ANDERSON, T. W. (2003): An Introduction to Multivariate Statistical Analysis. Wiley-Interscience.
ANDO, T.,ANDJ. BAI(2013): “Multifactor Asset Pricing with a Large Number of Observable Risk Factors and Unobservable Common and Group-specific Factors,” Working Paper.
ARUOBA, S. B., F. X. DIEBOLD, AND C. SCOTTI (2009): “Real-time Measurement of Business Condi-
tions,” Journal of Business and Economic Statistics, 27(4), 417–427.
BAI, J. (2003): “Inferential Theory for Factor Models of Large Dimensions,” Econometrica, 71, 135–171. BAI, J.,AND S. NG (2002): “Determining the Number of Factors in Approximate Factor Models,” Econo-
metrica, 70(1), 191–221.
(2006): “Confidence Intervals for Diffusion Index Forecasts and Inference for Factor-Augmented Regressions,” Econometrica, 74(4), 1133–1150.
BANBURA, M., AND G. RÜNSTLER(2011): “A Look into the Factor Model Black Box: Publication Lags
and the Role of Hard and Soft Data in Forecasting GDP,” International Journal of Forecasting, 27, 333– 346.
BEKAERT, G., R. J. HODRICK,AND X. ZHANG(2009): “International Stock Return Comovements,” Jour-
nal of Finance, 64, 2591–2626.
BOIVIN, J.,ANDS. NG(2006): “Are More Data Always Better for Factor Analysis?,” Journal of Economet-
rics, 132(1), 169–194.
BREITUNG, J., AND S. EICKMEIER(2014): “Analyzing Business and Financial Cycles Using Multi-level
Factor Models,” Deutsche Bundesbank, Research Centre, Discussion Papers 11/2014.
CARVALHO, V. M. (2010): “Aggregate Fluctuations and the Network Structure of Intersectoral Trade,”
Unpublished Manuscript.
CHEN, K. H., AND J. ROBINSON (1989): “Comparison of Factor Spaces of two Related Populations,”
Journal of Multivariate Analysis, 28, 190–203.
CHEN, P. (2010): “A Grouped Factor Model,” Working Paper.
(2012): “Common Factors and Specific Factors,” Working Paper.
DAUXOIS, J.,AND A. POUSSE(1975): “Une Extension de l’Analyse Canonique. Quelques Applications.,”
Ann. Inst. Henri Poincaré, XI, 355–379.
DAUXOIS, J., Y. ROMAIN, AND S. VIGUIER (1993): “Comparison of Two Factor Subspaces,” Journal of
DEMPSTER, A., N. LAIRD, AND D. RUBIN (1977): “Maximum Likelihood from Incomplete Data via di EM Algorithm,” Journal of the Royal Statistical Society Series B, 39(1), 1–38.
FLURY, B. N. (1984): “Common Principal Components in k Groups,” Journal of the American Statistical Association, 79, 892–898.
FLURY, B. N. (1988): Common Principal Components and Related Multivariate Models. Wiley, New York.
FOERSTER, A. T., P.-D. G. SARTE, ANDM. W. WATSON (2011): “Sectoral versus Aggregate Shocks: A
Structural Factor Analysis of Industrial Production,” Journal of Political Economy, 119(1), 1–38.
FORNI, M., AND L. REICHLIN (1998): “Let’s Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics,” The Review of Economic Studies, 65, 453–473.
FRALE, C., AND L. MONTEFORTE (2010): “FaMIDAS: A Mixed Frequency Factor Model with MIDAS
Structure,” Government of the Italian Republic (Italy), Ministry of Economy and Finance, Department of the Treasury Working Paper, 3.
GABAIX, X. (2011): “The Granular Origins of Aggregate Fluctuations,” Econometrica, 79, 733–772.
GOYAL, A., C. PÉRIGNON,ANDC. VILLA(2008): “How Common are Common Return Factors across the NYSE and Nasdaq?,” Journal of Financial Economics, 90, 252–271.
GREENWOOD, R.,ANDD. SCHARFSTEIN(2013): “The Growth of Finance,” Journal of Economic Perspec-
tives, 27(2), 3–28.
GREGORY, A. W.,AND A. C. HEAD(1999): “Common and Country-specific Fluctuations in Productivity,
Investment, and the Current Account,” Journal of Monetary Economics, 44(3), 423 – 451.
HALLIN, M.,AND R. LISKA(2007): “Determining the Number of Factors in the General Dynamic Factor
Model,” Journal of the American Statistical Association, 102(478), 603–617.
(2011): “Dynamic Factors in the Presence of Blocks,” Journal of Econometrics, 163, 29–41.
HORVATH, M. (1998): “Cyclicality and Sectoral Linkages: Aggregate Fluctuations from Independent Sec-
toral Shocks,” Review of Economic Dynamics, 1, 781–808.
JUNGBACKER, B., S. KOOPMAN, AND M. V. DER WEL (2011): “Maximum Likelihood Estimation for
Dynamic Factor Models with Missing Data,” Journal of Economic Dynamics and Control, 35(8), 1358 – 1368.
KING, R. G., AND M. WATSON(2002): “System Reduction and Model Solution Algorithms for Singular Linear Rational Expectations Models,” Computational Economics, 20, 57–86.
KORAJCZYK, R. A., AND R. SADKA (2008): “Pricing the Commonality across Alternative Measures of
Liquidity,” Journal of Financial Economics, 87(1), 45 – 72.
KOSE, A. M., C. OTROK,ANDC. H. WHITEMAN(2008): “Understanding the Evolution of World Business Cycles,” Journal of International Economics, 75, 110–130.
KRZANOWSKI, W. (1979): “Between-groups Comparison of Principal Components,” Journal of the Ameri- can Statistical Association, 74, 703–707.
LONG, J. B. J., AND C. I. PLOSSER(1983): “Real Business Cycles,” Journal of Political Economy, 91, 39–69.
MAGNUS, J. R.,AND H. NEUDECKER(2007): Matrix Differential Calculus with Applications in Statistics
and Econometrics. John Wiley and Sons: Chichester/New York.
MARCELLINO, M., AND C. SCHUMACHER(2010): “Factor MIDAS for Nowcasting and Forecasting with
Ragged-Edge Data: A Model Comparison for German GDP,” Oxford Bulletin of Economics and Statistics, 72(4), 518–550.
MARIANO, R. S., AND Y. MURASAWA (2003): “A New Coincident Index of Business Cycles Based on
Monthly and Quarterly Series,” Journal of Applied Econometrics, 18(4), 427–443.
MOENCH, E.,AND S. NG(2011): “A Hierarchical Factor Analysis of US Housing Market Dynamics,” The
Econometrics Journal, 14, C1–C24.
MOENCH, E., S. NG, AND S. POTTER (2013): “Dynamic Hierarchical Factor Models,” Review of Eco-
nomics and Statistics, 95, 1811–1817.
NUNES, L. C. (2005): “Nowcasting Quarterly GDP Growth in a Monthly Coincident Indicator Model,”
Journal of Forecasting, 24(8), 575–592.
SCHOTT, J. R. (1988): “Common Principal Components Subspaces in Two Groups,” Biometrika, 75, 229–
236.
(1991): “Some Tests for Common Principal Component Subspaces in Several Groups,” Biometrika, 78, 771–777.
SCHOTT, J. R. (1999): “Partial Common Principal Component Subspaces,” Biometrika, 86, 899–908.
(2005): Matrix Analysis for Statistic. Wiley, New York, 2 edn.
STOCK, J. H., AND M. W. WATSON (2002a): “Forecasting Using Principal Components from a Large
Number of Predictors,” Journal of the American Statistical Association, 97, 1167 – 1179.
(2002b): “Macroeconomic Forecasting Using Diffusion Indexes,” Journal of Business and Eco- nomic Statistics, 20, 147–162.
(2010): “Dynamic Factor Models,” in Oxford Handbook of Economic Forecasting, ed. by E. Gra- ham, C. Granger,andA. Timmerman, pp. 87–115. Michael P. Clements and David F. Hendry (eds), Oxford University Press, Amsterdam.
VIGUIER-PLA, S. (2004): “Factor-based Comparison of k Populations,” Statistics, 38(1), 1–15.
WANG, P. (2012): “Large Dimensional Factor Models with a Multi-Level Factor Structure: Identifcation, Estimation, and Inference,” Working Paper.
Chapter 4
Indirect Inference Estimation of Mixed
Frequency Stochastic Volatility State Space
Models using MIDAS Regressions and
ARCH Models
Abstract
We examine the relationship between MIDAS regressions and the estimation of state space models applied to mixed frequency data. While in some cases the binding function is known, in general it is not, and there- fore indirect inference is called for. The approach is appealing when we consider state space models which feature stochastic volatility, or other non-Gaussian and nonlinear settings where maximum likelihood meth- ods require computationally demanding approximate filters. The stochastic volatility feature is particularly relevant when considering high frequency financial series. In addition, we propose a filtering scheme which relies on a combination of reprojection methods and nowcasting MIDAS regressions with ARCH models. We assess the efficiency of our indirect inference estimator for the stochastic volatility model by comparing it with the Maximum Likelihood (ML) estimator in Monte Carlo simulation experiments. The ML estimate is computed with a simulation-based Expectation-Maximization (EM) algorithm, in which the smoothing distribution required in the E step is obtained via a particle forward-filtering/backward-smoothing algorithm. Our Monte Carlo simulations show that the Indirect Inference procedure is very appealing, as its statistical accuracy is close to that of MLE but the former procedure has clear advantages in terms of computational ef- ficiency. An application to forecasting quarterly GDP growth in the Euro area with monthly macroeconomic indicators illustrates the usefulness of our procedure in empirical analysis.
JEL Codes: C15, C31, C53, E37.
Keywords: Indirect Inference, Reprojection, Mixed-frequency Data, State Space Model, Stochastic Volatil- ity, GDP Forecasting.
4.1
Introduction
Econometric models that take into account the unbalanced nature of datasets have attracted substantial at- tention recently. Policy makers and practitioners alike need to assess in real-time the current state of the economy, with at best mixed frequency data at their disposal. For example, one of the key indicators of macroeconomic activity, the Gross Domestic Product (GDP), is released quarterly, while a range of leading and coincident indicators is timely available at a monthly or even higher frequency. Hence, we may want to construct a forecast of the current quarter GDP growth (a so called nowcast) based on the available higher frequency information.
Econometric models with mixed frequency data can be classified into two broad classes: (1) likelihood- based involving latent processes and (2) purely regression-based. The former category consists primarily of state space models, studied by Harvey and Pierse (1984), Harvey (1989), Zadrozny (1990), Bernanke, Gertler, and Watson (1997), Mariano and Murasawa (2003), Mittnik and Zadrozny (2005), Aruoba, Diebold, and Scotti (2009), Ghysels and Wright (2009), Kuzin, Marcellino, and Schumacher (2011), among others. The regression-based methods involve Mixed Data Sampling (MIDAS) regressions; see e.g. Ghysels, Santa- Clara, and Valkanov (2006), Andreou, Ghysels, and Kourtellos (2010). As one considers high frequency data, the issue of time-varying volatility becomes increasingly relevant. Dealing with stochastic volatility (SV) in state space models is doable but poses challenges both statistical and computational in nature. One possibility is to consider Bayesian approaches in this context, as done by Carriero, Clark, and Marcellino (2013) and Marcellino, Porqueddu, and Venditti (2015). However, when it comes to classical inference one typically relies on the Expectation-Maximization (EM) algorithm to compute numerically the ML estimate in a model with unobservable variables (Dempster, Laird, and Rubin (1977)). The likelihood function of the model involves a large-dimensional integral with respect to the latent factor paths as the latent factors appear in the conditional mean and volatility of the high frequency data series. This integral representation of the likelihood is impractical for the computation of the ML estimate.
If the objective is to estimate state space models with mixed frequency data - of which there are many examples - featuring stochastic volatility, using classical inference methods, is there perhaps a simpler way to do so? This is the contribution of our paper. We introduce indirect inference estimation procedures proposed by Gouriéroux, Monfort, and Renault (1993), Smith (1993) and Gallant and Tauchen (1996), to estimate the models of interest using MIDAS regressions augmented with ARCH-type models as well as mixed frequency Vector Autoregressive (VAR) models (see e.g. Ghysels (2014)) as auxiliary models. Same frequency data settings are a special case of mixed frequency ones. The analysis in this paper is therefore also applicable to standard state space models. Moreover, the idea of estimating SV-type models using ARCH-type auxiliary models has a long history starting with Engle and Lee (1999) and Pastorello, Renault, and Touzi (2000). Our paper combines insights from the literature on SV models with those from the mixed frequency data literature.
It is worth noting that in some specific cases we know the binding function between the state space model and the implied MIDAS regression, as discussed in Bai, Ghysels, and Wright (2013). However, these cases are rather too simple to be practical, so that the use of indirect inference is a natural way to tackle the unknown binding function. The methods we propose are fairly easy to implement and involve auxiliary model-based estimators involving MIDAS regressions combined with ARCH specifications for the errors. In addition, we filter latent variables, given observables, using reprojection methods proposed by Gallant and Tauchen (1998).
ence, via Monte Carlo simulations. To implement the former method in the mixed frequency SV model, we consider a simulation-based estimator relying on the EM algorithm. The smoothing distribution required in the Expectation step is computed via a particle forward-filtering/backward-smoothing algorithm. We com- pare the two estimation methods on the basis of (a) statistical criteria - mean/bias/quantiles of sampling distributions, (b) filtering accuracy - both conditional mean and volatility and (c) computational time. Our results show that there are clear advantages in terms of computational time to the new indirect inference procedure put forward in this paper, while the losses in statistical efficiency compared to MLE are very lim- ited. Even in the linear Gaussian case, we find our indirect inference methods remarkably accurate, when compared to the standard MLE based on the Kalman filter.
The paper is organized as follows. Section 4.2 introduces state space models with mixed frequency data and stochastic volatility. Section 4.3 defines our indirect inference estimator. This section covers the linear Gaussian state space model with mixed frequency data as a special case of the general specification, and discusses its relation with MIDAS regressions. This link yields useful insights to define the auxiliary model for indirect inference in the general SV case. Section 4.3 also describes the estimation of the SV model with ML via a simulation-based EM algorithm. Section 4.4 discusses filtering via reprojection, followed by Section 4.5 which reports the results of an extensive Monte Carlo study. Section 4.6 presents an empirical application of our model to the problem of forecasting at short horizons Euro-area quarterly GDP growth using monthly macroeconomic indicators. The dataset is the same as the one considered in the empirical study of Marcellino, Porqueddu, and Venditti (2015). Section 4.7 concludes the paper.