Density Forecasts in Panel Data Models - laurayuliu.com

1 Density Forecasts in Panel data Models :A Semiparametric Bayesian Perspective Laura Liu February 16, 2020 AbstractThis paper constructs individual-specific Density Forecasts for a Panel of firms or householdsusing a dynamic linear model with common and heterogeneous coefficients and cross-sectionalheteroskedasticity. The Panel considered in this paper features a large cross-sectional dimensionNbut short time seriesT. Due to the shortT, traditional methods have difficulty in disen-tangling the heterogeneous parameters from the shocks, which contaminates the estimates ofthe heterogeneous parameters. To tackle this problem, I assume that there is an underlyingdistribution of heterogeneous parameters, model this distribution nonparametrically allowing forcorrelation between heterogeneous parameters and initial conditions as well as individual-specificregressors, and then estimate this distribution by pooling the information from the whole cross-section together.

2 Theoretically, I prove that both the estimated common parameters and theestimated distribution of the heterogeneous parameters achieve posterior consistency, and thatthe Density Forecasts asymptotically converge to the oracle forecast. Methodologically, I develop asimulation-based posterior sampling algorithm specifically addressing the nonparametric densityestimation of unobserved heterogeneous parameters. Monte Carlo simulations and an empiricalapplication to young firm dynamics demonstrate improvements in Density Forecasts relative toalternative Codes:C11, C14, C23, C53, L25 Keywords:Bayesian, Semiparametric Methods, Panel data , Density Forecasts , PosteriorConsistency, Young Firm Dynamics First version: November 15, 2016. Latest version: I am indebted to my advisors, FrancisX. Diebold and Frank Schorfheide, for much help and guidance at all stages of this research project.

3 I also thankthe other members of my committee, Xu Cheng and Francis J. DiTraglia, for their advice and support. I furtherbenefited from many helpful discussions with St ephane Bonhomme, Evan Chan, Benjamin Connault, Hyungsik , Alexandre Poirier, and seminar participants at the University of Pennsylvania, the Federal Reserve Bank ofPhiladelphia, the Federal Reserve Board, the University of Virginia, Microsoft, the University of California, Berkeley,the University of California, San Diego (Rady), Boston University, the University of Illinois at Urbana Champaign,Princeton University, Libera Universit`a di Bolzano, University of Michigan, Universit e de Montr eal, Emory University,Tilburg University, Erasmus University Rotterdam, and Tinbergen Institute, as well as conference participants at the26th Annual Meeting of the Midwest Econometrics Group, NBER-NSF Seminar on Bayesian Inference in Economet-rics and Statistics, the 11th Conference on Bayesian Nonparametrics, Microeconometrics Class of 2017 Conference,Interactions Workshop 2017, First Italian Workshop of Econometrics and Empirical Economics: Panel data Modelsand Applications, the 2018 International Association for Applied Econometrics Conference, and the 2019 North Amer-ican Winter Meeting of the Econometric Society.

4 I would also like to acknowledge the Kauffman Foundation and theNORC data Enclave for providing researcher support and access to the confidential microdata. All remaining errorsare my own. Indiana University, IntroductionPanel data , such as a collection of firms or households observed repeatedly for a number of periods,are widely used in empirical studies and can be useful for forecasting individuals future outcomes,which is interesting and important in many applications, for example, PSID for income dynamics(Hirano, 2002; Gu and Koenker, 2017b) and bank balance sheet data for bank stress tests (Liuet al.,2019b). This paper constructs individual-specific Density Forecasts using a dynamic linear Panel datamodel with common and heterogeneous coefficients and cross-sectional this paper, I consider young firm dynamics as the empirical application.

5 For illustrativepurposes, let us consider a simple dynamic Panel data model as the baseline setup:yit performance= yi,t 1+ i skill+uit shock, uit N(0, 2),(1)wherei= 1, ,N, andt= 1, ,T+ the observed firm performance such as the logof employment, iis the unobserved skill of an individual firm, anduitis an shock. Skillis independent of the shock, and the shock is independent across firms and times. and 2arecommon across firms, where represents the persistence of the dynamic pattern and 2gives thesize of the shocks. Based on the observed Panel from period 0 to periodT, I am interested inforecasting the future performance of any specific firm in periodT+ 1,yi,T+ Panel considered in this paper features a large cross-sectional dimensionNbut short timeseriesT. For instance, the number of observations for each young firm is restricted by its estimates of the unobserved skill is facilitate good Forecasts ofyi,T+1s.

6 Because of the shortT, traditional methods have difficulty in disentangling the unobserved skill ifrom the shockuit,which contaminates the estimates of i, even ifNgoes to tackle this problem, I assume that iis drawn from an underlying skill distributionfandestimate this distribution by pooling the information from the whole cross-section. In terms ofmodelingf, the parametric Gaussian Density misses many features in real-world data , such asasymmetricity, heavy tails, and multiple peaks. For example, as good ideas are scarce, the skilldistribution of young firms may be highly skewed. In this sense, the challenge now is how we canmodelfmore carefully and flexibly. Here I estimatefvia a nonparametric Bayesian approachwhere the prior is constructed from a mixture model and allows for correlation between iand theinitial conditionyi0( a correlated random effects model ).

7 Conditional onf, we can, intuitively speaking, treat it as a prior distribution and combine itwith firm-specific data to obtain the firm-specific posterior. In a special case where the commonparameters are set to( , 2)= (0,1), the firm-specific posterior is characterized by Bayes theorem,p( i|f,yi,0:T) =p(yi,1:T| i)f( i|yi0) p(yi,1:T| i)f( i|yi0)d firm-specific posterior helps provide a better inference about the unobserved skill iof eachindividual firm and a better forecast of the firm-specific future performance, thanks to the underlyingdistributionfthat integrates the information from the whole Panel in an efficient and flexible that this is only an intuitive explanation why the skill distributionfis crucial. In the actualimplementation, the inferences of the correlated random effect distributionf, common parameters( , 2), and firm-specific skill iare all done is natural to construct Density Forecasts based on the firm-specific posterior.

8 In general,forecasting can be done in point, interval, or Density fashion, with Density Forecasts giving therichest insight regarding future outcomes. By definition, a Density forecast provides a predictivedistribution of firmi s future performance and summarizes all sources of uncertainties; hence, it ispreferable in the context of young firm dynamics and other applications with large uncertaintiesand nonstandard distributions. In particular, for the baseline model in (1), the Density forecastsreflect uncertainties arising from the future shockui,T+1, individual heterogeneity i, and estimationuncertainty of common parameters( , 2)and skill distributionf. Moreover, once the densityforecasts are obtained, one can easily recover the point and interval contributions of this paper are threefold.

9 First, I establish the theoretical properties of theproposed Bayesian predictor when the cross-sectional dimensionNtends to infinity. To begin, Iprovide conditions for identifying both the common parameters and the distribution of the individualheterogeneity. Then, I prove that estimates of both achieve posterior consistency in strong with previous literature on posterior consistency, there are several challenges in thepanel data framework: (1) a deconvolution problem disentangling unobserved individual effects andshocks, (2) an unknown common shock size in cross-sectional homoskedastic cases, (3) unknownindividual-specific shock sizes in cross-sectional heteroskedastic cases, (4) strictly exogenous andpredetermined variables (including lagged dependent variables) as covariates, and (5) correlatedrandom coefficients addressed by flexible conditional Density estimation.

10 Based on the posteriorconsistency of the estimates, the discrepancy between the proposed Density predictor and the oracleis arbitrarily small asymptotically. The oracle predictor is an (infeasible) benchmark defined asthe individual-specific posterior predictive distribution, assuming known common parameters anda known distribution of the heterogeneous , I develop a posterior sampling algorithm specifically addressing nonparametric densityestimation of the unobserved individual effects. For a random coefficients model , which is a specialcase where the individual effects are independent of the conditioning variables, thefpart becomesan unconditional Density estimation problem. I adopt a Dirichlet Process Mixture (DPM) prior forfand construct a posterior sampler building on the blocked Gibbs sampler proposed by Ishwaranand James (2001, 2002).