Lecture 31 – Structural VARs: I Let’s assume that yt (or ∆yt) is an n-dimensional I(0) process with a VAR(p) representation, i.e., yt = A0 + A1yt-1 + … + Apyt-p + εt where εt ~ w.n. (Ω). Note that the n elements of εt can be contemporaneously correlated (but not serially correlated). Note too that, for now, we are ruling out CI. Earlier this semester we talked about estimation, inference (e.g., causality tests), and forecasting with the VAR. In that discussion we also addressed practical issues like variable selection and lag length selection.In a major paper, that has had the sort of impact on macroeconometrics that Lucas’s work had on macroecomics, Sims (1980, Econometrica, “Macroeconomics and Reality”) made two important arguments. 1. The standard simultaneous equations modeling approach being used in empirical macroeconomics was not likely to be very useful because it relied on “incredible” identifying restrictions: exogeneity and exclusion restrictions (including restriction on dynamics, i.e., lag structures). While these restrictions might seem sensible when looked at from a partial equilibrium approach, they seem much less sensible from a general equilibrium approach. 2. Constructive alternative: Use the VAR as the basis for doing structural analyis. (Does not rely on exogeneity or exclusion restrictions.)How to use the VAR for structural analysis when the VAR itself is a reduced- form model? Innovation Accounting: the response of the system to structural shocks. It is straightforward to use the estimated VAR to compute the impulse response functions: dyi,t+s/dεj,t for i,j = 1,…,n and s = 0,1,2,… However, the ε’s are reduced-form shocks (evidenced by the nonzero contemporaneous correlations) and, therefore, these impulse response functions don’t have a natural economic interpretation. Structural VAR analysis – formulate the n ε’s in terms of n uncorrelated structural disturbances, say, v1t,…,vnt, and compute the impulse response functions: dyi,t+s/dvj,t for i,j = 1,…,n and s = 0,1,2,…Step back for more motivation – Recall the standard textbook simultaneous equation model of, e.g., supply and demand: Qt = α1 + α2Pt + α3It + v1t α2 < 0 (demand) Qt = β1 + β2Pt + β3Ct + v2t β2 > 0 (supply) where It = household sector income in t Ct = production cost index in t v1t = demand shock in t v2t = supply shock in t We assume that v1t and v2t are uncorrelated with one another. Q and P are jointly determined by this system given the v’s and the exogenous variables C and I. This a structural model – • each equation has a distinct and natural economic interpretation. • equations can include more than one endogenous variableWe wouldn’t directly estimate the supply and demand equations by OLS because of the endogenous r.h.s. variable in each of the two equations. Instead we might estimate its reduced-form by OLS and use it to recover the structural parameter estimates: Qt = a1 + a2It + a3Ct + ε1t Pt = b1 + b2It + b3Ct + ε2t This a reduced model – • explanatory variables are exogenous (or predetermined) • shocks are functions of the structural shocks • equations do not have natural economic interpretation.It is natural to estimate the reduced form by OLS (why not SUR?) and then untangle the reduced form parameter estimates to recover estimates of the structural parameters. This is feasible provided that the structure is identified by the reduced form. In this example it is because of our exogeneity and exclusion restrictions. (Why would these particular restrictions seem implausible in a dynamic macro setting?) How can we identify the structure from the reduced-form without using exogeneity and exclusion restrictions?Consider a dynamic bivariate structural system without exogeneity or exclusion restrictions: y1t = β12y2t + γ10 + γ11y1,t-1 + γ12y2,t-1 + v1ty2t = β21y1t γ20 + γ21y1,t-1 + γ22y2,t-1 + v2t where vt ~ i.i.d. (0,Σ), Σ a diagonal matrix. In this structural simultaneous equation model both variables are endogenous and appear in each equation; the lag structure is symmetric across equations. What distinguishes the equations from one another (other than the arbitrary normalizations)? The uncorrelated vt’s.Example – y1t = log real GDP y2t = log price level What kinds of orthogonal structural shocks would you expect to drive this system? Aggregate Supply Shocks (technology shocks) Aggregate Demand Shocks (monetary policy shocks) Let v1t = supply shock v2t = demand shock Then the first equation is the aggregate supply curve and the second equation is the aggregate demand curve.The reduced-form for this structural system? Rewrite the system as: Byt = Γ0 + Γ1yt-1 + vt where B = [Bij], B11=B22=1, B12=-β12, B21=-β21 This is sometimes called the structural form of the VAR(1). Then, premultiply both sides of the equation by B-1: yt = A0 + A1yt-1 + εt where A0 = B-1 Γ0, A1 = B-1 Γ1,and εt = B-1vt. This is the standard reduced-form VAR. We can fit this by OLS to estimate the A’s and then use the estimated A’s to see how the components of y respond over time to ε1 and ε2.But this is not very interesting since 1) there is no interesting economic interpretation to give to either ε (they are linear combinations of demand and supply shocks) and 2) the ε’s are correlated with one another so that, e.g., dy1t+s/dε1t│dε2t=0 doesn’t make much sense. Instead we would want to see how y1t and y2t respond over time to v1t and v2t shocks. εt = B-1vt So, let’s: • introduce a one-time v1 shock (holding v2 fixed) [How large? σ1] • use inverse(B-hat) to translate the v1 shock into an ε shock • use the A-hat to simulate the response of the y’s to the ε shock Problem – B is not identified! (There are 10 free structural parameters: B (2), Γ0 (2), Γ1 (4), σ12, σ22. There are 9 free reduced-form parameters: A0 (2), A1 (4), Σεε (3).)Solution – We need to find at least one additional restriction on the structural