Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors
Abstract
Consider the partly linear regression model
$$
y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n
$$
, where y
i
’s are
responses,
$$
x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T}
$$
are known and nonrandom design points,
$$
{\cal T}
$$
is a compact
set in the real line
$$
{\cal R}
$$
, β =
(β
1, ··· , β
p
)'
is an unknown parameter vector, g(·) is an unknown function and
{ε
i
} is a linear process,
i.e.,
$$
\varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} }
$$
, where
e
j
are i.i.d. random variables with zero
mean and variance
$$
\sigma ^{2}_{e}
$$
. Drawing upon B-spline estimation of g(·) and
least squares estimation of β, we construct estimators of the
autocovariances of {ε
i
}. The uniform
strong convergence rate of these estimators to their true values
is then established. These results not only are a compensation
for those of [23], but also have some application in modeling
error structure. When the errors {ε
i
} are
an ARMA process, our result can be used to develop a consistent
procedure for determining the order of the ARMA process and
identifying the non-zero coeffcients of the process. Moreover,
our result can be used to construct the asymptotically effcient
estimators for parameters in the ARMA error process.