Consider the partly linear regression model
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y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n
$$\end{document}, where yi’s are
responses, \documentclass[12pt]{minimal}
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x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T}
$$\end{document}are known and nonrandom design points, \documentclass[12pt]{minimal}
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{\cal T}
$$\end{document} is a compact
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{\cal R}
$$\end{document}, β =
(β1, ··· , βp)'
is an unknown parameter vector, g(·) is an unknown function and
{εi} is a linear process,
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\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 } }} }
$$\end{document} , where
ej are i.i.d. random variables with zero
mean and variance \documentclass[12pt]{minimal}
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\sigma ^{2}_{e}
$$\end{document}. Drawing upon B-spline estimation of g(·) and
least squares estimation of β, we construct estimators of the
autocovariances of {εi}. 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 {εi} 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.
