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LOCALLY STATIONARY FACTOR MODELS: IDENTIFICATION AND NONPARAMETRIC ESTIMATION

Published online by Cambridge University Press:  07 June 2011

Giovanni Motta*
Affiliation:
Maastricht University
Christian M. Hafner
Affiliation:
Université Catholique de Louvain
Rainer von Sachs
Affiliation:
Université Catholique de Louvain
*
*Address correspondence to Giovanni Motta, Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands; e-mail: dr.giovannimotta@gmail.com.

Abstract

In this paper we propose a new approximate factor model for large cross-section and time dimensions. Factor loadings are assumed to be smooth functions of time, which allows considering the model as locally stationary while permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. As is well known in the stationary case, this principal components estimator is consistent in approximate factor models if the eigenvalues of the noise covariance matrix are bounded. To show that this carries over to our locally stationary factor model is the main objective of our paper. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators. A simulation study illustrates the performance of these estimators.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Alexander, C.O. (2001) Orthogonal GARCH. In Alexander, C. O. (ed.), Mastering Risk, vol. 2, pp. 21–38. Prentice Hall.Google Scholar
Bai, J. (2003) Inferential theory for factor models of large dimensions. Econometrica 71(1), 135–171.Google Scholar
Bai, J. & Ng, S. (2002) Determining the number of factors in approximate factor models. Econometrica 70(1), 191–221.Google Scholar
Breitung, J. & Eickmeier, S. (2006) Dynamic factor models. In Hübler, O. and Frohn, J. (eds.), Modern Econometric Analysis. Springer.Google Scholar
Brockmann, M., Gasser, T., & Herrmann, E. (1993) Locally adaptive bandwidth choice for kernel regression estimators. Journal of the American Statistical Association 88, 1302–1309.Google Scholar
Chamberlain, G. & Rothschild, M. (1983) Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51(5), 1281–1304.Google Scholar
Chern, J.-L. & Dieci, L. (2000) Smoothness and periodicity of some matrix decompositions. SIAM Journal on Matrix Analysis and Applications 22(3), 772–792.Google Scholar
Dahlhaus, R. (1996) Asymptotic statistical inference for nonstationary processes with evolutionary spectra. In Robinson, P.M. & Rosenblatt, M. (eds.), Athens Conference on Applied Probability and Time Series Analysis, vol. II. Springer-Verlag.Google Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25(1), 1–37.Google Scholar
Dahlhaus, R. (2000) A likelihood approximation for locally stationary processes. Annals of Statistics 28(6), 1762–1794.Google Scholar
Dahlhaus, R. & Subba Rao, S.S. (2006) Statistical inference for time-varying arch processes. Annals of Statistics 34(3), 1075–1114.Google Scholar
Diebold, F.X. & Nerlove, M. (1989) The dynamics of exchange rate volatility: A multivariate latent factor arch model. Journal of Applied Econometrics 4, 1–21.Google Scholar
Engle, R.F., Ng, V., & Rothschild, M. (1990) Asset pricing with a factor-arch structure: Empirical estimates for Treasury bills. Journal of Econometrics 45, 213–237.Google Scholar
Fancourt, C.L. & Principe, J.C. (1998) Competitive principal components analysis for locally stationary time series. IEEE Transactions on Signal Processing 46, 3068–81.Google 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.Google Scholar
Forni, M., Hallin, M., Lippi, M., & Reichlin, L. (2005) The generalized dynamic factor model: One-sided estimation and forecasting. Journal of the American Statistical Association 100(471), 830–840.Google Scholar
Forni, M. & Lippi, M. (2001) The generalized dynamic factor model: Representation theory. Econometric Theory 17, 1113–1141.Google Scholar
Gasser, T., Kneip, A., & Köhler, W. (1991) A flexible and fast method for automatic smoothing. Journal of the American Statistical Association 86, 643–652.Google Scholar
Hafner, C.M., van Dijk, D., & Franses, P.H. (2006) Semiparametric modeling of correlation dynamics. In Fomby, T. & Hill, C. (eds.), Advances in Econometrics, vol. 20, part A, pp. 59–103. Emerald Group.Google Scholar
Härdle, W., Herwartz, H., & Spokoiny, V. (2004) Time inhomogeneous multiple volatility modeling. Journal of Financial Econometrics 1, 55–95.Google Scholar
Herrmann, E. (1997) Local bandwidth choice in kernel regression estimation. Journal of Computational and Graphical Statistics 6, 35–54.Google Scholar
Herzel, S., Stărică, C., & Tütüncü, R. (2006) A non-stationary paradigm for the dynamics of multivariate financial returns. In Bertail, P., Doukhan, P., & Soulier, P. (eds.), Statistics for Dependent Data, Lecture Notes in Statistics, vol. 187. Springer.Google Scholar
Kollo, T. & Neudecker, H. (1993) Asymptotics of eigenvalues and unit-length eigenvectors of sample variance and correlation matrices. Journal of Multivariate Analysis 47, 283–300.Google Scholar
Lintner, J. (1965) The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, 13–37.Google Scholar
Lütkepohl, H. (1996) Handbook of Matrices. Wiley.Google Scholar
Mikosch, T.. & Stărică, C. (2004) Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Review of Economics and Statistics 86, 378–390.Google Scholar
Phillips, P.C.B. & Moon, H.R. (1999) Linear regression limit theory for nonstationary panel data. Econometrica 67(5), 1057–1111.Google Scholar
Rodríguez-Poo, J.M. & Linton, O. (2001) Nonparametric factor analysis of residual time series. Test 10, 161–182.Google Scholar
Ross, S. (1976) The arbitrage theory of capital asset pricing. Journal of Economic Theory 13, 341–360.Google Scholar
Sharpe, W. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425–442.Google Scholar
Stărică, C. & Granger, C. (2005) Nonstationarities in stock returns. Review of Economics and Statistics 87, 503–522.Google Scholar
Stock, J.H. & Watson, M.W. (2002a) Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97(460), 1167–1179.Google Scholar
Stock, J.H. & Watson, M.W. (2002b) Macroeconomic forecasting using diffusion indexes. Journal of Business and Economic Statistics 20(2), 147–162.Google Scholar