Forecasting with Dynamic Panel Data Models

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1 Forecasting with Dynamic Panel Data Models Laura Liu 1 Hyungsik Roger Moon 2 Frank Schorfheide 3 1 University of Pennsylvania 2 University of Southern California 3 University of Pennsylvania, CEPR, and NBER Progress Report: March 2014

2 Introduction Goal: develop methods to generate forecasts from a panel data model of the form Y it = β X it + λ iw t + U it, t = 1,..., T ; i = 1,..., N. (1) Here X it may contain lags of Y it. We consider a large N and small T environment. Empirical context: Monitoring of banking sector; stress tests: forecasts of capital-asset ratios, charge-offs, etc. Why large N? Track individual banks or bank holding companies. Why small T? Mergers; changes in regulatory environments; lack of variation in Y s and X s in normal times.

3 Dynamic Panel Model Consider a simple dynamic panel data model: Y it = ρy it 1 + λ i + U it, (2) where U it iid (0, 1) and λ i represents the unobserved individual heterogeneity. For a given ρ, the optimal forecast of Y it +1 at time T is E(Y it +1 Y, ρ) = ρy it + E(λ i Y, ρ). In the dynamic panel literature, the focus has been to find a consistent estimate of ρ in the presence of the incidental parameters λ i to avoid the incidental parameter problems. Our interest is to have a good forecast that requires to use good estimates of both ρ and λ i s with small T panel.

4 Introduction Selection bias poses a challenge for short time span panel data: the usual panel data estimate of the fixed effects (QMLE) tends to over-predict (under-predict) the future capital-asset ratios for the banks with high (low) current capital-asset ratio.

5 Monte Carlo Illustration Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( λ i /(1 ρ), 1/(1 ρ 2 ) ). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE GMM (BB) QMLE Forecast errors: ( 2 Y (s) i,t +1 Ŷ c,(s) i,t +1 (ˆθ 0:T,< i> )). (3) Relatively large selection bias for top and bottom groups.

6 Efron (2011) s Motivation Our paper was inspired by work by Efron (2011). Consider the following question: Want to predict the US Masters golf tournament final scores (the average score after four rounds) after the first round. The first round score, Y i, consists of true skill, λ i, and (unpredicable) luck, U i. If the scores Y i are independent across i, the natural estimator of λ i appears to be Y i, the first round score. Question: Can we estimate λ i more precisely, by using the other players scores of the first round, (Y 1,..., Y N )? This question arises more generally in dynamic panel data models.

7 Introduction We employ an empirical Bayes approach to combine cross sectional and time series information together and thus obtain better forecasts for banks with extreme capital asset ratios. We estimate E(λ i Y, ρ) as the posterior mean of λ i.

8 Related Literature Tweedie s formula and its use: e.g., Robbins (1951), Brown (2008), Brown and Greenshtein (2009), Efron (2011), Gu and Koenker (2013). Consistent estimation of ρ in dynamic panel data models with fixed effects when T is small: IV/GMM: e.g. Anderson and Hsiao (1982), Arellano and Bond (1991), Arellano and Bond (1995), Blundell and Bond (1998), and Alvarez and Arellano (2003). Bayesian: e.g. Lancaster (2002) - (informational) orthogonal parameterization. Bayesian inference in panel data models Correlated random effect models PS: Maurice Tweedie = British medical physicist and statistician, born in 1919 and died in 1996.

9 Road Map 1 Introduction 2 Decision-theoretic Considerations 3 Two Empirical Bayes Predictors: Parametric Family of Distributions for λ i Nonparametric p(λ i ) and Tweedie s Formula 4 Simulations 5 Empirical Application 6 Conclusion.

10 Decision-Theoretic Considerations Simple model Y it = λ i + ρy i,t 1 + U it. Ŷ T +1 = [Ŷ1,T +1,..., ŶN,T +1] is vector of forecasts. Compound L 2 loss function: L N (Ŷ T +1, Y T +1 ) = 1 N N (Ŷ T +1 Y T +1) 2. (4) i=1 Expected compound loss: T E (ρ,λ) L N (Ŷ +1, Y T +1) (5) [ 1 N [ ) ] ] 2 = E (ρ,λ) E (ρ,λ) (Ŷi,T +1 Y i,t +1 Y 0:T N i=1 [ ] 1 N ) 2 = E (ρ,λ) (Ŷi,T +1 λ i ρy it + 1 N i=1

11 Decision-Theoretic Considerations Consider the class of additively separable forecasts Ŷ i,t +1 = ˆλ i + ˆρY it where ˆλ i and ˆρ are estimators of λ i and ρ. Decision space: { } ) D = (ˆλ 1 + ˆρY 1T,..., ˆλ N + ˆρY NT (ˆλ, ˆρ) F 0:T. (6) Find asymptotically optimal forecast in the class D that minimizes the expected compound loss (as N ): E (ρ,λ) [ LN (Ŷ T +1 opt, Y T +1 ) ] (7) [ inf E (ρ,λ) LN (Ŷ T +1, Y T +1 ) ] + o (1). Ŷ i,t +1 D

12 Decision-Theoretic Considerations Suppose ρ is known... Then forecast simplifies to Ŷi,T +1 = ˆλ i + ρy it. Finding an optimal forecast is equivalent to constructing an optimal estimator of λ: [ ] 1 N inf E λ (ˆλ i λ i ) 2. (8) ˆλ N i=1 This estimator is constructed from Z it = Z it (ρ) = Y it ρy i,t 1. For T = 1 see Robbins (1951, 1956).

13 Decision-Theoretic Considerations Suppose T = 1 and ˆλ i = g(z i1 ). Expected compound loss becomes integrated risk with empirical distribution of λ as prior: [ ] 1 N E λ (ˆλ i λ i ) 2 = 1 N (g(z) ) 2φ(z λi λi )dz N N i=1 i=1 [ ] (g(z) ) 2φ(z = λ λ)dz dg N (λ i ) Optimal estimator: = E GN [ (g(z) λi ) 2]. g G N (z) = λi φ(z λ i )dg N (λ i ) φ(z λi )dg N (λ i ). (9) To implement this estimator we need to generate an estimate of G N (λ i ) based on cross-sectional information.

14 Decision-Theoretic Considerations Idea: Approximate g G N (z) by ĝ (z) such that E GN [ (ĝ (z) λ i ) 2] E GN [ (g GN (z) λ i ) 2] + o(1). (10) There are some results in the statistics literature, e.g., Zhang (2003), Brown and Greenshtein (2009), and Jiang and Zhang (2009). TO DO: extend THEORETICAL results to panel data application. FOR NOW: we consider two different implementations of the basic idea: Treat G N parametrically (indexed by finite-dimensional hyperparameter): λ i N ( 0, ω 2) or λ i N ( φ 0 + φ 1Y i0, ω 2). Treat G N nonparametrically: use some general density p(λ i Y i0 ). Use cross-sectional information to estimate relevant features of G N.

15 Next Steps To fix ideas, we will consider the simple model: Y it = ρy i,t 1 + λ i + U it, U it (Y i,t 1, λ i ) N(0, 1), (11) For now we will assume that λ i is independent of Y i0. Step 1: parametric Bayesian analysis with family of priors λ i Y i0 N(0, ω 2 ). Step 2: treat p(λ) nonparametrically realizing that the Bayes estimator of λ i depends on p(λ i ) only through the marginal distribution of Z i (ρ) = 1 T T t=1 (Y it ρy i,t 1 ). Tweedie s Formula!.

16 Step 1: Parametric Analysis Y t is N 1; Y is N T. X is N T, λ is N 1. Likelihood function: p(ρ, λ Y 0:T ) (12) N p(yi 1:T ρ, λ i, Y i0 )p(λ i ) i=1 exp { 1 ( [ tr (Y X ρ λι 2 T )(Y X ρ λι T ) ] + ω 2 λ λ )}

17 Parametric Analysis: Posterior Distribution Posterior of λ ρ: where λ i (ρ, Y 0:T ) N ( µ λi (ρ), σ 2 λ), i = 1,..., N. (13) µ λ (ρ) = σλ(y 2 X ρ)ι T σλ 2 = (T + ω 2 ) 1.

18 Parametric Analysis: Forecasting The one-step-ahead predictive density for Y i,t +1 is given by p(y i,t +1 Y 0:T, ρ) = p(y i,t +1 Y it, ρ, λ i )p(λ i ρ, Y 0:T )dλ i. The mean of this predictive density can be written as (14) E[Y i,t +1 Y 0:T, ρ] = ρy it + E[λ i Y 0:T, ρ]. (15) Define the MLE of λ i conditional on ρ as Z i (ρ) = 1 T T (Y it ρy i,t 1 ). (16) i=1 Then the posterior mean of λ i can be decomposed as follows: E[λ i Y 0:T 1, ρ] = Z i (ρ) }{{} (1 + ω 2 T ) Z i(ρ). (17) MLE }{{} Bayes Correction

19 Parametric Analysis: Estimate Hyperparameters We will estimate the common parameters, e.g. ρ, jointly with the hyperparameter ω that serves as an index for p(λ) using the cross-sectional information: (ˆω, ˆρ ) = argmax ln p(y 1:T Y 0, ρ, ω), (18) where ln p(y 1:T Y 0, ρ, ω) = ln p(y 1:T Y 0, ρ, λ, ω)p(ρ, λ ω, Y 0 )d(ρ, λ). The posterior mean predictor with data-driven hyperparameter choice becomes (now making the dependence on ω explicit): E[λ i Y 0:T ) 1, ˆρ, ˆω] = Z i (ˆρ (1 + ˆω 2 T ) Z ) i (ˆρ. (19) Note: we could replace ˆρ by the posterior mean E[ρ Y 0:T, ˆω]. Generalization: condition on Y i0 : λ i N(φ 0 + φ 1 Y i0, ω 2 ).

20 Step 2: Tweedie s Formula Replace p(λ ω) by more general family of distributions p(λ). Recall Z i (ρ) = 1 T (Y i X i ρ)ι T. Our simple model implies that Z i (ρ) ρ N(λ i, 1/T ). Under our distributional assumptions we obtain q(z i (ρ) λ i ) = (2π/T ) 1/2 exp { T2 } (Z i(ρ) λ i ) 2. (20) Write the Gaussian density using the following exponential-family representation: where q(z i (ρ) λ i ) = exp {λ i TZ i (ρ) ψ(λ i )} q 0 (Z i (ρ)). (21) ψ(λ i ) = T 2 λ2 i and q 0 (Z i (ρ)) = (2π/T ) 1/2 exp { T } 2 Z i 2 (ρ)

21 Tweedie s Formula Posterior of λ conditional on ρ: p(λ Y 0:T, ρ) = N i=1 Now focus on the posterior of λ i and write exp {λ i TZ i (ρ) ψ(λ i )} p(λ i ) exp {λi TZ i (ρ) ψ(λ i )} p(λ i )dλ i (22) p(λ i Y 0:T, ρ) = exp {λ i TZ i (ρ) χ(z i (ρ))} p(λ i ) exp{ ψ(λ i )}. where χ(z i ) = ln exp {λ i TZ i (ρ) ψ(λ i )} p(λ i )dλ i. Since the posterior density integrates to one, we obtain 0 = exp {λ i TZ i χ(z i )} p(λ i ) exp{ ψ(λ i )}dλ i Z i = T λ i p(λ i Y 0:T, ρ)dλ i χ (Z i ).

22 Tweedie s Formula Tweedie s formula: E[λ i Y 0:T, ρ] = 1 T χ (Z i (ρ)). (23) Using the definition of q 0 (Z i ) we can write χ(z i ) = ln q(z i λ i )p(λ i )dλ i ln(2π/t ) + T 2 Z i 2. This leads to E[λ i Y 0:T, ρ] = Z i (ρ) + 1 }{{} T MLE ln q(z i ) Z i }{{} Bayes Correction (24) Zi =Z i (ρ). NOTE: we only need to estimate the marginal density of Z i. We do not need to estimate p(λ i )! Generalization: condition on Y i0.

23 Tweedie s Formula: Implementation First we find a consistent estimate of ρ, say ˆρ. Second, compute QMLE for λ: Z i (ˆρ) = 1 T T (Y it ˆρY it 1 ). (25) t=1 Third, nonparametric correction based on Tweedie s formula: ˆλ i = Z i (ˆρ)+ 1 T 1 ˆq(Z i (ˆρ)) ˆq (z) z=zi (ˆρ), (26) z where ˆp(z) is a nonparametric density estimate of Z i.

24 Tweedie s Formula: Implementation ˆρ Arellano & Bover (95) ( GMM (AB) ): Moment conditions based on Orthogonal Forward Demeaning: E ( W it ) U it = 0, where W it = ( ) Y i0,, Y i,t 1, U it = [ T t U T t+1 it U ] i,t+1+ +U it, T t t = 1,, T 1. Under homoskedasticity, one-step estimator as it s already an asymptotically efficient GMM estimator. Better finite sample properties than Arellano and Bond (91) estimator based on the first difference when ρ is close to 1. Blundell & Bond (98) ( GMM (BB) ): Moment conditions: E ( W it U it) = 0 E ( Y i,t 1 (λ i + U it ) ) = 0 where W it = ( ) Y i0,, Y i,t 2, t = 2,, T. Two-step estimator. Need E ( ) Y i,t 1 λ i = 0 or ( E (λ i Y i0 λ i 1 ρ initial condition. )) = 0. Stationary Better dealing with weak IV problem when ρ is close to 1 when the initial condition is stationary.

25 Tweedie s Formula: Implementation ˆq(Z) Lindsey s method: J ˆq Lindsey (z) = exp γ j z j j=0 Estimate γ i s by Poisson regression. Kernel smoothing: ˆq kernel (z) = 1 Nh N ( ) Zi z K h i=1 Note: in the application we use densities that are conditional on Y i0.

26 A Small Simulation Experiment Model: Y it = λ i + ρy i,t 1 + U it where U it iidn(0, 1). λ i Y i0 iidu[0, 1]. Y i0 distribution: Design 1 : Y i0 (λ i, ρ) N ( ) λi 1 ρ, 1 1 ρ 2. (27) Design 2 : Y i0 (λ i, ρ) N ( 0, 0.1 2). (28) Autoregressive coefficient: ρ = 0.8. N = 1, 000, T = 3.

27 Simulation: Performance Statistics Forecast errors: ( 2 Y (s) i,t +1 Ŷ c,(s) i,t +1 (ˆθ 0:T,< i> )). (29) We consider four different groups of observations: Bottom: 20 smallest Y it s (out of 1,000) Middle: 20 Y it s around the median Top: 20 largest Y it s All: all Y it s We compute mean-squared forecast errors and median forecast errors.

28 Monte Carlo: Design 1 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( λ i /(1 ρ), 1/(1 ρ 2 ) ). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE GMM (BB) QMLE AB and BB estimators perform very similarly. Relatively large selection bias for top and bottom groups.

29 Monte Carlo: Design 1 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( λ i /(1 ρ), 1/(1 ρ 2 ) ). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE Tweedie s Formula GMM (AB) Lindsey GMM (AB) Kernel Tweedie s formula is able to correct selection bias.

30 Monte Carlo: Design 1 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( λ i /(1 ρ), 1/(1 ρ 2 ) ). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE Tweedie s Formula GMM (AB) Lindsey GMM (AB) Kernel Empirical Bayes Forecast with Parametric Model Max of Marg. LH The parametric Bayes model works even better.

31 Monte Carlo: Design 2 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( 0, 0.1 2). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE GMM (BB) QMLE Under this design the GMM(BB) estimator is preferable. Relatively large selection bias for top and bottom groups.

32 Monte Carlo: Design 2 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( 0, 0.1 2). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (BB) QMLE Tweedie s Formula GMM (BB) Lindsey GMM (BB) Kernel Tweedie s formula is able to correct selection bias.

33 Monte Carlo: Design 2 Model: Y it = ρy i,t 1 + λ i + U it, t = 1,..., T ; i = 1,..., N. Design: T = 3, N = 1, 000, ρ = 0.8, λ i U[0, 1], Y i0 N ( 0, 0.1 2). Bottom 20 Middle 20 Top 20 All ˆρ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (BB) QMLE Tweedie s Formula GMM (BB) Lindsey GMM (BB) Kernel Empirical Bayes Forecast with Parametric Model Max of Marg. LH The parametric Bayes model works even better.

34 Application In the aftermath of the global financial crisis bank stress tests have become an important tool used by central banks and other regulators to conduct macroprudential regulation and supervision. Stress tests come in many flavors, one of them is to predict the evolution of bank balance sheets conditional on economic conditions. Bank-level forecasts can then be aggregated into industry-wide losses and revenues. Initially, we tried to focus on forecasts of charge-offs and revenues which can be mapped into forecasts of capital-asset ratios. However, charge-offs have very non-gaussian features and for now we switched to direct forecasts of capital-asset ratios. Stress tests condition on extreme counterfactual economic conditions, whereas in our forecast exercise we condition on actual economic conditions.

35 Application We follow Covas, Rump, and Zakrajsek (CRZ, 2013) in terms of capital-asset ratio definitions. Regulators pay attention to the so-called tier-1-common ratio: T1CR it = E it Deductions it RWA it. Tier-1 common equity is the highest quality component of bank capital. The denominator RWA is the Basel I risk-weighted assets. CRZ decompose the evolution of equity as [ E it = E i,t 1 + (1 τ) PPNR j it Assetsj it ] NCO l it Loans l it Equity Payouts it l where PPNR are net revenues and NCO are net charge-offs. j

36 Data Sources Bank balance sheet data are available through the Call Reports at quarterly frequency from the Federal Reserve Bank of Chicago. We multiply T1CR by 100 and take logs. We will relate T1CR to local economic conditions, e.g., house prices and unemployment. Thus, we will focus on small banks (assets less than 1 billion $). We use the Summary of Deposits data from the Federal Deposit Insurance Corporation to determine the local market for each bank. Currently: local market = state. We collect state-level housing price index (all transactions, not seasonally adjusted) from the Federal Housing Finance Agency; state-level unemployment rate (monthly data averaged to quarterly freq, seasonally adjusted) from the Bureau of Labor Statistics.

37 Model Specification Basic panel data model ln(100 T1CR it ) = λ i + β 1 ln(100 T1CR i,t 1 ) (30) U it iidn(0, σ 2 ). +β 2 UR it + β 3 ln HPI it + U it Parametric prior for λ i : λ i (T1CR i0, φ, ω 2 ) iidn ( φ 0 + φ 1 ln(100 T1CR i0 ), ω 2) (31) Sample period: t = 0 corresponds to 2008:Q1, t = T is 2009:Q4. Forecast period: t = T + 1 is 2010:Q1. Sample size is N = 6, 066.

38 log(100 T1CR) Data 08:Q1 08:Q2 08:Q3 08:Q4 09:Q1 09:Q2 09:Q3 09:Q4 10:Q

39 T1CR Data versus Unemployment and House Prices log(100t1cr) Unemployment Rate log(hpi) Note: capital asset ratios are averaged across time for each bank and across banks within the same state.

40 Parameter Estimates Parameter Max of GMM(AB) GMM(BB) Marg. LH ln(100 T1CR i,t 1 ) ln HPI it UR i,t ˆσ ˆφ ˆφ ˆω

41 Shrinkage Effects: Estimates of Z i (ˆρ) (blue, dashed) and ˆλ i (red, solid) Parametric GMM(AB)/Tweedie GMM(BB)/Tweedie The empirical Bayes procedures induce a substantial amount of shrinkage: ˆλ i densities are much more concentrated than Z i (ˆρ) densities.

42 Implicit Bias Correction: Parametric Bayesian Model The empirical Bayes procedures induces a bias correction for the bottom and top groups.

43 Forecast Results Bottom 2% Middle 2% Top 2% All ˆβ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE GMM (BB) QMLE GMM(BB) and GMM(AB) estimators perform similarly. Relatively large selection bias for top and bottom groups.

44 Forecast Results Bottom 2% Middle 2% Top 2% All ˆβ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE Tweedie s Formula GMM (AB) Lindsey GMM (AB) Kernel Tweedie s formula is able to correct the selection bias.

45 Forecast Results Bottom 2% Middle 2% Top 2% All ˆβ ˆλ i MSE Med MSE Med MSE Med MSE Med No Shrinkage GMM (AB) QMLE Tweedie s Formula GMM (AB) Lindsey GMM (AB) Kernel Empirical Bayes Forecast with Parametric Model Max of Marg. LH Similar performance of parametric approach and Tweedie s formula.

46 Conclusions To forecast dynamic panel data model, it s important to have a good estimates of the individual effects λ i. Selection bias: repeated positive shocks (U it ) lead to overestimation of their corresponding λ i s, especially when T is small. Shrinkage estimators can offset the selection bias and improve the forecasts: Empirical Bayes estimator of parametric model; essentially a random effects model. Plug-in predictor based on Tweedie s formula Both methods lead to improvements in forecast accuracy in simulations and in an application to capital-asset ratio forecasts. Work in progress... Many extensions.

Forecasting with Dynamic Panel Data Models

Forecasting with Dynamic Panel Data Models Laura Liu Federal Reserve Board Hyungsik Roger Moon University of Southern California USC Dornsife INET, and Yonsei Frank Schorfheide University of Pennsylvania