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

The asymptotic expressions of the covariance matrices for both the least square estimates _L α _T and Markov (best linear) estimates are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α ₁, …, 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, …, m ₀, 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 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 + λ ²)^m ⩾ C > 0, where g(λ) is the spectral density of the residual, and for every N > 0, converge uniformly in h, l ∈ (−N, N), then the following formula holds. The asymptotic equivalence of the least square estimates and the Markov estimates is also discussed. Translated by Wang Ting from the Chinese version of the paper published in Journal of Beijing Normal University (Natural Sciences), 1965, 1: 15–44