Massachusetts Institute of Technology Department of Economics Time Series Lecture 6: Additional Results for VAR s

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1 Massachusetts Institute of Technology Department of Economics Time Series Guido Kuersteiner Lecture 6: Additional Results for VAR s 6.1. Confidence Intervals for Impulse Response Functions There are three main methods to construct confidence intervals for VARs: Asymptotic Expansions based on delta method Bootstrap methods Bayesian Posterior distributions 6.2. Asymptotic Expansions Asymptotic Expansions were formally obtained by Lütkepohl JoE, It can be shown that µ vec(ˆπ π) d Σ Γ 1 n p 0 N 0, vech(ˆσ Σ) 0 2D + (Σ Σ)D + where D + =(D 0 D) 1 D 0 and D such that vec Σ = D vech Σ. Impluse coefficients C j are given as C j = J 0 A j JR 1 with Σ =(R 1 R 0 1 ). In other words C j = G(j, π, Σ) such that by the delta method vec G(j, π, Σ) vec G(j, π, Σ) ³ n vec( Ĉ j C j ) = n(vec (ˆπ π)) + n vec π 0 vech Σ 0 vech ³ˆΣ Σ = A 1 n(vec (ˆπ π)) + A2 n ³vech ³ˆΣ Σ. It then follows that n(vec Ĉ j C j ) d N(0,A 1 (Σ Γ 1 p )A A 2 D + (Σ Σ)D + A 0 2) A problem for applied work is that these bands collapse to zero very quickly as a function of j, thus underrepresenting the degree of uncertainty in finite samples. The reason is that C j becomes an increasingly non-linear function of π as j increases and thus the asymptotic approximation deteriorates for large j (see for example the size distortions reported in Killian 1998). An alternative procedure based on bootstrapping the VAR was proposed by Runkle in Before we discuss Runkle s procedure we first review results for non-parametric bootstrap of auto-regressions (and VARs) Bootstrap Based Confidence Intervals We first consider the problem of bootstrapping autoregressive models (Bose, 1988 Annals of Statistics).

2 Consider the model px y t = φ i y t i + ε t i=1 ε t iid y t can be multivariate OLS estimates: ˆφ obtain estimated residuals px ˆε t = y t ˆφ i y t i i=1 build the empirical distribution G n (x) = 1 n nx 1(ˆε t x) t=1 (puts mass 1 on each observation) n and draw random samples {ε t } n t=1 from F n(x) =G n (x + ε) with ε = 1 n empirical distribution has mean zero). Generate y t = px ˆφy t i + ε t and estimate ˆφ for each random sample. Construct an estimate ˆΩ based on cov(yt,yt k ) which can be computed as i=1 cov(y t,y t k) = 1 B BX (ytb,y t kb) b=1 np ε i (to guarantee that the where ytb is y t in the b-th bootstrap replication. Then sup P (n 1/2 ˆΩ 1/2 (ˆφ ˆφ) x) P (n 1/2 ˆΩ1/2 (ˆφ φ 0 ) x) = O(n 1/2 ) x almost surely. The problem with this procedure is that it will not work well near the unit circle where the distribution of ˆφ ceases to be pivotal and the nuisance parameters cannot be estimated consistently. Confidence intervals for ˆφ φ 0 will also not be accurate for confidence intervals of the impulse response function because of non linearities and bias of ˆφ. Runkle (1987) generates samples based on ˆφ and ˆε t in the same way as before to obtain estimates ˆφ. He then computes the impulses Ĉ j and constructs confidence bands hĉ (α) i kl,j, Ĉ (1 α) kl,j based on the simulated α and (1 α) percentiles of the empirical distribution of Ĉ j. t=1 2

3 The problem with this method seems to be that it is heavily affected by small sample bias that distorts the initial estimates of φ. The bootstrapped estimates are then biased again (usually towards the stationary region). As a consequence it can happen that the confidence bands constructed in this way do not contain the original parameter estimates. Killian (1998), Review of Economics and Statistics, proposes a boot-strapped bias correction to overcome this problem. We assume that the data allow for the reduced form representation y t = v + B 1 y t B p y t p + u t. For notation convenience let β = B1, 0..., Bp 0 0. Killian s procedure implements the following three steps: Step 1a Estimate VAR(p) and generate 1000 bootstrap replications ˆβ from yt =ˆv + ˆB 1 yt ˆB p yt p + u t Estimate the bias by ˆψ = E (ˆβ ˆβ) sp = 1 s ˆβ s ˆβ. Step 1b Shrink bias corrected estimate to the stationary region. Bias corrected estimate is s=1 β =2ˆβ 1 sx ˆβ s = s β ˆψ Step 2a Generate 2000 additional bootstrap samples using β as the parameter value in s=1 y t =ˆv + B 1 y t= B p y t p + u t and reestimate ˆβ. Now we again want to bias correct ˆβ. Killian uses ˆψ as a simple estimate of the bias. Step 2b Compute β = ˆβ ˆψ Step 3 Based on β compute impulse response functions C j = C j ( β, ˆσ ) Use the α and 1 α percentile points of the distribution of C j as confidence intervals. Killian reports that this method produces intervals with mostly correct size and only moderately inflated length, while the non-corrected method produces severe size distortions (the same size distortions also occur with the delta-method). 3

4 6.4. Bayesian Posteriors An alternative procedure advocated by Sims and Zha, Econometrica (1999), is based on the Bayesian posterior with flat priors. Let B(L)y t = u t B(L) m m matrix polynomial B 0 = I and Eu t u 0 t = Σ. The likelihood function for B, Σ is (conditional on initial observations) proportional to Use prior that is flat in Jeffrey s sense µ q(b,σ) = Σ T/2 1 exp 2 tr(s(b)σ 1 ) S(B) = Σû t û 0 t û t = B(L)y t Σ m+1/2 which leads to a joint posterior. Integrating out B form the joint posterior leads to the marginal posterior for Σ which becomes p(σ)α Σ (T +m V +1)/2 exp( 1 2 tr(s( ˆB)Σ 1 )) where V is the number of estimated coefficients per equation. This posterior is the p.d.f. of an inverted Wishart distribution with T V degrees of freedom. We can draw from this distribution by generating T V iid r.v. from x i = N(0,S(ˆβ) 1 ) forming their sample second moments and setting 1 T V TX V i x i x 0 i à 1 ˆΣ 1 X = x i x 0 T V i!. To generate a draw from the joint posterior of (B,Σ) one first generates ˆΣ andthendrawsb from the conditional normal i q(b, ˆΣ ) (Normalized so that it integrates to one). Based on ˆB one can then compute impulse responses Ĉ j. then can construct confidence intervals from If P (Ĉj <x) is the probability distribution of Ĉj 1 α = P (a <Ĉj C j <b)=p (Ĉj b<c j < Ĉj a). 4

5 6.5. Structural VAR s A special case of the reduced form model in the notes is the one where B 0 = I and B s =0, s>0, anda s =0 for s>p. Then the structural model becomes the following dynamical structural model A 0 y t = A 1 y t A p y t p + ε t The reduced form VAR is then obtained from the structural form by pre multiplying by A 1 0 where y t = π 1 y t π p y t p + u t π k = A 1 0 A K and u t = A 1 0 ε t From these relationships we see that the structural disturbances ε t can be obtained from the reduced form disturbances u t as ε t = A 0 u t If it happened to be the case that A is lower diagonal then the structural disturbances would just be the same as the orthogonal disturbances used for the calculation of the impulse response function. One way to achieve this is to formulate the model in a block recursive form. If the model cannot be expressed in block recursive form then one can still obtain impulse responses since ε t = A 0 u t is still uncorrelated Estimation of a structural VAR when there are no restrictions placed on the dynamics of the model Assume we have the structural VAR where ε t N(0,D) D diagonal matrix. Then A 0 y t = A 1 y t A p y t p + ε t = π 1 x t + ε t L(B 0,Dπ) = Tk 2 log 2π T log A D(A 1)1 0 1 X (y t π 0 x t ) 0 (A 0 D 1 A 0 2 0)(y t π 0 x t ) t such that vec ˆπ =(I (X 0 X) 1 X)vecY ˆΩ = 1 n Σˆε tˆε 0 t and L(B 0,D,ˆπ) = Tk 2 log 2π T log A D(A 1)0 0 T trace A0 D 1 A 0 ˆΩ. 0 The maximum satisfies  1 ˆD(  1 ) 0 = ˆΩ (see argument for maximizer of Ω in standard case). 5

6 The reduced form is identified if it is parametrized by n(n + 1)/2 free parameters (order condition). addition we need to be able to apply the implicit function theorem to the set of nonlinear equations In vech Ω =vech(a 1 0 D(A 1 0 )0 ) assume that A 0 = A(θ) and D = D(θ) then J(θ) = vec(a(θ) 1 D(θ)A(θ) 10 ) θ which follows from = 2(Ω A(θ) 1 vec A(θ) ) + A 1 (θ) A 1 (θ) vec D(θ) θ θ dω(θ) = A 1 daa 1 D(θ)A(θ) 10 A(θ) 1 D(θ)A(θ) 1 da(a 1 ) 0 + A 1 dd(a 1 ) 0 and vec ABC =(C 0 A)vecA. Then locally = A 1 daω ΩdA(A 1 ) 0 + A 1 dd(a 1 ) 0 vech Ω = D + J(θ)θ such that θ is given by (D + J(θ)) 1 vech Ω = θ if the inverse exists. A simple form of these restrictions where this condition is always satisfied is to let D be free and set 1 0 A 0 =... θ A 1 where θ A are free parameters. Under these restrictions it is possible to find matrices A 0 and D that satisfy vech Ω =vech(a 1 0 D(A 1 0 )0 ) because this decomposition of Ω is unique. 6