Spectral regression for cointegrated time series with long-memory innovations (Q2740045)

A two-dimensional time-series \((y_t,x_t)\) is considered, where NEWLINE\[NEWLINEy_t=\beta x_t+e_t,\quad x_t=\phi x_{t-1}+u_t,NEWLINE\]NEWLINE \(u_t\) and \(e_t\) are long-memory processes with orders \(d_u\) and \(d_e\), correspondingly (i.e., their spectral densities \(f(\lambda)\sim G\lambda^{-2d}\) as \(\lambda\to 0\)), \(0\leq d_e,d_u<1/2\), \(|\phi|\leq 1\). The author considers an estimator for \(\beta\) of the form NEWLINE\[NEWLINE\tilde\beta_M=\sum_{\tau=-n+1}^{n-1} k_M(\tau)c_{xy}(\tau) \big[\sum_{\tau=-n+1}^{n-1} k_M(\tau)c_{xx}(\tau)\big]^{-1},NEWLINE\]NEWLINE where \(c_{xx}\) and \(c_{xy}\) are sample autocovariance functions, \(k_M(\tau)=k(\tau/M)\) is a lag window with NEWLINE\[NEWLINE\int_{-1}^{1} k(v)=1,\quad 0\leq k(v)<C,\quad k(v)=0\;\text{if} |v|>1.NEWLINE\]NEWLINE It is shown (for \(d_u>d_e\)) that for a linear stationary \((e_t,x_t)\) with innovations satisfying some integrability condition NEWLINE\[NEWLINEM^{d_u-d_e}(\tilde\beta_M - \beta)=C_1+o_P(1)\;\text{if} |\phi|<1\;\text{and} n^{d_u-d_e}(\tilde\beta_M - \beta)=C_2+o_P(1)\;\text{if} |\phi|=1.NEWLINE\]NEWLINE Here \(C_1\) is a nonrandom constant and \(C_2\) is a random variable described in the article.