Dynamic Factor Models and Factor Augmented Vector Autoregressions. Lawrence J. Christiano

Transcription

1 Dynamic Factor Models and Factor Augmented Vector Autoregressions Lawrence J Christiano

2 Dynamic Factor Models and Factor Augmented Vector Autoregressions Problem: the time series dimension of data is relatively short the number of time series variables is huge DFM s and FAVARs take the position: there are many variables and, hence, shocks, but, the principle driving force of all the variables may be just a small number of shocks Factor view has a long-standing history in macro almost the definition of macroeconomics: a handfull of shocks - demand, supply, etc - are the principle economic drivers Sargent and Sims: only two shocks can explain a large fraction of the variance of US macroeconomic data 1977, Business Cycle Modeling Without Pretending to Have Too Much A-Priori Economic Theory, in New Methods in Business Cycle Research, ed by C Sims et al, Minneapolis: Federal Reserve Bank of Minneapolis

3 Why Work with a Lot of Data? Estimates of impulse responses to, say, a monetary policy shock, may be distorted by not having enough data in the analysis (Bernanke, et al (QJE, 2005)) Price puzzle: measures of inflation tend to show transitory rise to a monetary policy tightening shock in standard (small-sized) VARs One interpretation: Monetary authority responds to a signal about future inflation that is captured in data not included in a standard, small-sized VAR May suppose that core inflation is a factor that can only be deduced from a large number of different data May want to know (as in Sargent and Sims), whether the data for one country or a collection of countries can be characterized as the dynamic response to a few factors

4 Outline Describe Dynamic Factor Model Identification problem and one possible solution Derive the likelihood of the data and the factors Describe priors, joint distribution of data, factors and parameters Go for posterior distribution of parameters and factors Gibbs sampling, a type of MCMC algorithm Metropolis-Hastings could be used here, but would be very ineffi cient Gibbs exploits power of Kalman smoother algorithm and the type of fast direct sampling done with BVARS FAVAR

5 Dynamic Factor Model Let Y t denote an n 1 vector of observed data Y t related to κ n unobserved factors, f t, by measurement (or, observer) equation: y i,t = a i + vector of κ factor loadings idiosyncratic component of y {}}{ i,t λ i {}}{ f t + ξ i,t Law of motion of factors: f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) Idiosyncratic shock to y i,t ( measurement error ): ξ i,t = φ i,1 ξ i,t φ i,pi ξ i,t pi + u i,t, u i,t N ( ) 0, σ 2 i u i,t, i = 0,, n, drawn independently from each other and over time For convenience: p i = p, for all i, q p + 1

6 Notation for Observer Equation Observer equation: y i,t = a i + λ i f t + ξ i,t ξ i,t = φ i,1 ξ i,t φ i,pi ξ i,t pi + u i,t, u i,t N ( ) 0, σ 2 i Let θ i denote the parameters of the i th observer equation: σ 2 i φ i,1 θ }{{} i = a i λ, φ i =, i = 1,, n i (2+κ+p) 1 φ φ i,p i

7 Notation for Law of Motion of Factors Factors: f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) Let θ 0 denote the parameters of factors: All model parameters: [ ] Σ0 θ }{{} 0 = φ, φ 0 = 0 }{{} κ(q+1) κ κq κ θ = [θ 0, θ 1,, θ n ] φ 0,1 φ 0,q

8 DFM: Identification Problem in DFM y i,t = a i + λ i f t + ξ i,t f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) ξ i,t = φ i,1 ξ i,t φ i,p ξ i,t p + u i,t Suppose H is an arbitrary invertible κ κ matrix Above system is observationally equivalent to: where y i,t = a i + λ i f t + ξ i,t f t = φ 0,1 ft φ 0,q ft q + ũ 0,t N ( 0, Σ 0 ), f t = Hf t, λ i = λ i H 1, φ 0,j = Hφ 0,j H 1, Σ 0 = HΣ 0 H, Desirable to restrict model parameters so that there is no change of parameters that leaves the system observationally equivalent, yet has all different factors and parameter values

9 Geweke-Zhou (1996) Identification Note for any model parameterization, can always choose an H so that Σ 0 = I κ Find C such that CC = Σ 0 (there is a continuum of these), set H = C 1 Geweke-Zhou (1996) suggest the identifying assumption, Σ 0 = I κ But, this is not enough to achieve identification Exists a continuum of orthonormal matrices with property, CC = I κ Simple example: for κ = 2, for each ω [ π, π], [ C = cos (ω) sin (ω) sin (ω) cos (ω) ], 1 = cos 2 (ω) + sin 2 (ω) For each C, set H = C 1 = C That produces an observationally equivalent alternative parameterization, while leaving intact the normalization, Σ 0 = I κ, since HΣ 0 H = C C = C 1 C = I κ

10 Geweke-Zhou (1996) Identification Write: Λ = λ 1 λ κ λ κ+1 λ n = [ Λ1,κ Λ 2,κ ], Λ 1,κ κ κ Geweke-Zhou also require Λ 1,κ is lower triangular then, in simple example, only orthonormal matrix C that preserves lower triangular Λ 1,κ is lower triangular (ie, b = 0, a = ±1) Geweke-Zhou resolve identification problem with last assumption: diagonal elements of Λ 1,κ non-negative (ie, a = 1 in example)

11 Geweke-Zhou (1996) Identification Identifying restrictions: Λ 1,κ is lower triangular, Σ 0 = I κ Only first factor, f 1,t, affects first variable, y 1,t Only f 1,t and f 2,t affect y 2,t, etc Ordering of y it affects the interpretation of the factors Alternative identifications: Σ 0 diagonal and diagonal elements of Λ 1,κ equal to unity Σ 0 unrestricted (positive definite) and Λ 1,κ = I k

12 Next: Move In direction of using data to obtain posterior distribution of parameters and factors Start by going after the likelihood

13 Likelihood of Data and Factors System, i = 1,, n : Define: y i,t = a i + λ i f t + ξ i,t f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) ξ i,t = φ i,1 ξ i,t φ i,p ξ i,t p + u i,t φ i (L) = φ i,1 + + φ i,p L p 1, Lx t x t 1 Then, the quasi-differenced observer equation is: [1 φ i (L) L] y i,t = [1 φ i (1)] a i + λ i [1 φ i (L) L] f t u i,t {}}{ + [1 φ i (L) L] ξ i,t

14 Likelihood of Data and of Factors Quasi-differenced observer equation: y i,t = φ i (L) y i,t 1 + [1 φ i (1)] a i + λ i [1 φ i (L) L] f t + u i,t Consider the MATLAB notation: x t1 :t 2 x tt,, x t2 Note: y i,t, conditional on y i,t p:t 1, f t p:t, θ i, is Normal: p ( ) y i,t y i,t p:t 1, f t p:t, θ i ( ) N φ i (L) y i,t 1 + [1 φ i (1)] a i + λ i [1 φ i (L) L] f t, σ 2 i

15 Likelihood of Data and of Factors Independence of u i,t s implies the conditional density of Y t = [ y 1,t y n,t ] : n i=1 p ( y i,t y i,t p:t 1, f t p:t, θ i ) Density of f t conditional on f t q:t 1 : p ( f t f t q:t 1, θ 0 ) Conditional joint density of Y t, f t : n i=1 p ( y i,t y i,t p:t 1, f t p:t, θ i ) p ( ft f t q:t 1, θ 0 )

16 Likelihood of Data and of Factors Likelihood of Y p+1:t, f p+1:t, conditional on initial conditions: p ( Y p+1:t, f p+1:t Y 1:p, f p q:p, θ ) = [ T n p ( ) ( ) ] y i,t y i,t p:t 1, f t p:t, θ i p ft f t q:t 1, θ 0 t=p+1 i=1 Likelihood of initial conditions: p ( Y 1:p, f p q+1:p θ ) = p ( Y 1:p f p q+1:p, θ ) p ( f p q+1:p θ 0 ) Likelihood of Y 1:T, f p q:t conditional on parameters only, θ : T t=p+1 [ n i=1 p ( y i,t y i,t p:t 1, f t p:t, θ i ) p ( ft f t q:t 1, θ 0 ) ] p ( Y 1:p f p q+1:p, θ i, i = 1,, n ) p ( f p q+1:p θ 0 )

17 Joint Density of Data, Factors and Parameters Parameter priors: p (θ i ), i = 0,, n Joint density of Y 1:T, f p q:t, θ : T t=p+1 p ( ) n f t f t q:t 1, θ 0 p ( ) y i,t y i,t p:t 1, f t p:t, θ i i=1 ] [ n p ( y i,1:p f p q+1:p, θ ) p (θ i ) i=1 p ( f p q+1:p θ 0 ) p (θ0 ) From here on, drop the density of initial observations if T is not too small, then has no effect on results BVAR lecture notes describe an example of how to not ignore initial conditions; for general discussion, see Del Negro and Otrok (forthcoming, RESTAT, "Dynamic Factor Models with Time-Varying Parameters: Measuring Changes in International Business Cycles")

18 Outline Describe Dynamic Factor Model (done!) Identification problem and one possible solution Derive the likelihood of the data and the factors (done!) Describe priors, joint distribution of data, factors and parameters (done!) Go for posterior distribution of parameters and factors Gibbs sampling, a type of MCMC algorithm Metropolis-Hastings could be used here, but would be very ineffi cient Gibbs exploits power of Kalman smoother algorithm and the type of fast direct sampling done with BVARS FAVAR

19 Gibbs Sampling Idea is similar to what we did with the Metropolis-Hastings algorithm

20 Gibbs Sampling versus Metropolis-Hastings Metropolis-Hastings: we needed to compute the posterior distribution of parameters, θ, conditional on the data output of Metropolis-Hastings algorithm: sequence of values of θ whose distribution corresponds to the posterior distribution of θ given the data: P = [ θ (1) θ (M) ] Gibbs sampling algorithm: sequence of values of DFM model parameters, θ, and unobserved factors, f, whose distribution corresponds to the posterior distribution conditional on the data: [ P = θ (1) θ (M) ] f (1) f (M) Histogram of elements in individual rows of P represent marginal distribution of corresponding parameter or factor

21 Gibbs Sampling Algorithm Computes sequence: [ P = θ (1) θ (M) ] f (1) f (M) = [ P 1 P M ] Given P s 1 compute P s in two steps Step 1: draw θ (s) given P s 1 (direct sampling, using approach for BVAR) Step 2: draw f (s) given θ (s) (direct sampling, based on information from Kalman smoother)

22 Step 1: Drawing Model Parameters Parameters, θ observer equation: a i, λ i measurement error: σ 2 i, φ i law of motion of factors: φ 0 where the identification, Σ 0 = I, is imposed Algorithm must be adjusted if some other identification is used For each i : Draw a i, λ i, σ 2 i from Normal-Inverse Wishart, conditional on the φ (s 1) i s Draw φ i from Normal, given a i, λ i, σ 2 i

23 Drawing Observer Equation Parameters and Measurement Error Variance The joint density of Y 1:T, f p q:t, θ : [ T p ( ) f t f t q:t 1, θ 0 n p ( ) ] y i,t y i,t p:t 1, f t p:t, θ i t=p+1 i=1 p (θ 0 ) n i=1 p (θ i ), was derived earlier (but we have now dropped the densities associated with the initial conditions) Recall, p (A B) = p (A, B) p (B) = A p (A, B) p (A, B) da

24 Drawing Observer Equation Parameters and Measurement Error Variance Conditional density of θ i obtained by dividing joint density by itself, after integrating out θ i : p (θ i Y 1:T, f p q:t, { } ) p ( Y 1:T, f p q:t, θ ) θ j = j =i θ p ( Y i 1:T, f p q:t, θ ) dθ i p (θ i ) T t=p+1 p ( y i,t y i,t p:t 1, f t p:t, θ i ) here, we have taken into account that the numerator and denominator have many common terms We want to draw θ (s) i from this posterior distribution for θ i Gibbs sampling procedure: first, draw a i, λ i, σ 2 i taking the other elements of θ i from θ (s 1) i then, draw other elements of θ i taking a i, λ i, σ 2 i as given

25 Drawing Observer Equation Parameters and Measurement Error Variance The quasi-differenced observer equation: ỹ i,t f i,t {}}{{}}{ y i,t φ i (L) y i,t 1 = (1 φ i (1)) a i + λ i [1 φ i (L) L] f t + u i,t, or, Let so ỹ i,t = [1 φ i (1)] a i + λ i f i,t + u i,t A i = [ ] [ ] ai (1 φi (1)) λ, x i,t =, i f i,t ỹ i,t = A i x i,t + u i,t, where ỹ i,t and x t are known, conditional on φ (s 1) i

26 Drawing Observer Equation Parameters and Measurement Error Variance From the Normality of the observer equation error: Then, p ( ) y i,t y i,t p:t 1, f t p:t, θ i { 1 exp 1 ( yi,t [ φ i (L) y i,t 1 + A i x 2 } i,t]) σ i 2 σ 2 i { (ỹi,t A i x 2 } i,t) = 1 σ i exp 1 2 σ 2 i T p ( ) y i,t y i,t p:t 1, f t p:t, θ i t=p+1 { 1 σ T p i exp 1 2 T t=p+1 (ỹi,t A i x 2 } i,t) σ 2 i

27 Drawing Observer Equation Parameters and Measurement Error Variance As in the BVAR analysis, express in matrix terms: p ( { ) 1 y i y i,1:p, f 1:T, θ i σ T p exp 1 T (ỹi,t A i 2 x 2 } t) i t=p+1 σ 2 i { 1 = exp 1 [y i X i A i ] } [y i X i A i ] 2 σ 2 i where σ T p i f p+1:t = f (s 1), y i = ỹ i,p+1 ỹ i,t, X i = x i,p+1 x i,t, where f q p:p fixed (could set to unconditional mean of zero) Note: calculations are conditional on factors, f (s 1), from previous Gibbs sampling iteration

28 Including Dummy Observations As in the BVAR analysis, T dummy equations are one way to represent priors, p (θ i ) : p (θ i ) T t=p+1 p ( y i,t y i,t p:t 1, f t p:t, θ i ) Dummy observations (can include restriction that Λ 1,κ is lower triangular by suitable construction of dummies) ȳ i = X i A i + Ū i, u i,1 Ū i = u i, T Stack the dummies with the actual data: y }{{} i = (T p+ T) 1 [ ] yi ȳ, X i = i }{{} (T p+ T) (1+κ) [ Xi X i ]

29 Including Dummy Observations As in BVAR: = p ( ) y i y i,1:p, f 1:T, θ i p (λ i, a i σ 2 i ] ] 1 [y σ T+ exp T p 1 i X i A i [y i X i A i 2 σ 2 i i { 1 = σ T+ exp 1 S + (A i A i ) } X i X i (A i A i ) T p 2 σ 2 i i { } { 1 exp 1 S 2 σ 2 exp 1 (A i A i ) X i X i (A i A i ) 2 i σ 2 i where σ T+ T p i S = [y i X i A i ] [y i X i A i ], A i = ( X i X i) 1 X i y i ) }

30 Inverse Wishart Distribution Scalar version of Inverse Wishart distribution with (ie, m = 1 in BVAR discussion) : { } ( ) p σ 2 i = S ν/2 2 ν Γ [ ] ν σ 2 i ν+2 2 exp S 2 2σ 2, i degrees of freedom, ν, and shape, S (Γ denotes the Gamma function) Easy to verify (after collecting terms), that p ( ) y i y i,1:p, f 1:T, θ i p (λ i, a i σ 2 i ( = N A i, σ 2 ( i X i X ) ) 1 ) ( ) p σ 2 i IW (ν + T p + T (κ + 1), S + S ) Direct sampling from posterior of distribution: draw σ 2 i from IW Then, draw A i from N, given σ 2 i

31 Draw Distributed Lag Coeffi cients in Measurement Error Law of Motion Given λ i, a i, σ 2 i, draw φ i Observer equation and measurement error process: y i,t = a i + λ i f t + ξ i,t ξ i,t = φ i,1 ξ i,t φ i,p ξ i,t p + u i,t Conditional on a i, λ i and the factors, ξ i,t can be computed from ξ i,t = y i,t a i λ i f t, so the measurement error law of motion can be written, ξ i,t = A i x i,t + u i,t, A i = φ i = φ i,1 φ i,p, x i,t = ξ i,t 1 ξ i,t p

32 Draw Distributed Lag Coeffi cients in Measurement Error Law of Motion The likelihood of ξ i,t conditional on x i,t is ( ) ( ) p ξ i,t x i,t, φ i, σ 2 i = N A i x i,t, σ 2 i { = 1 exp 1 ( ξi,t A i x 2 } i,t), σ i 2 where σ 2 i, drawn previously, is for present purposes treated as known Then, the likelihood of ξ i,p+1,, ξ i,t is ( ) p ξ i,p+1:t x i,p+1, φ i, σ 2 i 1 (σ i ) T p exp { 1 2 T t=p+1 σ 2 i ( ξi,t A i x 2 } i,t) σ 2 i

33 Draw Distributed Lag Coeffi cients in Measurement Error Law of Motion where T ( ξi,t A i x ) 2 i,t = [yi X i A i ] [y i X i A i ], t=p+1 y i = ξ i,p+1 ξ i,t, X i = x i,p+1 x i,t,

34 Draw Distributed Lag Coeffi cients in Measurement Error Law of Motion If we impose priors by dummies, then ( ) p ξ i,p+1:t x i,p+1, φ i, σ 2 i p (φ i ) ] ] 1 [y exp T p (σ i ) 1 i X i A i [y i X i A i 2, where y and X i i represents the stacked data that includes dummies By Bayes rule, p ( φ i ξ i,p+1:t, x i,p+1, φ i, σ 2 i So, we draw φ i from N ( A i, σ 2 i ) = N σ 2 i ( A i, σ 2 i ( X i X ) 1 ) ( X i X ) 1 )

35 Draw Parameters in Law of Motion for Factors Law of motion of factors: f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) The factors, f p+1:t, are treated as known, and they correspond to f (s 1), the factors in the s 1 iteration of Gibbs sampling By Bayes rule: p ( φ 0 f p+1:t ) p ( fp+1:t φ 0 ) p ( φ0 ) The priors can be implemented by dummy variables direct application of the methods developed for inference about the parameters of BVARs Draw φ 0 from N

36 This Completes Step 1 of Gibbs Sampling Gibbs sampling computes sequence: [ P = θ (1) θ (M) ] f (1) f (M) = [ P 1 P M ] Given P s 1 compute P s in two steps Step 1: draw θ (s) given P s 1 (direct sampling) Step 2: draw f (s) given θ s (Kalman smoother) We now have θ (s), and must now draw factors This is done using the Kalman smoother

37 Drawing the Factors For this, we will put the DFM in the state-space form used to study Kalman filtering and smoothing In that previous state space form, the measurement error was assumed to be iid We will make use of the fact that we have all model parameters The DFM: y i,t = a i + λ i f t + ξ i,t f t = φ 0,1 f t φ 0,q f t q + u 0,t, u 0,t N (0, Σ 0 ) ξ i,t = φ i,1 ξ i,t φ i,p ξ i,t p + u i,t This can be put into our state space form (in which the errors in the observation equation are iid) by quasi-differencing the observer equation

38 Observer Equation Quasi differencing: ỹ i,t {}}{ [1 φ i (L) L] y i,t = constant {}}{ [1 φ i (1)] a i + λ i [1 φ i (L) L] f t + u i,t Then, a = H = [1 φ i (1)] a i ỹ 1,t f ṭ, ỹ t =, F t = [1 φ i (1)] a i ỹ n,t λ 1 λ 1 φ 1,1 λ 1 φ 1,p u 1,t, u t = λ n λ nφ n,1 λ nφ n,p u n,t f t p ỹ t = a + HF t + u t

39 Law of Motion of the State Here, the state is denoted by F t Law of motion: f t f t 1 f t 2 f t p LoM: = φ 0,1 φ 0,2 φ 0,q 0 κ (p+1 q) I κ 0 κ 0 κ 0 κ (p+1 q) 0 I κ 0 κ 0 κ (p+1 q) F t = ΦF t 1 + u t, u t N 0 0 I κ 0 κ (p+1 q) + f t 1 f t 2 f t 3 f t 1 p (0 κ(p+1) 1, V (p+1)κ (p+1)κ ) u 0,t 0 κ 1 0 κ 1 0 κ 1

40 State Space Representation of the Factors Observer equation: Law of motion of state: Kalman smoother provides: ỹ t = a + HF t + u t F t = ΦF t 1 + u t P [ F j ỹ 1,, ỹ T ], j = 1,, T, together with appropriate second moments Use this information to directly sample f (s) from the Kalman-smoother-provided Normal distribution, completing step 2 of the Gibbs sampler

41 Factor Augmented VARs (FAVAR) Favar s are DFM s which more closely resemble macro models There are observables that act like factors, hitting all variables directly Examples: the interest rate in the monetary policy rule, government spending, taxes, price of housing, world trade, international price of oil, uncertainty, etc The measurement equation: y i,t = a i + γ i y 0,t + λ i f t + ξ i,t, i = 1,, n, t = 1,, T, where y 0,t and γ i are m 1 and 1 m vectors, respectively The vectors, y 0,t and f t follow a VAR: [ ] [ ] [ ] ft y = ft 1 ft q Φ 0,1 0,t y + + Φ 0,q 0,t 1 y + u 0,t, 0,t q u 0,t N (0, Σ 0 )

42 Literature on FAVARs is Large Initial paper: Bernanke and Boivin (2005QJE), "Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach" Intention was to correct problems with conventional VAR-based estimates of the effects of monetary policy shocks Include a large number of variables: better capture the actual policy rule of monetary authorities, which look at lots of data in making their decisions include a lot of variables so that the FAVAR can be used to obtain a comprehensive picture of the effects of a monetary policy shock on the whole economy Bernanke, et al, include 119 variables in their analysis

43 Literature on FAVARs is Large Literature is growing: "Large Bayesian Vector Autoregressions," Banbura, Giannone, Reichlin (2010Journal of Applied Economicts), studies importance of including sectoral data to get better estimates of impulse response functions to policy shocks and a better estimate of their impact DFM have been taken in interesting directions, more suitable for multicountry settings, see, eg, Canova and Ciccarelli (2013,ECB WP1507) Time varying FAVARs: Eickmeier, Lemke, Marcellino, "Classical time-varying FAVAR models - estimation, forecasting and structural analysis," (2011Bundesbank Discussion Paper, no 04/2011) Argue that by allowing parameters to change over time, get better forecasts and characterize how the economy is changing