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CENTRAL LIMIT THEORY FOR COMBINED CROSS SECTION AND TIME SERIES WITH AN APPLICATION TO AGGREGATE PRODUCTIVITY SHOCKS

Published online by Cambridge University Press:  19 September 2022

Jinyong Hahn ,
Guido Kuersteiner  and
Maurizio Mazzocco
Jinyong Hahn
Affiliation:
University of California, Los Angeles
Guido Kuersteiner*
Affiliation:
University of Maryland
Maurizio Mazzocco
Affiliation:
University of California, Los Angeles
*
Address correspondence to Guido Kuersteiner, Department of Economics, University of Maryland, Tydings Hall 3145, College Park, MD 20742, USA; e-mail: [email protected].
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Abstract

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Combining cross-sectional and time-series data is a long and well-established practice in empirical economics. We develop a central limit theory that explicitly accounts for possible dependence between the two datasets. We focus on common factors as the mechanism behind this dependence. Using our central limit theorem (CLT), we establish the asymptotic properties of parameter estimates of a general class of models based on a combination of cross-sectional and time-series data, recognizing the interdependence between the two data sources in the presence of aggregate shocks. Despite the complicated nature of the analysis required to formulate the joint CLT, it is straightforward to implement the resulting parameter limiting distributions due to a formal similarity of our approximations with Murphy and Topel’s (1985, Journal of Business and Economic Statistics 3, 370–379) formula.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 1 , February 2024 , pp. 162 - 212
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We thank Peter Phillips and referees in previous rounds for many helpful suggestions.

References

REFERENCES

Ackerberg, D. A., Caves, K. & Frazer, G. (2015) Identification properties of recent production function estimators. Econometrica 83, 2411–2451.CrossRefGoogle Scholar
An, S. & Schorfheide, F. (2007) Left Bayesian analysis of DSGE models. Econometric Reviews 26, 113–172.CrossRefGoogle Scholar
Andrews, D.W.K. (2005) Cross-section regression with common shocks. Econometrica 73, 1551–1585.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008) Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76, 1481–1536.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.CrossRefGoogle Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81, 27–60.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Dedecker, J. & Merlevede, F. (2002) Necessary and sufficient conditions for the conditional central limit theorem. Annals of Probability 30, 1044–1081.CrossRefGoogle Scholar
Eagleson, G.K. (1975) Martingale convergence to mixtures of infinitely divisible laws. The Annals of Probability 3, 557–562.CrossRefGoogle Scholar
Feigin, P.D. (1985) Stable convergence of semimartingales. Stochastic Processes and their Applications 19, 125–134.CrossRefGoogle Scholar
Gänssler, P. & Häusler, E. (1979) Remarks on the functional central limit theorem for martingales. Z. fuer Wahrscheinlichkeitstheorie 50, 237–243.CrossRefGoogle Scholar
Gordin, M.I. (1969) The central limit theorem for stationary processes. Doklady Akademii Nauk SSSR 188, 739–741.Google Scholar
Gordin, M.I. (1973) Abstracts of Communication, T.1: A-K, International Conference on Probability Theory, Vilnius.Google Scholar
Hahn, J., Kuersteiner, G., & Mazzocco, M. (2020) Estimation with aggregate shocks. Review of Economic Studies 87(3), 1365–1398.CrossRefGoogle Scholar
Hahn, J., Kuersteiner, G., & Mazzocco, M. (2021) Joint time series and cross-section limit theory under mixingale assumptions. Econometric Theory, published online 11 August 2020. https://doi.org/10.1017/S0266466620000316.Google Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Heckman, J.J., Lochner, L., & Taber, C. (1998) Explaining rising wage inequality: Explorations with a dynamic general equilibrium model of labor earnings with heterogeneous agents. Review of Economic Dynamics 1, 1–58.CrossRefGoogle Scholar
Heckman, J.J. & Sedlacek, G. (1985) Heterogeneity, aggregation, and market wage functions: An empirical model of self-selection in the labor market. Journal of Political Economy 93, 1077–1125.CrossRefGoogle Scholar
Inoue, A. & Kilian, L. (2020) The uniform validity of impulse response inference in autoregressions. Journal of Econometrics 215(2), 450–472.CrossRefGoogle Scholar
Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. Annals of Statistics 38, 1478–1545.CrossRefGoogle Scholar
Jacod, J. & Protter, P. (2012) Discretization of Processes. Springer.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2002) Limit Theorems for Stochastic Processes, 2nd Edition. Springer.Google Scholar
Kallenberg, O. (1997) Foundations of Modern Probability. Springer.Google Scholar
Kuersteiner, G.M. & Prucha, I.R. (2013) Limit theory for panel data models with cross sectional dependence and sequential exogeneity. Journal of Econometrics 174, 107–126.CrossRefGoogle ScholarPubMed
Kuersteiner, G.M. & Prucha, I.R. (2020) Dynamic spatial panel models: Networks, common shocks, and sequential exogeneity. Econometrica 88, 2109–2146.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Kydland, F.E. & Prescott, E.C. (1982) Time to build and aggregate fluctuations. Econometrica 50, 1345–1370.CrossRefGoogle Scholar
Lee, D. & Wolpin, K.I. (2006) Intersectoral labor mobility and the growth of service sector. Econometrica 47, 1–46.CrossRefGoogle Scholar
Lee, D. & Wolpin, K.I. (2010) Accounting for wage and employment changes in the U.S. from 1968-2000: A dynamic model of labor market equilibrium. Journal of Econometrics 156, 68–85.CrossRefGoogle Scholar
Li, J. & Xiu, D. (2016) Generalized method of integrated moments for high-frequency data. Econometrica 84, 1613–1633.CrossRefGoogle Scholar
Long, J.B. & Plosser, C.I. (1983) Real business cycles. Journal of Political Economy 91, 39–69.CrossRefGoogle Scholar
McLeish, D.L. (1975a) A maximal inequality and dependent strong laws. Annals of Probability 3, 829–839.CrossRefGoogle Scholar
McLeish, D.L. (1975b) Invariance principles for dependent variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32, 165–178.CrossRefGoogle Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75, 1411–1452.CrossRefGoogle Scholar
Murphy, K.M. & Topel, R.H. (1985) Estimation and inference in two-step econometric models. Journal of Business and Economic Statistics 3, 370–379.Google Scholar
Olley, S. & Pakes, A. (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64, 1263–1297.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Regression theory for near integrated time series. Econometrica 56, 1021–1044.CrossRefGoogle Scholar
Phillips, P.C.B. (2014) On confidence intervals for autoregressive roots and predictive regression. Econometrica 82, 1177–1195.Google Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. & Moon, H.R. (1999) Linear regression limit theory for nonstationary panel data. Econometrica 67, 1057–1111.CrossRefGoogle Scholar
Phillips, P.C.B. & Ouliaris, S. (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.CrossRefGoogle Scholar
Phillips, P.C.B. & Sul, D. (2003) Dynamic panel estimation and homogeneity testing under cross section dependence. The Econometrics Journal 6, 217–259.CrossRefGoogle Scholar
Rényi, A. (1963) On stable sequences of events. Sankhya: The Indian Journal of Statistics, Series A 25, 293–302.Google Scholar
Ridder, G. & Moffitt, R. (2007) The econometrics of data combination. In Heckman, J.J. and Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6 (Part B). Elsevier.Google Scholar
Rootzen, H. (1983) Central limit theory for martingales via random change of time. In Holst, L. and Gut, A. (eds.), Essays in Honour of Carl Gustav Essen, pp. 154–190. Uppsala University.Google Scholar
Rosenzweig, M.R. & Udry, C. (2020) External validity in a stochastic world: Evidence from low-income countries. Review of Economic Studies 87(1), 343–381.CrossRefGoogle Scholar
Runkle, D.E. (1991) Liquidity constraints and the permanent-income hypothesis: Evidence from panel data. Journal of Monetary Economics 27, 73–98.CrossRefGoogle Scholar
Schorfheide, F. (2000) Loss function-based evaluation of DSGE models. Journal of Applied Econometrics 15, 645–670.CrossRefGoogle Scholar
Shea, J. (1995) Union contracts and the life-cycle/permanent-income hypothesis. American Economic Review 85, 186–200.Google Scholar
Smets, F. & Wouters, R. (2007) Shocks and frictions in US business cycles: A Bayesian DSGE approach. American Economic Review 97, 586–606.CrossRefGoogle Scholar
Stroock, D.W. & Varadhan, S.R.S. (1979) Multidimensional Diffusion Processes. Springer.Google Scholar
Tobin, J. (1950) A statistical demand function for food in the U.S.A. Journal of the Royal Statistical Society, Series A 113, 113–149.CrossRefGoogle Scholar
Vissing-Jørgensen, A. (2002) Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110, 825–853.CrossRefGoogle Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4, 210–230.CrossRefGoogle Scholar
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