Revisiting the Use of Generalized Least Squares in Time Series Regression Models
Pub. online: 21 July 2023
Type: Statistical Data Science
Open Access
Open Access
Received
3 January 2023
3 January 2023
Accepted
10 June 2023
10 June 2023
Published
21 July 2023
21 July 2023
Abstract
Linear regression models are widely used in empirical studies. When serial correlation is present in the residuals, generalized least squares (GLS) estimation is commonly used to improve estimation efficiency. This paper proposes the use of an alternative estimator, the approximate generalized least squares estimators based on high-order AR(p) processes (GLS-AR). We show that GLS-AR estimators are asymptotically efficient as GLS estimators, as both the number of AR lag, p, and the number of observations, n, increase together so that $p=o({n^{1/4}})$ in the limit. The proposed GLS-AR estimators do not require the identification of the residual serial autocorrelation structure and perform more robust in finite samples than the conventional FGLS-based tests. Finally, we illustrate the usefulness of GLS-AR method by applying it to the global warming data from 1850–2012.
References
Amemiya T (1973). Generalized least squares with an estimated autocovariance matrix. Econometrica, 41: 723–732. https://doi.org/10.2307/1914092
Bhansali RJ (1986). The criterion autoregressive transfer function of Parzen. Journal of Time Series Analysis, 7: 79–103. https://doi.org/10.1111/j.1467-9892.1986.tb00487.x
Bloomfield P, Nychka D (1992). Climate spectra and detecting climate change. Climate Change, 21: 275–287. https://doi.org/10.1007/BF00139727
Breusch T (1980). Useful invariance results for generalized regression models. Journal of Econometrics, 13: 327–340. https://doi.org/10.1016/0304-4076(80)90083-4
Chambers M (2013). Jackknife estimation of stationary autoregressive models. Journal of Econometrics, 172: 142–157. https://doi.org/10.1016/j.jeconom.2012.09.003
Chambers M, Ercolane JS, Taylor AMR (2014). Testing for seasonal unit roots by frequency domain regression. Journal of Econometrics, 178: 243–258. https://doi.org/10.1016/j.jeconom.2013.08.025
Choi I, Kurozumi E (2012). Model selection criteria for the leads-and-lags cointegrating regression. Journal of Econometrics, 169: 224–238. https://doi.org/10.1016/j.jeconom.2012.01.021
Cochrane D, Orcutt GH (1949). Application of least squares regression to relationships containing auto-correlated error terms. Journal of the American Statistical Association, 44: 32–61. https://doi.org/10.1080/01621459.1949.10483312
Engle R (1974). Specification of the disturbance for efficient estimation. Econometrica, 42: 135–146. https://doi.org/10.2307/1913690
Goldberger AS (1962). Best linear unbiased prediction in the generalized linear regression model. Journal of the American Statistical Association, 57: 369–375. https://doi.org/10.1080/01621459.1962.10480665
Hannan EJ, Rissanen J (1982). Recursive estimation of mixed autoregressive-moving average order. Biometrika, 69: 81–94. https://doi.org/10.1093/biomet/69.1.81
Harvey DI, Leybourne SJ, Taylor AMR (2010). Robust methods for detecting multiple level breaks in autocorrelated time series. Journal of Econometrics, 157: 342–358. https://doi.org/10.1016/j.jeconom.2010.02.003
Jones WTML, D P, Wright PB (1986). Global temperature variations between 1861 and 1984. Nature, 332: 430–434. https://doi.org/10.1038/322430a0
Kadiyala K (1970). Testing for the independence of regression disturbances. Econometrica, 38: 97–117. https://doi.org/10.2307/1909244
Kiefer N, Vogelsang T, Bunzel H (2000). Simple robust testing of regression hypotheses. Econometrica, 68: 695–714. https://doi.org/10.1111/1468-0262.00128
Kiefer N, Vogelsang TJ (2005). A new asymptotic theory for heteroscedasticity-autocorrelation robust tests. Econometric Theory, 21: 1130–1164. https://doi.org/10.1017/S0266466605050565
Kiefer NM, Vogelsang TJ (2002). Heteroscedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation. Econometrica, 70: 2093–2095. https://doi.org/10.1111/1468-0262.00366
King M (1983). Testing for autoregressive against moving average errors in the linear regression model. Journal of Econometrics, 21: 35–51. https://doi.org/10.1016/0304-4076(83)90118-5
Koreisha S, Fang Y (2001). Generalized least squares with misspecified serial correlation structures. Journal of the Royal Statistical Society, Series B, 63: 515–531. https://doi.org/10.1111/1467-9868.00296
Koreisha S, Pukkila T (1985). Properties of predictors in misspecified autoregressive time series model. Journal of the American Statistical Association, 80: 941–950. https://doi.org/10.1080/01621459.1985.10478208
Kunitomo N, Yamamoto T (1985). Properties of predictors in misspecified autoregressive time series model. Journal of the American Statistical Association, 80: 941–950. https://doi.org/10.1080/01621459.1985.10478208
Lee J, Lund R (2004). Revisiting simple linear regression with autocorrelated errors. Biometrika, 91: 240–245. https://doi.org/10.1093/biomet/91.1.240
Newey W, West K (1987). A simple positive semi-definite, heteroskedastic and autocorrelation consistent covariance matrix. Econometrica, 55: 703–708. https://doi.org/10.2307/1913610
Phillips PCB (2007). Regression with slowly varying regressors and nonlinear trends. Econometric Theory, 23: 557–614. https://doi.org/10.1017/S0266466607070491
Politis DN (2011). Higher-order accurate, positive semi-definite estimation of large-sample covariance and spectral density matrices. Econometric Theory, 27: 703–744. https://doi.org/10.1017/S0266466610000484
Pukkila T, Koreisha S, Kallinen A (1990). The identification of ARMA models. Biometrika, 77: 537–549. https://doi.org/10.1093/biomet/77.3.537
Sun Y, Philips PCB, Jin S (2011). Power maximization and size control in heteroscedasticity and autocorrelation robust tests with exponentiated kernels. Econometric Theory, 27: 1320–1368. https://doi.org/10.1017/S0266466611000077
Thursby J (1987). OLS or GLS in the presence of specification error? Journal of Econometrics, 35: 359–374. https://doi.org/10.1016/0304-4076(87)90033-9
Wu WB, Woodroofe M, Mentz G (2001). Isotonic regression: Another look at the changepoint problem. Biometrika, 88: 793–804. https://doi.org/10.1093/biomet/88.3.793
Zinde-Walsh V, Galberaith J (1991). Estimation of a linear regression model with stationary ARMA($p,q$) errors. Journal of Econometrics, 47: 333–357. https://doi.org/10.1016/0304-4076(91)90106-N
