Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

This paper studies Quasi Maximum Likelihood estimation of dynamic factor models for large panels of time series. Specifically, we consider the case in which the autocorrelation of the factors is explicitly accounted for and therefore the factor model has a state-space form. Estimation of the factors and their loadings is implemented by means of the Expectation Maximization algorithm, jointly with the Kalman smoother. We prove that, as both the dimension of the panel $n$ and the sample size $T$ diverge to infinity, the estimated loadings, factors, and common components are $\min(\sqrt n,\sqrt T)$-consistent and asymptotically normal. Although the model is estimated under the unrealistic constraint of independent idiosyncratic errors, this mis-specification does not affect consistency. Moreover, we give conditions under which the derived asymptotic distribution can still be used for inference even in case of mis-specifications. Our results are confirmed by a MonteCarlo simulation exercise where we compare the performance of our estimators with Principal Components.

[1]  M. Hallin,et al.  Dynamic Factor Models with Infinite-Dimensional Factor Space: Asymptotic Analysis , 2015 .

[2]  J. Stock,et al.  Forecasting Using Principal Components From a Large Number of Predictors , 2002 .

[3]  A. E. Maxwell,et al.  Factor Analysis as a Statistical Method. , 1964 .

[4]  J. Bai,et al.  Determining the Number of Factors in Approximate Factor Models , 2000 .

[5]  Julius S. Bendat,et al.  Stationary Random Processes , 2012 .

[6]  Geoffrey E. Hinton,et al.  Parameter estimation for linear dynamical systems , 1996 .

[7]  D. Giannone,et al.  Money, Credit, Monetary Policy and the Business Cycle in the Euro Area: What Has Changed since the Crisis? , 2012, SSRN Electronic Journal.

[8]  Marek Jarociński,et al.  An Inflation-Predicting Measure of the Output Gap in the Euro Area , 2016, SSRN Electronic Journal.

[9]  M. Hallin,et al.  The Generalized Dynamic-Factor Model: Identification and Estimation , 2000, Review of Economics and Statistics.

[10]  Melvin J. Hinich,et al.  Time Series Analysis by State Space Methods , 2001 .

[11]  K. Jöreskog.  A general approach to confirmatory maximum likelihood factor analysis , 1969 .

[12]  Hyun Hak Kim,et al.  Methods for backcasting, nowcasting and forecasting using factor‐MIDAS: With an application to Korean GDP , 2018 .

[13]  M. Hallin,et al.  Determining the Number of Factors in the General Dynamic Factor Model , 2007 .

[14]  Michael H. Neumann,et al.  Probability and moment inequalities for sums of weakly dependent random variables, with applications , 2007 .

[15]  S. Koopman,et al.  Likelihood-based Analysis for Dynamic Factor Models , 2008 .

[16]  Yuan Liao,et al.  Efficient Estimation of Approximate Factor Models via Regularized Maximum Likelihood , 2012, 1209.5911.

[17]  Davide Delle Monache,et al.  Efficient matrix approach for classical inference in state space models , 2019 .

[18]  Gabriele Fiorentini,et al.  A Spectral EM Algorithm for Dynamic Factor Models , 2015, Journal of Econometrics.

[19]  David H. Small,et al.  Nowcasting: the real time informational content of macroeconomic data releases , 2008 .

[20]  Sastry G. Pantula,et al.  Computational algorithms for the factor model , 1986 .

[21]  M. Solari.  The “Maximum Likelihood Solution” of the Problem of Estimating a Linear Functional Relationship , 1969 .

[22]  R. Tibshirani,et al.  Covariance‐regularized regression and classification for high dimensional problems , 2009, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[23]  J. Merikoski,et al.  Inequalities for spreads of matrix sums and products. , 2004 .

[24]  Kunpeng Li,et al.  STATISTICAL ANALYSIS OF FACTOR MODELS OF HIGH DIMENSION , 2012, 1205.6617.

[25]  S. Hörmann,et al.  Prediction of Singular VARs and an Application to Generalized Dynamic Factor Models , 2020, Journal of Time Series Analysis.

[26]  Michael E. Tipping,et al.  Probabilistic Principal Component Analysis , 1999 .

[27]  G. Thomson.  Some points of mathematical technique in the factorial analysis of ability. , 1936 .

[28]  Domenico Giannone,et al.  Exploiting the Monthly Data Flow in Structural Forecasting , 2015 .

[29]  S. Koopman,et al.  Forecasting the U.S. Term Structure of Interest Rates Using a Macroeconomic Smooth Dynamic Factor Model , 2012 .

[30]  Borus Jungbacker,et al.  Maximum Likelihood Estimation for Dynamic Factor Models with Missing Data , 2011 .

[31]  Robert Kohn,et al.  Fixed interval estimation in state space models when some of the data are missing or aggregated , 1983 .

[32]  Massimiliano Marcellino,et al.  Factor Midas for Nowcasting and Forecasting with Ragged-Edge Data: A Model Comparison for German GDP , 2008 .

[33]  W. Barnett,et al.  Real-Time Nowcasting of Nominal GDP Under Structural Breaks , 2014 .

[34]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[35]  Michele Modugno,et al.  Now-casting inflation using high frequency data , 2013 .

[36]  D. Giannone,et al.  Now-Casting and the Real-time Data Flow , 2012, SSRN Electronic Journal.

[37]  M. S. Bartlett,et al.  The statistical conception of mental factors. , 1937 .

[38]  H. Rauch.  Solutions to the linear smoothing problem , 1963 .

[39]  Thomas J. Sargent,et al.  Business cycle modeling without pretending to have too much a priori economic theory , 1976 .

[40]  Jianqing Fan,et al.  Large covariance estimation by thresholding principal orthogonal complements , 2011, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[41]  M. Barigozzi,et al.  Measuring the Output Gap using Large Datasets , 2019, Review of Economics and Statistics.

[42]  R. Mariano,et al.  A New Coincident Index of Business Cycles Based on Monthly and Quarterly Series , 2002 .

[43]  Dorothy T. Thayer,et al.  EM algorithms for ML factor analysis , 1982 .

[44]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[45]  G. Goodwin,et al.  Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems , 1984 .

[46]  E. J. Hannan,et al.  Multiple time series , 1970 .

[47]  R. Sundberg.  Maximum Likelihood Theory for Incomplete Data from an Exponential Family , 2016 .

[48]  Brian D. O. Anderson,et al.  Generalized linear dynamic factor models - a structure theory , 2008, 2008 47th IEEE Conference on Decision and Control.

[49]  Piotr Fryzlewicz,et al.  Simultaneous multiple change-point and factor analysis for high-dimensional time series , 2016, Journal of Econometrics.

[50]  Qi Li,et al.  Quasi Maximum Likelihood Analysis of High Dimensional Constrained Factor Models , 2018, Journal of Econometrics.

[51]  Forecas,et al.  Forecasting with Approximate Dynamic Factor Models : the role of non-pervasive shocks , 2011 .

[52]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[53]  Ivan Petrella,et al.  Speculation in the Oil Market , 2011 .

[54]  F. Diebold,et al.  Improving GDP Measurement: A Measurement-Error Perspective , 2013 .

[55]  Riccardo Lucchetti,et al.  A replication of “A quasi-maximum likelihood approach for large, approximate dynamic factor models” (Review of Economics and Statistics, 2012) , 2020 .

[56]  Vasja Sivec,et al.  Monetary, Fiscal and Oil Shocks: Evidence Based on Mixed Frequency Structural FAVARs , 2015 .

[57]  H. White.  A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity , 1980 .

[58]  Marco Lippi,et al.  Opening the Black Box: Structural Factor Models Versus Structural Vars , 2003 .

[59]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[60]  B. Poignard,et al.  Statistical analysis of sparse approximate factor models , 2020 .

[61]  A. Basilevsky,et al.  Factor Analysis as a Statistical Method. , 1964 .

[62]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[63]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[64]  John T. Scott.  Factor Analysis and Regression , 1966 .

[65]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[66]  P. D. Jong.  Smoothing and Interpolation with the State-Space Model , 1989 .

[67]  In Choi,et al.  EFFICIENT ESTIMATION OF FACTOR MODELS , 2011, Econometric Theory.

[68]  J. Robins,et al.  Double/Debiased Machine Learning for Treatment and Structural Parameters , 2017 .

[69]  Tengyao Wang,et al.  A useful variant of the Davis--Kahan theorem for statisticians , 2014, 1405.0680.

[70]  A. Harvey,et al.  Estimation Procedures for Structural Time Series Models , 1990 .

[71]  Matteo Luciani.  Monetary Policy and the Housing Market: A Structural Factor Analysis , 2010 .

[72]  C. Spearman.  General intelligence Objectively Determined and Measured , 1904 .

[73]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[74]  Jean Boivin,et al.  Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach , 2003 .

[75]  E. Hannan.  The Estimation of the Order of an ARMA Process , 1980 .

[76]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[77]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[78]  J. Breitung,et al.  GLS Estimation of Dynamic Factor Models , 2011 .

[79]  Ricardo Reis,et al.  Relative Goods&Apos; Prices, Pure Inflation, and the Phillips Correlation , 2007 .

[80]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[81]  J. Bai,et al.  Principal components estimation and identification of static factors , 2013 .

[82]  J. Davidson.  Stochastic Limit Theory , 1994 .

[83]  Domenico Giannone,et al.  Unspanned Macroeconomic Factors in the Yield Curve , 2014 .

[84]  Herman Rubin,et al.  Statistical Inference in Factor Analysis , 1956 .

[85]  J. Bai,et al.  Inferential Theory for Factor Models of Large Dimensions , 2003 .

[86]  Maximo Camacho,et al.  Introducing the Euro-Sting: Short-Term Indicator of Euro Area Growth , 2009 .

[87]  Serena Ng,et al.  Determining the Number of Primitive Shocks in Factor Models , 2007 .

[88]  Marco Lippi,et al.  The Generalized Dynamic Factor Model , 2002 .

[89]  Borus Jungbacker,et al.  Smooth Dynamic Factor Analysis with Application to the U.S. Term Structure of Interest Rates , 2012 .

[90]  H. M. Walker,et al.  The Factorial Analysis of Human Ability. , 1940 .

[91]  Yasuo Amemiya,et al.  The asymptotic distributions of some estimators for a factor analysis model , 1987 .

[92]  P. Doukhan,et al.  A new weak dependence condition and applications to moment inequalities , 1999 .

[93]  R. Sundberg.  An iterative method for solution of the likelihood equations for incomplete data from exponential families , 1976 .

[94]  Danny Quah,et al.  A Dynamic Index Model for Large Cross Sections , 1993 .

[95]  Kunpeng Li,et al.  Maximum Likelihood Estimation and Inference for Approximate Factor Models of High Dimension , 2016, Review of Economics and Statistics.

[96]  R. Giacomini,et al.  Anchoring the Yield Curve Using Survey Expectations , 2013, SSRN Electronic Journal.

[97]  Xiao-Li Meng,et al.  Maximum likelihood estimation via the ECM algorithm: A general framework , 1993 .

[98]  R. Engle,et al.  Alternative Algorithms for the Estimation of Dynamic Factor , 1983 .

[99]  Serena Ng,et al.  Are more data always better for factor analysis , 2006 .

[100]  Matteo Luciani,et al.  Nowcasting Norway , 2013 .

[101]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[102]  Michele Modugno,et al.  Nowcasting Business Cycles: A Bayesian Approach to Dynamic Heterogeneous Factor Models , 2015 .

[103]  G. Thomson,et al.  Methods of Estimating Mental Factors , 1938, Nature.

[104]  Domenico Giannone,et al.  Comparing Alternative Predictors Based on Large‐Panel Factor Models , 2006 .

[105]  Michele Modugno,et al.  Maximum Likelihood Estimation of Factor Models on Data Sets with Arbitrary Pattern of Missing Data , 2010, SSRN Electronic Journal.

[106]  Roman Vershynin,et al.  High-Dimensional Probability , 2018 .

[107]  Peng Wang,et al.  Identification and Bayesian Estimation of Dynamic Factor Models , 2015 .

[108]  C. B.,et al.  The Factorial Analysis of Human Ability , 1939, Nature.

[109]  Marco Del Negro,et al.  99 Luftballons: Monetary Policy and the House Price Boom Across U.S , 2007 .

[110]  Domenico Giannone,et al.  Conditional Forecasts and Scenario Analysis with Vector Autoregressions for Large Cross-Sections , 2014, SSRN Electronic Journal.

[111]  Serena Ng,et al.  Matrix Completion, Counterfactuals, and Factor Analysis of Missing Data , 2019, 1910.06677.

[112]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[113]  GourierouxMonfort.  Statistics and Econometric Models, Volume 2 , 1996 .

[114]  Thomas Drechsel,et al.  Tracking the Slowdown in Long-Run GDP Growth , 2017, Review of Economics and Statistics.

[115]  Xiao-Li Meng,et al.  On the global and componentwise rates of convergence of the EM algorithm , 1994 .

[116]  T. W. Anderson,et al.  The asymptotic normal distribution of estimators in factor analysis under general conditions , 1988 .

[117]  Rolf Sundberg,et al.  Exploratory factor analysis - Parameter estimation and scores prediction with high-dimensional data , 2016, J. Multivar. Anal..

[118]  Matteo Barigozzi,et al.  Measuring US Aggregate Output and Output Gap Using Large Datasets , 2018 .

[119]  D. Cochrane,et al.  Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms , 1949 .

[120]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[121]  Catherine Doz,et al.  A Quasi–Maximum Likelihood Approach for Large, Approximate Dynamic Factor Models , 2006, Review of Economics and Statistics.

[122]  Mark W. Watson,et al.  Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics , 2016 .

[123]  Catherine Doz,et al.  A Two-Step Estimator for Large Approximate Dynamic Factor Models Based on Kalman Filtering , 2007 .

[124]  Pilar Poncela,et al.  More is not always better: Kalman filtering in dynamic factor models , 2015 .

[125]  I. Prucha,et al.  Limit Theory for Panel Data Models with Cross Sectional Dependence and Sequential Exogeneity. , 2013, Journal of econometrics.