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Root-\(N\)-consistent estimation of partially linear time series models. (English) Zbl 0953.62094
A partially linear model for a random vector \((X,Y',Z')'\) of dimension \((1+p+q)+1\), is represented by \(E(Y |X,Z)= X'\gamma+ \theta(Z)\) almost surely, where \(\gamma\) is a parameter vector and \(\theta\) is a function. This paper considers the \(\sqrt {n}\)-consistent estimation of \(\gamma\) with time series data \((Y_t, X_t', Z_t')'\), \(t=1,2,\dots, n\).
The authors show that in this case, if conditions similar to those used with i.i.d. data by P.M. Robinson [Econometrica 56, No. 4, 931-954 (1988; Zbl 0647.62100)] hold, \(\sqrt{n}\)-consistency still holds, provided \(q\leq 3\) and a second order kernel is used. This result plus other tools can also be used to generalize a consistent test for testing a partially linear model versus a nonparametric regression model in the time-series framework.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
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