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Simple regressions with linear time trends. (English) Zbl 0974.62068

The author considers the following problem. Let \(x_t\) be a scalar trend stationary time series \(x_t =\gamma_x+\mu_x t+x_t^0\), \(\mu_x \neq 0\), \(t=1, 2,\ldots,T.\) The purely stochastic deviations from the linear trend are weakly stationary with zero mean. The scalar series \(y_t\) is given by \(y_t = a+b x_t +u_t\), \(t=1, 2,\ldots,T\), where the error \(u_t\) is again a zero-mean weakly stationary process that may be correlated with \(x_t^0.\) The problem is to find the limiting distribution of the ordinary least squares (OLS) estimator of \(a\) and \(b\). Let \[ \hat b-b=\sum(x_t-\bar x)u_t/\sum (x_t-\bar x)^2, \]
\[ \omega_u^2=V(u_t)+2\sum_{\tau=1}^{\infty} \text{cov} (u_t, u_{t-\tau}),\quad T^{-1}\sum (u_t -\bar u)^2 \Rightarrow V(u_t), \]
\[ \hat u_t=y_t -\bar y - \hat b(x_t -\bar x),\quad s^2 =T^{-1}\sum \hat u_t^2, \]
\[ R^2=1-s^2/T^{-1} \sum (y_t - \bar y)^2,\quad \hat\rho_u =(T^{-1}\sum_{t=2}^T \hat u_t \hat u_{t-1})/s^2, \]
\[ t_b=(\hat b -b) \bigl\{\sum (x_t - \bar x)^2 \bigr\}^{1/2}/s. \] The main result of this paper is the following. Under some assumptions on the OLS regression \(y_t = \hat b x_t + \hat u_t\), \(t=1, 2, \ldots, T\), the following assertion is true: \[ T^{1.5}(\hat b -b) \Rightarrow N\biggl(0, 12 \omega_u^2/\mu_x^2\biggr),\quad t_b \Rightarrow N\biggl(0, \omega_u^2/\sigma_u^2\biggr), \]
\[ s^2 {\buildrel P \over \longrightarrow} \sigma_u^2,\quad \hat{\rho_u} {\buildrel P \over \longrightarrow} \text{cov} (u_t, u_{t-1})/\sigma_u^2, \quad T^2 (1-R^2) {\buildrel P \over \longrightarrow} 12 \sigma_u^2/\mu_x^2 b^2, \] as \(T\to \infty\) with \(\sigma_u^2 = V(u_t)\).
The author concludes that when a linear trend dominates the stochastic components the rates of convergence and the limit distributions of OLS statistics are exactly the same as in the case of cointegrated regressions with drift. In particular, the asymptotic standard normal \(t\) statistics are available. The asymptotic inference requires no distinction between simple regressions of trend stationary series of cointegrated variables with drifts.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators
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