Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-14T00:05:54.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Ols Bias in a Nonstationary Autoregression

Published online by Cambridge University Press:  11 February 2009

Karim M. Abadir
Karim M. Abadir
Affiliation:
The American University in Cairo
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical formula is derived to approximate the finite sample bias of the ordinary least-squares (OLS) estimator of the autoregressive parameter when the underlying process has a unit root. It is found that the bias is expressible in terms of parabolic cylinder functions which are easy to compute. Numerical evaluation of the formula reveals that the approximation is very accurate. The derived formula inspires a heuristic approximation, obtained by leastsquares fitting of the asymptotic bias. More importantly, the formula proves analytically that the bias declines at a rate which is slower than the consistency rate, thus explaining some previous simulation findings. A case where the bias increases with the sample size is also given.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 1 , January 1993 , pp. 81 - 93
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abadir, K.M.Some expansions for the parabolic cylinder function. Technical paper #1, Department of Economics and Political Science, American University in Cairo, 1991.Google Scholar
2.Abadir, K.M.The limiting distribution of the t ratio under a unit root (mimeo), American University in Cairo, 1991.Google Scholar
3.Abadir, K.M.The limiting distribution of the autocorrelation coefficient under a unit root. Annals of Statistics (1992) (forthcoming).Google Scholar
4.Abramowitz, M. & Stegun, I.A.. Handbook of Mathematical Functions. New York: Dover, 1965.Google Scholar
5.Banerjee, A., Dolado, J.J., Hendry, D.F. & Smith, G.W.. Exploring equilibrium relationships in econometrics through static models: some Monte Carlo evidence. Oxford Bulletin of Economics and Statistics 48 (1986): 253–277.CrossRefGoogle Scholar
6.Bleistein, N. & Handelsman, R.A.. Asymptotic Expansions of Integrals. New York: Dover, 1975.Google Scholar
7.De Bruijn, N.G.Asymptotic Methods in Analysis. New York: Dover, 1981.Google Scholar
8.Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (1979): 427–431.Google Scholar
9.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
10.Erdélyi, A.Asymptotic Expansions. New York: Dover, 1956.Google Scholar
11.Erdélyi, A. (ed.) Higher transcendental functions, Vols. 1–3. New York: McGraw-Hill, 1953.Google Scholar
12.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
13.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.CrossRefGoogle Scholar
14.Fuller, W.A.Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
15.Gradshteyn, I.S. & Ryzhik, I.M.. Table of Integrals, Series, and Products. New York: Academic Press, 1980.Google Scholar
16.Granger, C.W.J.The typical spectral shape of an economic variable. Econometrica 34 (1966): 150–161.CrossRefGoogle Scholar
17.Grubb, D. & Symons, J.. Bias in regressions with a lagged dependent variable. Econometric Theory 3 (1987): 371–386.CrossRefGoogle Scholar
18.Le Breton, A. & Pham, D.T.. On the bias of the least squares estimator for the first order autoregressive process. Annals of the Institute of Statistical Mathematics 41 (1989): 555–563.CrossRefGoogle Scholar
19.Phillips, P.C.B. A note on the saddlepoint approximation in the first order non-circular autoregression. Cowles Foundation Discussion Paper No. 487, Yale University, March 1978.Google Scholar
20.Phillips, P.C.B. Exact small sample theory in the simultaneous equations model. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Vol. 1. Amsterdam: North-Holland, 1983.Google Scholar
21.Phillips, P.C.B.Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535–547.CrossRefGoogle Scholar
22.Phillips, P.C.B.Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3 (1987): 45–68.CrossRefGoogle Scholar
23.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
24.Shenton, L.R. & Johnson, W.L.. Moments of a serial correlation coefficient. Journal of the Royal Statistical Society series B 27 (1965): 308–320.Google Scholar
25.Stock, J.H.Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 55 (1987): 1035–1056.CrossRefGoogle Scholar
26.Tsui, A.K. & Ali, M.M. Exact moments of the least squares estimator in a first-order nonstationary autoregressive model. Proceeding of the American Statistical Association, Business and Economic Statistics Section (1989): 220–225.Google Scholar
27.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.CrossRefGoogle Scholar
28.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case II. Annals of Mathematical Statistics 30 (1959): 831–834.CrossRefGoogle Scholar
29.White, J.S.Asymptotic expansions for the mean and variance of the serial correlation coefficent. Biometrika 48 (1961): 85–94.CrossRefGoogle Scholar