Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-22T08:42:51.719Z Has data issue: false hasContentIssue false
  • English
  • Français

ROBUST INFERENCE IN AUTOREGRESSIONS WITH MULTIPLE OUTLIERS

Published online by Cambridge University Press:  01 December 2009

Giuseppe Cavaliere  and
Iliyan Georgiev
Giuseppe Cavaliere*
Affiliation:
Università di Bologna
Iliyan Georgiev
Affiliation:
Universidade Nova de Lisboa
*
*Address correspondence to Giuseppe Cavaliere, Department of Statistical Sciences, University of Bologna, Via Belle Arti 41, I-40126 Bologna, Italy; e-mail: giuseppe.cavaliere@unibo.it.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider robust methods for estimation and unit root (UR) testing in autoregressions with infrequent outliers whose number, size, and location can be random and unknown. We show that in this setting standard inference based on ordinary least squares estimation of an augumented Dickey–Fuller (ADF) regression may not be reliable, because (a) clusters of outliers may lead to inconsistent estimation of the autoregressive parameters and (b) large outliers induce a jump component in the asymptotic distribution of UR test statistics. In the benchmark case of known outlier location, we discuss why the augmentation of the ADF regression with appropriate dummy variables not only ensures consistent parameter estimation but also gives rise to UR tests with significant power gains, growing with the number and the size of the outliers. In the case of unknown outlier location, the dummy-based approach is compared with a robust, mixed Gaussian, quasi maximum likelihood (QML) approach, novel in this context. It is proved that, when the ordinary innovations are Gaussian, the QML and the dummy-based approach are asymptotically equivalent, yielding UR tests with the same asymptotic size and power. Moreover, as a by-product of QML the outlier dates can be consistently estimated. When the innovations display tails fatter than Gaussian, the QML approach ensures further power gains over the dummy-based method. Simulations show that the QML ADF-type t-test, in conjunction with standard Dickey–Fuller critical values, yields the best combination of finite-sample size and power.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 6: Newbold Conference Special Issue , December 2009 , pp. 1625 - 1661
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Abadir, K. & Lucas, A. (2004) A comparison of minimum MSE and maximum power for the nearly integrated non-Gaussian model. Journal of Econometrics 119, 45–71.Google Scholar
Bai, J. & Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78.CrossRefGoogle Scholar
Balke, N.S. & Fomby, T.B. (1991) Infrequent permanent shocks and the finite-sample performance of unit root tests. Economics Letters 36, 269–273.Google Scholar
Balke, N.S. & Fomby, T.B. (1994) Large shocks, small shocks and economic fluctuations: Outliers in macroeconomic time series. Journal of Applied Econometrics 9, 181–200.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bohn-Nielsen, H. (2004) Cointegration analysis in the presence of outliers. Econometrics Journal 7, 249–271.Google Scholar
Burridge, P. & Taylor, A.M.R. (2006) Additive outlier detection via extreme-value theory. Journal of Time Series Analysis 27, 685–701.Google Scholar
Cavaliere, G. & Georgiev, I. (2009) Supplement to “Robust inference in autoregressions with multiple level shifts”, downloadable fromwww2.stat.unibo.it/cavaliere/io/.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.Google Scholar
Chang, I., Tiao, G.C., & Chen, C. (1988) Estimation of time series parameters in the presence of outliers. Technometrics 30, 193–204.CrossRefGoogle Scholar
Doornik, J. (2001) Ox: An Object-Oriented Matrix Programming Language. Timberlake Consultants.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Franses, P.H. & Haldrup, N. (1994) The effects of additive outliers on tests for unit roots and cointegration. Journal of Business & Economic Statistics 12, 471–478.Google Scholar
Franses, P.H. & Lucas, A. (1998) Outlier detection in cointegration analysis. Journal of Business & Economic Statistics 16, 459–468.Google Scholar
Fuller, W. (1976) Introduction to Statistical Time Series. Wiley.Google Scholar
Georgiev, I. (2008) Asymptotics for cointegrated processes with infrequent stochastic level shifts and outliers. Econometric Theory 24, 587–615.Google Scholar
Gourieroux, C. & Monfort, A. (1995) Statistics and Econometric Models. Cambridge University Press.Google Scholar
Jansson, M. (2008) Semiparametric power envelopes for tests of the unit root hypothesis. Econometrica 76, 1103–1142.Google Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7, 201–212.Google Scholar
Kurtz, T. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Lanne, M., Lütkepohl, H., & Saikkonen, P. (2002) Unit root tests in the presence of innovational outliers. In Klein, I., & Mittnik, S. (eds.), Contributions to Modern Econometrics, pp. 151–167. Kluwer Academic Publishers.Google Scholar
Leipus, R. & Viano, M.-C. (2003) Long memory and stochastic trend. Statistics and Probability Letters 61, 177–190.Google Scholar
Leybourne, S. & Newbold, P. (2000a) Behaviour of the standard and symmetric Dickey-Fuller type tests when there is a break under the null hypothesis. Econometrics Journal 3, 1–15.CrossRefGoogle Scholar
Leybourne, S. & Newbold, P. (2000b) Behavior of Dickey–Fuller t-tests when there is a break under the alternative hypothesis. Econometric Theory 16, 779–789.Google Scholar
Lucas, A. (1995) An outlier robust unit root test with an application to the extended Nelson–Plosser data. Journal of Econometrics 66, 153–173.Google Scholar
Lucas, A. (1997) Cointegration testing using pseudo likelihood ratio tests. Econometric Theory 13, 149–169.CrossRefGoogle Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.Google Scholar
Ng, S. & Perron, P. (1995) Unit root tests in ARMA models with data dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90, 268–281.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.CrossRefGoogle Scholar
Perron, P. (1990) Testing for a unit root in a time series with a changing mean. Journal of Business & Economic Statistics 8, 153–162.Google Scholar
Perron, P. (2006) Dealing with structural breaks. In Patterson, K. & Mills, T.C. (eds.), Palgrave Handbook of Econometrics, vol. 1: Econometric Theory, pp. 278–352. Palgrave Macmillan.Google Scholar
Perron, P. & Ng, S. (1998) An autoregressive spectral density estimator at frequency zero for nonstationarity tests. Econometric Theory 14, 560–603.Google Scholar
Perron, P. & Rodriguez, G. (2003) Searching for additive outliers in non-stationary time series. Journal of Time Series Analysis 24, 193–220.Google Scholar
Perron, P. & Vogelsang, T.J. (1992) Nonstationarity and level shifts with an application to purchasing power parity. Journal of Business & Economic Statistics 10, 301–320.Google Scholar
Phillips, P.C.B. (1987) Toward a unified asymptotic theory for autoregression. Biometrika 74, 535–547.Google Scholar
Phillips, P.C.B. (1990) Time series regression with a unit root and infinite-variance errors. Econometric Theory 6, 44–62.Google Scholar
Rothenberg, T.J. & Stock, J.H. (1997) Inference in a nearly integrated autoregressive model with nonnormal innovations. Journal of Econometrics 80, 269–286.Google Scholar
Stock, J.H. (1994) Unit roots, structural breaks and trends. In Engle, R. & McFadden, D. (eds.), Handbook of Econometrics, vol. 4, pp. 2740–2841. North-Holland.Google Scholar
Tsay, R. (1988) Outliers, level shifts, and variance changes in time series. Journal of Forecasting 7, 1–20.Google Scholar
Vogelsang, T.J. & Perron, P. (1998) Additional tests for a unit root allowing for a break in the trend function at an unknown time. International Economic Review 39, 1073–1100.Google Scholar