On the estimation of the regression coefficients of a continuous parameter process with stationary residual

Abstract

The asymptotic expressions of the covariance matrices for both the least square estimates _Lα_T and Markov (best linear) estimates $\bar \alpha _T $$ are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α₁, …, α_{m 0})′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ_ν(t), t ⩾ 0, ν = 1, …, m₀, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f(λ). Under these assumptions, it is proven that $\mathop {\lim }\limits_{T \to \infty } D_T E_0 \left( {L\alpha _T L\alpha _T^* } \right)D_T = 2\pi \left[ {B\left( 0 \right)} \right]^{ - 1} \int_{ - \infty }^\infty {f( - \lambda )d\alpha (\lambda )\left[ {B\left( 0 \right)} \right]^{ - 1} } ,$ where the matrices D_T, B(0), α(λ) are defined in Section 3.

Under the assumptions mentioned above, if, furthermore, there exist some positive integer m and a constant C such that g(λ)(1 + λ²)^m ⩾ C > 0, where g(λ) is the spectral density of the residual, and for every N > 0, $\frac{{\lim _{T \to \infty } \smallint _0^T \overline {\varphi _\mu ^{(k)} (t + h)} \varphi _\nu ^{(j)} (t + l)dt}}{{\sqrt {\Phi _\mu (T)\Phi _\nu (T)} }},0 \leqslant k,j \leqslant m$ converge uniformly in h, l ∈ (−N, N), then the following formula holds. $\mathop {\lim }\limits_{T \to \infty } D_T E_0 \hat \alpha _T \hat \alpha _T^ * D_T = 2\pi \left[ {\int_{ - \infty }^\infty {\frac{1}{{g( - \lambda )}}d\alpha (\lambda )} } \right]^{ - 1} .$

The asymptotic equivalence of the least square estimates and the Markov estimates is also discussed.

Keywords

regression coefficients / parameter process / stationary residual / 60J27 / 60G10

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