VAR and SVAR WITH STATA

Introduction to VARs and Structural VARs: Estimation & Tests Using Stata Bar-Ilan University 26/5/2009 Avichai Snir Background: VAR • Background: • Structural simultaneous equations – Lack of Fit with the data – Lucas Critique (1976) • VAR: Vector Auto Regressions – Simple – Non Structural – All Variables are treated identically – Better Fit with the Data Simple VAR: Sims (1980) • Symmetric – Lags of the dependent variables – Same Number of Lags y1,t = α0 + α1y1,t −1 + α2 y2,t + α3 y3,t −1 + α4 y1,t −2 + α2 y2,t + α3 y3,t −2 + + ε1,t −1 −2 y2,t = β0 + β1y1,t −1 + β2 y2,t + β3 y3,t −1 + β4 y1,t −2 + β2 y2,t + β3 y3,t −2 + + ε 2,t −1 −2 y31,t = γ + γ1y1,t −1 + γ y2,t + γ y3,t −1 + γ y1,t −2 + γ y2,t + γ y3,t −2 + + ε3,t −1 σ i, j E (ε i ,t , ε j ,τ ) =  0 i, j ∈ (1,2,3) −2 if t = τ if t ≠ τ Simple VAR: Matrix Form • In Matrix Form: y t = α + Γt −1y t −1 + Γt − y t − + + ε t or Simply : [I − Γ(L )]y t = Α + ε t • y t is a vector of the Dependent Variables • Γt − i is a Matrix of Coefficients • Γ(L ) is a Matrix in Lagged Variables • ε t is a Vector of White Noise Errors • Α is a Matrix of exogenous variables (constant,…) ε  ε ,1   ε 1,2  ε  1,3 :   ε  ,1  ε ,2 ε ' × =  t t  ε ,1  :   ε ,1  ε 3,2   ε ,1  : σ ,1 σ ,1      :  σ ,1      :   σ ,1   :            × ε ,1 ε , ε ,          σ 1,2 Covariance Matrix ( σ ,1 ε , ε , ε , σ 0 1,2 0 0 ,3 : σ 0 ,1 ,1 : 0 ,1 σ σ σ ε , ε , ε , 1,3 σ 1,3 : σ ,2 σ σ ,2 0 : 3,2 3,2 σ σ σ σ ,3 ,3 σ ,3      :       :        )= Contemporary Variance Matrix  σ 1,1 σ 1,2 σ 1,3    Ω =  σ 2,1 σ 2,2 σ 2,3  σ  σ σ , , 3,3   Issues Before Estimation • Stationarity: – Constant expected value – Constant Variance – Constant Covariances • Granger Exogeneity: – Order of variables • Lag Length – Optimal lag length Testing Stationarity • We have data on Canada 1966Q1-2002Q1 – GDP – Consumer Price Index (CPI) – Household Consumption (consumption) Consumption CPI GDP Descriptor 36.91 18.04 62.53 37.47 18.24 64.58 38.46 18.41 65.47 1966Q1 1966Q2 1966Q3 Declare: Time Series • Define and format: time variable – date(var_name,”dmy”) or – Quarterly(var_name, “yq”) – format: format var_name %d • Declare database as time series – Menu: statistics time series setup & utilities declare dataset to be time series data Declaring Time series We can write it: yt = α + α1 yt −1 + α 2π t −1 + ε t π t − β1 yt = β + β yt + β 3π t −1 + υt In Matrix Form:  yt   α 0t   yt −1   ε t   +       =  +      β − π π υ   t   β   t −1   t  OR: −1 −1 −1   α 0t     yt −1   0 εt   yt    +    =   +           π t   − β1   β   − β1   π t −1   − β1  υt  Inverting the Matrix gives 0    β −   −1  0  =  β 1   So we can substitute this in the equations: We find: yt = α + α1 yt −1 + α 2π t −1 + ε t π t = (β1α + β ) + (β1α1 + β ) yt −1 + (β1α + β3 )π t −1 + (β1ε t + υt ) So we can write in VAR form: yt = α + α1 yt −1 + α 2π t −1 + ε t π t = θ + θ1 yt −1 + θ 2π t −1 + ηt Almost there • After estimating the VAR we can find: (β1α + β ) = θ (β1α1 + β ) = θ1 (β1α + β3 ) = θ So we have three equations and four unknowns… Hakuna Matata • We also have the covariance matrix:  σ ε ,ε   σ ε ,η  σ ε ,η   σ ε ,ε = ση ,η   β1σ ε ,ε β1σ ε ,ε   2 β1σ ε ,ε + υt  ( • So we have a fourth equation: β1σ ε ,ε = σ ε ,η ) Run the VAR • Note that because we assume that the “real” covariance matrix has the triangular form:  σ ε ,ε   β1σ ε ,ε    σ ε ,ε  • We can use the OIRF that Stata gives us (Cholesky factorization) to watch the Structural impulse functions Run the VAR (1 lag) Study the Impulse Responses v arbas ic , inf lat ion, inf lat ion v arbas ic , inf lat ion, y v arbas ic , y , inf lat ion v arbas ic , y , y 01 00 01 00 0 s tep 95% CI orthogonali zed irf Graphs by irf name, impuls e v ariable, and res pons e v ariable Get the coefficients α1 α2 α0 θ1 θ2 θ0 Get the Errors matrix σ ε ,ε β1σ ε ,ε We find: β1 = cov(ε ,η) σε ,ε 0.000007018 = = 0.086 0.00008145 β0 = θ0 − β1α0 = 0.0005972− 0.086× 0.0574 = −0.0043 β2 = θ1 − β1α1 = 0.195− 0.086× 0.622 = 0.142 β3 = θ2 − β1α2 = 0.625− 0.086ì 0.142 = 0.614 Conclusion ã Enough restrictions ã Exact Identification • Possible to deduce the Structural Parameters To test a restricted Model • Run a non restricted model • Test the null by using the LR test on the difference between the restricted and unrestricted model λ = T (ln Wres − ln Wunres ) → χ (M ) Wres − restricted cov ariance matrix Wunres − unrestricted cov ariance matrix M − Number of restrictions Caveat • With the data we used, it is likely that the variables are cointegrated (consumption and GDP) • One should (theoretically) check for that option ... multivariate time series Basic Vector Autoregression Model • Choose – Variables – Lag Length Granger Test: Running VAR Testing in Stata • Statistics multivariate time series var diagnostics and. .. AIC We go with the LR and AIC and say (why not?) BIC Run Simple VAR • We run a simple VAR (not structural, no assumptions on order of variables) between Household Consumption, Inflation and GDP... − for AIC, T for BIC, Tests in Stata • Menu: Statistics multivariate time series var diagnostics and tests Lag-Order Selection statistics Running test • Choose Variables • Choose maximum lags