Abstract
Estimates of the regression coefficients which are unbiased and linear in the observations are discussed in this paper. The residual is assumed to be a stationary process. Two specific estimates are discussed, the least-squares estimate and the Markov estimate. I call the estimate which is computed under the assumption that the residual is an orthogonal process the least-squares estimate. The Markov estimate is the linear unbiased estimate with minimal covariance matrix. The basic assumptions made in the paper are discussed in section 2 and are held to throughout the paper. In section 3 some remarks about the approximation of a continuous positive definite matrix-valued function by finite trigonometric forms are made. These remarks are used in section 4 to obtain the main results about the asymptotic behavior of the covariance matrices of the least-squares and Markov estimates. The next section discusses the many interesting cases in which the least-squares estimate is asymptotically as good as the Markov estimate. The first really systematic discussion of some of these problems was given by U. Grenander [1]. Further work was carried out by U. Grenander and M. Rosenblatt in [2], [3], and [4]. The author considers some of these problems in the case of a vector-valued time series in [5]. Some of the results of this paper are a generalization of some of those obtained in [5].
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References
U. Grenander, “On the estimation of regression coefficients in the case of an autocorrelated disturbance,” Annals of Math. Stat., Vol. 25 (1954), pp. 252–272.
U. Grenander and M. Rosenblatt, “An extension of a theorem of G. Szego and its application to the study of stochastic processes,” Trans. Amer. Math. Soc., Vol. 76 (1954), pp. 112–126.
U. Grenander, “Regression analysis of time series with stationary residuals,” Proc. Nat. Acad. Sci. Vol. 40 (1954), pp. 812–816.
U. Grenander, A monograph on time series analysis, to be published.
M, Rosenblatt, ”On the estimation of regression coefficients of a vector-valued time series with a stationary residual,” to be published in Annals of Math, Stat,
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Davis, R.A., Lii, KS., Politis, D.N. (2011). Some Regression Problems in Time Series Analysis. In: Davis, R., Lii, KS., Politis, D. (eds) Selected Works of Murray Rosenblatt. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8339-8_14
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