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Locally Stationary Factor Models Giovanni Motta Supervisors: Professors Rainer von Sachs & Christian M. Hafner Abstract Linear factor models have attracted considerable interest over recent years especially in the econometrics literature. The intuitively appealing idea to explain a panel of economic variables by a few common factors is one of the reasons for their popularity. From a statistical viewpoint, the need to reduce the cross-section dimension to a much smaller factor space dimension is obvious considering the large data sets available in economics and finance. The traditional approach of fixing either the time dimension T or the cross-section dimension N and letting the other dimension go to infinity is likely to be inappropriate in situations where both dimensions are large. The large factor model literature, including Stock and Watson (2002a, 2002b), Forni et al. (2000, 2005), Forni and Lippi (2001), Bai and Ng (2002) and Bai(2003), use the concept introduced by Chamberlain and Rothschild (1983) of simultaneous asymptotics, where both N and T go to infinity with rates that are rather flexible. One of the characteristics of the traditional factor model is that the process is stationary in the time dimension. This appears restrictive, given the fact that over long time periods it is unlikely that e.g. factor loadings remain constant. For example, in the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), typical empirical results show that factor loadings are time-varying, which in the CAPM is caused by time-varying second moments. In this thesis we propose two new approximate factor models for large cross-section and time dimensions. In the first model factor loadings are assumed to be smooth functions of time, which allows to consider the model as locally stationary while permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically covariance matrix. As is wellknown 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 one of the main objective of this thesis. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators of the time varying covariance matrix, the loadings and the factors. We generalize to the locally stationary case the results given by Bai (2003) in the stationary framework. A simulation study illustrates the performance of these estimators. This approach can be generalized even further. In the second approach we propose a factor model with infinite dynamics characterized by a locally stationary behavior à la Dahlhaus (1997) and nonorthogonal idiosyncratic components as in the static approximate factor model of Chamberlain and Rothschild (1983). This model generalizes the dynamic (but stationary) factor model of Forni et al. (2000), as well as the nonstationary (but static) factor model of Motta et al. (2006). In the stationary (dynamic) case, Forni et al. (2000) show that the common components are estimated by the eigenvectors of a consistent estimator of the spectral density matrix, which is a matrix depending only on the frequency. In the locally stationary framework the dynamics of the model is explained by a time-varying spectral density matrix. This operator is a function of time as well as of the frequency. We show that the common components of a locally stationary dynamic factor model can be estimated consistently by the eigenvectors of a consistent estimator of the time-varying spectral density matrix. References Bai, J. (2003). Inferential theory for factor models of large dimension. Econometrica 71(11), 135-171. Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica 70(1), 191-221. Chamberlain, G. and Rothschild, M. (1983). Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets. Econometrica 51(5), 1281-1304. Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. The Annals of Statistics 25, 1-37. Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2000). The generalized dynamic factor model: Identification and estimation. The Review of Economics and Statistics 82, 540-554. Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005). The generalized dynamic factor model: One-sided estimation and forecasting. Journal of the American Statistical Association 100(471), 830-840. Forni, M. and Lippi, M. (2001). The generalized dynamic factor model: Representation theory. Econometric Theory 17, 1113-1141. Lintner, J. (1965). The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics 47, 13-37. Motta, G., Hafner, C. and von Sachs, R. (2006). Locally stationary factor models: Identification and nonparametric estimation. Discussion Paper 0624, Institut de Statistique, UCL. Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, 425-442. Stock, J. H. and Watson, M. W. (2002a). Macroeconomic forecasting using diffusion indexes. Journal of Business and Economic Statistics 20(2), 147-162. Stock, J. H. and Watson, M. W. (2002b). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97(460), 1167-1179. 2