Low Frequency and Weighted Likelihood Solutions for Mixed Frequency Dynamic Factor Models

The multivariate analysis of a panel of economic and financial time series with mixed frequencies is a challenging problem. The standard solution is to analyze the mix of monthly and quarterly time series jointly by means of a multivariate dynamic model with a monthly time index: artificial missing values are inserted for the intermediate months of the quarterly time series. In this paper we explore an alternative solution for a class of dynamic factor models that is specified by means of a low frequency quarterly time index. We show that there is no need to introduce artificial missing values while the high frequency (monthly) information is preserved and can still be analyzed. We also provide evidence that the analysis based on a low frequency specification can be carried out in a computationally more efficient way. A comparison study with existing mixed frequency procedures is presented and discussed. Furthermore, we modify the method of maximum likelihood in the context of a dynamic factor model. We introduce variable-specific weights in the likelihood function to let some variable equations be of more importance during the estimation process. We derive the asymptotic properties of the weighted maximum likelihood estimator and we show that the estimator is consistent and asymptotically normal. We also verify the weighted estimation method in a Monte Carlo study to investigate the effect of differen t choices for the weights in different scenarios. Finally, we empirically illustrate the new developments for the extraction of a coincident economic indicator from a small panel of mixed frequency economic time series.

[1]  Real-Time Measurement of Business Conditions, Second Version , 2008 .

[2]  Roberto Golinelli,et al.  The use of monthly indicators to forecast quarterly GDP in the short run: an application to the G7 countries , 2007 .

[3]  P. Bougerol.  Kalman filtering with random coefficients and contractions , 1993 .

[4]  P. Billingsley,et al.  The Lindeberg-Lévy theorem for martingales , 1961 .

[5]  K. Wohlrabe,et al.  Forecasting with mixed-frequency time series models , 2009 .

[6]  Massimiliano Marcellino,et al.  Realtime nowcasting with a Bayesian mixed frequency model with stochastic volatility , 2012, Journal of the Royal Statistical Society. Series A,.

[7]  Eric Ghysels,et al.  State Space Models and MIDAS Regressions , 2013 .

[8]  F. Blasques,et al.  Maximum Likelihood Estimation for Generalized Autoregressive Score Models , 2014 .

[9]  R. Rao.  Relations between Weak and Uniform Convergence of Measures with Applications , 1962 .

[10]  Adrian Pagan,et al.  Estimation, Inference and Specification Analysis. , 1996 .

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

[12]  T. Eisner,et al.  Ergodic Theorems , 2019, Probability.

[13]  J. Stock,et al.  A Probability Model of the Coincident Economic Indicators , 1988 .

[14]  R. Golinelli,et al.  Bridge models to forecast the euro area GDP , 2004 .

[15]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[16]  Forecasting growth in current quarter real GNP , 1989 .

[17]  R. Mehra.  On the identification of variances and adaptive Kalman filtering , 1970 .

[18]  Siem Jan Koopman,et al.  Time Series Analysis by State Space Methods , 2001 .

[19]  Thomas Mikosch,et al.  Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach , 2006 .

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

[21]  Massimiliano Marcellino,et al.  U-Midas: Midas Regressions with Unrestricted Lag Polynomials , 2012, SSRN Electronic Journal.

[22]  Siem Jan Koopman,et al.  Fast Filtering and Smoothing for Multivariate State Space Models , 2000 .

[23]  E. Ghysels,et al.  Série Scientifique Scientific Series Predicting Volatility: Getting the Most out of Return Data Sampled at Different Frequencies , 2022 .

[24]  H. White,et al.  A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , 1988 .