underlies this approach is inconsistent with the properties of actual expectations of investors and survey forecasters. I document large and persistent errors in the way investors and professional forecasters form expectations about the future short rate over the business cycle. Rather than pursuing a risk-premium interpretation, I show that predictive regressions of bond excess returns estimated with real-activity variables as predictors forecast changes in the short rate that investors in real time did not expect to occur. Using survey forecasts of the federal funds rate (FFR) going back to the 1980s, I study the properties of short-rate forecast errors. Ex post short-rate forecast errors are defined as FFR observed at time $$t$$ minus its survey expectation at time $$t-h$$, where $$h$$ is the forecast horizon. Figure 1 illustrates motivation for my main analysis. The figure superimposes FFR forecast errors at the four-quarter horizon against contemporaneous survey forecast errors of unemployment (panel A), and against realized annual excess returns on a two-year Treasury bond (panel B). FFR forecast errors display a systematic pattern over the business cycle. They become strongly negative during and after NBER-dated recessions when forecasters underestimate the extent of monetary easing. They also comove negatively with forecast errors of unemployment, suggesting that economic downturns were largely unexpected a year before they happened. Unexpected short-rate declines translate into large positive realized returns on Treasury bonds (panel B of Figure 1), and are thus good news for bond investors. Figure 1. View largeDownload slide Forecast errors of FFR and bond excess returns Forecast errors of FFR are superimposed against contemporaneous forecast errors of unemployment, both at the four-quarter horizon (panel A) and against contemporaneous annual excess return on a two-year Treasury bond (panel B). Forecast error is defined as FFR realized at time $$t$$ minus its survey expectation at time $$t-h$$, $$FFR_{t}-E^s_{t-h}(FFR_t)$$, $$h=4$$ quarters, and analogously for unemployment. Expectations of FFR are from the Blue Chip Financial Forecasts, and expectations of unemployment are from the Survey of Professional Forecasters. Figure 1. View largeDownload slide Forecast errors of FFR and bond excess returns Forecast errors of FFR are superimposed against contemporaneous forecast errors of unemployment, both at the four-quarter horizon (panel A) and against contemporaneous annual excess return on a two-year Treasury bond (panel B). Forecast error is defined as FFR realized at time $$t$$ minus its survey expectation at time $$t-h$$, $$FFR_{t}-E^s_{t-h}(FFR_t)$$, $$h=4$$ quarters, and analogously for unemployment. Expectations of FFR are from the Blue Chip Financial Forecasts, and expectations of unemployment are from the Survey of Professional Forecasters. I show that real-time expectations of the FFR are consistent with forecasters overextrapolating the current short rate, while ignoring the predictive information that past real activity contains about the short rate going forward. An econometrician who understands this relationship can ex post predict short-rate forecast errors of real-time forecasters. I construct a measure of a gap between the information set of econometrician and investors—an expectations wedge—and show that it has a persistent and cyclical pattern over the business cycle, widening in economic downturns. Approaching recessions, real-time forecasters expect the next year’s short-rate level to be more than 200 basis points (bps) higher than what an econometrician would predict using lagged variables. The wedge between the short-rate expectations of real-time forecasters and the econometrician predicts realized bond excess returns with a declining strength across maturities. I show that ex post predictability of short-rate forecast errors naturally leads to this pattern as a consequence of short-rate mean reversion. I estimate that a 100 bps unexpected (from investors’ perspective) decline in the short rate over a four-quarter period translates into about 80–85 bps unexpected decline in the two-year yield, 48–55 (20–40) bps decline in the five-year (ten-year) yield, depending on the estimation approach and sample period. I obtain these estimates using survey measures of forecast errors in longer-maturity yields as well as a dynamic model of short-rate expectations. Seeking to understand the economic determinants of the size and persistence of short-rate forecast errors, I find that only 18% of their variance is driven by inflation shocks and 82% by real-rate shocks (of which about 26% can be explained by unemployment shocks, and the remaining 56% by other shocks, most importantly monetary policy). Accordingly, evidence of persistent errors in short-rate expectations can also be detected in monetary policy shocks extracted from interest rate futures at the frequency of Federal Open Market Committee (FOMC) meetings. These shocks capture innovations relative to the information set of investors just prior to the FOMC meeting, and under the FIRE assumption, they should not be predictable. In practice, they are in fact strongly predictable by measures of real activity. This suggests that from 1980s, the Fed has reacted to economic downturns by easing more aggressively than expected by the public. At the same time, I do not find evidence that Fed officials are better able than the public to predict the path of the economy and the associated path of the short rate, except a couple of quarters ahead. Ex post predictability of forecast errors does not imply that people make “obvious” mistakes that could be easily fixed in real time. Even when conducting a quasi-real-time estimation, an econometrician uses ex post knowledge of a statistical relationship that would have been much harder to uncover in real time. I support this interpretation with narrative evidence from the transcripts of the FOMC meetings that describe the challenges of real-time forecasting. Although forecasters at the Fed realized that their past forecasts were repeatedly off target in a particular direction, they were struggling to correct these errors in real time. I document that actual short-rate forecasts produced by the Fed display properties similar to those of the private sector. My findings have important implications for the way we interpret the estimates of the Treasury risk premiums. Recent empirical literature documents an incremental predictive power for bond excess returns of various auxiliary predictors in additional to yields (see Duffee 2012 for review). I refer to this feature of the data as “excess” predictability, that is, predictability achieved with variables other than current yields. Excess predictability is surprising in that today’s yield curve reflects investors’ expectations of future short rates and of future excess returns, and thus should already subsume all information necessary to estimate bond risk premiums. So why doesn’t the yield curve summarize all predictive information about future Treasury bond returns? Potential explanations fall into three categories that differ by assumptions they require about the information set of investors versus econometrician. First, there can be hidden or unspanned factors along the lines of the models in Duffee (2011) and in Joslin, Priebsch, and Singleton (2014). These models assume that the information set of econometrician equals that of investors. As such, a variable can forecast future returns without being revealed by current yields only when it affects short-rate expectations and term premiums in an exactly offsetting manner at each maturity. The second possibility is that current yields do in fact contain all relevant information about risk premium required by investors but this information is obscured by measurement noise (Cochrane and Piazzesi 2005; Duffee 2011). Cieslak and Povala (2015) use this argument to justify the predictive power of inflation expectations for bond excess returns. They show that, while expected inflation is a key driver of the level of nominal interest rates and is spanned by current yields, augmenting yields-only predictive regressions with a proxy for investors’ inflation expectations helps uncover bond risk premium variation when measurement error is present. In this case, the information set of an econometrician and of investors differ only by noise. The third possibility, and the focus of the current paper, is that there is a systematic gap between the actual information set of investors and that assumed by an econometrician. Interest rate dynamics as perceived by real-time investors may persistently differ from the dynamics embedded in data samples that researchers examine. Investors do not impound some information into prices because they do not recognize it to be relevant at the time they form their expectations. Those different hypotheses are not mutually exclusive in that the explanation for excess predictability depends on the particular variables that the econometrician assumes as part of his/her information set. I highlight the different properties of the estimated bond risk premiums when real and nominal variables are used as predictors in addition to yields. Expected inflation helps forecast excess bond returns across the full range of maturities, while the significance of real variables is located at short maturities and dissipates as the maturity increases. Conditioning on expected inflation produces bond risk premium estimates that vary at a frequency higher than the business cycle, a feature supported by survey-based risk-premium estimates. Instead, conditioning on real-activity measures implies that fitted risk premiums at short maturities are strongly countercyclical. The traditional risk-premium interpretation of those estimates would therefore necessitate that investors require a particularly high compensation for holding short-term Treasury bonds in recession, countering the perception of such bonds as safe assets. I argue that the predictive power of real-activity variables for bond returns found in the literature derives from errors in short-rate expectations. Survey data have been used to study expectations formation in financial markets, for example, foreign exchange (Frankel and Froot 1987), bonds, and stocks (Froot 1989; Bacchetta, Mertens, and van Wincoop 2009). This research shows that forecast errors are predictable with past information. Nagel (2012) and Greenwood and Shleifer (2013) document investors’ tendency to extrapolate past asset returns. Using survey expectations of interest rates, Piazzesi, Salomao, and Schneider (2015) argue that bond risk premiums implied by survey forecasts of longer-maturity yields are more persistent and less volatile than those obtained with statistical approaches such as the Cochrane-Piazzesi regressions. Relatedly, Cieslak and Povala (2015) show that survey expectations of longer-maturity yields provide a more noisy measure of the bond risk premium than the risk premium constructed in real time using yields-plus-expected-inflation regressions. My work complements those studies with a different focus that lies in identifying expectational errors in short-rate expectations. I study the properties and sources of these errors, assess their effect on bond return predictability across the maturity structure, decompose them into inflation and real-rate components, and link them to monetary policy shocks. I argue that short-rate expectations of survey participants and investors closely trace each other, as evidenced by comparing survey forecasts of the FFR with the Fed fund futures. I show that the pattern of short-rate forecast errors could be predicted ex post with proxies of real activity that forecasters had access to in real time. A growing body of research in finance and macroeconomics emphasizes the role of expectations formation by relaxing the FIRE assumption (see Mankiw and Reis 2011; Woodford 2013 for an overview). A large theoretical literature studies nominal expectations frictions and their implications for monetary policy (e.g., Orphanides and Williams 2005; Woodford 2010). Using survey data, Coibion and Gorodnichenko (2012, 2015) provide empirical support for such frictions. Angeletos and La’O (2012) introduce real-side imperfections. Empirical evidence for the real-side frictions is relatively less developed. One exception is the recent work by Greenwood and Hanson (2015), who show that expectational errors of firms lead to capital overinvestment, causing investment boom and bust cycles and excess volatility in prices that cannot be rationalized as a risk premium. In a related way, my results point to significant errors in expectations about the real-rate dynamics, which is the key input in savings and investment decisions. 1. Data 1.1 Survey forecasts To measure short-rate expectations, I use forecasts of the FFR from the Blue Chip Financial Forecasts (BCFF) survey. It is the longest consistently compiled monthly survey of the FFR forecasts.1 From BCFF, I also obtain forecasts of longer-maturity yields. Consistent time series of forecasts back to the 1980s can be constructed for horizons up to four quarters ahead. Internet Appendix A contains details about the BCFF survey. I also use survey forecasts of inflation and unemployment. Forecasts of CPI inflation (seasonally adjusted annualized rate of change in total CPI) are available monthly from the BCFF. Unemployment forecasts are from the Survey of Professional Forecasters (SPF), as unemployment is not part of the BCFF. SPF reports forecasts for quarterly average seasonally adjusted unemployment rate, and is released in the middle month of each quarter. Both inflation and unemployment forecasts are available up to (at least) four quarters ahead, consistent with the horizons of FFR forecasts. I use the median of individual forecasts, but the results are essentially unchanged if the mean forecast is used instead. 1.2 Macro data The daily effective FFR is from the FRED database at the St. Louis Fed. For quarterly (monthly) sampling, I average the daily FFR within a quarter (month). To make sure that my main predictive results are not affected by revisions to macro series or by publication lags, I use monthly vintages of nonfarm payroll employment from the real-time macroeconomic database at the Philadelphia Fed. I construct an annual (year-on-year) employment growth rate $$\Delta \textit{EMP}_{t-1,t}=100\left(\frac{\textit{EMP}_t}{\textit{EMP}_{t-1}}-1\right)$$ using the vintage available in month $$t$$ rather than employment realized in month $$t$$.2 From FRED, I also obtain the unemployment rate, Chicago Fed National Activity Index (CFNAI), and core and total CPI. I denote the year-on-year inflation rate of core CPI as $$\Delta \textit{CPI}^c_{t-1,t}=100\left(\frac{\textit{CPI}_t^c}{\textit{CPI}^c_{t-1}}-1\right)$$ and analogously for total CPI inflation, $$\Delta CPI_{t-1,t}$$.3 1.3 Interest rate data I use zero-coupon nominal Treasury yields from Gürkaynak Sack, and Wright (2006) available on the Federal Reserve Board’s (FRB) Web site. The data are sampled at the end of month. FFR and interest rates are expressed in percent per annum. 1.4 Sample period The BCFF forecasts of the FFR are available since March 1983, forecasts of inflation—since June 1984, and forecasts of Treasury yields with maturities of one, two, three, and ten years—since January 1988. The main empirical analysis is based on data from June 1984 through August 2012, a period when the FFR was the main operating target of the Fed. The sample covers 327 months (109 quarters): The first annual bond return (and the four-quarter-ahead forecast error) is realized in June 1985 conditioning on data from June 1984, the last one is realized in August 2012 conditioning on data from August 2011. The start of the sample is determined by the availability of the CPI inflation forecasts in the BCFF. Whenever I rely on survey forecasts of Treasury yields other than the FFR, the sample covers 284 months beginning in January 1988 when these forecasts are first released. The end of the sample is when the zero lower bound becomes binding for FFR expectations.4 2. Bond Risk Premia and Bond Return Predictability 2.1 Yield curve identities Assume interest rates are continuously compounded and expressed in percent per period. One-period nominal short rate is denoted with $$i_t$$. The realized $$h$$-period excess return on a zero-coupon bond with $$n$$ periods to maturity is $$rx^{(n)}_{t+h} = n y_t^{(n)}-(n-h) y_{t+h}^{(n-h)}-h y_t^{(h)}$$, where $$y_t^{(n)}$$ is a $$n$$-period yield. Rearranging the definition of one-period return on a two-period bond, $$rx_{t+1}^{(2)} = -i_{t+1} + 2y_{t}^{(2)}-i_{t}$$, the two-period yield is \begin{equation} y_{t}^{(2)}=\frac{1}{2}\left( i_{t} +i_{t+1}\right) +\frac{1}{2}rx_{t+1}^{(2)}. \end{equation} (1) Since (1) holds ex post realization-by-realization it also holds ex ante in expectations: \begin{equation} y_{t}^{(2)}=\frac{1}{2}E_{t}\left( i_{t}+ i_{t+1}\right) +\frac{1}{2}E_{t}\left( rx_{t+1}^{(2)}\right). \end{equation} (2) By iterative argument, one obtains an expression for the $$n$$-period yield: \begin{equation} y_{t}^{(n)}= {\frac{1}{n}E_t\left(\sum_{k=0}^{n-1} i_{t+k} \right)} + \underbrace{\frac{1}{n}E_t\left( \sum_{k=0}^{n-2}rx_{t+k+1}^{(n-k)}\right)}_{\text{term premium, } tp_t^{(n)}}. \end{equation} (3) The current $$n$$-period yield is a sum of investors’ expectations about the average short rate (the expectations hypothesis component) and average expected excess returns to be earned over the life of the bond (the term premium component). Suppose that all information used by investors to forecast future short rates and excess returns is summarized in a vector $$x_t$$. Since yields are conditional expectations, the current yield curve can be fully described as a function of $$x_t$$, $${y}_t={f}(x_t,J)$$, where $$y_t$$ is a vector of yields with $$J$$ different maturities observed at time $$t$$. If the mapping $$f(\cdot)$$ is invertible, the current yield curve reflects the state vector used by investors to form expectations. Therefore, as long as investors’ expectations are FIRE, the yield curve contains all information relevant for forecasting future yields and returns. This argument implies that there is no immediate reason to use variables other than current yields for forecasting future yields or bond returns, a point emphasized by Cochrane and Piazzesi (2005) and by Duffee (2011). I first review the empirical evidence in the literature documenting predictability of bond excess returns with variables other than current yields. Then I discuss possible explanations for excess predictability. 2.2 Overview of empirical evidence on bond return predictability The usual approach to measuring the risk premium variation in financial assets is through predictive regressions. A general specification that embeds different regressions estimated in the bond return predictability literature is: \begin{equation} rx_{t+1}^{(n)} = \alpha + \gamma_1'\text{yields}_{t} + \gamma_2'V_t + \varepsilon_{t+1}, \end{equation} (4) where $$rx_{t+1}^{(n)}$$ is an annual holding period excess return on an $$n$$-year zero-coupon bond, and $$V_t$$ contains auxiliary predictors. Cochrane and Piazzesi (2005) project excess returns on a set of current forward rates, which is equivalent to projections on yields or the principal components (PCs) of yields (i.e., $$\gamma_2=0$$). Subsequent authors augment the yields-only specification with auxiliary variables including a range of financial and realized macro variables (Ludvigson and Ng 2009), output gap (Cooper and Priestley 2009), proxies for long-horizon inflation expectations or trend inflation (Cieslak and Povala 2015), CFNAI and four-quarter-ahead expected inflation from the BCFF survey (Joslin, Priebsch, and Singleton 2014). Those auxiliary variables are found to contain predictive information about future returns in addition to the information in current yields. To review this evidence, I estimate predictive regressions of annual bond excess returns on current yields and additional predictors. Auxiliary predictors are selected to summarize the findings of previous studies. In regression (4), time and maturities are measured in years. As common in the literature, I predict bond returns with an annual holding period using data sampled at the monthly frequency, so each month I forecast excess return over the next twelve months. I report Newey and West (1987),t-statistics with 18 lags as well as more conservative reverse-regression t-statistics (Hodrick 1992). Table 1 presents the results. Panel A contains yields-only regressions using the first three principal components (PCs) of yields.5 The remaining panels extend this baseline specification by including, in addition to yield PCs, auxiliary regressors one at a time: $$CFNAI_t$$,6 year-on-year growth in employment $$(\Delta {EMP}_{t-1,t})$$, BCFF survey forecast of CPI inflation four quarters ahead $$(E_t^s(\Delta {CPI}_{t,t+1}))$$, and trend inflation $$(\tau_t^{CPI})$$.7 The first two regressors measure real economic activity: $$CFNAI_t$$ is a broader measure but it is revised, whereas $$\Delta EMP_{t-1,t}$$ is entirely real time.8 The two nominal variables aim to reflect real-time inflation expectations of investors. Table 1 Predictive regressions of bond returns with auxiliary variables (1) (2) (3) (4) (5) $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=20Y$$ $$rx^{S}$$ A. Only yield PCs: $$rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \varepsilon_{t+1}$$ SD $$\hat{rx}/n\, (\%)$$ 0.33 0.45 0.46 0.45 0.25 $$\bar{R}^2$$ 0.21 0.21 0.24 0.26 0.17 B. Chicago Fed National Activity Index, $$V_t=CFNAI_t$$ $$CFNAI_t$$ –0.24 –0.18 –0.038 0.064 –0.27 (–3.25) (–1.96) (–0.53) (1.05) (–4.50) [–2.81] [–1.50] [–0.42] [0.64] [–3.70] $$\bar{R}^2$$ 0.31 0.24 0.24 0.26 0.35 C. Year-on-year growth in employment, $$V_t =\Delta EMP_{t-1,t}$$ $$\Delta EMP_{t-1,t}$$ –0.45 –0.35 –0.031 0.19 –0.53 (–3.14) (–1.88) (–0.18) (1.33) (–4.14) [–2.58] [–1.45] [–0.28] [0.79] [–3.31] $$\bar{R}^2$$ 0.31 0.24 0.24 0.27 0.38 D. Four-quarter-ahead inflation forecast from BCFF, $$V_t=E_t^s{ (\Delta CPI_{t,t+1})}$$ $$E_t^s(\Delta CPI_{t,t+1})$$ –0.63 –0.92 –0.95 –1.03 –0.15 (–2.19) (–2.55) (–2.93) (–3.24) (–0.58) [–2.09] [–2.27] [–2.22] [–2.08] [–0.67] $$\bar{R}^2$$ 0.29 0.30 0.35 0.41 0.17 E. Trend inflation, $$V_t= \tau_t^{CPI}$$ $$\tau_t^{CPI}$$ –0.81 –1.39 –1.60 –1.59 –0.080 (–3.64) (–4.82) (–6.31) (–7.68) (–0.44) [–2.34] [–3.25] [–4.28] [–4.59] [0.014] $$\bar{R}^2$$ 0.32 0.40 0.52 0.57 0.17 N (months) 327 327 327 327 327 (1) (2) (3) (4) (5) $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=20Y$$ $$rx^{S}$$ A. Only yield PCs: $$rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \varepsilon_{t+1}$$ SD $$\hat{rx}/n\, (\%)$$ 0.33 0.45 0.46 0.45 0.25 $$\bar{R}^2$$ 0.21 0.21 0.24 0.26 0.17 B. Chicago Fed National Activity Index, $$V_t=CFNAI_t$$ $$CFNAI_t$$ –0.24 –0.18 –0.038 0.064 –0.27 (–3.25) (–1.96) (–0.53) (1.05) (–4.50) [–2.81] [–1.50] [–0.42] [0.64] [–3.70] $$\bar{R}^2$$ 0.31 0.24 0.24 0.26 0.35 C. Year-on-year growth in employment, $$V_t =\Delta EMP_{t-1,t}$$ $$\Delta EMP_{t-1,t}$$ –0.45 –0.35 –0.031 0.19 –0.53 (–3.14) (–1.88) (–0.18) (1.33) (–4.14) [–2.58] [–1.45] [–0.28] [0.79] [–3.31] $$\bar{R}^2$$ 0.31 0.24 0.24 0.27 0.38 D. Four-quarter-ahead inflation forecast from BCFF, $$V_t=E_t^s{ (\Delta CPI_{t,t+1})}$$ $$E_t^s(\Delta CPI_{t,t+1})$$ –0.63 –0.92 –0.95 –1.03 –0.15 (–2.19) (–2.55) (–2.93) (–3.24) (–0.58) [–2.09] [–2.27] [–2.22] [–2.08] [–0.67] $$\bar{R}^2$$ 0.29 0.30 0.35 0.41 0.17 E. Trend inflation, $$V_t= \tau_t^{CPI}$$ $$\tau_t^{CPI}$$ –0.81 –1.39 –1.60 –1.59 –0.080 (–3.64) (–4.82) (–6.31) (–7.68) (–0.44) [–2.34] [–3.25] [–4.28] [–4.59] [0.014] $$\bar{R}^2$$ 0.32 0.40 0.52 0.57 0.17 N (months) 327 327 327 327 327 The table presents regressions of annual bond excess returns on three yield PCs ($$PC_t$$) and an auxiliary regressor $$V_t$$. The regression is specified as \[ rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \gamma_2 V_t + \varepsilon_{t+1}, \] where the dependent variable is excess return for an annual holding period on a $$n$$-year bond. Regressions are estimated at monthly frequency, so each month I forecast excess returns earned over the following twelve months. For comparison of coefficients across maturities, excess returns are duration-standardized, $$rx_{t+1}^{(n)}/n$$, and explanatory variables are standardized to have unit standard deviation. $$rx^{S}_{t+1}$$ in the last column is the short-maturity factor defined as a residual from regressing annual excess return on a two-year bond onto one-year excess return on a twenty-year bond, both duration standardized. Panel A uses only PCs as regressors. The row “SD of $$\widehat{rx}/n\, (\%)$$” reports the unconditional standard deviation of the fitted value from the regression. Panels B–E show results for different variables $$V_t$$, and only $$\gamma_2$$ coefficient is reported. The data covers the period 1984M6–2012M8 for a total of 327 months (first annual return is realized in June 1985 and last in August 2012). t-statistics are reported for two types of standard errors: (1) Newey-West adjustment with 18 lags (in parentheses) and (2) based on Hodrick’s (1992) reverse regressions (in brackets). Hodrick’s correction relies on predicting monthly returns with annual averages of predictors rather than predicting annual overlapping returns with month-$$t$$ value of a predictor. I follow the delta method implementation of reverse regressions by Wei and Wright (2013). Table 1 Predictive regressions of bond returns with auxiliary variables (1) (2) (3) (4) (5) $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=20Y$$ $$rx^{S}$$ A. Only yield PCs: $$rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \varepsilon_{t+1}$$ SD $$\hat{rx}/n\, (\%)$$ 0.33 0.45 0.46 0.45 0.25 $$\bar{R}^2$$ 0.21 0.21 0.24 0.26 0.17 B. Chicago Fed National Activity Index, $$V_t=CFNAI_t$$ $$CFNAI_t$$ –0.24 –0.18 –0.038 0.064 –0.27 (–3.25) (–1.96) (–0.53) (1.05) (–4.50) [–2.81] [–1.50] [–0.42] [0.64] [–3.70] $$\bar{R}^2$$ 0.31 0.24 0.24 0.26 0.35 C. Year-on-year growth in employment, $$V_t =\Delta EMP_{t-1,t}$$ $$\Delta EMP_{t-1,t}$$ –0.45 –0.35 –0.031 0.19 –0.53 (–3.14) (–1.88) (–0.18) (1.33) (–4.14) [–2.58] [–1.45] [–0.28] [0.79] [–3.31] $$\bar{R}^2$$ 0.31 0.24 0.24 0.27 0.38 D. Four-quarter-ahead inflation forecast from BCFF, $$V_t=E_t^s{ (\Delta CPI_{t,t+1})}$$ $$E_t^s(\Delta CPI_{t,t+1})$$ –0.63 –0.92 –0.95 –1.03 –0.15 (–2.19) (–2.55) (–2.93) (–3.24) (–0.58) [–2.09] [–2.27] [–2.22] [–2.08] [–0.67] $$\bar{R}^2$$ 0.29 0.30 0.35 0.41 0.17 E. Trend inflation, $$V_t= \tau_t^{CPI}$$ $$\tau_t^{CPI}$$ –0.81 –1.39 –1.60 –1.59 –0.080 (–3.64) (–4.82) (–6.31) (–7.68) (–0.44) [–2.34] [–3.25] [–4.28] [–4.59] [0.014] $$\bar{R}^2$$ 0.32 0.40 0.52 0.57 0.17 N (months) 327 327 327 327 327 (1) (2) (3) (4) (5) $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=20Y$$ $$rx^{S}$$ A. Only yield PCs: $$rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \varepsilon_{t+1}$$ SD $$\hat{rx}/n\, (\%)$$ 0.33 0.45 0.46 0.45 0.25 $$\bar{R}^2$$ 0.21 0.21 0.24 0.26 0.17 B. Chicago Fed National Activity Index, $$V_t=CFNAI_t$$ $$CFNAI_t$$ –0.24 –0.18 –0.038 0.064 –0.27 (–3.25) (–1.96) (–0.53) (1.05) (–4.50) [–2.81] [–1.50] [–0.42] [0.64] [–3.70] $$\bar{R}^2$$ 0.31 0.24 0.24 0.26 0.35 C. Year-on-year growth in employment, $$V_t =\Delta EMP_{t-1,t}$$ $$\Delta EMP_{t-1,t}$$ –0.45 –0.35 –0.031 0.19 –0.53 (–3.14) (–1.88) (–0.18) (1.33) (–4.14) [–2.58] [–1.45] [–0.28] [0.79] [–3.31] $$\bar{R}^2$$ 0.31 0.24 0.24 0.27 0.38 D. Four-quarter-ahead inflation forecast from BCFF, $$V_t=E_t^s{ (\Delta CPI_{t,t+1})}$$ $$E_t^s(\Delta CPI_{t,t+1})$$ –0.63 –0.92 –0.95 –1.03 –0.15 (–2.19) (–2.55) (–2.93) (–3.24) (–0.58) [–2.09] [–2.27] [–2.22] [–2.08] [–0.67] $$\bar{R}^2$$ 0.29 0.30 0.35 0.41 0.17 E. Trend inflation, $$V_t= \tau_t^{CPI}$$ $$\tau_t^{CPI}$$ –0.81 –1.39 –1.60 –1.59 –0.080 (–3.64) (–4.82) (–6.31) (–7.68) (–0.44) [–2.34] [–3.25] [–4.28] [–4.59] [0.014] $$\bar{R}^2$$ 0.32 0.40 0.52 0.57 0.17 N (months) 327 327 327 327 327 The table presents regressions of annual bond excess returns on three yield PCs ($$PC_t$$) and an auxiliary regressor $$V_t$$. The regression is specified as \[ rx_{t+1}^{(n)}/n = \gamma_0 + \gamma_1'PC_t + \gamma_2 V_t + \varepsilon_{t+1}, \] where the dependent variable is excess return for an annual holding period on a $$n$$-year bond. Regressions are estimated at monthly frequency, so each month I forecast excess returns earned over the following twelve months. For comparison of coefficients across maturities, excess returns are duration-standardized, $$rx_{t+1}^{(n)}/n$$, and explanatory variables are standardized to have unit standard deviation. $$rx^{S}_{t+1}$$ in the last column is the short-maturity factor defined as a residual from regressing annual excess return on a two-year bond onto one-year excess return on a twenty-year bond, both duration standardized. Panel A uses only PCs as regressors. The row “SD of $$\widehat{rx}/n\, (\%)$$” reports the unconditional standard deviation of the fitted value from the regression. Panels B–E show results for different variables $$V_t$$, and only $$\gamma_2$$ coefficient is reported. The data covers the period 1984M6–2012M8 for a total of 327 months (first annual return is realized in June 1985 and last in August 2012). t-statistics are reported for two types of standard errors: (1) Newey-West adjustment with 18 lags (in parentheses) and (2) based on Hodrick’s (1992) reverse regressions (in brackets). Hodrick’s correction relies on predicting monthly returns with annual averages of predictors rather than predicting annual overlapping returns with month-$$t$$ value of a predictor. I follow the delta method implementation of reverse regressions by Wei and Wright (2013). Columns 1–4 of Table 1 contain estimates for excess returns on bonds with maturities of two, five, ten, and twenty years. For comparison of coefficients across maturities, excess returns are standardized by bond duration, $$rx_{t+1}^{(n)}/n$$, and predictors are standardized to have a unit standard deviation. While each of the auxiliary predictors contains information about future returns beyond current yields, two distinct predictability patterns emerge. Real-activity measures are strongly significant at shorter maturities, increasing predictive $$R^2$$ by more than 10% relative to the PCs-only regressions at the two-year maturity. However, their predictive power declines with maturity, losing significance at maturities beyond five years. In contrast, expected inflation proxies add to the explained variation of excess returns both at short and at long maturities, and their significance generally increases with maturity. To summarize the factor structure in realized bond excess returns and the distinct predictive power of real and nominal variables, I construct a short-end return factor, denoted $$rx_{t+1}^{S}$$, as a residual from regressing annual excess return on a two-year bond onto annual excess return on a twenty-year bond, both duration-standardized: $$rx_{t+1}^{S} \equiv rx_{t+1}^{(2)}/2 - \hat a - \hat b \cdot rx_{t+1}^{(20)}/20$$ where $$\hat a, \hat b$$ are OLS regression coefficients. Based on the intuition that bond risk premiums move on a single factor (Cochrane and Piazzesi 2005), and that long-term bonds should be more exposed to risk premium shocks than short-term bonds, the short-end return factor $$rx_{t+1}^{S}$$ aims to capture the independent variation in realized excess returns at short maturities.9 The last column of Table 1 illustrates the differences among auxiliary predictors in their ability to forecast $$rx_{t+1}^{S}$$. Measures of expected inflation are insignificant, but the significance of the real-activity predictors increases compared to the excess return on the two-year bond in Column 1. This implies that real-activity variables uncover a predictable element of bond excess returns at short maturity that is uncorrelated with sources of return predictability at the long end of the yield curve. In particular, estimates with real-activity proxies imply risk premiums at the short end of the yield curve that are more countercyclical than those estimated with either the yields-only or yields-plus-expected-inflation regressions. To illustrate the different implications of real and nominal predictors, Figure 2 superimposes the risk premium estimates for a two-year bond from panels C and D of Table 1. Estimates obtained with yields plus employment growth generate risk premiums that rise around recessions, while estimates using yields plus expected inflation vary at a frequency higher than the business cycle. Using the fitted value from regression with employment growth as predictor, one would conclude that in nine months after Lehman Brothers’ collapse, investors required more than 100 bps as compensation to hold a two-year Treasury bond (average fitted excess return from October 2008 through June 2009). An analogous number based on regression with expected inflation as predictor would be 8 bps. It is a priori unclear why short-term bonds would earn a particularly high risk premium in recessions. Next, I turn to an alternative interpretation of this evidence. Figure 2. View largeDownload slide Comparison of fitted bond risk premiums on a two-year bond The figure shows the fitted bond risk premium on a two-year bond from two regressions using: (1) three PCs and annual change in unemployment as regressors and (2) three PCs and expected inflation as regressors. These regressions correspond to panels C and D in Table 1, Column 1. Figure 2. View largeDownload slide Comparison of fitted bond risk premiums on a two-year bond The figure shows the fitted bond risk premium on a two-year bond from two regressions using: (1) three PCs and annual change in unemployment as regressors and (2) three PCs and expected inflation as regressors. These regressions correspond to panels C and D in Table 1, Column 1. 2.3 Unexpected returns and short-rate forecast errors Rather than from the risk-premium variation, some of the predictive evidence can stem from differences between the information sets of investors and the econometrician. I argue that these differences are important for understanding why measures of real activity predict bond excess returns, and why this predictability is mostly present at short maturities. In short, I connect this result to forecast errors that investors make about the evolution of the short rate, and that turn out to be predictable ex post. The yield curve Equation (3) is a tautology and is economically void unless one makes an assumption about expectations $$E_t(\cdot)$$. In particular, given current yield $$y_t^{(n)}$$, this equation holds for any model of expectations formation and conditioning information set, as long as it contains $$y_t^{(n)}$$. Since yields reflect real-time expectations of investors, a predictor can have no effect on the current yield curve if the econometrician uses a variable that is not included in the time-$$t$$ information set of investors. Such a variable can, however, predict investors’ forecast errors. A more nuanced possibility is that investors have access to the particular variable at time $$t$$ but either deem it unimportant for the yield curve or the relationship with the yield curve is difficult to estimate in real time. We want to link the excess return that investors earn on their bond investment to their expectations about future short rates. Write an $$h$$-period return on an $$n$$-period bond as \begin{align} rx_{t+h}^{(n)} = -(n-h)\left[y_{t+h}^{(n-h)}-y_t^{(n-h)}\right] + h\left[f_t^{(n-h,h)} - y_t^{(h)}\right], \end{align} (5) where the $$f_t^{(n-h,h)}$$ is the $$h$$-period forward rate between time $$t+n-h$$ and time $$t+n$$. Unexpected returns on bond investments are perfectly negatively correlated with yield forecast errors; that is, if yields decline more than expected, investors earn unexpectedly high returns: \begin{align} rx_{t+h}^{(n)} - E_t(rx_{t+h}^{(n)}) = -(n-h)\left[y_{t+h}^{(n-h)}-E_t(y_{t+h}^{(n-h)})\right]. \end{align} (6) From Equation (3), an unexpected change in the $$n$$-period yield can be decomposed as \begin{align} y_{t+h}^{(n)} - E_t(y_{t+h}^{(n)}) &= \frac{1}{n} \sum_{k=0}^{n-1} \left(E_{t+h} \left(i_{t+h+k}\right) - E_{t} \left(i_{t+h+k}\right) \right) + (tp_{t+h}^{(n)} - E_t(tp_{t+h}^{(n)})) \nonumber \\ & = (i_{t+h}-E_t(i_{t+h})) + \sum_{k=1}^{n-1} \frac{n-k}{n}(E_{t+h}-E_t)(\Delta i_{t+h+k})\notag\\ &\quad{}+ (tp_{t+h}^{(n)} - E_t(tp_{t+h}^{(n)})) , \end{align} (7) where $$(i_{t+h}-E_t(i_{t+h}))$$ is the forecast error about the short rate between $$t$$ and $$t+h$$, $$\Delta i_{t+h+k}$$ is a future one-period change in the short rate beyond time $$t+h$$, $$\Delta i_{t+h+k}= i_{t+h+k} - i_{t+h+k-1}$$, and $$(E_{t+h}-E_t)(\Delta i_{t+h+k})$$ denotes an expectations update between time $$t$$ and $$t+h$$; $$(tp_{t+h}^{(n)} - E_t(tp_{t+h}^{(n)}))$$ is the unexpected change in the term premium from $$t$$ to $$t+h$$. How much of variation in yield forecast errors at $$n$$-maturity and, by extension, in unexpected bond returns can be justified by the short-rate forecast errors alone? With a mean-reverting short rate, the first and second term in (7) are negatively correlated, and thus, the contribution of $$(i_{t+h}-E_t(i_{t+h}))$$ to the variation $$(y^{(n)}_{t+h}-E_t(y^{(n)}_{t+h}))$$ declines with $$n$$.10 If indeed short-rate forecast errors are ex post predictable, predictability of bond excess returns induced by this fact should be most visible at short maturities and decay as maturity increases. To empirically evaluate the expectations errors story, I first analyze the short-rate expectations of survey forecasters and investors. Then I implement decomposition (7) within a parametric model, which I estimate with survey expectations of the short rate and macro variables to study the effect of short-rate forecast errors across the term structure. 3. Properties of Short-Rate Expectations I use the FFR as a proxy for the nominal short rate. I study the properties and the accuracy of survey-based expectations of the FFR, and compare them to investors’ expectations of the short rate from the Fed funds futures market. I document ex post predictability of short-rate forecast errors by variables available to forecasters and to investors in real time. Finally, I characterize the differences between the expectations of real-time forecasters and the econometrician. 3.1 Survey conventions Forecasters in BCFF predict an average effective FFR within a given calendar quarter. Because of the difference in the frequency of the survey (monthly) and of the outcome variable (quarterly), it is important to clarify the notation. Unless otherwise noted, I measure time in years. Notation $$E_t^s(FFR_{t+\frac{h}{4}})$$ means that a survey forecast is formed in month $$t$$$$(E_t^s(\cdot))$$ and the outcome variable ($$FFR_{t+\frac{h}{4}}$$) is the within-quarter average FFR realized $$h$$ calendar quarters ahead from month $$t$$. For example, for a survey conducted in February 1990, $$E^s_{t}(FFR_{t+1})$$ represents the expected average effective FFR four quarters ahead, that is, in the first quarter of 1991. Forecast errors are changes in the FFR that forecasters did not expect at the time of their forecast: \begin{equation} FE_t(FFR_{t+\frac{h}{4}}) = FFR_{t+\frac{h}{4}} - E_t^s(FFR_{t+\frac{h}{4}}), \end{equation} (8) where the forecast error, conditional on a forecast made in month $$t$$, becomes known $$h$$ calendar quarters after month $$t$$. I use a similar notation for the survey forecasts of an $$n$$-year nominal Treasury yield, $$E_t^s(y_{t+h/4}^{(n)})$$. The design of the survey implies a shrinking forecast horizon: for example, both in January and in February 1990, the four-quarter-ahead forecast pertains to the same average value of the FFR realized in the first quarter of 1991. In practice, this has little impact on forecasts beyond one quarter ahead. When shrinking horizon may be a concern, I rely on quarterly sampling of forecasts, in which case I use the survey from the middle month of each quarter. Correspondingly, in those cases, I also sample realized macro series in the middle month of each quarter. 3.2 Summary statistics for FFR survey forecasts Figure 3 displays the term structures of FFR forecasts at different points in time, conditioning on the value of the FFR at the time of the forecast. The forecast horizon ranges from the current quarter (nowcast) to four quarters ahead. The distance between points along each term structure and the solid line depicts the magnitude of the forecast error. The visual impression from the graph suggests that forecast errors are large and negative following easing decisions by the Fed, and are positive but smaller following tightening decisions. In months in which the Fed eases policy, forecasters overestimate the FFR observed one (four) quarters later by an average of 28 (130) bps $$(t = -3.14 (-3.91))$$. In contrast, in months in which the Fed tightens, forecasters are essentially on target: the average forecast error realized four quarters later is 16 bps and is not significantly different from zero. Figure 3. View largeDownload slide Conditional term-structures of FFR survey forecasts The figure plots the term structures of FFR forecasts in the BCFF survey. The forecasts are for the current quarter up to four quarters ahead. The plot shows forecasts made in the middle month of each quarter. The shaded areas are NBER-dated recessions. Figure 3. View largeDownload slide Conditional term-structures of FFR survey forecasts The figure plots the term structures of FFR forecasts in the BCFF survey. The forecasts are for the current quarter up to four quarters ahead. The plot shows forecasts made in the middle month of each quarter. The shaded areas are NBER-dated recessions. Table 2, panel A, provides summary statistics. Forecast errors are on average negative ranging from $$-10$$ bps one quarter ahead to $$-59$$ bps four quarters ahead. At the four-quarter horizon, agents overpredict (underpredict) the future level of FFR by as much as 440 (255) bps. Table 2 Properties of FFR forecasts $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ A. Summary statistics for FFR forecast errors: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ Mean –0.10 –0.25 –0.42 –0.59 t-stat (–1.66) (–1.98) (–2.19) (–2.29) SD 0.44 0.77 1.08 1.36 Min, Max [ $$-$$2.39, 0.95] [ $$-$$3.52, 1.38] [ $$-$$4.33, 1.85] [ $$-$$4.40, 2.55] N (quarters) 109 109 109 109 B. Predicting FFR changes: $$\Delta FFR_{t+h/4} = \beta_0 + \beta_1 E_t^s(\Delta FFR_{t+h/4}) + u_{t+h/4}$$ $$\beta_1$$ 1.03 1.18 1.26 1.23 (5.62) (4.71) (4.28) (4.48) $$p$$-val $$(\beta_1=1)$$ [0.87] [0.46] [0.38] [0.40] $$\beta_0$$ –0.10 –0.26 –0.46 –0.65 (–2.06) (–2.78) (–3.28) (–3.35) $$R^2$$ 0.27 0.27 0.25 0.22 N (quarters) 109 109 109 109 C. Comparison with time-series models: RMSEs (in %) BCFF survey 0.45 0.81 1.15 1.48 RW 0.53 0.92 1.27 1.57 AR(1) 0.53 0.93 1.26 1.55 AR(2) 0.48 0.87 1.23 1.54 N (quarters) 109 109 109 109 $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ A. Summary statistics for FFR forecast errors: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ Mean –0.10 –0.25 –0.42 –0.59 t-stat (–1.66) (–1.98) (–2.19) (–2.29) SD 0.44 0.77 1.08 1.36 Min, Max [ $$-$$2.39, 0.95] [ $$-$$3.52, 1.38] [ $$-$$4.33, 1.85] [ $$-$$4.40, 2.55] N (quarters) 109 109 109 109 B. Predicting FFR changes: $$\Delta FFR_{t+h/4} = \beta_0 + \beta_1 E_t^s(\Delta FFR_{t+h/4}) + u_{t+h/4}$$ $$\beta_1$$ 1.03 1.18 1.26 1.23 (5.62) (4.71) (4.28) (4.48) $$p$$-val $$(\beta_1=1)$$ [0.87] [0.46] [0.38] [0.40] $$\beta_0$$ –0.10 –0.26 –0.46 –0.65 (–2.06) (–2.78) (–3.28) (–3.35) $$R^2$$ 0.27 0.27 0.25 0.22 N (quarters) 109 109 109 109 C. Comparison with time-series models: RMSEs (in %) BCFF survey 0.45 0.81 1.15 1.48 RW 0.53 0.92 1.27 1.57 AR(1) 0.53 0.93 1.26 1.55 AR(2) 0.48 0.87 1.23 1.54 N (quarters) 109 109 109 109 D. Comparison with Fed fund futures (in %) 3-month horizon 6-month horizon FUT BCFF,m BCFF,q FUT BCFF,m BCFF,q Mean –0.09 –0.12 –0.12 –0.28 –0.28 –0.28 SD 0.33 0.34 0.33 0.62 0.64 0.64 Min –1.64 –1.49 –1.84 –2.16 –2.17 –2.29 Max 0.77 0.74 0.75 1.03 1.09 0.87 RMSE 0.34 0.36 0.35 0.67 0.69 0.69 N (months) 273 273 273 273 273 273 D. Comparison with Fed fund futures (in %) 3-month horizon 6-month horizon FUT BCFF,m BCFF,q FUT BCFF,m BCFF,q Mean –0.09 –0.12 –0.12 –0.28 –0.28 –0.28 SD 0.33 0.34 0.33 0.62 0.64 0.64 Min –1.64 –1.49 –1.84 –2.16 –2.17 –2.29 Max 0.77 0.74 0.75 1.03 1.09 0.87 RMSE 0.34 0.36 0.35 0.67 0.69 0.69 N (months) 273 273 273 273 273 273 Panel A reports summary statistics for FFR forecast errors in the BCFF survey. FFR is reported as a percentage per annum. Panel B presents projections of the realized FFR change on the expected change. The results are presented at the quarterly frequency (as forecasters predict average FFR within a quarter), using forecasts from the middle month in a quarter. Results for monthly sampling of survey forecasts are reported in Internet Appendix Table A-3. Row “$$p$$-val $$(\beta_1=1)$$” reports the $$p$$-value for the test that $$\beta_1=1$$. In panels A and B, Newey-West t-statistics with six quarterly lags are in parentheses. Panel C compares out-of-sample RMSEs from univariate models (random walk, and AR(1) and AR(2)) with those from survey forecasts. The models are estimated at the quarterly frequency. The burn-in period for AR specifications (120 quarters) is selected to minimize the sum of RMSEs across horizons, and the order of the AR(2) model is determined by Schwarz information criterion on the post-1984 sample. Out-of-sample forecasts start in 1984Q3 with the last forecast made in 2011Q3. Panel D compares survey forecast errors with the returns on the Fed fund futures, $$r^{FUT}_{t+h/12} =FFR_{t+h/12} - {FUT}_{t}^{h}$$, where $${FFR}_{t+h/12}$$ is the within-month average daily effective FFR $$h$$ months ahead from month $$t$$ and $${FUT}_{t}^{h}$$ is the end-of-month rate from the contract (100 $$-$$ futures price) expiring in $$h$$ months relative to month $$t$$. For comparison with futures, survey forecast errors are constructed relative to both within-month (“BCFF,m”) and within-quarter (“BCFF,q”) average FFR. In panels A, B, and C, the sample period is 1984Q3–2011Q3. Futures and survey data in panel D are sampled monthly and span the 1988M12–2011M8 period. Table 2 Properties of FFR forecasts $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ A. Summary statistics for FFR forecast errors: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ Mean –0.10 –0.25 –0.42 –0.59 t-stat (–1.66) (–1.98) (–2.19) (–2.29) SD 0.44 0.77 1.08 1.36 Min, Max [ $$-$$2.39, 0.95] [ $$-$$3.52, 1.38] [ $$-$$4.33, 1.85] [ $$-$$4.40, 2.55] N (quarters) 109 109 109 109 B. Predicting FFR changes: $$\Delta FFR_{t+h/4} = \beta_0 + \beta_1 E_t^s(\Delta FFR_{t+h/4}) + u_{t+h/4}$$ $$\beta_1$$ 1.03 1.18 1.26 1.23 (5.62) (4.71) (4.28) (4.48) $$p$$-val $$(\beta_1=1)$$ [0.87] [0.46] [0.38] [0.40] $$\beta_0$$ –0.10 –0.26 –0.46 –0.65 (–2.06) (–2.78) (–3.28) (–3.35) $$R^2$$ 0.27 0.27 0.25 0.22 N (quarters) 109 109 109 109 C. Comparison with time-series models: RMSEs (in %) BCFF survey 0.45 0.81 1.15 1.48 RW 0.53 0.92 1.27 1.57 AR(1) 0.53 0.93 1.26 1.55 AR(2) 0.48 0.87 1.23 1.54 N (quarters) 109 109 109 109 $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ A. Summary statistics for FFR forecast errors: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ Mean –0.10 –0.25 –0.42 –0.59 t-stat (–1.66) (–1.98) (–2.19) (–2.29) SD 0.44 0.77 1.08 1.36 Min, Max [ $$-$$2.39, 0.95] [ $$-$$3.52, 1.38] [ $$-$$4.33, 1.85] [ $$-$$4.40, 2.55] N (quarters) 109 109 109 109 B. Predicting FFR changes: $$\Delta FFR_{t+h/4} = \beta_0 + \beta_1 E_t^s(\Delta FFR_{t+h/4}) + u_{t+h/4}$$ $$\beta_1$$ 1.03 1.18 1.26 1.23 (5.62) (4.71) (4.28) (4.48) $$p$$-val $$(\beta_1=1)$$ [0.87] [0.46] [0.38] [0.40] $$\beta_0$$ –0.10 –0.26 –0.46 –0.65 (–2.06) (–2.78) (–3.28) (–3.35) $$R^2$$ 0.27 0.27 0.25 0.22 N (quarters) 109 109 109 109 C. Comparison with time-series models: RMSEs (in %) BCFF survey 0.45 0.81 1.15 1.48 RW 0.53 0.92 1.27 1.57 AR(1) 0.53 0.93 1.26 1.55 AR(2) 0.48 0.87 1.23 1.54 N (quarters) 109 109 109 109 D. Comparison with Fed fund futures (in %) 3-month horizon 6-month horizon FUT BCFF,m BCFF,q FUT BCFF,m BCFF,q Mean –0.09 –0.12 –0.12 –0.28 –0.28 –0.28 SD 0.33 0.34 0.33 0.62 0.64 0.64 Min –1.64 –1.49 –1.84 –2.16 –2.17 –2.29 Max 0.77 0.74 0.75 1.03 1.09 0.87 RMSE 0.34 0.36 0.35 0.67 0.69 0.69 N (months) 273 273 273 273 273 273 D. Comparison with Fed fund futures (in %) 3-month horizon 6-month horizon FUT BCFF,m BCFF,q FUT BCFF,m BCFF,q Mean –0.09 –0.12 –0.12 –0.28 –0.28 –0.28 SD 0.33 0.34 0.33 0.62 0.64 0.64 Min –1.64 –1.49 –1.84 –2.16 –2.17 –2.29 Max 0.77 0.74 0.75 1.03 1.09 0.87 RMSE 0.34 0.36 0.35 0.67 0.69 0.69 N (months) 273 273 273 273 273 273 Panel A reports summary statistics for FFR forecast errors in the BCFF survey. FFR is reported as a percentage per annum. Panel B presents projections of the realized FFR change on the expected change. The results are presented at the quarterly frequency (as forecasters predict average FFR within a quarter), using forecasts from the middle month in a quarter. Results for monthly sampling of survey forecasts are reported in Internet Appendix Table A-3. Row “$$p$$-val $$(\beta_1=1)$$” reports the $$p$$-value for the test that $$\beta_1=1$$. In panels A and B, Newey-West t-statistics with six quarterly lags are in parentheses. Panel C compares out-of-sample RMSEs from univariate models (random walk, and AR(1) and AR(2)) with those from survey forecasts. The models are estimated at the quarterly frequency. The burn-in period for AR specifications (120 quarters) is selected to minimize the sum of RMSEs across horizons, and the order of the AR(2) model is determined by Schwarz information criterion on the post-1984 sample. Out-of-sample forecasts start in 1984Q3 with the last forecast made in 2011Q3. Panel D compares survey forecast errors with the returns on the Fed fund futures, $$r^{FUT}_{t+h/12} =FFR_{t+h/12} - {FUT}_{t}^{h}$$, where $${FFR}_{t+h/12}$$ is the within-month average daily effective FFR $$h$$ months ahead from month $$t$$ and $${FUT}_{t}^{h}$$ is the end-of-month rate from the contract (100 $$-$$ futures price) expiring in $$h$$ months relative to month $$t$$. For comparison with futures, survey forecast errors are constructed relative to both within-month (“BCFF,m”) and within-quarter (“BCFF,q”) average FFR. In panels A, B, and C, the sample period is 1984Q3–2011Q3. Futures and survey data in panel D are sampled monthly and span the 1988M12–2011M8 period. Table 2, panel B, projects realized changes in the FFR on changes predicted by the forecasters. The results are reported at the quarterly frequency; doing so avoids distortions due to the shrinking forecast horizon within the quarter (Appendix Table A-3 provides results at the monthly frequency). The slope coefficient of the regression is not significantly different from unity across horizons, implying that forecasts are efficient in the sense of Mincer and Zarnowitz (1969); that is, they are uncorrelated with both current predictions and current realizations of the FFR. Table 2, panel C, compares forecast precision of the survey against out-of-sample forecasts from three univariate models: a random walk model and two autoregressive (AR) models with one and two quarterly lags, respectively. Prior parameters for the AR models are selected using a burn-in period that minimizes the out-of-sample root-mean-square error (RMSE); the choice of the AR(2) model is determined by the Schwarz information criterion applied to post-1984 data. Those choices endow statistical models with information that survey participants did not have in real time, making the comparison conservative. Nevertheless, surveys have the lowest RMSE across all forecast horizons. On average across horizons, the RMSEs for the random walk and AR(1) are about 12% higher, and for AR(2) about 6% higher than those implied by survey forecasts. Finally, Table 2, panel D, compares survey forecasts of the FFR with investors’ expectations impounded in the Fed fund futures rates. Futures payoff is determined by the average effective FFR in a given calendar month. In analogy to the survey, I define the forecast error as $$r^{FUT}_{t+{h}/{12}} ={FFR}_{t+{h}/{12}} - {FUT}_{t}^{h}$$, where $${FFR}_{t+{h}/{12}}$$ is the within-month average FFR $$h$$ months ahead from month $$t$$ and $${FUT}_{t}^{h}$$ is the end-of-month rate implied by the futures contract expiring in $$h$$ months from month $$t$$. I calculate returns for $$h=3$$ months and $$h=6$$ months.11 The properties of futures- and survey-based expectations are very similar. At the six-month horizon, RMSE for the futures contract is 67 bps and for the survey it is 69.5 bps. Both survey forecast errors and futures returns are the largest (in absolute value) during monetary easings, reaching around -200 bps at the 6-month horizon. Further details on this comparison are provided in the Internet Appendix (see Figures A-2 and A-3). Taken together, the results indicate that survey expectations provide a good proxy for how market participants form expectations about the short rate. 3.3 Ex post predictability of short-rate forecast errors To compare short-rate forecasts of an econometrician and survey forecaster, I estimate two regressions: \begin{align} FFR_{t+\frac{h}{4}} &= \rho_0^e + \rho_1^e FFR_t + \rho^e_{2} \Delta EMP_{t-1,t} + \varepsilon^e_{t+\frac{h}{4}} \\ \end{align} (9) \begin{align} E_t^s(FFR_{t+\frac{h}{4}}) &= \rho^s_0 + \rho^s_{1} FFR_t + \rho^s_{2} \Delta EMP_{t-1,t} + \varepsilon_{t,\frac{h}{4}}^{s} . \end{align} (10) Equation (9) is a predictive regression of future FFR on time-$$t$$ FFR and annual employment growth. Equation (10) is a regression of the survey forecast on the same explanatory variables. Results are reported in Table 3, panels A and B, for horizons $$h$$ between one and four quarters ahead and with data sampled quarterly. Table 3 Econometric forecasts versus survey expectations of FFR A. Dependent variable: $$FFR_{t+h/4}$$ B. Dependent variable: $$E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ 0.88 0.75 0.62 0.50 0.93 0.90 0.87 0.84 (24.10) (11.10) (6.41) (4.10) (45.18) (23.28) (16.41) (14.02) $$\Delta EMP_{t-1,t}$$ 0.18 0.36 0.52 0.65 0.10 0.15 0.15 0.12 (3.04) (3.13) (3.24) (3.35) (2.92) (2.78) (2.23) (1.58) Constant 0.22 0.49 0.79 1.09 0.17 0.33 0.56 0.84 (1.62) (1.98) (2.24) (2.43) (2.33) (2.47) (2.99) (3.90) $$\bar{R}^2$$ 0.97 0.91 0.83 0.76 0.99 0.98 0.97 0.96 N (quarters) 109 109 109 109 109 109 109 109 A. Dependent variable: $$FFR_{t+h/4}$$ B. Dependent variable: $$E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ 0.88 0.75 0.62 0.50 0.93 0.90 0.87 0.84 (24.10) (11.10) (6.41) (4.10) (45.18) (23.28) (16.41) (14.02) $$\Delta EMP_{t-1,t}$$ 0.18 0.36 0.52 0.65 0.10 0.15 0.15 0.12 (3.04) (3.13) (3.24) (3.35) (2.92) (2.78) (2.23) (1.58) Constant 0.22 0.49 0.79 1.09 0.17 0.33 0.56 0.84 (1.62) (1.98) (2.24) (2.43) (2.33) (2.47) (2.99) (3.90) $$\bar{R}^2$$ 0.97 0.91 0.83 0.76 0.99 0.98 0.97 0.96 N (quarters) 109 109 109 109 109 109 109 109 C. Dependent variable: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ –0.055 –0.15 –0.25 –0.34 (–1.76) (–2.62) (–3.10) (–3.34) $$p$$-val $$(\rho^e_1-\rho^s_1=0)$$ (0.079) (0.009) (0.002) (0.001) $$\Delta EMP_{t-1,t}$$ 0.074 0.21 0.37 0.53 (2.38) (2.65) (2.90) (3.17) $$p$$-val $$(\rho^e_2-\rho^s_2=0)$$ (0.017) (0.008) (0.004) (0.002) Constant 0.051 0.16 0.24 0.25 (0.46) (0.82) (0.86) (0.72) $${\bar R}^2$$ 0.03 0.09 0.15 0.18 N (quarters) 109 109 109 109 C. Dependent variable: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ –0.055 –0.15 –0.25 –0.34 (–1.76) (–2.62) (–3.10) (–3.34) $$p$$-val $$(\rho^e_1-\rho^s_1=0)$$ (0.079) (0.009) (0.002) (0.001) $$\Delta EMP_{t-1,t}$$ 0.074 0.21 0.37 0.53 (2.38) (2.65) (2.90) (3.17) $$p$$-val $$(\rho^e_2-\rho^s_2=0)$$ (0.017) (0.008) (0.004) (0.002) Constant 0.051 0.16 0.24 0.25 (0.46) (0.82) (0.86) (0.72) $${\bar R}^2$$ 0.03 0.09 0.15 0.18 N (quarters) 109 109 109 109 D. $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$ and $$\widehat{E}^f_t(FFR_{t+h/4}) = \alpha'V_t$$, $$h=4Q$$ (1) (2) (3) (4) (5) $$\widehat{E}^f_t(FFR_{t+h/4})$$ –0.19 0.66 0.72 0.72 0.89 (–0.43) (2.94) (2.64) (2.68) (3.98) $$\chi^2(1) \, (\beta_e=1)$$ 7.09 2.39 1.07 1.09 0.23 p-val $$(\beta_e=1)$$ 0.008 0.12 0.30 0.30 0.63 $$E_t^s(FFR_{t+h/4})$$ 1.09 0.34 0.28 0.28 0.11 (2.59) (1.49) (1.06) (1.06) (0.56) Constant –0.19 –0.18 –0.15 –0.15 –0.08 (–0.46) (–0.44) (–0.35) (–0.35) (–0.20) $$\bar R^2$$ 0.73 0.77 0.77 0.77 0.82 N (quarters) 109 109 109 109 109 D. $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$ and $$\widehat{E}^f_t(FFR_{t+h/4}) = \alpha'V_t$$, $$h=4Q$$ (1) (2) (3) (4) (5) $$\widehat{E}^f_t(FFR_{t+h/4})$$ –0.19 0.66 0.72 0.72 0.89 (–0.43) (2.94) (2.64) (2.68) (3.98) $$\chi^2(1) \, (\beta_e=1)$$ 7.09 2.39 1.07 1.09 0.23 p-val $$(\beta_e=1)$$ 0.008 0.12 0.30 0.30 0.63 $$E_t^s(FFR_{t+h/4})$$ 1.09 0.34 0.28 0.28 0.11 (2.59) (1.49) (1.06) (1.06) (0.56) Constant –0.19 –0.18 –0.15 –0.15 –0.08 (–0.46) (–0.44) (–0.35) (–0.35) (–0.20) $$\bar R^2$$ 0.73 0.77 0.77 0.77 0.82 N (quarters) 109 109 109 109 109 Panel A presents predictive regressions of future FFR, $$FFR_{t+h/4}$$, on time $$t$$ predictors, for forecast horizon $$h$$ ranging from one through four quarters ahead (regression (9)): $$FFR_{t+{h}/{4}} = \rho_0^e + \rho_1^e FFR_t + \rho^e_{2} \Delta EMP_{t-1,t} + \varepsilon^e_{t+{h}/{4}}$$. Panel B presents analogous regressions using survey expectations of the FFR as the dependent variable (regression (10)): $$E_t^s(FFR_{t+{h}/{4}}) = \rho^s_0 + \rho^s_{1} FFR_t + \rho^s_{2} \Delta EMP_{t-1,t} + \varepsilon_{t,{h}/{4}}^{s}.$$ Panel C regresses forecast errors on $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$, and tests whether the coefficients in regressions (9) and (10) are the same. Panel D estimates regression (11): $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$, where $$\widehat{E}_t^{f}(FFR_{t+{h}/{4}})$$ is the econometrician’s forecast of the FFR, $$\widehat{E}^f_t(FFR_{t+\frac{h}{4}}) = \alpha'V_t$$, and $$V_t$$ is a vector of instruments. The full set of instruments includes current quarter $$FFR_t$$, annual growth rate of employment $$\Delta EMP_{t-1,t}$$, year-on-year core inflation $$\Delta CPI^c_{t-1,t}$$, level of unemployment $$UNE_t$$, and $$CFNAI_t$$. Column 1 uses $$FFR_t$$ as the only instrument; Column 2 uses $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$, and so on, until in Column 5 all instruments are used. Regressions in all panels are estimated at the quarterly frequency. The sample covers the period 1984Q3–2011Q3. t-statistics in parentheses use Newey-West adjustment with six quarterly lags. Table 3 Econometric forecasts versus survey expectations of FFR A. Dependent variable: $$FFR_{t+h/4}$$ B. Dependent variable: $$E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ 0.88 0.75 0.62 0.50 0.93 0.90 0.87 0.84 (24.10) (11.10) (6.41) (4.10) (45.18) (23.28) (16.41) (14.02) $$\Delta EMP_{t-1,t}$$ 0.18 0.36 0.52 0.65 0.10 0.15 0.15 0.12 (3.04) (3.13) (3.24) (3.35) (2.92) (2.78) (2.23) (1.58) Constant 0.22 0.49 0.79 1.09 0.17 0.33 0.56 0.84 (1.62) (1.98) (2.24) (2.43) (2.33) (2.47) (2.99) (3.90) $$\bar{R}^2$$ 0.97 0.91 0.83 0.76 0.99 0.98 0.97 0.96 N (quarters) 109 109 109 109 109 109 109 109 A. Dependent variable: $$FFR_{t+h/4}$$ B. Dependent variable: $$E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ 0.88 0.75 0.62 0.50 0.93 0.90 0.87 0.84 (24.10) (11.10) (6.41) (4.10) (45.18) (23.28) (16.41) (14.02) $$\Delta EMP_{t-1,t}$$ 0.18 0.36 0.52 0.65 0.10 0.15 0.15 0.12 (3.04) (3.13) (3.24) (3.35) (2.92) (2.78) (2.23) (1.58) Constant 0.22 0.49 0.79 1.09 0.17 0.33 0.56 0.84 (1.62) (1.98) (2.24) (2.43) (2.33) (2.47) (2.99) (3.90) $$\bar{R}^2$$ 0.97 0.91 0.83 0.76 0.99 0.98 0.97 0.96 N (quarters) 109 109 109 109 109 109 109 109 C. Dependent variable: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ –0.055 –0.15 –0.25 –0.34 (–1.76) (–2.62) (–3.10) (–3.34) $$p$$-val $$(\rho^e_1-\rho^s_1=0)$$ (0.079) (0.009) (0.002) (0.001) $$\Delta EMP_{t-1,t}$$ 0.074 0.21 0.37 0.53 (2.38) (2.65) (2.90) (3.17) $$p$$-val $$(\rho^e_2-\rho^s_2=0)$$ (0.017) (0.008) (0.004) (0.002) Constant 0.051 0.16 0.24 0.25 (0.46) (0.82) (0.86) (0.72) $${\bar R}^2$$ 0.03 0.09 0.15 0.18 N (quarters) 109 109 109 109 C. Dependent variable: $$FE_t(FFR_{t+h/4}) = FFR_{t+h/4} - E_t^s(FFR_{t+h/4})$$ $$h=1Q$$ $$h=2Q$$ $$h=3Q$$ $$h=4Q$$ $$FFR_t$$ –0.055 –0.15 –0.25 –0.34 (–1.76) (–2.62) (–3.10) (–3.34) $$p$$-val $$(\rho^e_1-\rho^s_1=0)$$ (0.079) (0.009) (0.002) (0.001) $$\Delta EMP_{t-1,t}$$ 0.074 0.21 0.37 0.53 (2.38) (2.65) (2.90) (3.17) $$p$$-val $$(\rho^e_2-\rho^s_2=0)$$ (0.017) (0.008) (0.004) (0.002) Constant 0.051 0.16 0.24 0.25 (0.46) (0.82) (0.86) (0.72) $${\bar R}^2$$ 0.03 0.09 0.15 0.18 N (quarters) 109 109 109 109 D. $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$ and $$\widehat{E}^f_t(FFR_{t+h/4}) = \alpha'V_t$$, $$h=4Q$$ (1) (2) (3) (4) (5) $$\widehat{E}^f_t(FFR_{t+h/4})$$ –0.19 0.66 0.72 0.72 0.89 (–0.43) (2.94) (2.64) (2.68) (3.98) $$\chi^2(1) \, (\beta_e=1)$$ 7.09 2.39 1.07 1.09 0.23 p-val $$(\beta_e=1)$$ 0.008 0.12 0.30 0.30 0.63 $$E_t^s(FFR_{t+h/4})$$ 1.09 0.34 0.28 0.28 0.11 (2.59) (1.49) (1.06) (1.06) (0.56) Constant –0.19 –0.18 –0.15 –0.15 –0.08 (–0.46) (–0.44) (–0.35) (–0.35) (–0.20) $$\bar R^2$$ 0.73 0.77 0.77 0.77 0.82 N (quarters) 109 109 109 109 109 D. $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$ and $$\widehat{E}^f_t(FFR_{t+h/4}) = \alpha'V_t$$, $$h=4Q$$ (1) (2) (3) (4) (5) $$\widehat{E}^f_t(FFR_{t+h/4})$$ –0.19 0.66 0.72 0.72 0.89 (–0.43) (2.94) (2.64) (2.68) (3.98) $$\chi^2(1) \, (\beta_e=1)$$ 7.09 2.39 1.07 1.09 0.23 p-val $$(\beta_e=1)$$ 0.008 0.12 0.30 0.30 0.63 $$E_t^s(FFR_{t+h/4})$$ 1.09 0.34 0.28 0.28 0.11 (2.59) (1.49) (1.06) (1.06) (0.56) Constant –0.19 –0.18 –0.15 –0.15 –0.08 (–0.46) (–0.44) (–0.35) (–0.35) (–0.20) $$\bar R^2$$ 0.73 0.77 0.77 0.77 0.82 N (quarters) 109 109 109 109 109 Panel A presents predictive regressions of future FFR, $$FFR_{t+h/4}$$, on time $$t$$ predictors, for forecast horizon $$h$$ ranging from one through four quarters ahead (regression (9)): $$FFR_{t+{h}/{4}} = \rho_0^e + \rho_1^e FFR_t + \rho^e_{2} \Delta EMP_{t-1,t} + \varepsilon^e_{t+{h}/{4}}$$. Panel B presents analogous regressions using survey expectations of the FFR as the dependent variable (regression (10)): $$E_t^s(FFR_{t+{h}/{4}}) = \rho^s_0 + \rho^s_{1} FFR_t + \rho^s_{2} \Delta EMP_{t-1,t} + \varepsilon_{t,{h}/{4}}^{s}.$$ Panel C regresses forecast errors on $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$, and tests whether the coefficients in regressions (9) and (10) are the same. Panel D estimates regression (11): $$FFR_{t+{h}/{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+{h}/{4}}) + \beta_s E_t^s(FFR_{t+{h}/{4}}) + u_{t+{h}/{4}}$$, where $$\widehat{E}_t^{f}(FFR_{t+{h}/{4}})$$ is the econometrician’s forecast of the FFR, $$\widehat{E}^f_t(FFR_{t+\frac{h}{4}}) = \alpha'V_t$$, and $$V_t$$ is a vector of instruments. The full set of instruments includes current quarter $$FFR_t$$, annual growth rate of employment $$\Delta EMP_{t-1,t}$$, year-on-year core inflation $$\Delta CPI^c_{t-1,t}$$, level of unemployment $$UNE_t$$, and $$CFNAI_t$$. Column 1 uses $$FFR_t$$ as the only instrument; Column 2 uses $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$, and so on, until in Column 5 all instruments are used. Regressions in all panels are estimated at the quarterly frequency. The sample covers the period 1984Q3–2011Q3. t-statistics in parentheses use Newey-West adjustment with six quarterly lags. Compared to the survey forecast in (10), estimates of $$(9)$$ deliver a lower coefficient on $$FFR_t$$ and a higher coefficients on $$\Delta EMP_{t-1,t}$$. This holds true across all horizons. For example, at the four-quarter horizon, the loadings on $$FFR_t$$ are $$\rho_1^e=0.50$$$$(t = 4.1)$$ and $$\rho_1^s= 0.84$$$$(t = 14)$$, whereas the loadings on $$\Delta EMP_{t-1,t}$$ are $$\rho_2^e=0.65$$$$(t = 3.4)$$ and $$\rho_2^s = 0.12$$$$(t = 1.6)$$. This means that a 1-percentage-point slowdown in employment growth over the past year translates into a 53 bps decline in the FFR over the next year that survey forecasters did not expect at time of their forecast. Survey expectations behave as if the forecaster was extrapolating FFR from recent past while underestimating or even ignoring the predictive power of employment growth for the future FFR. This leads to a pattern in coefficients in (10) similar to an omitted variable bias.12 To test whether the differences between the coefficients in (9) and (10) are statistically significant, panel C of Table 3 reports a regression of survey forecast errors, $$FE_t(FFR_{t+\frac{h}{4}})=FFR_{t+\frac{h}{4}}-E_t^s(FFR_{t+\frac{h}{4}})$$, on the instruments $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$. The t-statistics test whether $$\rho_1^e - \rho_1^s=0$$ and $$\rho_2^e-\rho_2^s=0$$.13 For horizons beyond one quarter, the test rejects that the coefficients are equal at the 1% level for both $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$; for the one-quarter horizon the rejection is at the 8% level for the coefficient on $$FFR_t$$ and at the 2% level for $$\Delta EMP_{t-1,t}$$. The adjusted $$R^2$$’s show that up to 18% of variation in forecast errors can be ex post predicted with just two variables. The results suggest that survey expectations of the short rate deviate in a significant way from expectations obtained using predictive regressions. While overextrapolation visible in surveys can arise as a consequence of an omitted variable bias on the part of the forecaster, overextrapolation can be a sign of more complex belief formation. Fuster Laibson, and Mendel (2010) discuss beliefs—natural expectations—where forecasters do not sufficiently account for the fact that good times (or bad times) will not last forever. More broadly, faced with complex underlying dynamics, forecasters may base their expectations on simpler intuitive models that deviate from the truth in a significant way but still imply a small utility loss (Cochrane 1989).14 3.4 Is survey forecast subsumed by the econometrician’s information set? The results so far tell us little about possible misspecification on the side of the econometrician estimating regression (9). In particular, it is possible that survey forecasters use some variables that are useful for predicting the short rate but that the specification (9) ignores. To test whether surveys contain information about future short rate over and above the econometrician’s model, I exploit the regression \begin{equation} FFR_{t+\frac{h}{4}} = \beta_0 + \beta_e \widehat{E}_t^{f}(FFR_{t+\frac{h}{4}}) + \beta_s E_t^s(FFR_{t+\frac{h}{4}}) + u_{t+\frac{h}{4}}, \end{equation} (11) where $$\widehat{E}_t^{f}(FFR_{t+\frac{h}{4}})$$ is the fitted value from a regression of $$FFR_{t+\frac{h}{4}}$$ on instruments of econometrician’s choice: \begin{equation} FFR_{t+\frac{h}{4}} = \widehat{E}_t^{f}(FFR_{t+\frac{h}{4}}) + \varepsilon_{t+\frac{h}{4}}^e, \quad \widehat{E}^f_t(FFR_{t+\frac{h}{4}}) = \alpha'V_t. \end{equation} (12) $$V_t$$ is a vector of instruments and superscript $$f$$ in $$\widehat{E}_t^{f}(\cdot)$$ indicates that the regression is estimated over the full sample. If the survey forecast does not provide useful information that the econometrician’s model misses, then $$E(E_t^s(FFR_{t+\frac{h}{4}})\varepsilon_{t+\frac{h}{4}}^e)=0$$ holds, and $$\beta_e$$ in (11) should be equal to one. I test whether $$\beta_e=1$$ for different sets of instruments, $$V_t$$.15 The results are presented in panel D of Table 3. I focus on forecast horizon of four quarters ahead, $$h=4$$, and sample data quarterly. Columns 1–5 correspond to an expanding sets of instruments used to form $$\widehat{E}_t^{f}(FFR_{t+1})$$. The full set of instruments includes current quarter $$FFR_t$$, annual growth rate of employment $$\Delta EMP_{t-1,t}$$, year-on-year core inflation $$\Delta CPI^c_{t-1,t}$$, level of unemployment $$UNE_t$$, and $$CFNAI_t$$. Column 1 assumes that econometrician’s model only uses $$FFR_t$$ to forecast future short rate, Column 2 assumes that it uses both $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$ corresponding to specification (9), and so on, until in Column 5 all instruments are used. Column 1 confirms that survey forecasters use more information than just current $$FFR_t$$; that is, their expectations are not purely extrapolative. However, Column 2 shows that including just the $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$ in $$\widehat{E}^f_t(FFR_{t+1})$$ makes the survey forecast insignificant, and the null of $$\beta_e=1$$ can only be rejected with a p-value of 0.12.16 Adding more instruments in Columns 3 through 5, increases $$\beta_e$$ toward unity and raises the p-values for the test of $$\beta_e=1$$ further up to 0.63 in Column 5. This is to be expected, as by extending the information set, the econometrician is less likely to omit variables that convey information about the future short rate. While many variables could be used as potential predictors of FFR, with many predictors one worries about overfitting. Thus, in the remainder of this section, I rely on a parsimonious choice of instruments, $$FFR_t$$ and $$\Delta EMP_{t-1,t}$$, to analyze the difference between the econometrician’s and survey short-rate forecasts over time. 3.5 Wedge between short-rate expectations of survey forecaster and econometrician: Full-sample and rolling estimates An important question is whether one could have used employment growth in real time to form better forecasts of FFR than the survey forecasters did. If this is the case, then the results indicate systematic expectations errors that may persist in the future. To cast light on this question, I estimate the model in Equation (9) over a rolling window. Since forecaster can update their FFR predictions at a monthly frequency, this is the frequency that I focus on here. (None of my results depends on the choice of monthly or quarterly sampling.) For rolling estimates, I use 5 years of monthly data to form a twelve-month-ahead forecast of the FFR, $$\widehat{E}^{r}_t(FFR_{t+1})$$. So each month, I forecast FFR realized twelve months later, $$FFR_{t+1}$$.17 I obtain analogous forecasts using a full-sample regression, $$\widehat{E}_t^f(FFR_{t+1})$$. I then compare the econometric forecasts with the survey forecast four quarters ahead, $$E_t^{s}(FFR_{t+1})$$.18 Figure 4, panel A, displays full-sample and rolling estimates of expected FFR and superimposes them against survey expectations. Absent expectations frictions and/or statistical biases, econometric and survey expectations should differ just by an unpredictable noise component. However, the graph suggests that survey expectations systematically lag behind those of the econometrician. For example, during the 2007–2009 recession, econometric and survey forecasts start diverging already in the early summer of 2008. In June 2008, the rolling forecast predicts FFR in twelve months to be at 1.55%, whereas the survey predicts 2.75%. In October 2008, the rolling forecast reaches zero, whereas the survey is still at 1.165%.19 Figure 4. View largeDownload slide Short-rate expectations: Econometrician versus survey forecasters Panel A plots the expected FFR fitted from full-sample and rolling predictive regressions, as well as the corresponding survey forecast. The regression predicts FFR one-year-ahead $$FFR_{t+1}=\rho_0^e + \rho_1^e FFR_t + \rho_2^e \Delta EMP_{t-1,t}+\varepsilon_{t+1}$$, and is estimated at the monthly frequency, so each month one predicts (average within-month) FFR twelve months ahead. The rolling regressions are estimated on five years of monthly data. Survey expectations are sampled monthly for the horizons of four quarters ahead, $$E_t^s(FFR_{t+1})$$. Panel B reports the difference between the econometric and survey expectations, as defined in Equation (13). Figure 4. View largeDownload slide Short-rate expectations: Econometrician versus survey forecasters Panel A plots the expected FFR fitted from full-sample and rolling predictive regressions, as well as the corresponding survey forecast. The regression predicts FFR one-year-ahead $$FFR_{t+1}=\rho_0^e + \rho_1^e FFR_t + \rho_2^e \Delta EMP_{t-1,t}+\varepsilon_{t+1}$$, and is estimated at the monthly frequency, so each month one predicts (average within-month) FFR twelve months ahead. The rolling regressions are estimated on five years of monthly data. Survey expectations are sampled monthly for the horizons of four quarters ahead, $$E_t^s(FFR_{t+1})$$. Panel B reports the difference between the econometric and survey expectations, as defined in Equation (13). To trace out the discrepancy between the econometric and survey forecast over time, I define a “wedge” variable as \begin{equation} \text{wedge}_t^{j} = \widehat{E}_t^{j}(FFR_{t+1}) - E_t^s(FFR_{t+1}), \quad j=\{f,r\}. \end{equation} (13) I construct this difference using full-sample (wedge$$_t^f$$) as well as rolling (wedge$$_t^r$$) estimates. Their correlation is 0.55 in levels and 0.65 in one-month changes. Figure 4, panel B, shows that the rolling estimate is more volatile than the full-sample estimate (standard deviation of 98 vs. 82 bps), and it declines faster in economic downturns. The wedge reflects the gap between the information set of the econometrician and the survey forecaster. Since the wedge is persistent and widens most visibly in easing episodes around recessions, those are periods where most of the ex post predictability of short-rate forecast errors arises. A regression of survey forecast error $$FE_t(FFR_{t+1})$$ on the wedge delivers the following estimates at the monthly frequency ($$N=327$$ months):20 \begin{align} FE_t(FFR_{t+1})&=\underset{{(-1.48)}}{-0.29} + \underset{\substack{(7.79)\\ [ 0.69; 1.22]}}{0.87}\,\text{wedge}_t^r, \;\; R^2=0.38,\\ \end{align} (14) \begin{align} FE_t(FFR_{t+1})&=\underset{(-0.67)}{-0.20} + \underset{\substack{(3.16)\\ [0.26; 0.94]}}{0.69}\,\text{wedge}_t^f,\;\; R^2=0.17. \end{align} (15) Clearly, the wedge predicts forecast errors by construction as long as the econometrician uses predictive information that the survey forecasters omit.21 However, regressions (14) and (15) conveniently summarize the differences between the full-sample and rolling estimates. The rolling estimate is a stronger predictor of forecast errors. This stems from the fact that the rolling estimate detects economic downturns, and the associated short rate declines, sooner than the full-sample estimate. Its predictive power is consistent with the presence of systematic errors in short-rate expectations from mid-1980s. Both for the full-sample and rolling estimate, the wedge reaches the most negative level in the early 1990s, and is somewhat smaller (in absolute value) in the two subsequent recessions. One interpretation is that after Volcker’s disinflation in the early 1980s, and with Alan Greenspan becoming the Federal Reserve Chair in 1987, it took markets time to understand the Fed’s response to employment in an environment of relatively stable inflation expectations. Relatedly, I discuss the properties of monetary policy shocks over this period in Section 6. 4. Implications for Bond Return Predictability I now turn to linking the short-rate forecast errors predictability to the predictability of bond excess returns. I first document that the wedge between the econometrician’s and survey short-rate expectations predicts bonds excess returns at short maturities. I then discuss the empirical relationship between short-rate forecast errors and unexpected excess returns (and yield forecast errors) across maturities. Suppose that part of return predictability is indeed induced by ex post predictability of short-rate forecast errors. Then one should empirically find that (1) excess bond returns can be predicted by the difference between econometric and survey short-rate expectations, (2) the predictive power declines with bond maturity, and the speed of the decline is dictated by the speed of short-rate mean-reversion. I establish these two results below. 4.1 Predictive regressions of bond excess returns with the expectations wedge Table 4 presents predictive regressions of bond excess returns for maturities of two and ten years in panels A and B, respectively. As in Table 1, I estimate the regressions at the monthly frequency and forecast excess returns over the following twelve months. I report both the unconstrained bi-variate regressions using econometric and survey expectations of the short rate as predictors, as well as constrained univariate regressions using their difference, the wedge variable. All variables are constructed as in Section 3.5. Econometric expectations are obtained from full-sample regressions in Columns 1 and 2 and from rolling regressions in Columns 3 and 4. Columns 5–7 control for bond risk premium variation using the risk-premium measure from Cieslak and Povala (2015), the cycle factor $$\widehat{cf}_t$$, obtained from yields-plus-$$\tau_t^{CPI}$$ predictive regressions of excess returns. Table 4 Predictability of bond excess returns by the wedge between survey and econometrician’s expectations of FFR (1) (2) (3) (4) (5) (6) (7) A. Dependent variable: Annual excess return on two-year Treasury bond, $$rx_{t+1}^{(2)}$$ $$E_t^s(FFR_{t+1})$$ 0.94 0.99 (4.27) (7.80) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –0.85 (–4.39) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –0.84 (–8.41) Wedge$$^{f}_t$$ –0.94 –0.65 (–4.11) (–3.17) Wedge$$^{r}_t$$ –0.88 –0.74 (–9.26) (–5.58) $$\widehat{cf}_t$$ 1.08 0.66 0.81 (4.41) (2.95) (3.00) Constant –0.053 0.36 –0.11 0.60 0.45 0.23 0.28 (–0.11) (1.32) (–0.28) (3.08) (1.81) (0.86) (1.25) $$\bar{R}^2$$ 0.31 0.28 0.42 0.36 0.26 0.35 0.50 N (months) 327 327 327 327 327 327 327 B. Dependent variable: Annual excess return on ten-year Treasury bond, $$rx_{t+1}^{(10)}$$ $$E_t^s(FFR_{t+1})$$ 3.83 3.27 (1.88) (2.72) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –3.87 (–2.32) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –3.03 (–2.79) Wedge$$^{f}_t$$ –3.83 0.58 (–1.88) (0.35) Wedge$$^{r}_t$$ –3.11 –1.55 (–3.01) (–1.66) $$\widehat{cf}_t$$ 9.46 9.84 8.90 (7.80) (8.10) (6.84) Constant 3.51 3.33 3.37 4.52 1.28 1.47 0.92 (0.95) (1.63) (1.04) (3.25) (0.95) (0.86) (0.71) $$\bar{R}^2$$ 0.11 0.11 0.11 0.11 0.49 0.49 0.51 N (months) 327 327 327 327 327 327 327 (1) (2) (3) (4) (5) (6) (7) A. Dependent variable: Annual excess return on two-year Treasury bond, $$rx_{t+1}^{(2)}$$ $$E_t^s(FFR_{t+1})$$ 0.94 0.99 (4.27) (7.80) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –0.85 (–4.39) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –0.84 (–8.41) Wedge$$^{f}_t$$ –0.94 –0.65 (–4.11) (–3.17) Wedge$$^{r}_t$$ –0.88 –0.74 (–9.26) (–5.58) $$\widehat{cf}_t$$ 1.08 0.66 0.81 (4.41) (2.95) (3.00) Constant –0.053 0.36 –0.11 0.60 0.45 0.23 0.28 (–0.11) (1.32) (–0.28) (3.08) (1.81) (0.86) (1.25) $$\bar{R}^2$$ 0.31 0.28 0.42 0.36 0.26 0.35 0.50 N (months) 327 327 327 327 327 327 327 B. Dependent variable: Annual excess return on ten-year Treasury bond, $$rx_{t+1}^{(10)}$$ $$E_t^s(FFR_{t+1})$$ 3.83 3.27 (1.88) (2.72) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –3.87 (–2.32) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –3.03 (–2.79) Wedge$$^{f}_t$$ –3.83 0.58 (–1.88) (0.35) Wedge$$^{r}_t$$ –3.11 –1.55 (–3.01) (–1.66) $$\widehat{cf}_t$$ 9.46 9.84 8.90 (7.80) (8.10) (6.84) Constant 3.51 3.33 3.37 4.52 1.28 1.47 0.92 (0.95) (1.63) (1.04) (3.25) (0.95) (0.86) (0.71) $$\bar{R}^2$$ 0.11 0.11 0.11 0.11 0.49 0.49 0.51 N (months) 327 327 327 327 327 327 327 The table presents predictive regressions of annual excess returns on bonds with two- and ten-year maturity on survey $$E_t^s()$$ and econometric $$\widehat{E}_t()$$ short-rate forecasts. Results are reported using both full-sample and rolling forecast by the econometrician. The econometrician predicts the within-month average FFR twelve-month-ahead relative to month $$t$$, that is, $$FFR_{t+1}$$. Columns 1 and 2 run unconstrained regressions on survey and econometricians’ forecasts; Columns 2 and 4 run constrained regressions on the expectations wedge, that is, the difference between the econometric and survey forecasts defined in Equation (13). Columns 5 though 7 control for the risk premium variation with the cycle factor ($$\widehat{cf}_t$$) obtained following Cieslak and Povala (2015). Regressions are estimated at monthly frequency, so each month I forecast excess return over the next twelve months. t-statistics using Newey-West adjustment with 18 monthly lags are reported in the parentheses. The data covers the period 1984M6–2012M8 for a total of 327 months (the first annual return is realized in June 1985 and the last in August 2012). Table 4 Predictability of bond excess returns by the wedge between survey and econometrician’s expectations of FFR (1) (2) (3) (4) (5) (6) (7) A. Dependent variable: Annual excess return on two-year Treasury bond, $$rx_{t+1}^{(2)}$$ $$E_t^s(FFR_{t+1})$$ 0.94 0.99 (4.27) (7.80) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –0.85 (–4.39) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –0.84 (–8.41) Wedge$$^{f}_t$$ –0.94 –0.65 (–4.11) (–3.17) Wedge$$^{r}_t$$ –0.88 –0.74 (–9.26) (–5.58) $$\widehat{cf}_t$$ 1.08 0.66 0.81 (4.41) (2.95) (3.00) Constant –0.053 0.36 –0.11 0.60 0.45 0.23 0.28 (–0.11) (1.32) (–0.28) (3.08) (1.81) (0.86) (1.25) $$\bar{R}^2$$ 0.31 0.28 0.42 0.36 0.26 0.35 0.50 N (months) 327 327 327 327 327 327 327 B. Dependent variable: Annual excess return on ten-year Treasury bond, $$rx_{t+1}^{(10)}$$ $$E_t^s(FFR_{t+1})$$ 3.83 3.27 (1.88) (2.72) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –3.87 (–2.32) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –3.03 (–2.79) Wedge$$^{f}_t$$ –3.83 0.58 (–1.88) (0.35) Wedge$$^{r}_t$$ –3.11 –1.55 (–3.01) (–1.66) $$\widehat{cf}_t$$ 9.46 9.84 8.90 (7.80) (8.10) (6.84) Constant 3.51 3.33 3.37 4.52 1.28 1.47 0.92 (0.95) (1.63) (1.04) (3.25) (0.95) (0.86) (0.71) $$\bar{R}^2$$ 0.11 0.11 0.11 0.11 0.49 0.49 0.51 N (months) 327 327 327 327 327 327 327 (1) (2) (3) (4) (5) (6) (7) A. Dependent variable: Annual excess return on two-year Treasury bond, $$rx_{t+1}^{(2)}$$ $$E_t^s(FFR_{t+1})$$ 0.94 0.99 (4.27) (7.80) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –0.85 (–4.39) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –0.84 (–8.41) Wedge$$^{f}_t$$ –0.94 –0.65 (–4.11) (–3.17) Wedge$$^{r}_t$$ –0.88 –0.74 (–9.26) (–5.58) $$\widehat{cf}_t$$ 1.08 0.66 0.81 (4.41) (2.95) (3.00) Constant –0.053 0.36 –0.11 0.60 0.45 0.23 0.28 (–0.11) (1.32) (–0.28) (3.08) (1.81) (0.86) (1.25) $$\bar{R}^2$$ 0.31 0.28 0.42 0.36 0.26 0.35 0.50 N (months) 327 327 327 327 327 327 327 B. Dependent variable: Annual excess return on ten-year Treasury bond, $$rx_{t+1}^{(10)}$$ $$E_t^s(FFR_{t+1})$$ 3.83 3.27 (1.88) (2.72) $$\widehat{E}_t^{f}(FFR_{t+1})$$ –3.87 (–2.32) $$\widehat{E}_t^{r}(FFR_{t+1})$$ –3.03 (–2.79) Wedge$$^{f}_t$$ –3.83 0.58 (–1.88) (0.35) Wedge$$^{r}_t$$ –3.11 –1.55 (–3.01) (–1.66) $$\widehat{cf}_t$$ 9.46 9.84 8.90 (7.80) (8.10) (6.84) Constant 3.51 3.33 3.37 4.52 1.28 1.47 0.92 (0.95) (1.63) (1.04) (3.25) (0.95) (0.86) (0.71) $$\bar{R}^2$$ 0.11 0.11 0.11 0.11 0.49 0.49 0.51 N (months) 327 327 327 327 327 327 327 The table presents predictive regressions of annual excess returns on bonds with two- and ten-year maturity on survey $$E_t^s()$$ and econometric $$\widehat{E}_t()$$ short-rate forecasts. Results are reported using both full-sample and rolling forecast by the econometrician. The econometrician predicts the within-month average FFR twelve-month-ahead relative to month $$t$$, that is, $$FFR_{t+1}$$. Columns 1 and 2 run unconstrained regressions on survey and econometricians’ forecasts; Columns 2 and 4 run constrained regressions on the expectations wedge, that is, the difference between the econometric and survey forecasts defined in Equation (13). Columns 5 though 7 control for the risk premium variation with the cycle factor ($$\widehat{cf}_t$$) obtained following Cieslak and Povala (2015). Regressions are estimated at monthly frequency, so each month I forecast excess return over the next twelve months. t-statistics using Newey-West adjustment with 18 monthly lags are reported in the parentheses. The data covers the period 1984M6–2012M8 for a total of 327 months (the first annual return is realized in June 1985 and the last in August 2012). Realized excess returns load positively on survey expectations and negatively on econometric expectations of the short rate, with coefficients of roughly the same magnitude (Columns 1 and 3). This is to be expected if the predictability stems from fitting the unexpected return component, which is a mirror reflection of unexpected yield changes (Equation (6)). The magnitude of loadings shows that the unconstrained estimates effectively recover the expectations wedge. Indeed, regressions using the expectations wedge (Columns 2 and 4) deliver the same results as the unconstrained specification. (I provide more discussion of the offsetting coefficients in Appendix C.2.) Comparing panels A and B, the predictive power of the expectations wedge is statistically and economically strong at the two-year maturity but dissipates at the ten-year maturity. Controlling for risk premium variation with the cycle factor, at the two-year maturity, the expectations wedge remains strongly significant with the coefficient declining by about 15% (in absolute value) for the rolling estimate.22 4.2 Effect of short-rate forecast error predictability across maturities Having documented the differential predictive power of the expectations wedge for bond excess returns at short and long maturities, it is important to understand whether these results conform with how short-rate expectations propagate across the term structure. How much of variation in yield forecast errors and, therefore, in unexpected bond returns across maturities can be explained by the variation in short-rate forecast errors alone? I address this question by studying the empirical properties of decomposition (7). With a mean-reverting short rate, the contribution of short-rate forecast errors to the variation in forecast errors of longer-term yields should decline with maturity. To test this prediction, I proxy for yield forecast errors across maturities using survey expectations of longer-maturity yields from the BCFF.23 Table 5, panel A, reports regressions of forecast errors for yields with maturities of one, two, three, and ten years on the contemporaneous FFR forecast errors, all at the four-quarter horizon. The regressions are estimated at the monthly frequency starting in January 1988 and are determined by the availability of longer-maturity yield forecasts. The estimates show declining coefficients and $$R^2$$’s across maturities. An investor overpredicting the next year’s FFR by 100 bps, would also overpredict the level of the two-, five-, and ten-year yields by 80, 48, and 21 bps, respectively. Table 5 Forecast errors of the short rate and longer-maturity yields A. Regressions of yield forecast errors, $$FE_t(y^{(n)}_{t+1})=y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$, on contemporaneous FFR forecast errors (1) (2) (3) (4) $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$FE_t(FFR_{t+1})$$ 0.94 0.80 0.48 0.21 (19.42) (11.45) (5.71) (2.43) Constant –0.12 –0.29 –0.38 –0.26 (–1.35) (–2.49) (–2.82) (–1.78) $$R^2$$ 0.85 0.72 0.39 0.11 N (months) 284 284 284 284 B. Predictability of yield forecast errors: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})=-(rx_{t+1}^{(n+1)}-E^s_t(rx_{t+1}^{(n+1)}))/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Dependent variable: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ Wedge$$^{r}_t$$ 0.87 0.78 0.47 0.24 (6.00) (5.37) (3.35) (1.91) Constant –0.28 –0.41 –0.45 –0.28 (–1.34) (–2.12) (–2.71) (–1.85) $$R^2$$ 0.42 0.39 0.21 0.077 N (months) 284 284 284 284 A. Regressions of yield forecast errors, $$FE_t(y^{(n)}_{t+1})=y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$, on contemporaneous FFR forecast errors (1) (2) (3) (4) $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$FE_t(FFR_{t+1})$$ 0.94 0.80 0.48 0.21 (19.42) (11.45) (5.71) (2.43) Constant –0.12 –0.29 –0.38 –0.26 (–1.35) (–2.49) (–2.82) (–1.78) $$R^2$$ 0.85 0.72 0.39 0.11 N (months) 284 284 284 284 B. Predictability of yield forecast errors: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})=-(rx_{t+1}^{(n+1)}-E^s_t(rx_{t+1}^{(n+1)}))/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Dependent variable: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ Wedge$$^{r}_t$$ 0.87 0.78 0.47 0.24 (6.00) (5.37) (3.35) (1.91) Constant –0.28 –0.41 –0.45 –0.28 (–1.34) (–2.12) (–2.71) (–1.85) $$R^2$$ 0.42 0.39 0.21 0.077 N (months) 284 284 284 284 Dependent variable: realized annual bond return $$rx_{t+1}^{(n+1)}/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Wedge$$^{r}_t$$ –0.88 –0.78 –0.49 –0.24 –0.15 –0.090 –0.068 –0.035 (–9.43) (–7.93) (–4.80) (–2.65) (–1.90) (–1.26) (–1.27) (–0.89) $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ –0.83 –0.89 –0.90 –0.88 (–12.73) (–15.43) (–21.07) (–26.41) Constant 0.43 0.44 0.47 0.44 0.20 0.079 0.069 0.19 (2.47) (2.59) (3.18) (3.41) (2.02) (0.91) (0.94) (3.06) $$R^2$$ 0.44 0.39 0.23 0.085 0.85 0.87 0.85 0.83 N (months) 284 284 284 284 284 284 284 284 Dependent variable: realized annual bond return $$rx_{t+1}^{(n+1)}/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Wedge$$^{r}_t$$ –0.88 –0.78 –0.49 –0.24 –0.15 –0.090 –0.068 –0.035 (–9.43) (–7.93) (–4.80) (–2.65) (–1.90) (–1.26) (–1.27) (–0.89) $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ –0.83 –0.89 –0.90 –0.88 (–12.73) (–15.43) (–21.07) (–26.41) Constant 0.43 0.44 0.47 0.44 0.20 0.079 0.069 0.19 (2.47) (2.59) (3.18) (3.41) (2.02) (0.91) (0.94) (3.06) $$R^2$$ 0.44 0.39 0.23 0.085 0.85 0.87 0.85 0.83 N (months) 284 284 284 284 284 284 284 284 Panel A reports regressions of survey forecast errors for yields with maturities of one, two, five, and ten years on the FFR forecast errors. Forecast errors are computed for the four-quarter-ahead forecast horizon. Panel B shows the predictability of longer-maturity yield forecast errors with the short-rate expectations wedge defined in Equation (13). The bottom part of panel B shows regressions of realized excess returns whose maturities correspond to the survey-based yield forecast errors above. The returns are scaled so that the magnitude of the coefficients is directly comparable with yield forecast errors (see Equation (6)). The regressions are estimated at the monthly frequency. The sample period is 1988M1–2011M8, when the survey forecasts of longer-maturity yields are available in BCFF. Newey-West t-statistics with 18 lags are reported in parentheses. Table 5 Forecast errors of the short rate and longer-maturity yields A. Regressions of yield forecast errors, $$FE_t(y^{(n)}_{t+1})=y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$, on contemporaneous FFR forecast errors (1) (2) (3) (4) $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$FE_t(FFR_{t+1})$$ 0.94 0.80 0.48 0.21 (19.42) (11.45) (5.71) (2.43) Constant –0.12 –0.29 –0.38 –0.26 (–1.35) (–2.49) (–2.82) (–1.78) $$R^2$$ 0.85 0.72 0.39 0.11 N (months) 284 284 284 284 B. Predictability of yield forecast errors: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})=-(rx_{t+1}^{(n+1)}-E^s_t(rx_{t+1}^{(n+1)}))/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Dependent variable: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ Wedge$$^{r}_t$$ 0.87 0.78 0.47 0.24 (6.00) (5.37) (3.35) (1.91) Constant –0.28 –0.41 –0.45 –0.28 (–1.34) (–2.12) (–2.71) (–1.85) $$R^2$$ 0.42 0.39 0.21 0.077 N (months) 284 284 284 284 A. Regressions of yield forecast errors, $$FE_t(y^{(n)}_{t+1})=y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$, on contemporaneous FFR forecast errors (1) (2) (3) (4) $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$FE_t(FFR_{t+1})$$ 0.94 0.80 0.48 0.21 (19.42) (11.45) (5.71) (2.43) Constant –0.12 –0.29 –0.38 –0.26 (–1.35) (–2.49) (–2.82) (–1.78) $$R^2$$ 0.85 0.72 0.39 0.11 N (months) 284 284 284 284 B. Predictability of yield forecast errors: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})=-(rx_{t+1}^{(n+1)}-E^s_t(rx_{t+1}^{(n+1)}))/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Dependent variable: $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ Wedge$$^{r}_t$$ 0.87 0.78 0.47 0.24 (6.00) (5.37) (3.35) (1.91) Constant –0.28 –0.41 –0.45 –0.28 (–1.34) (–2.12) (–2.71) (–1.85) $$R^2$$ 0.42 0.39 0.21 0.077 N (months) 284 284 284 284 Dependent variable: realized annual bond return $$rx_{t+1}^{(n+1)}/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Wedge$$^{r}_t$$ –0.88 –0.78 –0.49 –0.24 –0.15 –0.090 –0.068 –0.035 (–9.43) (–7.93) (–4.80) (–2.65) (–1.90) (–1.26) (–1.27) (–0.89) $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ –0.83 –0.89 –0.90 –0.88 (–12.73) (–15.43) (–21.07) (–26.41) Constant 0.43 0.44 0.47 0.44 0.20 0.079 0.069 0.19 (2.47) (2.59) (3.18) (3.41) (2.02) (0.91) (0.94) (3.06) $$R^2$$ 0.44 0.39 0.23 0.085 0.85 0.87 0.85 0.83 N (months) 284 284 284 284 284 284 284 284 Dependent variable: realized annual bond return $$rx_{t+1}^{(n+1)}/n$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ $$n=1Y$$ $$n=2Y$$ $$n=5Y$$ $$n=10Y$$ Wedge$$^{r}_t$$ –0.88 –0.78 –0.49 –0.24 –0.15 –0.090 –0.068 –0.035 (–9.43) (–7.93) (–4.80) (–2.65) (–1.90) (–1.26) (–1.27) (–0.89) $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ –0.83 –0.89 –0.90 –0.88 (–12.73) (–15.43) (–21.07) (–26.41) Constant 0.43 0.44 0.47 0.44 0.20 0.079 0.069 0.19 (2.47) (2.59) (3.18) (3.41) (2.02) (0.91) (0.94) (3.06) $$R^2$$ 0.44 0.39 0.23 0.085 0.85 0.87 0.85 0.83 N (months) 284 284 284 284 284 284 284 284 Panel A reports regressions of survey forecast errors for yields with maturities of one, two, five, and ten years on the FFR forecast errors. Forecast errors are computed for the four-quarter-ahead forecast horizon. Panel B shows the predictability of longer-maturity yield forecast errors with the short-rate expectations wedge defined in Equation (13). The bottom part of panel B shows regressions of realized excess returns whose maturities correspond to the survey-based yield forecast errors above. The returns are scaled so that the magnitude of the coefficients is directly comparable with yield forecast errors (see Equation (6)). The regressions are estimated at the monthly frequency. The sample period is 1988M1–2011M8, when the survey forecasts of longer-maturity yields are available in BCFF. Newey-West t-statistics with 18 lags are reported in parentheses. Figure 5 illustrates the contribution of short-rate forecast errors to the variation in realized excess returns at the two-year maturity, $$rx_{t+1}^{(2)}$$. Panel A presents the survey-based decomposition of $$rx_{t+1}^{(2)}$$ into the expected and unexpected components. Using Equations (5) and (6), we have $$rx_{t+1}^{(2)} = [f_t^{(1,1)}-E_t^s(y_{t+1}^{(1)})] - [y_{t+1}^{(1)}-E_t^s(y_{t+1}^{(1)})]$$, where the first term is the expected return (risk premium) and the second term is the unexpected return (time and maturities are measured in years). The unexpected return has a strong countercyclical pattern, whereas the survey-based expected return moves on a frequency higher than the business cycle; in that sense it resembles the risk premium estimates from yields-plus-expected inflation regressions (Figure 2). Panel B of Figure 5 superimposes the unexpected return on the two-year bond against (the negative of) the FFR forecast error at the four-quarter horizon, confirming their high correlation. The plot makes clear that the short-rate forecast error dominates in contributing to the variation in the unexpected return relative to the other two terms in decomposition (7). Figure 5. View largeDownload slide Decomposing realized excess return on a two-year Treasury bond Panel A shows the decomposition of the realized excess return on a two-year bond into an expected return (risk premium) and unexpected return (forecast error): $$rx_{t+1}^{(2)}=E_t^s(rx_{t+1}^{(2)})+ Urx_{t+1}^{s,(2)}$$, where $$E_t^s(rx_{t+1}^{(2)})=f_t^{(1,1)}-E_t^s(y_{t+1}^{(1)})$$ and $$Urx_{t+1}^{s,(2)}=-[y_{t+1}^{(1)}-E_t^s(y_{t+1}^{(1)})]$$, and $$E_t^s(y_{t+1}^{(1)})$$ is the four-quarter-ahead forecast of the one-year yield from the BCFF survey. Panel B superimposes the unexpected return $$Urx_{t+1}^{s,(2)}$$ with the (negative of) the FFR forecast error at the four-quarter horizon. Figure 5. View largeDownload slide Decomposing realized excess return on a two-year Treasury bond Panel A shows the decomposition of the realized excess return on a two-year bond into an expected return (risk premium) and unexpected return (forecast error): $$rx_{t+1}^{(2)}=E_t^s(rx_{t+1}^{(2)})+ Urx_{t+1}^{s,(2)}$$, where $$E_t^s(rx_{t+1}^{(2)})=f_t^{(1,1)}-E_t^s(y_{t+1}^{(1)})$$ and $$Urx_{t+1}^{s,(2)}=-[y_{t+1}^{(1)}-E_t^s(y_{t+1}^{(1)})]$$, and $$E_t^s(y_{t+1}^{(1)})$$ is the four-quarter-ahead forecast of the one-year yield from the BCFF survey. Panel B superimposes the unexpected return $$Urx_{t+1}^{s,(2)}$$ with the (negative of) the FFR forecast error at the four-quarter horizon. To map these results into bond return predictability, in Table 5, panel B, I project yield forecast errors onto the expectations wedge. For brevity, I only report results using the rolling estimate of the wedge. Starting from the identity $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})=-(rx_{t+1}^{(n+1)}-E^s_t(rx_{t+1}^{(n+1)}))/n$$, if part of return predictability is induced by ex post predictable short-rate forecast errors, and if this predictability is well captured by the difference between econometric and survey expectations, then the wedge should predict $$y_{t+1}^{(n)}-E_t^s(y_{t+1}^{(n)})$$ with a declining coefficient across maturities. Additionally, predictive regressions of (appropriately scaled) excess bond returns, $$rx_{t+1}^{(n+1)}/n$$, should recover these coefficients as well. Both predictions are confirmed in the data. The wedge predicts unexpected yield changes (realized excess returns) with a coefficient of 0.87 (-0.88) for $$n=1$$ years and 0.24 ($$-$$0.24) for $$n=10$$ years. I also show that controlling for yield forecast errors drives out the significance of the wedge for bond return predictability, as should be the case if predictability stems from the ex ante unexpected return component rather than the risk premium. 5. Decomposing Short-Rate Forecast Errors I estimate a dynamic model of short-rate expectations. The model has two goals. First, it allows me to study the contribution of short-rate forecast errors to yield changes across maturities in a setting where risk premium is zero by construction. The model-based results corroborate the interpretation of the regression-based results presented in Section 4. Second, I use the model to analyze the economic sources of short-rate forecast errors. I find that about 82% of variation in short-rate forecast errors at the four-quarter horizon is unrelated to inflation shocks and therefore pertains to real-rate shocks. Within those 82%, 26% can be explained by shocks to unemployment, consistent with Figure 1 in the Introduction, and the remaining 56% by other factors driving the short rate, most importantly monetary policy shocks.24 5.1 Setting I specify an empirical model in the spirit of a monetary vector autoregression (VAR) (e.g., Stock and Watson 2001), which I estimate to match survey expectations of inflation, FFR and unemployment at different horizons. While the model imposes a parametric structure on how the short rate and macro variables are related, it is not intended to produce the best forecasts of the short rate. Rather, it serves to provide a framework for a joint interpretation of expectations of key variables and their shocks as perceived by the survey forecasters in real time. Vector $$X_t$$ describes the state of the economy and includes CPI inflation, unemployment and the nominal short rate: $$X_t = ( \pi_t, u_t, i_t)'$$. I follow a large literature that highlights the need to account for persistent variation in macroeconomic series: in inflation (Stock and Watson 2007; Cogley, Primiceri, and Sargent 2010), and in the real rate (e.g., Laubach and Williams 2003). I assume that $$X_t$$ has a trend ($$X_t^*$$) and a cycle ($$X_t^c$$) component: $$X_t = X_t^* + X_t^c$$. The cycle $$X_t^c$$ is interpreted as deviations of macroeconomic variables from their long-run means, and evolves as a VAR(2) at quarterly frequency: \begin{equation} X_t^c = \sum_{j=1}^2 \Phi_j X_{t-j}^c + \Sigma_c \varepsilon_t^c. \end{equation} (16) In equation (16) and in the remainder of Section 5, one period denotes one quarter. Vector $$\varepsilon_t^c = \left(\varepsilon_t^{\pi^c}, \varepsilon_t^{u}, \varepsilon^{i^c}_t\right)'$$ represents iid normally distributed structural shocks. I impose a recursive identification scheme assuming that $$\Sigma_c$$ is lower triangular. The long-run mean of inflation, $$\pi_t^{*}$$, and the long-run mean of the real rate, $$r_t^{*}$$, are random walks \begin{align} \pi_t^* &= \pi_{t-1}^* + \sigma_{\pi^*} \varepsilon_t^{\pi^*} \\ \end{align} (17) \begin{align} r_t^* &= r^*_{t-1} + \sigma_{r^*} \varepsilon_t^{r^*}, \end{align} (18) with shocks $$\varepsilon_t^{\pi^*}$$, $$\varepsilon_t^{r^*}$$ iid normally distributed. I further assume that the long-run mean of the nominal short-rate satisfies \begin{align} i^*_t = r_t^* + \pi_t^*. \end{align} (19) This specification implies that interest rates and inflation have a unit root component as well as a mean-reverting component. The unit root assumption is for empirical convenience and parsimony. From an economic standpoint, it is reasonable to assume that interest rates are stationary. However, in small samples it is not possible to distinguish a highly persistent but a stationary series from a unit root (e.g., Campbell and Perron 1991; Duffee 2012 in the context of interest rates). I collect state variables in vector $$Z_t= \left(X_t^{c\prime}, X_{t-1}^{c\prime},\pi_t^*, r_t^*\right)'$$: \begin{equation} Z_t = A_ZZ_{t-1} + \Sigma_Z \varepsilon_t^Z, \end{equation} (20) where $$\varepsilon_t^Z=\left(\varepsilon_t^{c\prime},\varepsilon_t^{\pi^*},\varepsilon_t^{r^*}\right)'$$. The first six elements of $$Z_t$$ include elements of $$X_t^c$$ written in a companion form and the remaining two elements collect inflation and real-rate trends. 5.2 Estimation I estimate the model with maximum likelihood via Kalman filter, with quarterly sampling of the data over the 1984Q3–2011Q3 sample. There are 15 measurements, denoted $$M_t$$, which include current CPI inflation (quarterly rate), civilian unemployment rate, the FFR, as well as survey forecasts of those variables for horizons from one to four quarters ahead. Rather than actual realizations of macro variables, I use forecasts for the current quarter (nowcasts). Doing so ensures that the measurements for the realized macro variables are consistent with variables being predicted in the survey and are in the information set of forecasters, avoiding problems related to a publication lag of inflation and unemployment. I assume that all measurements are measured with zero-mean normally distributed iid noise, $$\eta_t$$. Measurements can be expressed as linear functions in $$Z_t$$: \begin{align} M_t = \bar{\mathcal{M}} + \mathcal{H}Z_t + \eta_t, \end{align} (21) with appropriately specified parameters $$\bar{\mathcal{M}}$$ and $$\mathcal{H}$$. The current quarter FFR is obtained $$FFR_t=i^*_t + e_3'Z_t + \eta_t^{FFR}$$, where vector $$e_3$$ is a unit vector with third element equal to one and zeros elsewhere. Standard errors for the parameters are constructed with Newey-West adjustment (with six quarterly lags), treating maximum likelihood scores as moment conditions. For nonlinear functions of the parameters, I obtain standard errors by Monte Carlo simulation, drawing parameters from a multivariate normal distribution with mean equal to parameter point estimates and with the Newey-West-adjusted variance-covariance matrix. Internet Appendix D provides details on the model specification and parameter estimates. 5.3 Effect of short-rate forecast errors across the term structure The model in (16)–(19) allows to study how unexpected changes in the short rate affect the term structure of interest rates across maturities. In particular, it allows to verify the previous regression-based results in Table 5 and the diminishing effect of short-rate expectations errors across maturities. It is important to note that while the argument in Table 5 relies on using survey expectations of longer-maturity yields (which in addition to short-rate expectations are also affected by term premiums), the estimates from the model exploit only expectations of the FFR at horizons from one to four quarters ahead. Since the model abstracts from the variation in the term premiums, it allows to study the pure expectations hypothesis component in longer-maturity yields, that is, $$EH_t^{(n)} \equiv \frac{1}{n} \sum_{k=0}^{n-1} E_t \left(i_{t+k}\right).$$ Using model estimates, I perform the decomposition in Equation (7) assuming that unexpected changes in the term premium are zero. I then recover the model-implied coefficients from regression of unexpected yield changes at different maturities, $$(y_{t+h}^{(n)}-E_t(y_{t+h}^{(n)}))$$, on the short-rate forecast errors, $$(i_{t+h}-E_t(i_{t+h}))$$, in analogy to those reported in Table 5, panel A. For a yield with maturity of $$n$$ periods, the regression coefficient \begin{align} \beta^{(n)} = 1 + \frac{Cov(i_{t+h}-E_t(i_{t+h}),\sum_{k=1}^{n-1} \frac{n-k}{n}(E_{t+h}-E_t)(\Delta i_{t+h+k}))}{Var(i_{t+h}-E_t(i_{t+h}))} \end{align} (22) can be calculated explicitly based on model estimates. Internet Appendix D.4 provides the details. The $$\beta^{(n)}$$ coefficient has the interpretation that, given a 100 bps unexpected change in the short rate from $$t$$ to $$t+h$$, the $$n$$-period rate should unexpectedly change by $$100\beta^{(n)}$$ bps over the same time interval. The mean-reversion in the short rate (around trend components) makes the covariance term in (22) negative; thus, one expects $$\beta^{(n)}$$ to decline with $$n$$, consistent with the discussion in Section 2.2. Figure 6 presents the model-implied coefficients for horizon $$h=4$$ quarters and maturities $$n$$ between four and forty quarters (expressed in years in the graph) together with a 5% confidence interval. Superimposed against model-implied numbers are regression estimates based on survey forecasts of longer-maturity yields from Table 5, panel A, also with 5% confidence intervals.25 Even though the two sets of estimates are obtained using different information and methods, they line up closely with each other, and show a similar decay across maturities. This suggests that the previous regression-based results are not significantly distorted by the term premiums embedded in forecasts of longer-maturity yields. The model implies that given a 100 bps unexpected decline in the short rate over a four-quarter period, the two-, five- and ten-year yield should also unexpectedly decline by 85, 55, and 40 bps, respectively. Figure 6. View largeDownload slide Impact of short-rate forecast errors across maturities The coefficients measure the effect of the short-rate forecast error on the forecast error about longer-term yields. Model-implied coefficients (stars) are obtained from the model in Section 5, Equation (22), and are plotted together with a 95% confidence interval. Survey-based coefficients (diamonds) are obtained from regressions in Table 5, panel A, plotted also with a 95% confidence interval. Figure 6. View largeDownload slide Impact of short-rate forecast errors across maturities The coefficients measure the effect of the short-rate forecast error on the forecast error about longer-term yields. Model-implied coefficients (stars) are obtained from the model in Section 5, Equation (22), and are plotted together with a 95% confidence interval. Survey-based coefficients (diamonds) are obtained from regressions in Table 5, panel A, plotted also with a 95% confidence interval. 5.4 Economic sources of short-rate forecast errors Short-rate forecast errors reflect a combined effect of unexpected changes in the economy and in the stance of monetary policy over a particular forecast horizon. Given their size and persistence, an important question is whether these errors pertain to the inflation or the real-rate component of the short rate. I use the model to answer this question. 5.4.1 Forecast error variance decomposition The model has two nominal shocks: cyclical inflation shock $$\varepsilon^{\pi^c}_t$$ and trend inflation shock $$\varepsilon^{\pi^*}_t$$. The remaining three shocks, cyclical short-rate shock $$\varepsilon^{i^c}_t$$, unemployment shock $$\varepsilon_t^{u}$$, and trend real-rate shock $$\varepsilon^{r^*}_t$$ can be interpreted as shocks to the real interest rate. Using model estimates, I decompose FFR forecast errors at a given horizon into forecast errors due to each of the shocks. The forecast error for the state vector is \begin{equation} FE_t(Z_{t+h}) = Z_{t+h} - E_t(Z_{t+h}) = \sum_{k=1}^h A_Z^{h-k} \Sigma_Z \varepsilon^Z_{t+k}. \end{equation} (23) Defining $$\widetilde{M}_t=\bar{\mathcal{M}} + \mathcal{H}Z_t$$ as the true, that is, measurement-error-free, vector of macro variables, and their survey forecasts, one obtains the forecast error for FFR as \begin{equation} FE_t(\widetilde{FFR}_{t+h}) = e_3'\mathcal{H}\,FE_t(Z_{t+h}). \end{equation} (24) I denote variance-covariance matrix of $$Z_t$$ forecast errors due to all shocks as $$Var(FE_t(Z_{t+h}))$$ and variance-covariance matrix due to $$j$$-th as shock $$Var(FE_t(Z_{t+h})|e_j'\varepsilon^Z)$$. Then the fraction of FFR forecast error variance generated by $$j$$-th shock is given by the variance ratio: \begin{equation} \mathcal{VRM}_{h,FFR}^{j} = \frac{e_3'\mathcal{H}\, Var(FE_t(Z_{t+h})|e_j'\varepsilon^Z) \, \mathcal{H}' e_3}{e_3'\mathcal{H}\,Var(FE_t(Z_{t+h})) \,\mathcal{H}' e_3}. \end{equation} (25) 5.4.2 Results Figure 7, panel A, displays variance decomposition of FFR forecast errors at horizon of four quarters ($$h=4$$ in Equation (25)). The largest contributors are the cyclical shocks to the short rate ($$\varepsilon^{i^c}_t$$, 43% of the overall variance) and shocks to unemployment ($$\varepsilon_t^{u}$$, above 26% of the variance). The effect of $$\varepsilon^{i^c}_t$$ summarizes monetary policy shocks as well as other shocks that drive the short rate (and the Fed’s reaction function) that the model does not explicitly account for. The contribution of trend real-rate shocks $$\varepsilon^{r^*}_t$$ is 13%. Importantly, inflation shocks, cyclical $$\varepsilon_t^{\pi^c}$$ and trend $$\varepsilon_t^{\pi^*}$$, jointly account for only 18% of the overall FFR forecast error variance.26Figure 7, panel B, plots the combined contributions of inflation and real-rate shocks to FFR forecast errors over time. The graph makes clear that most of forecast error variation at the business cycle frequency stems from the real-rate shocks. As a result, real-rate forecast errors account for the entire predictability of short-rate forecast errors documented in Section 3.3 (see Internet Appendix Table A-7 for details on this result). Figure 7. View largeDownload slide Contribution of inflation and real-rate shocks to short-rate forecast errors The figure presents a decomposition of FFR forecast errors at the horizon of four-quarters ahead. Panel A displays unconditional variance ratios for FFR forecast errors from Equation (25), corresponding to each of the model’s shocks: $$(\varepsilon_{t}^{\pi^c},\varepsilon_t^{u},\varepsilon_t^{i^c},\varepsilon_t^{\pi^*},\varepsilon_t^{r^*})^\prime$$. One-standard-error bars are included. Panel B shows the decomposition of the short-rate forecast errors over time. Survey-based FFR forecast error is plotted together with its decomposition into a component stemming from inflation shocks $$(\varepsilon_{t}^{\pi^c},\varepsilon_t^{\pi^*})$$ and real-rate shocks $$(\varepsilon_t^{u},\varepsilon_t^{i^c},\varepsilon_t^{r^*})$$. Figure 7. View largeDownload slide Contribution of inflation and real-rate shocks to short-rate forecast errors The figure presents a decomposition of FFR forecast errors at the horizon of four-quarters ahead. Panel A displays unconditional variance ratios for FFR forecast errors from Equation (25), corresponding to each of the model’s shocks: $$(\varepsilon_{t}^{\pi^c},\varepsilon_t^{u},\varepsilon_t^{i^c},\varepsilon_t^{\pi^*},\varepsilon_t^{r^*})^\prime$$. One-standard-error bars are included. Panel B shows the decomposition of the short-rate forecast errors over time. Survey-based FFR forecast error is plotted together with its decomposition into a component stemming from inflation shocks $$(\varepsilon_{t}^{\pi^c},\varepsilon_t^{\pi^*})$$ and real-rate shocks $$(\varepsilon_t^{u},\varepsilon_t^{i^c},\varepsilon_t^{r^*})$$. 6. Short-Rate Expectations of the Public and the Fed 6.1 Ex post predictability of monetary policy shocks As suggested by the above decomposition, a large part of short-rate forecast errors stems from shocks other than shocks to inflation and unemployment. Monetary policy shocks are part of this residual variation. Thus, to further distill sources of predictability in short-rate forecast errors, I analyze the predictability of monetary policy shocks extracted from the Fed fund futures contracts at the frequency of FOMC meetings. By capturing updates to investors’ short-rate expectations within a narrow window at each FOMC meeting, these shocks are innovations relative to the information set of investors just prior to the meeting. As such, they represent a subset of shocks driving survey-based FFR forecast errors. Under FIRE, shocks should not be predictable by any predetermined variables. Instead, if some variables contain information about the monetary policy component of the short rate that is not in the information set of investors, those variables should predict monetary policy shocks. To verify this intuition, I consider monetary policy shocks from Kuttner (2001) and Swanson (2017). Swanson (2017) updates the estimates of Gürkaynak Sack, and Swanson (2005, GSS). These studies differ in the range of futures’ maturities that they use and details of the identification strategy.27 Following GSS, shocks fall into two categories—shocks to the current Fed’s interest rate target (target shocks) and shocks to the future interest rate path (path shocks)—which provide a way to distinguish between the effects of Fed actions versus the effects of Fed communication of their own expectations about the policy path. Kuttner (2001) captures only the target shocks by focusing on the futures contract with the shortest maturity, whereas Swanson/GSS separate these two components by including contracts with longer maturities. Table 6 provides the summary statistics for these shocks. Table 6 Predicting monetary policy shocks (1) (2) (3) (4) Kuttner Kuttner (sched) Swanson (target) Swanson (path) A. Summary statistics for monetary policy shocks (monthly, in bps) Sample 1989M6–2008M6 1991M7-2008M12 Mean –2.54 –0.54 0.04 0.04 SD 7.89 4.28 5.84 4.10 Min –57.39 –34.43 –36.62 –15.81 Max 12.14 12.14 12.38 22.13 B. Dependent variable: Monthly shocks, $$\varepsilon_{t+1/12}^{MP}$$ $$FFR_t$$ –1.53 –0.29 –0.96 –0.13 (–6.41) (–1.89) (–3.21) (–0.79) $$\Delta EMP_{t-1,t}$$ 2.42 0.30 1.50 0.61 (5.83) (0.95) (3.93) (2.69) Constant 1.19 0.39 2.07 –0.19 (1.07) (0.70) (2.52) (–0.27) $$\bar{R}^2$$ 0.11 0.00 0.05 0.01 N (months) 228 228 210 210 C. Dependent variable: Cumulative 12-month shocks, $$\sum_{i=1}^{12}\varepsilon_{t+i/12}^{MP} $$ $$FFR_t$$ –21.34 –4.29 –14.62 –1.92 (–7.95) (–3.26) (–3.78) (–1.44) $$\Delta EMP_{t-1,t}$$ 23.53 4.47 13.43 5.50 (8.80) (2.12) (4.36) (2.18) Constant 37.62 7.29 45.03 2.03 (3.41) (1.50) (3.79) (0.34) $$\bar{R}^2$$ 0.69 0.24 0.45 0.09 N (months) 216 216 198 198 (1) (2) (3) (4) Kuttner Kuttner (sched) Swanson (target) Swanson (path) A. Summary statistics for monetary policy shocks (monthly, in bps) Sample 1989M6–2008M6 1991M7-2008M12 Mean –2.54 –0.54 0.04 0.04 SD 7.89 4.28 5.84 4.10 Min –57.39 –34.43 –36.62 –15.81 Max 12.14 12.14 12.38 22.13 B. Dependent variable: Monthly shocks, $$\varepsilon_{t+1/12}^{MP}$$ $$FFR_t$$ –1.53 –0.29 –0.96 –0.13 (–6.41) (–1.89) (–3.21) (–0.79) $$\Delta EMP_{t-1,t}$$ 2.42 0.30 1.50 0.61 (5.83) (0.95) (3.93) (2.69) Constant 1.19 0.39 2.07 –0.19 (1.07) (0.70) (2.52) (–0.27) $$\bar{R}^2$$ 0.11 0.00 0.05 0.01 N (months) 228 228 210 210 C. Dependent variable: Cumulative 12-month shocks, $$\sum_{i=1}^{12}\varepsilon_{t+i/12}^{MP} $$ $$FFR_t$$ –21.34 –4.29 –14.62 –1.92 (–7.95) (–3.26) (–3.78) (–1.44) $$\Delta EMP_{t-1,t}$$ 23.53 4.47 13.43 5.50 (8.80) (2.12) (4.36) (2.18) Constant 37.62 7.29 45.03 2.03 (3.41) (1.50) (3.79) (0.34) $$\bar{R}^2$$ 0.69 0.24 0.45 0.09 N (months) 216 216 198 198 Monetary policy shocks in Columns 1 and 2 are from Ken Kuttner’s Web site (Kuttner (2001)), and in Columns 3 and 4—from Swanson (2017), who updates the estimates of Gürkaynak, Sack and Swanson (2005). Shocks are identified at the FOMC meeting frequency and converted into monthly frequency following the approach of Romer and Romer (2004). Swanson’s shocks are standardized so that the first (fourth) futures contract loads with a unit coefficient on the target (path) shock over the 1991M7–2008M12 sample period. Columns 1, 3 and 4 contain all FOMC announcement days (both scheduled and unscheduled); Column 2 uses Kuttner’s shocks only on scheduled FOMC announcement days. Panel A reports summary statistics in basis points. Panels B and C report the predictability of shocks by lagged predictors. Panel B displays results for monthly shocks realized in month $$t+1/12$$ and panel B for shocks accumulated over a year from $$t+1/12$$ to $$t+1$$. t-statistics in parentheses are Newey-West adjusted with 18 lags. All shocks are in basis points. Table 6 Predicting monetary policy shocks (1) (2) (3) (4) Kuttner Kuttner (sched) Swanson (target) Swanson (path) A. Summary statistics for monetary policy shocks (monthly, in bps) Sample 1989M6–2008M6 1991M7-2008M12 Mean –2.54 –0.54 0.04 0.04 SD 7.89 4.28 5.84 4.10 Min –57.39 –34.43 –36.62 –15.81 Max 12.14 12.14 12.38 22.13 B. Dependent variable: Monthly shocks, $$\varepsilon_{t+1/12}^{MP}$$ $$FFR_t$$ –1.53 –0.29 –0.96 –0.13 (–6.41) (–1.89) (–3.21) (–0.79) $$\Delta EMP_{t-1,t}$$ 2.42 0.30 1.50 0.61 (5.83) (0.95) (3.93) (2.69) Constant 1.19 0.39 2.07 –0.19 (1.07) (0.70) (2.52) (–0.27) $$\bar{R}^2$$ 0.11 0.00 0.05 0.01 N (months) 228 228 210 210 C. Dependent variable: Cumulative 12-month shocks, $$\sum_{i=1}^{12}\varepsilon_{t+i/12}^{MP} $$ $$FFR_t$$ –21.34 –4.29 –14.62 –1.92 (–7.95) (–3.26) (–3.78) (–1.44) $$\Delta EMP_{t-1,t}$$ 23.53 4.47 13.43 5.50 (8.80) (2.12) (4.36) (2.18) Constant 37.62 7.29 45.03 2.03 (3.41) (1.50) (3.79) (0.34) $$\bar{R}^2$$ 0.69 0.24 0.45 0.09 N (months) 216 216 198 198 (1) (2) (3) (4) Kuttner Kuttner (sched) Swanson (target) Swanson (path) A. Summary statistics for monetary policy shocks (monthly, in bps) Sample 1989M6–2008M6 1991M7-2008M12 Mean –2.54 –0.54 0.04 0.04 SD 7.89 4.28 5.84 4.10 Min –57.39 –34.43 –36.62 –15.81 Max 12.14 12.14 12.38 22.13 B. Dependent variable: Monthly shocks, $$\varepsilon_{t+1/12}^{MP}$$ $$FFR_t$$ –1.53 –0.29 –0.96 –0.13 (–6.41) (–1.89) (–3.21) (–0.79) $$\Delta EMP_{t-1,t}$$ 2.42 0.30 1.50 0.61 (5.83) (0.95) (3.93) (2.69) Constant 1.19 0.39 2.07 –0.19 (1.07) (0.70) (2.52) (–0.27) $$\bar{R}^2$$ 0.11 0.00 0.05 0.01 N (months) 228 228 210 210 C. Dependent variable: Cumulative 12-month shocks, $$\sum_{i=1}^{12}\varepsilon_{t+i/12}^{MP} $$ $$FFR_t$$ –21.34 –4.29 –14.62 –1.92 (–7.95) (–3.26) (–3.78) (–1.44) $$\Delta EMP_{t-1,t}$$ 23.53 4.47 13.43 5.50 (8.80) (2.12) (4.36) (2.18) Constant 37.62 7.29 45.03 2.03 (3.41) (1.50) (3.79) (0.34) $$\bar{R}^2$$ 0.69 0.24 0.45 0.09 N (months) 216 216 198 198 Monetary policy shocks in Columns 1 and 2 are from Ken Kuttner’s Web site (Kuttner (2001)), and in Columns 3 and 4—from Swanson (2017), who updates the estimates of Gürkaynak, Sack and Swanson (2005). Shocks are identified at the FOMC meeting frequency and converted into monthly frequency following the approach of Romer and Romer (2004). Swanson’s shocks are standardized so that the first (fourth) futures contract loads with a unit coefficient on the target (path) shock over the 1991M7–2008M12 sample period. Columns 1, 3 and 4 contain all FOMC announcement days (both scheduled and unscheduled); Column 2 uses Kuttner’s shocks only on scheduled FOMC announcement days. Panel A reports summary statistics in basis points. Panels B and C report the predictability of shocks by lagged predictors. Panel B displays results for monthly shocks realized in month $$t+1/12$$ and panel B for shocks accumulated over a year from $$t+1/12$$ to $$t+1$$. t-statistics in parentheses are Newey-West adjusted with 18 lags. All shocks are in basis points. Analogous to regressions in Table 3, panel C, I analyze the predictability of monetary policy shocks by $$FFR_t$$ and the growth rate of employment, $$\Delta EMP_{t-1,t}$$. In panel A of Table 6, I project shocks observed in month $$t+\frac{1}{12}$$, denoted $$\varepsilon_{t+1/12}^{MP}$$, on month $$t$$ predictors. In panel B, I report similar regressions for cumulative twelve-month shocks realized over the course of the following year, $$\sum_{i=1}^{12}\varepsilon_{t+i/12}^{MP}$$. The two variables predict 10.5% of the variation in the next month’s Kuttner’s shocks, and 69% in the cumulative twelve-month shocks. A one-percentage-point decline in employment growth over the past year predicts a cumulative easing shock of $$23.5$$ bps over the next year ($$t = 8.8$$). Similar estimates are obtained with Swanson/GSS target shocks (over slightly different sample period) albeit the economic magnitude is smaller at $$13.4$$ bps. These results can be compared with the previous survey-based estimates. Recall that a 100-basis-point decline in $$\Delta EMP_{t-1,t}$$ predicts a 53-basis-point decline in the FFR over next four quarters that was unexpected by survey forecasters at time $$t$$ (last column in Table 3, panel C). Based on variance decomposition in Section 5.4, of those 53 bps, about 23 bps ($$=$$53bps$$\times$$43%) is due to cyclical short-rate shocks ($$\varepsilon^{i^c}_t$$), which are the model-based interpretation of monetary policy shocks (see Figure 7, panel A). Thus, survey-based results fall in a similar range as those using high-frequency identified monetary policy shocks. The above results invite the question whether the Fed is better able to predict the future path of the short rate than the public. Two results suggest that this may not be the case. First, contrary to target shocks, the predictability of path shocks is much weaker (Column 4 of Table 6). This is important because path shocks are news that the Fed transmits to the public about their own expectations of how monetary policy should evolve going forward. Second, the strong predictability of target shocks stems in large part from unscheduled FOMC announcement days (compare Columns 1 and 2 of Table 6). Out of 24 unscheduled meetings during the 1989–2008 period (13% of all meetings), all but one decisions were easings, constituting 46% of all easing moves in this period. Typically, unscheduled monetary policy decisions happen as the Fed responds to unexpected economic events. For example, before 1994 unscheduled surprise easings reflected the Fed’s response to a weak employment report (Bernanke and Kuttner 2005); that is, Fed eased more than the public expected in response to negative economic news. Post-1994, surprise easing came in response to turbulence in financial markets (Cieslak, Morse, and Vissing-Jorgensen 2016). This suggests that the ex post predictability of FFR forecast errors arises, at least in part, due the Fed acting more aggressively than expected in response to surprise economic events. In fact, policy makers may not be better forecasters than the public, and themselves may exhibit persistent forecast errors. Narrative evidence presented next supports this view. 6.2 Narrative evidence on the nature of policy makers’ forecast errors The challenges related to forecasting the business cycle and the associated path of interest rates in real time are well appreciated among monetary policy makers and Fed economists. Greenspan recognizes that the “success of monetary policy depends importantly on the quality of forecasting” and writes, “As the transcripts of FOMC meetings attest, making monetary policy is an especially humbling activity. In hindsight, the paths of inflation, real output, stock prices, and exchange rates may have seemed preordained, but no such insight existed as we experienced it at the time. (...) our ability to anticipate was limited. From time to time the FOMC made decisions, some to move and some not to move, that we came to regret” (Greenspan 2004, p. 40). A similar concern about forecast accuracy is expressed by Bernanke (2015): “The accuracy of both central bank and private-sector forecasters has been extensively studied and the results are not impressive. Unfortunately, beyond a quarter or two, the course of the economy is extremely hard to forecast. That said, careful projections are essential for coherent monetary policymaking (...)” (Bernanke 2015, Kindle locations 904–907). Transcripts of the FOMC meetings provide a real-time record of the difficulties that forecasters face. The FOMC members and staff economists at the Fed seem aware of the systematic nature of forecast errors at different stages of the business cycle. For example, in the second half of the 1990s, the FOMC members struggled to reconcile the strong consumer demand with the lack of inflationary pressures. While with hindsight it became evident that the U.S. economy had experienced a sequence of positive supply-side shocks, in real time it was much harder to realize that such development was taking place, let alone to anticipate it (e.g