Hostname: page-component-5d59c44645-l48q4 Total loading time: 0 Render date: 2024-03-02T18:43:52.781Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  08 February 2005

Guido M. Kuersteiner
Boston University


Infinite order vector autoregressive (VAR) models have been used in a number of applications ranging from spectral density estimation, impulse response analysis, and tests for cointegration and unit roots, to forecasting. For estimation of such models it is necessary to approximate the infinite order lag structure by finite order VARs. In practice, the order of approximation is often selected by information criteria or by general-to-specific specification tests. Unlike in the finite order VAR case these selection rules are not consistent in the usual sense, and the asymptotic properties of parameter estimates of the infinite order VAR do not follow as easily as in the finite order case. In this paper it is shown that the parameter estimates of the infinite order VAR are asymptotically normal with zero mean when the model is approximated by a finite order VAR with a data dependent lag length. The requirement for the result to hold is that the selected lag length satisfies certain rate conditions with probability tending to one. Two examples of selection rules satisfying these requirements are discussed. Uniform rates of convergence for the parameters of the infinite order VAR are also established.Very helpful comments by the editor and two referees led to a substantial improvement of the manuscript. I am particularly indebted to one of the referees for pointing out an error in the proofs. All remaining errors are my own. Financial support from NSF grant SES−0095132 is gratefully acknowledged.

Research Article
© 2005 Cambridge University Press

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.)



Abadir, K.M., K. Hadri, & E. Tzavalis (1999) The influence of VAR dimensions on estimator biases. Econometrica 67, 163181.Google Scholar
Akaike, H. (1969) Power spectrum estimation through autoregressive model fitting. Annals of the Institute of Statistical Mathematics 21, 407419.Google Scholar
An, H.-Z., Z.-G. Chen, & E.J. Hannan (1982) Autocorrelation, autoregression and autoregressive approximation. Annals of Statistics 10, 926936.Google Scholar
Andrews, D.W. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.Google Scholar
Bekker, P.A. (1994) Alternative approximations to the distributions of instrumental variable estimators. Econometrica 62, 657681.Google Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489502.Google Scholar
den Haan, W.J. & A.T. Levin (2000) Robust Covariance Matrix Estimation with Data-Dependent VAR Prewhitening Order. NBER Technical Working paper 255.
Donald, S.G. & W.K. Newey (2001) Choosing the number of instruments. Econometrica 69, 11611191.Google Scholar
Eastwood, B.J. & A.R. Gallant (1991) Adaptive rules for seminonparametric estimators that achieve asymptotic normality. Econometric Theory 7, 307340.Google Scholar
Goncalves, S. & L. Kilian (2003) Asymptotic and Bootstrap Inference for AR(Infinity) Processes with Conditional Heteroskedasticity. Mimeo, Université de Montreal.
Hahn, J., J. Hausman, & G.M. Kuersteiner (2000) Bias Corrected Instrumental Variables Estimation for Dynamic Panel Models with Fixed Effects. Manuscript, MIT.
Hahn, J. & G. Kuersteiner (2002) Asymptotically unbiased inference for a dynamic panel model with fixed effects. Econometrica 70, 16391657.Google Scholar
Hahn, J. & G. Kuersteiner (2003) Bias Reduction for Dynamic Nonlinear Panel Models with Fixed Effects. Mimeo, UCLA.
Hall, A. (1994) Testing for a unit root in time series with pretest data-based model selection. Journal of Business & Economic Statistics 12, 461470.Google Scholar
Hannan, E. & M. Deistler (1988) The Statistical Theory of Linear Systems. Wiley.
Hannan, E. & L. Kavalieris (1984) Multivariate linear time series models. Advanced Applications in Probability 16, 492561.Google Scholar
Hannan, E. & L. Kavalieris (1986) Regression, autoregression models. Journal of Time Series Analysis 7, 2749.Google Scholar
Inoue, A. & L. Kilian (2002) Bootstrapping smooth functions of slope parameters and innovation variances in VAR(Infinity) models. International Economic Review 43, 309331.Google Scholar
Kilian, L. (1998) Small-sample confidence intervals for impulse response functions. Review of Economics and Statistics 80, 218230.Google Scholar
Kuersteiner, G.M. (2002) Rate-Adaptive GMM Estimators for Linear Time Series Models. Manuscript, MIT.
Leeb, H. & B.M. Pötscher (2003) The finite sample distribution of post-model-selection estimators and uniform versus nonuniform approximations. Econometric Theory 19, 100142.Google Scholar
Lewis, R. & G. Reinsel (1985) Prediction of multivariate time series by autoregressive model fitting. Journal of Multivariate Analysis 64, 393411.Google Scholar
Lütkepohl, H. & D. Poskitt (1996) Testing for causation using infinite order vector autoregressive processes. Econometric Theory 12, 6187.Google Scholar
Lütkepohl, H. & P. Saikkonen (1997) Impulse response analysis in infinite order cointegrated vector autoregressive processes. Journal of Econometrics 81, 127157.Google Scholar
Ng, S. & P. Perron (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, 268281.Google Scholar
Ng, S. & P. Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 15191554.Google Scholar
Paparoditis, E. (1996) Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. Journal of Multivariate Analysis 57, 277296.Google Scholar
Parzen, E. (1974) Some recent advances in time series modeling. IEEE Transactions on Automatic Control AC19, 723730.Google Scholar
Pötscher, B.M. (1991) Effects of model selection on inference. Econometric Theory 7, 163185.Google Scholar
Saikkonen, P. & H. Lütkepohl (1996) Infinite-order cointegrated vector autoregressive processes: Estimation and inference. Econometric Theory 12, 814844.Google Scholar
Saikkonen, P. & R. Luukkonen (1997) Testing cointegration in infinite order vector autoregressive processes. Journal of Econometrics 81, 93126.Google Scholar
Sargan, J. (1975) Asymptotic theory and large models. International Economic Review 16, 7591.Google Scholar
Sen, P.K. (1979) Asymptotic properties of maximum likelihood estimators based on conditional specification. Annals of Statistics 7, 10191033.Google Scholar
Shibata, R. (1980) Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics 8, 147164.Google Scholar
Shibata, R. (1981) An optimal autoregressive spectral estimate. Annals of Statistics 9, 300306.Google Scholar