Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-18T22:32:05.844Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  24 May 2018

Anders Bredahl Kock*
University of Oxford Aarhus University and CREATES
Haihan Tang*
Fudan University
*Address correspondence to Anders Bredahl Kock, Department of Economics, University of Oxford, Aarhus University and CREATES, Manor Road, Oxford OX1 3UQ, UK; e-mail:
Haihan Tang, Fanhai International School of Finance and School of Economics, Fudan University, 220 Handan Road, Yangpu District, Shanghai, 200433, China; e-mail:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish oracle inequalities for a version of the Lasso in high-dimensional fixed effects dynamic panel data models. The inequalities are valid for the coefficients of the dynamic and exogenous regressors. Separate oracle inequalities are derived for the fixed effects. Next, we show how one can conduct uniformly valid inference on the parameters of the model and construct a uniformly valid estimator of the asymptotic covariance matrix which is robust to conditional heteroskedasticity in the error terms. Allowing for conditional heteroskedasticity is important in dynamic models as the conditional error variance may be nonconstant over time and depend on the covariates. Furthermore, our procedure allows for inference on high-dimensional subsets of the parameter vector of an increasing cardinality. We show that the confidence bands resulting from our procedure are asymptotically honest and contract at the optimal rate. This rate is different for the fixed effects than for the remaining parts of the parameter vector.

Copyright © Cambridge University Press 2018 


Financial support from the Center for Research in the Econometric Analysis of Time Series (grant DNRF78) is gratefully acknowledged.



Andersen, T.B., Bentzen, J., Dalgaard, C.-J., & Selaya, P. (2012) Lightning, IT diffusion, and economic growth across U.S. states. The Review of Economics and Statistics 94(4), 903924.CrossRefGoogle Scholar
Arellano, M. (2003) Panel Data Econometrics. Oxford University Press.CrossRefGoogle Scholar
Baltagi, B. (2008) Econometric Analysis of Panel Data, vol. 1. Wiley.Google Scholar
Belloni, A., Chen, D., Chernozhukov, V., & Hansen, C. (2012) Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica 80(6), 23692429.Google Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2014) Inference on treatment effects after selection among high-dimensional controls. The Review of Economic Studies 81(2), 608650.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., Hansen, C., & Kozbur, D. (2015) Inference in high-dimensional panel models with an application to gun control. Journal of Business & Economic Statistics 34(4), 590605.CrossRefGoogle Scholar
Bickel, P.J., Ritov, Y., & Tysbakov, A.B. (2009) Simultaneous analysis of Lasso and Dantzig selector. The Annals of Statistics 37(4), 17051732.CrossRefGoogle Scholar
Bonhomme, S. & Manresa, E. (2015) Grouped patterns of heterogeneity in panel data. Econometrica 83(3), 11471184.CrossRefGoogle Scholar
Bühlmann, P. & van de Geer, S. (2011) Statistics for High-Dimensional Data. Springer.CrossRefGoogle Scholar
Caner, M. & Kock, A.B. (2018) Asymptotically honest confidence regions for high dimensional parameters by the de-sparsified conservative Lasso. Journal of Econometrics 203(1), 143168.CrossRefGoogle Scholar
Caner, M. & Zhang, H.H. (2014) Adaptive elastic net for generalized methods of moments. Journal of Business & Economic Statistics 32(1), 3047.CrossRefGoogle ScholarPubMed
Davidson, J. (2000) Econometric Theory. Blackwell Publishers.Google Scholar
De Neve, J.-E., Christakis, N.A., Fowler, J.H., & Frey, B.S. (2012) Genes, economics, and happiness. Journal of Neuroscience, Psychology, and Economics 5(4), 193.CrossRefGoogle ScholarPubMed
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica: Journal of the Econometric Society 50(4), 9871007.CrossRefGoogle Scholar
Fan, X., Grama, I., & Liu, Q. (2012) Large deviation exponential inequalities for supermartingales. Electronic Communications in Probability 17(59), 18.CrossRefGoogle Scholar
Friedman, J., Hastie, T., & Tibshirani, R. (2010) Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software 33(1), 122.CrossRefGoogle ScholarPubMed
Galvao, A.F. & Montes-Rojas, G.V. (2010) Penalized quantile regression for dynamic panel data. Journal of Statistical Planning and Inference 140(11), 34763497.CrossRefGoogle Scholar
Horn, R.A. & Johnson, C.R. (1990) Matrix Analysis. Cambridge University Press.Google Scholar
Islam, N. (1995) Growth empirics: A panel data approach. The Quarterly Journal of Economics 110(4), 11271170.CrossRefGoogle Scholar
Javanmard, A. & Montanari, A. (2014) Confidence intervals and hypothesis testing for high-dimensional regression. Journal of Machine Learning Research 15, 28692909.Google Scholar
Knight, K. & Fu, W. (2000) Asymptotics for Lasso-type estimators. The Annals of Statistics 28(5), 13561378.Google Scholar
Kock, A.B. (2013) Oracle efficient variable selection in random and fixed effects panel data models. Econometric Theory 29, 115152.CrossRefGoogle Scholar
Kock, A.B. (2016) Oracle inequalities, variable selection and uniform inference in high-dimensional correlated random effects panel data models. Journal of Econometrics 195(1), 7185.CrossRefGoogle Scholar
Koenker, R. (2004) Quantile regression for longitudinal data. Journal of Multivariate Analysis 91(1), 7489.CrossRefGoogle Scholar
Leeb, H. & Pötscher, B.M. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 2159.CrossRefGoogle Scholar
Lesigne, E. & Volny, D. (2001) Large deviations for martingales. Stochastic Processes and Their Applications 96(1), 143159.CrossRefGoogle Scholar
Li, K.-C. (1989) Honest confidence regions for nonparametric regression. The Annals of Statistics 17(3), 10011008.CrossRefGoogle Scholar
Lu, X. & Su, L. (2016) Shrinkage estimation of dynamic panel data models with interactive fixed effects. Journal of Econometrics 190(1), 148175.CrossRefGoogle Scholar
McLeish, D.L. (1974) Dependent central limit theorems and invariance principles. The Annals of Probability 2(4), 620628.CrossRefGoogle Scholar
Merlevède, F., Peligrad, M., & Rio, E. (2011) A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probability Theory and Related Fields 151(3–4), 435474.CrossRefGoogle Scholar
Negahban, S.N., Ravikumar, P., Wainwright, M.J., & Yu, B. (2012) A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers. Statistical Science 27(4), 538557.CrossRefGoogle Scholar
Nickell, S. (1981) Biases in dynamic models with fixed effects. Econometrica 49(6), 14171426.CrossRefGoogle Scholar
Pötscher, B.M. (2009) Confidence sets based on sparse estimators are necessarily large. Sankhyā: The Indian Journal of Statistics, Series A 71(1), 118.Google Scholar
Su, L., Shi, Z., & Phillips, P.C. (2016) Identifying latent structures in panel data. Econometrica 84(6), 22152264.CrossRefGoogle Scholar
Team, R.C. (2000) R Language Definition. R foundation for statistical computing.Google Scholar
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B 58, 267288.Google Scholar
van de Geer, S., Bühlmann, P., Ritov, Y., & Dezeure, R. (2014) On asymptotically optimal confidence regions and tests for high-dimensional models. The Annals of Statistics 42(3), 11661202.CrossRefGoogle Scholar
van de Geer, S.A., & Bühlmann, P. (2009) On the conditions used to prove oracle results for the lasso. Electronic Journal of Statistics 3, 13601392.CrossRefGoogle Scholar
van der Vaart, A. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer.CrossRefGoogle Scholar
Wooldridge, J.M. (2010) Econometric Analysis of Cross Section and Panel Data. MIT press.Google Scholar
Yuan, M. (2010) High dimensional inverse covariance matrix estimation via linear programming. Journal of Machine Learning Research 11, 22612286.Google Scholar
Zhang, C.-H. & Zhang, S.S. (2014) Confidence intervals for low dimensional parameters in high dimensional linear models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76(1), 217242.CrossRefGoogle Scholar