An EM-Algorithm for Maximum-Likelihood Estimation of Mixed Frequency VARs

Transcription

1 An EM-Algorithm for Maximum-Likelihood Estimation of Mixed Frequency VARs Jürgen Antony, Pforzheim Business School and Torben Klarl, Augsburg University EEA 2016, Geneva

2 Introduction frequent problem in time series analysis: data set contains series sampled/aggregated at different frequencies trivial (unsatisfactory) solution is analysis at lowest common frequency loss of information at high frequency, outdated not longer acceptable from scientific point of view problem demands estimation methods capable of dealing with mixed frequency data this contribution: vector autoregressions (VARs) data set with several series at different frequencies VAR analysis (e.g. IRA) at highest frequency in the data set

3 Introduction An example: ECB s BLS and ESRB data non-financial equity markets bond markets credit standards large enterprises credit standards small/medium enterprises Financial stress indicators and credit standards tightening Note: ECB s European Systemic Risk Board financial stress indicators (left scale) and quarterly EUROSYSTEM s BLS results (right scales). Financial stress indicators on non-financial equity and bond markets. BLS results on net percentage of Euro area banks tightening credit standards for large and small/medium enterprises (uniformly distributed over the three month of the respective quarter).

4 Introduction Structure of the talk 1. Review of Existing Methods 2. New EM-Algortihm for ML estimation 3. Application: ECB s Bank Lending Survey 4. Discussion

5 Literature existing approaches in mixed frequency VAR modeling MIDAS VAR ML of state space representation (Kalman filter, traditional optimization) ML of state space representation (Kalman filter, optimization via EM) existing approaches have their benefits and costs no argument against existing approaches from theoretical perspective however, there seem to exist quite strong limitations in practice: statistical inference

6 MIDAS VAR Literature MIxed DAta Sampling approaches originally developed for ADL models with mixed frequency data (Ghysels et al. 2004, 2005, 2006; Ghysels et al. 2007; Ghysels et al. 2009) approach extended to VARs in Ghysels (2011, 2015) idea: infer unobserved high frequency realizations by weighted averages of observed low frequency data weights usually implemented by (hyper)-parameterized kernel functions problem is reduced by estimating a few parameters shaping the kernel function in some cases asymptotically equivalent to Kalman filter approaches drawback: statistical inference distribution of estimators depends on nuisance parameters (kernel functions); Hansen (1996) construction of confidence bands in e.g. IR-analysis cumbersome

7 Literature ML-Estimation applying the Kalman filter formulates the mixed frequency problem in state-space form state equation: high frequency VAR observation equation: relates high frequency date to observed counterparts maximization of log-likelihood by usual gradient methods hardly used in MF-VAR problems drawback: Mittnik and Zadrozny (2005) elaborate lengthy on the trouble they experienced finding the likelihood s maximum maximization via EM-algorithm (Shumway and Stoffer 1982, Mariano and Murasawa 2010, VECM Seong et al. 2013) more reliable in practice still Kalman filter implies drawbacks as convergence to the likelihood s maximum is not guaranteed inference demands bootstrapping which needs an extremely reliable optimization algorithm

8 Literature General observation from literature we identified 14 empirical contributions (journals) estimating VARs with MF data only 1 provides inferential analysis 13 presented results without any comments on statistical significance obviously, something prevents authors from inferential analysis motivation for this contribution developing an EM algorithm with proper convergence properties such an algorithm would allow for bootstrapping and inferential analysis how to do this: find a way avoiding the Kalman filter

9 EM-Algorithm EM-algorithm aims at maximizing the log-likelihood of the MF-VAR in state-space form splits up the problem of jointly estimating the high frequency VAR s parameters and the high but unobserved data series E-step: calculation of expected log-likelihood given a set of the model s parameters M-step: maximization of the expected log-likelihood resulting in a set of new model parameters iterate over E- and M-step: Dempster et al. (1977) show that iteration converges and yields ML estimates at least one series in the VAR needs to be observable at the high frequency

10 EM-Algorithm State equation y t = Φ 1y t 1 + Φ 2y t Φ py t p + ε t, (1) y t : n 1 column vector t = 1,..., T Φ i, i = 1, 2,..., p: n n parameter matrices ε t: Gaussian n 1, E(ε tε t) = Ω, uncorrelated over time

11 EM-Algorithm Observation equation x 0 = Ā0x + ν, (2) x = [y T y T 1... y p+1] x 0 contains only observable elements in x Ā 0 design matrix (problem dependent) ν = [ν T ν T 1... ν p+1]; ν t N(0, R) measurement error, uncorrelated over time θ(k): VAR s parameters at kth iteration of the algorithm θ(k) = {Φ i, Ω, R} k

12 EM-Algorithm Design matrix Ā0 x 0 = Ā0x + ν Ā 0 case dependent corresponding to MF problem consider a VAR with n = 2 variables variable 1 observed every month, i.e. y 1,t variable 2 observed aggregated once a quarter (e.g. GDP growth), i.e. y 2,t, y 2,t 1,... unobservable we only observe y 2,t + y 2,t 1 + y 2,t 2 for t, t 1, t 2 defining a particular quarter

13 EM-Algorithm Design matrix Ā0: x0 = Ā0x... y 1,t y 1,t 1 y 1,t 2 y 2,t + y 2,t 1 + y 2,t 2... }{{} x 0 = }{{} Ā 0... y 1,t y 2,t y 1,t 1 y 2,t 1 y 1,t 2 y 2,t 2... }{{} x

14 EM-Algorithm Replacing the Kalman filter with linear projection usually, one would project x into x 0 by applying the Kalman filter (e.g. Mariano and Murasawa 2010) but this is just one possibility; another one is the linear projection in (3) E θ(k) (xx ) = V θ(k) (x x 0) + E θ(k) (x x 0)[E θ(k) (x x 0)] ( ) 1 E θ(k) (x x 0) = Γ θ(k) x,x 0 Γ θ(k) x 0,x 0 x0, (3) V θ(k) (x x 0) = Γ θ(k) x,x Γ θ(k) x,x 0 = Γ θ(k) x,x Ā 0, Γx θ(k) 0,x 0 = Ā0Γθ(k) x,x ( ) 1 ( ) Γ θ(k) x,x 0 Γ θ(k) x 0,x 0 Γ θ(k) x,x 0 Ā 0 + R k, Γ θ(k) x,x unconditional covariance matrix of x

15 EM-Algorithm E-step builds expected likelihood based on E θ(k) (xx ) M-step maximizes wrt to Φ = [Φ 1 Φ 2... Φ p], Ω, R Π k+1 = Ω k+1 = 1 T R k+1 = 1 T [ T ] [ T ] 1 E θ(k) y tx t E θ(k) x tx t, t=1 t=1 T [ Eθ(k) (y ty t ) E θ(k) (y tx t )Π k Π ke θ(k) (x ty t ) t=1 T t=1 +Π ke θ(k) (x tx t )Π k ], [ x0,tx 0,t x 0,tE θ(k) (x t) A 0 A 0E θ(k) (x t)x 0,t + A 0E θ(k) (x tx t )A ] 0. algorithm iterates over E- and M-step until convergence in θ(k) is achieved

16 ECB s Quarterly BLS Bank Lending Survey conducted quarterly, asking Euro-area banks about their behavior in lending to non-financial corporations available data (details in ECB 2015) quarterly from 2003 Q1 up to 2015 Q2 representative sample of currently 140 banks we investigate answers on Over the past three months, how have your bank s credit standards as applied to the approval of loans or credit lines to enterprises changed? Please note that we are asking about the change in credit standards, rather than about their level answers: tightened considerably, tightened somewhat, remained basically unchanged, eased somewhat or eased considerably quantitative measure: percentage respondents answering with tightened considerably or tightened somewhat

17 ECB s Quarterly BLS we are interested in the interrelation between credit standards and development on Euro area financial markets theoretical background: bank capital channel (van den Heuvel 2007) we try to quantify a reduced form of bank capital channel Euro-area financial market conditions: ECB s European Systemic Risk Board Financial Stress indicators (see Hollo et al for details) equity markets (volatility based) bond markets (spread based) money markets (spread/volatility based) financial intermediaries (financial equities volatility) FOREX markets (volatility based) 5 indicators at monthly frequency

18 ECB s Quarterly BLS non-financial equity markets bond markets credit standards large enterprises credit standards small/medium enterprises 8 financial equity markets foreign exchange markets money markets credit standards large enterprises credit standards small/medium enterprises Financial stress indicators and credit standards tightening Note: ECB s European Systemic Risk Board financial stress indicators (left scales) and quarterly EUROSYSTEM s BLS results (right scales). Financial stress indicators on non-financial equity and bond markets (left panel) and indicators on financial equity, foreign exchange and money markets (right panel). BLS results on net percentage of Euro area banks tightening credit standards for large and small/medium enterprises (uniformly distributed over the three month of the respective quarter).

19 ECB s Quarterly BLS VAR in n = 7 variables two low frequency variables: net percentage banks tightening credit standards for large and small/medium enterprises 5 high frequency variables: ESRB financial stress indicators lag length p = 1 sample of 150 months: 01/ /2015 equivalent to 50 quarters statistical inference via bootstrapping (see Efron and Tibsherani 1993, Stoffer and Wall 2004) with 5,000 replications

20 ECB s Quarterly BLS Results IRA composite financial market shock large enterprises small/medium enterprises Credit standards and composite financial shock Note: Impulse responses after composite financial market shock (scaled to a one standard deviation bond market shock). Monthly net percentage of banks tightening credit standards for large (left panel) and small/medium sized enterprises (right panel) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

21 ECB s Quarterly BLS Results IRA financial market shocks large enterprises small/medium enterprises Credit standards and financial shocks - bond markets Note: Impulse responses after orthogonal one standard deviation shock on bond (upper panels) and money markets (lower panels). Monthly net percentage of banks tightening credit standards for large (left panels) and small/medium sized enterprises (right panels) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

22 ECB s Quarterly BLS Results IRA financial market shocks large enterprises small/medium enterprises Credit standards and financial shocks - money markets Note: Impulse responses after orthogonal one standard deviation shock on bond (upper panels) and money markets (lower panels). Monthly net percentage of banks tightening credit standards for large (left panels) and small/medium sized enterprises (right panels) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

23 ECB s Quarterly BLS Results IRA financial market shocks large enterprises small/medium enterprises Credit standards and financial shocks - non-financial equity markets Note: Impulse responses after orthogonal one standard deviation shock on non-financial (upper panels) and financial equity (lower panels). Monthly net percentage of banks tightening credit standards for large (left panels) and small/medium sized enterprises (right panels) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

24 ECB s Quarterly BLS Results IRA financial market shocks large enterprises small/medium enterprises Credit standards and financial shocks - financial equity markets Note: Impulse responses after orthogonal one standard deviation shock on non-financial (upper panels) and financial equity (lower panels). Monthly net percentage of banks tightening credit standards for large (left panels) and small/medium sized enterprises (right panels) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

25 ECB s Quarterly BLS Results IRA financial market shocks large enterprises small/medium enterprises Credit standards and financial shocks - FOREX markets Note: Impulse responses after orthogonal one standard deviation shock on foreign exchange markets. Monthly net percentage of banks tightening credit standards for large (left panels) and small/medium sized enterprises (right panels) in percentage points. Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

26 ECB s Quarterly BLS estimated VAR can be used to predict unobservable high frequency series via E θ(k) x x 0 at final parameter estimates % 95% 90% 85% 80% 70% 60% median Predicted monthly values on credit standard tightening (large enterprises) Note: Predicted monthly values for net percentage of banks tightening credit standards for large enterprises Confidence intervals obtained after bootstrapping with 5,000 replications using bias correction as in Kilian (1998).

27 Discussion Methodological contribution new EM-algorithm for ML estimation of MFVARs settled within existing methods using linear projections instead of the Kalman filter benefit of new approach: gain in convergence properties (stability) of the EM-approach opens up possibility of using bootstrapping methods for statistical inference Economic contribution quantification of bank capital channel in reduced form response of banks credit standards to financial market shocks most adverse reaction to bond market shocks equity market shocks imply high uncertainty

28 Discussion Economic contribution response of banks credit standards to financial market shocks composite financial shocks lead to significant credit standard tightening by banks reaction faster and more pronounced for large enterprises Future research MF problem not only important in time series analysis spatial analysis knows similar problems different variables sampled over different geographical units approach could serve for spatial interpolation