Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-19T07:39:58.518Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMAL MULTISTEP VAR FORECAST AVERAGING

Published online by Cambridge University Press:  23 March 2020

Jen-Che Liao  and
Wen-Jen Tsay
Jen-Che Liao*
Affiliation:
Fu Jen Catholic University
Wen-Jen Tsay
Affiliation:
Academia Sinica
*
Address correspondence to Jen-Che Liao, Department of Economics, Fu Jen Catholic University, 510 Zhongzheng Road, Xinzhuang, New Taipei City 24205, Taiwan; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This article proposes frequentist multiple-equation least-squares averaging approaches for multistep forecasting with vector autoregressive (VAR) models. The proposed VAR forecast averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multistep forecast averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecast averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step-ahead forecast averaging, whereas for direct multistep forecast averaging, the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. To present our methodology, we investigate the finite-sample behavior of the proposed averaging procedures under model misspecification via simulation experiments.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 6 , December 2020 , pp. 1099 - 1126
Copyright
© Cambridge University Press 2020

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.)

Footnotes

This article was previously circulated under the title “Multivariate Least Squares Forecasting Averaging by Vector Autoregressive Models.” We thank the Editor Peter Phillips, the Co-Editor Robert Taylor, and two anonymous referees for their constructive comments on earlier versions of this article. We appreciate helpful comments and suggestions from Le-Yu Chen, Yi-Ting Chen, Graham Elliott, Bruce Hansen, Chu-An Liu, Chor-Yiu (CY) Sin, and participants of the 2016 Cross-Strait Dialogue III, the 2016 Taiwan Economics Research workshop, IAAE 2017, SETA 2019, CMES 2019, and econometrics seminars at several universities. We appreciate research support from Institute of Economics at Academia Sinica. We assume responsibility for any errors in the article.

References

REFERENCES

Andersson, M.K. & Karlsson, S. (2007) Bayesian Forecast Combination for VAR Models, Working Papers 2007:13. Orebro University, School of Business.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors. Journal of Econometrics 47, 359–377.CrossRefGoogle Scholar
Basu, S. & Michailidis, G. (2015) Regularized estimation in sparse high-dimensional time series models. Annals of Statistics 43, 1535–1567.CrossRefGoogle Scholar
Bates, J.M. & Granger, C.W.J. (1969) The combination of forecasts. Journal of the Operational Research Society 20, 451–468.CrossRefGoogle Scholar
Bhansali, R.J. (1996) Asymptotically efficient autoregressive model selection for multistep prediction. Annals of the Institute of Statistical Mathematics 48, 577–602.CrossRefGoogle Scholar
Bhansali, R.J. (1997) Direct autoregressive predictors for multistep prediction: Order selection and performance relative to the plug in predictors. Statistica Sinica 7, 425–449.Google Scholar
Bhansali, R.J. (1999) Parameter estimation and model selection for multistep prediction of time series: A review. In Gosh, S. (ed.). Asymptotics, Nonparametrics and Time Series, pp. 201–225. Marcel Dekker.Google Scholar
Chen, R., Yang, L., & Hafner, C. (2004) Nonparametric multistep-ahead prediction in time series analysis. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 66, 669–686.CrossRefGoogle Scholar
Cheng, T.-C.F., Ing, C.-K., & Yu, S.-H. (2015) Toward optimal model averaging in regression models with time series errors. Journal of Econometrics 189, 321–334.CrossRefGoogle Scholar
Cheng, X. & Hansen, B.E. (2015) Forecasting with factor-augmented regression: A frequentist model averaging approach. Journal of Econometrics 186, 280–293.CrossRefGoogle Scholar
Chevillon, G. (2007) Direct multi-step estimation and forecasting. Journal of Economic Surveys 21, 746–785.CrossRefGoogle Scholar
Chevillon, G. & Hendry, D.F. (2005) Non-parametric direct multi-step estimation for forecasting economic processes. International Journal of Forecasting 21, 201–218.Google Scholar
Clark, T.E., & McCracken, (2010) Averaging forecasts from VARs with uncertain instabilities. Journal of Applied Econometrics 25, 5–29.CrossRefGoogle Scholar
Clemen, R.T. (1989) Combining forecast: A review and annotated bibliography. International Journal of Forecasting 5, 559–581.CrossRefGoogle Scholar
Diebold, F.X., & Lopez, J.A. (1996) Forecast evaluation and combination. In Statistical Methods in Finance. Handbook of Statistics, vol. 14, pp. 241–268. Elsevier.Google Scholar
Doan, T., Litterman, R., & Sims, C. (1984) Forecasting and conditional projection using realistic prior distributions. Econometric Reviews 3, 1–100.Google Scholar
Elliott, G. & Timmermann, A. (2016) Economic Forecasting. Princeton University Press.Google Scholar
Findley, D.F. & Wei, C.-Z. (2002) AIC, overfitting principles, and the boundedness of moments of inverse matrices for vector autotregressions and related models. Journal of Multivariate Analysis 83, 415–450.CrossRefGoogle Scholar
Fujikoshi, Y. & Satoh, K. (1997) Modified AIC and C p in multivariate linear regression. Biometrica 84, 707–716.CrossRefGoogle Scholar
Gao, Y., Zhang, X., Wang, S., & Zou, G. (2016) Model averaging based on leave-subject-out cross-validation. Journal of Econometrics 192, 139–151.CrossRefGoogle Scholar
Granger, C. (1989) Combining forecasts twenty years later. Journal of Forecast 8, 167–173.CrossRefGoogle Scholar
Hansen, B.E. (2007) Least squares model averaging. Econometrica 75, 1175–1189.Google Scholar
Hansen, B.E. (2008) Least-squares forecast averaging. Journal of Econometrics 146, 342–350.CrossRefGoogle Scholar
Hansen, B.E. (2016a) Efficient shrinkage in parametric models. Journal of Econometrics 190, 115–132.CrossRefGoogle Scholar
Hansen, B.E. (2016b) Stein Combination Shrinkage for Vector Autoergressions, Discussion paper. University of Wisconsin.Google Scholar
Hansen, B.E. & Racine, J.S. (2012) Jackknife model averaging. Journal of Econometrics 167, 38–46.CrossRefGoogle Scholar
Hendry, D. & Clements, M. (2004) Pooling of forecasts. Econometric Journal 7, 1–31.Google Scholar
Hsu, N.-J., Hung, H.-L., & Chang, Y.-M. (2008) Subset selection for vector autoregressive processes using Lasso. Computational Statistics & Data Analysis 52, 3645–3657.CrossRefGoogle Scholar
Ing, C.-K. (2003) Multistep prediction in autoregressive processes. Econometric Theory 19, 254–279.CrossRefGoogle Scholar
Ing, C.-K. & Wei, C.-Z. (2003) On same-realization prediction in an infinite-order autoregressive process. Journal of Multivariate Analysis 85, 130–155.CrossRefGoogle Scholar
Ing, C.-K. & Wei, C.-Z. (2005) Order selection for same-realization predictions in autoregressive processes. Annals of Statistics 33, 2423–2474.CrossRefGoogle Scholar
Kabaila, P. (2002) On variable selection in linear regression. Econometric Theory 18, 913–925.Google Scholar
Kunitomo, N. & Yamamoto, T. (1985) Properties of predictors in misspecified autoregressive time series models. Journal of the American Statistical Association 80, 941–950.CrossRefGoogle Scholar
Lee, W. & Liu, Y. (2012) Simultaneous multiple response regression and inverse covariance matrix estimation via penalized Gaussian maximum likelihood. Journal of Multivariate Analysis 111, 241–255.CrossRefGoogle ScholarPubMed
Leeb, H. & Pötscher, B.M. (2009) Model selection. In Andersen, T.G., Davis, R.A., Kreib, J.-P., and Mikosch, T. (eds.).The Handbook of Financial Time Series, pp. 889–925. Springer.CrossRefGoogle Scholar
Lewis, R. & Reinsel, G. (1985) Prediction of multivariate time Series by autoregressive model fitting. Journal of Multivariate Analysis 16, 393–411.CrossRefGoogle Scholar
Li, K.-C. (1987) Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: Discrete index set. Annals of Statistics 15, 958–975.CrossRefGoogle Scholar
Litterman, R.B. (1986) Forecasting with Bayesian vector autoregressions: Five years of experience. Journal of Business & Economic Statistics 4, 25–38.Google Scholar
Liu, Q., Okui, R., & Yoshimura, A. (2016) Generalized least squares model averaging. Econometric Reviews 35, 1692–1752.CrossRefGoogle Scholar
Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis. Springer.CrossRefGoogle Scholar
Marcellino, M., Stock, J., & Watson, M. (2006) A comparison of direct and iterated multistep AR method for forecasting macroeconomic time series. Journal of Econometrics 135, 499–526.CrossRefGoogle Scholar
McQuarrie, A.D. & Tsai, C.-L. (1998) Regression and Time Series Model Selection. World Scientific Publishing Co. Pte. Ltd. CrossRefGoogle Scholar
Pesaran, M.H., Pick, A., & Timmermann, A. (2011) Variable selection, estimation and inference for multi-period forecasting problems. Journal of Econometrics 164, 173–187.CrossRefGoogle Scholar
Phillips, P.C.B. (1995) Automated forecasts of Asia-Pacific economic activity. Asia-Pacific Economic Review 1, 92–102.Google Scholar
Phillips, P.C.B. (1996) Econometric model determination. Econometrica 64, 763–812.CrossRefGoogle Scholar
Reid, D.J. (1968) Combining three estimates of gross domestic product. Economica 35, 431–444.CrossRefGoogle Scholar
Reid, D.J. (1969) A Comparative Study of Time Series Prediction Techniques on Economic Data, Ph.d. Thesis. University of Nottingham, Nottingham, UK.Google Scholar
Ren, Y., Xiao, Z., & Zhang, X. (2013) Two-step adaptive model selection for vector autoregressive processes. Journal of Multivariate Analysis 116, 349–364.CrossRefGoogle Scholar
Ren, Y. & Zhang, X. (2010) Subset selection for vector autoregressive processes via adaptive Lasso. Statistics and Probability Letters 80, 1705–1712.Google Scholar
Schorfheide, F. (2005) VAR forecasting under misspecification. Journal of Econometrics 128, 99–136.CrossRefGoogle Scholar
Shao, J. (1997) An asymptotic theory for linear model selection. Statistica Sinica 7, 221–264.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, 147–164.CrossRefGoogle Scholar
Shibata, R. (1981) An optimal selection of regression variables. Biometrika 68, 45–54.CrossRefGoogle Scholar
Shibata, R. (1983) Asymptotic mean efficiency of a selection of regression variables. Annals of the Institute of Statistical Mathematics 35, 415–423.CrossRefGoogle Scholar
Sims, C.A. (1980) Macroeconomics and reality. Econometrica 48, 1–48.CrossRefGoogle Scholar
Sparks, R., Coutsourides, D., & Troskie, L. (1983) The multivariate C p . Communications in Statistics—Theory and Methods 12, 1775–1793.CrossRefGoogle Scholar
Stock, J. & Watson, M. (2006) Forecast with many predictors. In Handbook of Economic Forecasting, pp. 515–554. Elsevier Press.CrossRefGoogle Scholar
Tiao, G.C. & Box, G.E.P. (1981) Modeling multiple times series with applications. Journal of the American Statistical Association 76, 802–816.Google Scholar
Timmermann, A. (2006) Forecast combinations. In Handbook of Economic Forecasting, pp. 135–196. Elsevier Press.CrossRefGoogle Scholar
Varmuza, K. & Filzmoser, P. (2009) Introduction to Multivariate Statistical Analysis in Chemometrics. CRC Press.Google Scholar
Wan, A.T., Zhang, X., & Zou, G. (2010) Least squares model averaging by Mallows criterion. Journal of Econometrics 156, 277–283.CrossRefGoogle Scholar
Yanagihara, H. & Satoh, K. (2010) An unbiased criterion for multivariate ridge regression. Journal of Multivariate Analysis 101, 1226–1238.CrossRefGoogle Scholar
Zellner, A. (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association 57, 348–368.CrossRefGoogle Scholar
Zhang, X., Wan, A.T., & Zou, G. (2013) Model averaging by Jackknife criterion in models with dependent data. Journal of Econometrics 174, 82–94.CrossRefGoogle Scholar
Supplementary material: PDF

Liao and Tsay supplementary material

Liao and Tsay supplementary material

Download Liao and Tsay supplementary material(PDF)
PDF 680.7 KB