Abstract
This paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically, they enable to derive the optimal kernel associated with a particular covariance structure of the measurement error, where optimality has to be intended in the Gauss-Markov sense. For local polynomial regression it is shown that there is a class of covariance structures, associated with non-invertible moving average processes of given orders which yield the Epanechnikov and the Henderson kernels as the optimal kernels.
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References
Aitken A.C. (1935) On the least squares and linear combinations of observations. Proceedings of the Royal Society of Edinburgh, A 55: 42–48
Amemiya T. (1985) Advanced econometrics. Harvard University Press, Cambridge
Anderson T.W. (1948) On the theory of testing serial correlation. Skandinavisk Aktuarietidskrift 31: 88–116
Anderson T.W. (1971) The statistical analysis of time series. Wiley, New York
Baksalary J.K., Kala R. (1983) On equalities between BLUEs, WLSEs and SLSEs. The Canadian Journal of Statistics 11: 119–123
Baksalary J.K., Van Eijnsbergeren A.C (1988) A comparison of two criteria for ordinary-least-squares estimators to be best linear unbiased estimators. The American Statistician 42: 205–208
Benedetti J.K. (1977) On the nonparametric estimation of regression functions. Journal of the Royal Statistical Society, Series B 39: 248–253
Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 64, 368, 829–836.
Dagum E.B. (1980) The X-11-ARIMA seasonal adjustment method. Statistics Canada, Ottawa
Epanechnikov V.A. (1969) Nonparametric estimation of a multivariate probability density. Theory of Probability and Applications 14: 153–158
Fan J., Gjibels I. (1996) Local polynomial modelling and its applications. Chapman and Hall, New York
Findley D.F., Monsell B.C., Bell W.R., Otto M.C., Chen B. (1998) New capabilities and methods of the X12-ARIMA seasonal adjustment program. Journal of Business and Economic Statistics 16: 2
Grenander U., Rosenblatt M. (1957) Statistical analysis of stationary time series. Wiley, New York
Gross J., Trenkler G. (1997) When do linear transforms of ordinary least squares and Gauss-Markov estimator coincide? Sankhyā 59-A(2): 175–178
Gross J., Trenkler G. (1998) On the equality of linear statistics in the general Gauss-Markov Models. In: Mukherjee S.P., Basu S.K., Sinha B.K. (eds) Frontiers in probability and statistics. Narosa Publishing House, New Delhi, pp 189–194
Gross J., Trenkler G., Werner H.J. (2001) The equality of linear transforms of the ordinary least squares estimator and the best linear unbiased estimator. Sankhyā 63-A(1): 118–127
Hannan E.J. (1970) Multiple time series. Wiley, New York
Henderson R. (1916) Note on graduation by adjusted average. Transaction of the Actuarial Society of America 17: 43–48
Horn R.A., Olkin I. (1996) When does A * A = B * B and why does one want to know? The American Mathematical Monthly 103(6), 470–482
Hoskins, W. D., Ponzo, P. J. (1972). Some properties of a class of band matrices. Mathematics of Computation, 26, (118), 393–400.
Jaeger A., Krämer W. (1998) A final twist on the equality of OLS and GLS. Statistical Papers 39: 321–324
Kenny P.B., Durbin J. (1982) Local trend estimation and seasonal adjustment of economic and social time series. Journal of the Royal Statistical Society A 145(I): 1–41
Krämer W. (1980) A note on the equality of ordinary least squares and Gauss-Markov estimates in the general linear model. Sankhyā, A 42: 130–131
Krämer W. (1986) Least-squares regression when the independent variable follows an ARIMA process. Journal of the American Statistical Association 81: 150–154
Krämer W., Hassler U. (1998) Limiting efficiency of OLS vs. GLS when regressors are fractionally integrated. Economics Letters 60(3): 285–290
Kruskal W. (1968) When are Gauss-Markov and least squares estimators identical? A coordinate-free approach. The Annals of Mathematical Statistics 39: 70–75
Ladiray, D., Quenneville, B. (2001). Seasonal adjustment with the X-11 method. Lecture Notes in Statistics. New York: Springer.
Loader C. (1999) Local regression and likelihood. Springer, New York
Lowerre J. (1974) Some relationships between BLUEs, WLSEs and SLSEs. Journal of the American Statistical Association 69: 223–225
Macaulay F.R. (1931) The smoothing of time series. National Bureau of Economic Research, New York
Magnus J.R., Neudecker H. (2007) Matrix differential calculus with applications in statistics and econometrics (3rd ed.). Wiley, Chichester
McAleer M. (1992) Efficient estimation: The Rao-Zyskind condition, Kruskal’s theorem and ordinary least squares. Economic Record 68: 65–72
McElroy F.W. (1967) A necessary and sufficient condition that ordinary least-squares estimators be best linear unbiased. Journal of the American Statistical Association 62: 1302–1304
Meyer C.D. (2000) Matrix analysis and applied linear algebra. SIAM, Philadelphia
Mukherjee, B. N., Maiti, S. S. (1988). On conditions for equality of OLSE, GLSE and MLE in the analysis of covariance structures. Calcutta Statistical Association Bullettin, 171–191.
Müller H.G. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. The Annals of Statistics 12(2): 766–774
Müller H.G. (1987) Weighted local regression and kernel methods for nonparametric curve fitting. Journal of the American Statistical Association 82: 231–238
Nadaraya E.A. (1964) On estimating regression. Theory of Probability and its Applications 9: 141–142
Phillips P.C.B. (1992) Geometry of the equivalence of OLS and GLS in the linear model. Econometric Theory 8(1): 158–159
Phillips P.C.B., Park J.Y. (1988) Asymptotic equivalence of OLS and GLS in regressions with integrated regressors. Journal of the American Statistical Association 83: 111–115
Priestley M.B., Chao M.T. (1972) Nonparametric function fitting. Journal of the Royal Statistical Society, Series B 34: 384–392
Puntanten S., Styan G.P.H. (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator. The American Statistician 43(3): 153–161
Tian Y., Weins D.P. (2006) On equality and proportionality of ordinary least squares, weighted least squares and best linear unbiased estimators in the general linear model. Statistics and Probability Letters 76: 1265–1272
Wallis K. (1983) Models for X-11 and X-11 forecast procedures for preliminary and revised seasonal adjustments. In: Zellner A. (eds) Applied time series analysis of economic data. Bureau of the Census, Washington DC, pp 3–11
Wand, M. P., Jones, M. C. (1995). Kernel smoothing. In Monographs on Statistics and Applied Probability, 60. London: Chapman and Hall.
Watson G.S. (1964) Smooth regression analysis. Sankhyā, A 26: 359–372
Watson G.S. (1967) Linear least squares regression. The Annals of Mathematical Statistics 38: 1679–1699
Zyskind G. (1967) On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. The Annals of Mathematical Statistics 38: 1092–1119
Zyskind G. (1969) Parametric argumentations and error structures under which certain simple least squares and analysis of variance procedures are also best. Journal of the American Statistical Association 64: 1353–1368
Zyskind G., Martin F.B. (1969) On best linear estimation and a general Gauss-Markoff theorem in linear models with arbitrary non-negative structure. SIAM Journal of Applied Mathematics 17: 1190–1202
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Luati, A., Proietti, T. On the equivalence of the weighted least squares and the generalised least squares estimators, with applications to kernel smoothing. Ann Inst Stat Math 63, 851–871 (2011). https://doi.org/10.1007/s10463-009-0267-8
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DOI: https://doi.org/10.1007/s10463-009-0267-8