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INSTRUMENTAL VARIABLES INFERENCE IN A SMALL-DIMENSIONAL VAR MODEL WITH DYNAMIC LATENT FACTORS

Published online by Cambridge University Press:  10 November 2022

Federico Carlini  and
Patrick Gagliardini
Federico Carlini
Affiliation:
LUISS University Rome
Patrick Gagliardini*
Affiliation:
Università della Svizzera italiana
*
Address correspondence to Patrick Gagliardini, Faculty of Economics, Università della Svizzera italiana, Lugano, Switzerland; e-mail: [email protected].
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Abstract

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We study semiparametric inference in a small-dimensional vector autoregressive (VAR) model of order p augmented by unobservable common factors with a dynamic described by a VAR process of order q. This state-space specification is useful to measure separately the direct causality effects and the responses to dynamic common factors. We show that the state-space parameters are identifiable from the autocovariance function of the observed process. We estimate the model by means of a multistep procedure in closed-form, which combines an eigenvalue–eigenvector matrix decomposition and a linear instrumental variable estimation allowing for Hansen–Sargan specification tests. We study the asymptotic and finite-sample properties of the parameter estimators and of rank tests for selecting the number of unobservable factors and VAR orders. In an empirical illustration, we investigate the dynamic common factors and the spillover effects that explain the co-movements among the log daily realized volatilities of four European stock market indices.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 4 , August 2024 , pp. 705 - 751
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We are grateful to the Editor, a Co-Editor, three anonymous referees, M. Deistler, C. Gourieroux, S. Johansen, L. Mancini, P. Santucci de Magistris, O. Scaillet, and participants at the 5th Empirical Finance Workshop in Cergy, the ESEM 2018 Conference in Cologne, the SFI 2019 Research Day in Gerzensee, the First Rome Workshop of Time Series and Financial Econometrics, and seminars at LUISS University, Durham University, and Geneva University for useful comments. We thank F. Diebold and K. Yilmaz for kindly providing us with their datasets used in a previous version of the paper circulated under the title of “Vector Autoregressive Model with Dynamic Factors.” The first version of this paper has been written while F. Carlini was a postdoctoral fellow at the Faculty of Economics of the Università della Svizzera italiana, Lugano, Switzerland. We gratefully acknowledge the Swiss National Science Foundation for Grant 105218-162633.

References

REFERENCES

Acemoglu, D., Ozdaglar, A., & Tahbaz-Salehi, A. (2015) Systemic risk and stability in financial networks. American Economic Review 105(2), 564–608.CrossRefGoogle Scholar
Acharya, V., Pedersen, L., Philippon, T., & Richardson, M. (2017) Measuring systemic risk. Review of Financial Studies 30, 2–47.CrossRefGoogle Scholar
Adrian, T. & Brunnermeier, M. (2016) CoVaR. American Economic Review 106, 1705–1741.CrossRefGoogle Scholar
Ait-Sahalia, Y., Cacho-Diaz, J., & Laeven, R. (2015) Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics 117, 585–606.CrossRefGoogle Scholar
Akaike, H. (1974) Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Annals of the Institute of Statistical Mathematics 26, 363–387.CrossRefGoogle Scholar
Allen, F. & Gale, D. (2000) Financial contagion. Journal of Political Economy 108(1), 1–33.CrossRefGoogle Scholar
Al-Sadoon, M.M. (2017) A unifying theory of tests of rank. Journal of Econometrics 199(1), 49–62.CrossRefGoogle Scholar
Anderson, T.W. (1951a) The asymptotic distribution of certain characteristic roots and vectors. In Neyman, J. (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability . University of California Press, pp. 103–130.CrossRefGoogle Scholar
Anderson, T.W. (1951b) Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327–351.CrossRefGoogle Scholar
Andrews, D. & Monahan, C. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.CrossRefGoogle Scholar
Bai, J., Li, K., & Lu, L. (2016) Estimation and inference of FAVAR models. Journal of Business & Economic Statistics 34(4), 620–641.CrossRefGoogle Scholar
Bernanke, B., Boivin, J., & Eliasz, P. (2005) Measuring the effects of monetary policy: A factor-augmented vector autoregressive (FAVAR) approach. Quarterly Journal of Economics 120, 387–422.Google Scholar
Billio, M., Getmansky, M., Lo, A.W., & Pelizzon, L. (2012) Econometric measures of connectedness and systemic risk in the finance and insurance sectors. Journal of Financial Economics 104(3), 535–559.CrossRefGoogle Scholar
Brownlees, C. & Engle, R. (2017) SRISK: A conditional capital shortfall measure of systemic risk. Review of Financial Studies 30, 48–79.CrossRefGoogle Scholar
Chan, J., Eisenstat, E., & Koop, G. (2016) Large Bayesian VARMAs. Journal of Econometrics 192, 374–390.CrossRefGoogle Scholar
Chan, J., Leon-Gonzalez, R., & Strachan, R.W. (2018) Invariant inference and efficient computation in the static factor model. Journal of the American Statistical Association 113(522), 819–828.CrossRefGoogle Scholar
Chen, Q. & Fang, Z. (2019) Improved inference on the rank of a matrix. Quantitative Economics 10(4), 1787–1824.CrossRefGoogle Scholar
Cragg, J. & Donald, S. (1996) On the asymptotic properties of LDU-based tests of the rank of a matrix. Journal of the American Statistical Association 91, 1301–1309.CrossRefGoogle Scholar
Darolles, S., Dubecq, S., & Gouriéroux, C. (2014). Contagion Analysis in the Banking Sector. Working paper.CrossRefGoogle Scholar
Darolles, S. & Gourieroux, C. (2015) Contagion Phenomena with Applications in Finance . Elsevier.Google Scholar
De Jong, R.M. & Davidson, J. (2000) Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Econometrica 68(2), 407–423.CrossRefGoogle Scholar
Diebold, F.X. & Yılmaz, K. (2014) On the network topology of variance decompositions: Measuring the connectedness of financial firms. Journal of Econometrics 182(1), 119–134.CrossRefGoogle Scholar
Eisenberg, L. & Noe, T. (2001) Systemic risk in financial systems. Management Science 47, 236–249.CrossRefGoogle Scholar
Elliott, M., Golub, B., & Jackson, M.O. (2014) Financial networks and contagion. American Economic Review 104(10), 3115–3153.CrossRefGoogle Scholar
Engle, R., Ito, T., & Lin, W.-L. (1990) Meteor showers or heat waves? Heteroskedastic intra-day volatility in foreign exchange market. Econometrica 58, 525–542.CrossRefGoogle Scholar
Foerster, A.T., Sarte, P.-D.G., & Watson, M.W. (2011) Sectoral versus aggregate shocks: A structural factor analysis of industrial production. Journal of Political Economy 119(1), 1–38.CrossRefGoogle Scholar
Forbes, K.J. & Rigobon, R. (2002) No contagion, only interdependence: Measuring stock market comovements. The Journal of Finance 57(5), 2223–2261.CrossRefGoogle Scholar
Forni, M., Giannone, D., Lippi, M., & Reichlin, L. (2009) Opening the black box: Structural factor models with large cross sections. Econometric Theory 25(5), 1319–1347.CrossRefGoogle Scholar
Forni, M., Hallin, M., Lippi, M., & Reichlin, L. (2000) The generalized dynamic-factor model: Identification and estimation. Review of Economics and Statistics 82(4), 540–554.CrossRefGoogle Scholar
Forni, M. & Lippi, M. (2001) The generalized dynamic factor model: Representation theory. Econometric Theory 82(4), 1113–1141.CrossRefGoogle Scholar
Gagliardini, P. & Gouriéroux, C. (2019) Identification by Laplace transforms in nonlinear time series and panel models with unobserved stochastic dynamic effects. Journal of Econometrics 208(2), 613–637.CrossRefGoogle Scholar
Gallant, R., Giacomini, R., & Ragusa, G. (2017) Bayesian estimation of state space models using moment conditions. Journal of Econometrics 201, 198–211.CrossRefGoogle Scholar
Glover, K. & Willems, J. (1974) Parametrizations of linear dynamical systems: Canonical forms and identifiability. IEEE Transactions on Automatic Control 19(6), 640–646.CrossRefGoogle Scholar
Hallin, M. & Lippi, M. (2013) Factor models in high dimensional time series. A time domain approach. Stochastic Processes and their Applications 123(7), 2678–2695.CrossRefGoogle Scholar
Hamilton, J. D. (1994) State-space models. In Engle, R. F., McFadden, D. L. (eds.), Handbook of Econometrics , vol. 4. Elsevier, pp. 3041–3080.Google Scholar
Hannan, E. (1969) The identification of vector mixed autoregressive-moving average system. Biometrika 56(1), 223–225.Google Scholar
Hannan, E. (1971) The identification problem for multiple equation systems with moving average errors. Econometrica 39(5), 751–756.CrossRefGoogle Scholar
Hannan, E. (1975) The estimation of ARMA models. Annals of Statistics 3, 975–981.CrossRefGoogle Scholar
Hannan, E. & Heyde, C. (1972) On limit theorems for quadratic functions of discrete time series. The Annals of Mathematical Statistics 43, 2058–2066.CrossRefGoogle Scholar
Hannan, E.J. & Deistler, M. (1988) The Statistical Theory of Linear Systems . SIAM.Google Scholar
Hansen, L.P. & Singleton, K.J. (1991) Computing semi-parametric efficiency bounds for linear time series models. In Nonparametric and Semiparametric Methods in Econometrics and Statistics , pp. 387–412.Google Scholar
Herrndorf, N. (1984) A functional central limit theorem for weakly dependent sequences of random variables. The Annals of Probability 12(1), 141–153.CrossRefGoogle Scholar
Izenman, A.J. (1975) Reduced-rank regression for the multivariate linear model. Journal of Multivariate Analysis 5(2), 248–264.CrossRefGoogle Scholar
King, M.A. & Wadhwani, S. (1990) Transmission of volatility between stock markets. The Review of Financial Studies 3(1), 5–33.CrossRefGoogle Scholar
Kleibergen, F. & Paap, R. (2006) Generalized reduced rank tests using the singular value decomposition. Journal of Econometrics 133(1), 97–126.CrossRefGoogle Scholar
Komunjer, I. & Ng, S. (2011) Dynamic identification of dynamic stochastic general equilibrium models. Econometrica 79(6), 1995–2032.Google Scholar
Koop, G., Pesaran, M.H., & Potter, S.M. (1996) Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74(1), 119–147.CrossRefGoogle Scholar
Long, J.B.J. & Plosser, C.I. (1983) Real business cycles. Journal of Political Economy 91(1), 39–69.CrossRefGoogle Scholar
Manski, C.F. (1993) Identification of endogenous social effects: The reflection problem. The Review of Economic Studies 60(3), 531–542.CrossRefGoogle Scholar
Masson, P. (1999) Contagion: Macroeconomic models with multiple equilibria. Journal of International Money and Finance 18(4), 587–602.CrossRefGoogle Scholar
Mokkadem, A. (1988) Mixing properties of ARMA processes. Stochastic Processes and their Applications 29(2), 309–315.CrossRefGoogle Scholar
Newey, W. & McFadden, D. (1994). Large sample estimation and hypothesis testing. In Engle, R. F., McFadden, D. L. (eds.), Handbook of Econometrics , chapter 36, vol. 4. Elsevier.Google Scholar
Newey, W. & West, K. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.CrossRefGoogle Scholar
Pesaran, M.H. & Pick, A. (2007) Econometric issues in the analysis of contagion. Journal of Economic Dynamics and Control 31(4), 1245–1277.CrossRefGoogle Scholar
Pesaran, M.H. & Shin, Y. (1998) Generalized impulse response analysis in linear multivariate models. Economics Letters 58(1), 17–29.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Poetscher, B. (1983) Order estimation in ARMA-models by Lagrangian multiplier tests. Annals of Statistics 11, 872–885.Google Scholar
Reinsel, G. & Velu, R. (1998). Multivariate Reduced-Rank Regression . Lecture Notes in Statistics, Springer.CrossRefGoogle Scholar
Robin, J.-M. & Smith, R.J. (2000) Tests of rank. Econometric Theory 16(2), 151–175.CrossRefGoogle Scholar
Robinson, P.M. (1973) Generalized canonical analysis for time series. Journal of Multivariate Analysis 3(2), 141–160.CrossRefGoogle Scholar
Schennach, S.M. (2014) Entropic latent variable integration via simulation. Econometrica 82(1), 345–385.Google Scholar
Shiu, J.-L. & Hu, Y. (2013) Identification and estimation of nonlinear dynamic panel data models with unobserved covariates. Journal of Econometrics 175(2), 116–131.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (2005) Understanding changes in international business cycle dynamics. Journal of the European Economic Association 3(5), 968–1006.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (2018) Identification and estimation of dynamic causal effects in macroeconomics using external instruments. The Economic Journal 128(610), 917–948.CrossRefGoogle Scholar
Trevino, I. (2020) Informational channels of financial contagion. Econometrica 88(1), 297–335.CrossRefGoogle Scholar
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