VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Jacek Suda, Banque de France October 11, 2013
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Outline Outline: 1 Dynamic Structural Models and VAR 2 Identification 3 Multivariate Wold Form and Forecasting 4 Impulse Response Functions 5 Variance Decomposition 6 Identification Short-run restrictions Long-run restrictions 7 Granger Causality
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Example: Money Demand Let y 1t = real money balance = M P, y 2t = real GNP. Money Demand y 1t = γ 10 + β 12 y 2t + γ 11 y 1,t 1 + γ 12 y 2,t 1 + ε 1t. ε 1t encompasses all other factors, β 12 is a short-run elasticity of real money balances, ( M P ) d, with respect to real income, lagged terms allow for different long-run elasticity.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Example: Money Supply Money Supply y 2t = γ 20 + β 21 y 1t + γ 21 y 1,t 1 + γ 22 y 2,t 1 + ε 2t. β 21 is a short-run impact of money on output, Estimating Money Demand or Money Supply by OLS: ˆβ 12 OLS and ˆβ OLS 21 are inconsistent due to endogeneity (simultaneity). ˆβ OLS ij picks up correlation between income and money.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Aggregate Demand and Supply If both equations estimated together ε 1t and ε 2t may include omitted variables (e.g. interest rate). In a fully specified structural model, ε 1t and ε 2t are orthogonal, exogenous shocks: ε 1t = money demand shock, ε 2t = productivity shock. If ε 2t exogenous shock, change in ε 2t will affect y 2t and, through β 12, will affect y 1t. This allows us to identify β 12. Identification problem when shock ε 2t affects ε 1t : change in y 1t will due to change in y 2t and in ε 1t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast VAR We want to make inference on causality from correlation. Using VAR, we 1 get correlation structure of data, 2 can do macro modelling without pretending to have too much a priori theory (Sims, 1980), 3 given the structure, make identification assumptions what shocks are (timing as identification tool).
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Structural VAR(1): System Write down supply and demand equation as a system of equation: [ ] [ ] [ ] [ ] [ ] 1 β12 y1t γ10 γ11 γ = + 12 y1,t 1 + β 21 1 y 2t γ 20 γ 21 γ 22 y 2,t 1 Or, in matrix notation B 2 2 y t2 1 = Γ 02 1 + Γ 12 2 y t 12 1 + ε t2 1. [ ε1t ε 2t ]. Similar to AR(1) but in a vector form.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Structural VAR(1): Shocks Shocks [ ε1t ε 2t ] [ ( 0 iid 0 ) ( σ 2, 1 0 0 σ2 2 ) ] = iid(0, D). Exogenous shocks to each variable diagonal variance covariance matrix, D = Diag.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Reduced-form VAR(1) Solve structural model. If β 12 β 21 1 B 1 exists and where y t = B 1 Γ 0 + B 1 Γ 1 y t 1 + B 1 ε t y t = C 2 1 + Φ 2 2 y t 1 + e t, e t = B 1 1 ε t = 1 β 12 β 21 [ ] [ ] ε1t + β 12 ε 2t e1t = forecast errors ε 2t + β 21 ε 1t e 2t
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Forecast errors e t = B 1 1 ε t = 1 β 12 β 21 [ ] [ ] ε1t + β 12 ε 2t e1t = ε 2t + β 21 ε 1t e 2t 1 1 β 12β 21 overall feedback effect. Total effect of shock ε 2t on money: shock to income money income money. Any forecast error has the form of the linear combination of structural shocks.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Forecast errors Forecast errors: E[e t ] = 0 E[e t e t] = E[B 1 ε t ε t(b 1 ) ] = B 1 E[ε t ε t](b 1 ) [ = B 1 D(B 1 ) ω11 ω Ω = 12 ω 12 ω 22 ]. Ω is not diagonal as both forecast error are affected by both shocks. If eigenvalue(φ) < 1 then reduced form VAR can be consistently estimated by OLS (equation by equation). Similar to SUR (Seemingly Unrelated Regressions): a special case where all xs are the same for each equation. Estimation of system of equations with OLS is equivalent to CMLE to SUR. Problem with reduced form: identification.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Identification Structural VAR(1) has 10 parameters: (γ 10, γ 20, γ 11, γ 12, γ 21, γ 22, σ 2 1, σ 2 2, β 12, β 21 ) Reduced form VAR(1) has 9 parameters: (c 1, c 2, φ 11, φ 12, φ 21, φ 22, ω 11, ω 12, ω 22 ) Cannot identify β 12, β 21 from ω 12. All we have is correlation of income with forecast and money with forecast. Infinite number of structural VARs that are consistent with reduced from VAR. Need additional restrictions to identify the model (e.g. short-run, long-run, sign restrictions).
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Multivariate Wold Form Take covariance stationary process, {Y t }. The multivariate Wold Form For VAR(1) and [ y1t y 2t y t = µ + ψ j ε t j, j=0 y t2 1 = [ y1t y 2t ]. {ε t } WN. y t = C + Φy t 1 + e t, e t iid(0, Ω), ] [ ] [ ] [ ] c1 φ11 φ = + 12 y1,t 1 + c 2 φ 21 φ 22 y 2t 1 [ e1t e 2t ].
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Multivariate Wold Form In lag notation, (I ΦL)y t = C + e t y t = µ + (I ΦL) 1 e t, µ = (I Φ) 1 C, as LC = C where y t = µ + Ψ(L)e t, Ψ(L) = Ψ k L k, Ψ 0 = I, Ψ k = Φ k, k=0 Ψ(L) = I + ΦL + Φ 2 L 2 +...
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast Forecast for VAR(1) From Wold Form: y t+s = µ + e t+s + Φe t+s 1 +... + Φ s e t + Φ s+1 e t 1 +... Then, y t+s t = µ + Φ s e t + Φ s+1 e t 1 +... = µ + Φ s (e t + Φe t 1 +...) = µ + Φ s (y t µ) So the forecast is just the deviation of the series from their long-run unconditional means. Note: Ψ s [1, 1] y 1,t+s e 1t, because E[e 1t e 2t ] = ω 12 0. y 1,t+s e 1t = univariate effect e t on y 1,t+s, ceteris paribus. In the univariate case Ψ s = yt+s e t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR(1) Reduced-form Identification Wold Form Forecast MSE Mean square error MSE(y t+s s, y t+s ) = E[(e t+s + Φe t+s 1 +... + Φ s 1 e t+1 ) (e t+s + Φe t+s 1 +... + Φ s 1 e t+1 ) ] = E[e t+s e t+s] + ΦE[e t+s e t+s]φ +... + Φ s 1 E[e t+s e t+s](φ s 1 = Ω + ΦΩΦ +... + Φ s 1 Ω(Φ s 1 ) s 1 = Φ k Ω(Φ k ). k=0 lim MSE = Φ k Ω(Φ k ) s = k=0 Ψ k ΩΨ k = var(y t ) k=0
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Impulse Response Functions Given a reduced-form VAR and an identification assumption for B, solve the structural VAR By t = Γ 0 + Γ 1 y t 1 + ε t, where Γ 0 = Bc, Γ 1 = BΦ, ε t = Be t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Impulse Response Functions Solve for vector MA in terms of structural shocks: where µ = (B Γ 1 L) 1 Γ 0. (B Γ 1 L)y t = Γ 0 + ε t y t = (B Γ 1 L) 1 (Γ 0 + ε t ) = µ + θ(l)ε t, From Wold Form: θ(l)ε t = Ψ(L)e t = Ψ(L)B 1 ε t θ(l) = Ψ(L)B 1 = B 1 + Ψ 1 B 1 L + Ψ 2 B 1 L 2 +...
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Impulse Response Functions Therefore, [ ] y1t = y 2t [ µ1 Note θ 0 = B 1 I. ] [ ] [ ] [ ] [ ] θ11,0 θ + 12,0 ε1t θ11,1 θ + 12,1 ε1,t 1 +.. µ 2 θ 21,0 θ 22,0 ε 2t θ 21,1 θ 22,1 ε 2,t 1 Then, θ 11,s = y1,t+s ε 1t. For n variable system we have n 2 impulse response functions.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Impulse Response Functions θ 11,s = y1,t+s ε 1t 11,s 1.0 IRF 0.8 0.6 0.4 0.2 0 1 2 3 4 s, Time
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Cumulative Response Define Cumulative Response Function θ ij,s = s θ ij,k, k=0 θ ij (1) = long-run cumulative impact of shock j on variable i. θ ij (1) = θ ij,1 + θ ij,2 + θ ij,3 +... θ ij (1) = lim s θ ij,s
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Cumulative Response IRF, ij,s 0.4 0.3 0.2 0.1 0.1 10 20 30 40 50 0.2 0.3 3.5 CumulativeReponse, ij,s 3.0 2.5 2.0 1.5 1.0 0.5 10 20 30 40 50
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Cumulative Response 0.4 IRF, ij,s 0.2 5 10 15 20 25 30 35 0.2 3.0 CumulativeReponse, ij,s, ij 1 0 2.5 2.0 1.5 1.0 0.5 0.0 5 10 15 20 25 30 35
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p Variance decomposition For MSE at different horizons, what share is due to each structural shocks? Variance decomposition: p i,j (s) = σ2 j (θij,0 2 +... + θ2 ij,s 1 ), MSE(y i,t+s t, y i,t+s ) with i denoting series, and j denoting shocks. MSE i = MSE(y i,t+s t, y i,t+s ) = n σj 2 (θij,0 2 +... + θij,s 1), 2 j=1 n variables, n shocks.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p VAR(2) Consider the two equation two-lag system x 1t = a 11 x 1,t 1 + a 12 x 1,t 2 + a 13 x 2,t 1 + a 14 x 2,t 2 + e 1,t x 2t = a 21 x 1,t 1 + a 22 x 1,t 2 + a 23 x 2,t 1 + a 24 x 2,t 2 + e 2,t. Defining the vector y t = x 1,t x 2,t x 1,t 1 x 2,t 1, the system can be represented in a VAR(2) matrix form y t = Ay t 1 + e t, where A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 1 0 0 0 0 1 0 0, e t = e 1,t e 2,t 0 0
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Impulse Response Functions Cumulative Response Variance decomposition VAR(p VAR(p) More generally, the VAR(p) system x t = c + Φ 1 x t 1 + Φ 2 x t 2 + Φ 3 x t 3 +... + Φ p x t p + e t, e t iid(0, Ω), can be written as a VAR(1) y t = Ay t 1 + v t
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions IDENTIFICATION
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions SHORT-RUN RESTRICTION
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions VAR(1) Recall: Structural VAR(1) By t = Γ 0 + Γ 1 y t 1 + ε t, Reduced-form VAR(1) y t = B 1 Γ 0 + B 1 Γ 1 y t 1 + B 1 ε t, = C + Φy t 1 + e t. Need n(n 1) 2 restrictions to identify structural VAR.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Error term Recall that e t = B 1 ε t, where e t is a forecast error and ε t is a structural shock. e t are correlated, ε t are not correlated. It is linear relationship.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Short-run Restrictions Assumptions: [ ] σ 2 ε t (0, D), D = 1 0 0 σ2 2, [ ] ω11 ω e t (0, Ω), Ω = 12. ω 12 ω 22 Suppose B 1 is lower triangular, then B 1 and D can be identified from Cholesky decomposition of Ω. Cholesky decomposition: For any positive definite symmetric matrix there exist unique decomposed, triangular factorization Ω = PP = TΛT, where Λ is a diagonal matrix with positive elements and T is lower diagonal matrix with 1s on diagonal, P is a lower diagonal matrix. Cholesky decomposition is unique.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Short-run Restrictions Therefore, if B 1 is lower triangular then take and so that T = B 1, Λ = D, var(e t ) = Ω = B 1 D(B 1 ) Reduced form VAR and Cholesky decomposition Structural VAR if B 1 lower triangular. What does it mean that B 1 is lower triangular?
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Example: Sims, 1980 Consider VAR for y t, π t, i t (output growth, inflation, and interest rate). Policy rule: i t = β 31 y t + β 32 π t + ε 3t, where β 31 y t + β 32 π t is a reaction function and ε 3t is a policy shock. Inflation: π t = β 21 y t + γ 23 i t 1 + ε 2t, with ε 2t being, for example, oil price shock. Policy variable affects inflation with a lag an assumption. Output growth: y t = γ 13 i t 1 + ε 1t, with ε 1t denoting a productivity/supply shock. Timing assumption: it may take a while to have change in i affecting output
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Example: Structural VAR VAR: 1 0 0 β 21 1 0 β 31 β 32 1 y t π t i t = γ 11 γ 12 γ 13 γ 21 γ 22 γ 23 γ 31 γ 32 γ 33 y t 1 π t 1 i t 1 + ε 1t ε 2t ε 3t. We say that y t does not respond to shocks in i t and π t. Sims on γ s: Why put 0 restrictions if they are not obvious from the model we can estimate it and see if they are really zero.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Example: Structural VAR(1) VAR(1): Bx t = Γ 1 x t 1 + ε t, β 13 = β 23 = 0 output and inflation respond to policy shocks with a lag, β 12 = 0 output responds to oil price shock with a lag, β 13 = β 23 = 0 forecast error for output growth is a productivity shock.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Example: Reduced-form VAR(1) Structural Bx t = Γ 1 x t 1 + ε t implies a reduced form VAR: Since B and B 1 are lower triangular, x t = B 1 Γ 1 x t 1 + B 1 ε t = Φx t 1 + e t. implies e t = B 1 ε t e 1t = ε 1t e 2t = ε 2t + β 21 ε 1t e 3t = ε 3t + β 22 ε 2t + β 32 ε 1t. It is recursive identification. We can identify ε 2t knowing ε 1t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Wold-causal ordering Wold-causal ordering: All variables can be endogenous but y t is causally prior to π t and i t, π t is causally prior to i t. Ordering is what defines the impact. We put y t first, π t second and i t last. If we put different order, like π t last, we say that interest rate affect inflation. Sims: Fed doesn t observe GDP, it has only a lagged value.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Example: Kilian(2009, AER) Lutz Killian (2009, American Economic Review): Not All Oil Price Shocks are Alike: Disentangling Demand and Supply Shocks in the Crude Oil Market. Oil shocks: large increases and declines in the price of oil, receive a lot of attention. Many recent recessions were preceded by an increase in the price of oil but oil usage is actually a relatively small input compared to GDP. Kilian asks what is an oil price shock and are there different kinds of oil price shocks? (instead of what are the effects of an oil price shock? ) Paper uses VAR analysis to distinguish between shocks to oil prices due to global demand, shocks due to oil supply, and shocks due to speculation in the oil price market.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Kilian(2009): Model Three variable monthly VAR in the growth rate of oil production, real global economic activity, and the real price of oil: zt = ( prod t, rea t, rpo t ). VAR structure A 0 z t = α + 24 i=1 A i z t i + ε t, where ε t are structural shocks and A 0 is lower triangular A 0 = a 0 0 b c 0 d e f Identifying assumptions 1 Oil production does not respond within the month to world demand and oil prices. 2 World demand is affected within the month by oil production, but not by oil prices. 3 Oil prices respond immediately to oil production and world demand.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Kilian(2009): Shocks Since A 0 is lower-triangular, so is A 1 0. Reduced-form VAR 24 z t = A 1 0 α + A 1 0 A iz t i + A 1 0 ε t i=1 Reduced form shocks A 1 0 ε t = e t = e prod t e rea t e rpo t = a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 ε prod t ε rea t ε rpo t
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Kilian(2009): Findings Main Lesson: How the economy reacts to an oil price shock will depend on the origins of that shock. Shocks to oil supply have limited effects on oil prices and have been of negligible importance in driving oil prices over time. Both global demand and speculative oil price shocks can have significant effects on oil prices, but speculative oil price shocks have limited effects on global economic activity. Speculative oil-market shocks have accounted for most of the month-to-month movements in oil prices. The steady increase in oil prices from 2000 onwards was almost solely due to strong global demand.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Short-run Restrictions Example Kilian(2009, AER) Short-run restrictions Short-run restrictions Short-run restrictions: Use the recursive identification method. Construct a set of uncorrelated structural shocks directly from the reduced-form shocks. Assumes that certain shocks having effects on only some variables at time t or, alternatively, that some variables only having effects on some variables at time t. Corresponds to assuming that B is lower triangular in VAR(1) By t = Γ 0 + Γ 1 y t 1 + ε t. Causal ordering of variables in y t defines the impact.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ LONG-RUN RESTRICTION
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Too strong assumptions? We need a story to tell why B is lower triangular. Ordering x t = y t π t i t assumes that output is causally prior to inflation and interest rate: if there is a shock to interest rate, it will take time to be reflected in π t and y t. We want simultaneity in out model (hence VAR) but we assume it away in the first period for identification purposes. Additionally: p i t: commodity prices changes are connected to/directly reflected in interest rate, and both i t and p i t are determined simultaneously.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Too strong assumptions? Problems: 1 short-run identifications may have some limitations can t be done it in some cases, 2 we want assumptions on identification that will not just assume answer we want data to be decisive, not model/identification selected. Economic theory gives very little guidance. Need method of identification that allows for general B matrix, not only lower triangular. Long run identification: impose more plausible restrictions, does not assume Keynesian or classical approach.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Long run identification Long-run identification approach: use these theoretically-inspired long-run restrictions to identify shocks and impulse responses. Economic theory usually tells us a lot more about what will happen in the longer-run, rather than exactly what will happen today. For instance, theory tells us that whatever positive aggregate demand shocks do in the short-run, in the long-run while they should have no effect on output, they have a positive effect on the price level.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Structural VAR Structural model: By t = Γ 0 + Γ 1 y t 1 + ε t, ε t iid(0, D). Vector moving average representation: (B Γ 1 L) y t = Γ 0 + ε t y t = (B Γ 1 L) 1 Γ 0 + (B Γ 1 L) 1 ε t, = µ + θ(l)ε t, θ 0 = B 1, θ 1 = Γ 1 B 1,... = µ + θ 0 ε t + θ 1 ε t 1 + θ 2 ε t 2 +... Elements in θ i tell us what the structure of shocks is. We can identify Γ s but not B.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Cumulative impact of the shock Impulse Response Function: θ s,11 = y 1,t+s ε 1t Cumulative impact: θ s,11 = s j=0 y 1,t+j ε 1t = s θ j,11. j=0
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Cumulative impact of the shock IRF, ij,s IRF, ij,s 0.4 0.4 0.3 0.2 0.2 0.1 0.1 10 20 30 40 50 0.2 5 10 15 20 25 30 35 0.2 0.3 3.5 CumulativeReponse, ij,s 3.0 CumulativeReponse, ij,s, ij 1 0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 10 20 30 40 50 0.0 5 10 15 20 25 30 35
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ VAR When we think about long-run restriction we think about long-run effects of the shock. For stationary VAR model IRF(s) 0 as s but it does not have to be true for the long run cumulative response.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Blanchard and Quah, AER 1989 Example: Blanchard and Quah Let y t denotes GDP growth (stationary) and u t unemployment [ ] yt x t =. Then the structural VAR model is u t Bx t = Γ 0 + Γ 1 x t 1 + ε t, ε = [ ε AS t ε AD t ] iid(0, D).
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Assumptions Assumption: AS and AD shocks are exogenous, uncorrelated. They drive fluctuations in each series. [ ] σ 2 D = AS 0 0 σad 2. Assumption: All dynamics are in the VAR, not in the structure of shocks. A vector MA version of the model: x t = µ + θ(l)ε t vector MA. Long-run variance: Λ = var(x t ) = var(θ(1)ε t ) = θ(1)dθ(1), which reflects the cumulative effects of shocks. But shocks to y t do not have same periodic persistence as shocks to u t they have long run effect. We can identify long-run variance from reduced form model.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Reduced form MA Reduced form: Wold form: x t = µ + Ψ(L)e t Λ = Ψ(1)ΩΨ(1), Estimates of each of these variances: y t = ĉ + ˆΦy t 1 + e t, e t iid(0, ˆΩ), y t = (I ˆΦ) 1 ĉ + (I ˆΦL) 1 e t, = ˆµ + e t + ˆΨe t 1 + ˆΨ 2 e t 2 +..., Ψ 1 = Φ, Ψ 2 = Φ 2, Ψ j = Φ j, ˆΨ(1) = I + ˆΨ 1 + ˆΨ 2 +... Blanchard and Quah: Aggregate shocks have no long-run effect on level of output.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Structural MA Structural MA [ ] yt u t = [ µ1 ] [ ] [ ] θ0,11 θ + 0,12 ε AS t µ 1 θ 0,21 θ 0,22 ε AD t [ ] [ ] θ1,11 θ + 1,12 ε AS t 1 θ 1,21 θ 1,22 ε AD t 1 [ ] [ ] θj,11 θ +... + j,12 ε AS t j +... θ j,21 θ j,22 ε AD t j Average of past shocks. Supply shocks and demand shocks affect both output growth and unemployment...... if short-run restriction: θ 0,12 = 0.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Long-run restriction Long-run assumption: shocks in 1902 does not have any effect on output growth today accumulation of the shocks conveyed only to some level. Both shocks to unemployment and output growth die out. Cumulative impacts lim s lim s θ j,xy 0 as j. θ j,11 = θ 11 (1), cumulative impact of AS shocks on output j=1 θ j,12 = θ 12 (1), cumulative impact of AD shocks on output. j=1 Assumption: θ 12 (1) = 0. AD shock has no long-run impact on the level of output, y t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Cholesky decomposition Estimate ˆΛ: ˆΛ = ˆΨ(1)ˆΩ ˆΨ(1), Do a Cholesky decomposition of ˆΛ: ˆΛ = ˆθ(1)ˆDˆθ(1), with ˆθ(1) lower triangular and ˆD diagonal matrices.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ VAR From the structural form: x t = µ + Ψ(L)e t, x t = µ + Θ(L)ε t. Since we have e t = B 1 ε t, Ψ(L)B 1 = θ(l). It is true for every VAR model, any L. As this hold in general, then it also holds Ψ(1)B 1 = θ(1) ˆB 1 = ˆΨ(1) 1 ˆθ(1).
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ Identification We have ˆθ(1) identified because of Cholesky decomposition Recall ˆΛ = ˆB is unique with  being lower triangular. Imposing long-run restriction makes ˆθ(1) lower triangular. Since ˆθ(1) is lower triangular we get a ˆθ(1) from the unique decomposition of Λ = θ(1)dθ(1). In both long-run and and short-run identifications, we construct a lower triangular matrix so we can use Cholesky decomposition.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Long run identification Structural VAR Cumulative impact of the shock Cumulativ More on identification See Galí (1999, American Economic Review), "Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?" for long-run restrictions. Other identification restrictions: 1 Identification by sign restrictions 2 Identification from heteroskedasticity 3 DSGE priors 4 Identification through regional/multicountry restrictions 5 Natural experiment approach See Recent Developments in Structural VAR Modeling NBER Summer Institute lecture by Stock and Watson
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) GRANGER CAUSALITY
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) Granger Causality Two things behind the notion of Granger causality: 1 The cause occurs before the effects. 2 The cause contains information about the effect that is unique and is in no other variable. In some case if we have one direction, Granger causality allows us to do inference about causality. In some cases, however, it may seem we have Granger causality but it may be driven by the lack of relevant variables in the regression. It doesn t require structural assumptions.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) Granger(AER) Granger(AER) causality: Definition Process {y 2t } Granger causes {y 1t } if Mean Square Error of (linear prediction) Ê[y 1,t+s ỹ 1t ] Mean Square Error of Ê[y 1,t+s ỹ 1t, ỹ 2t ], (ỹ 1t = {y 1,t, y 1,t 1, y 1,t 2,...}). y 2t contains marginal predictive power above and beyond of what can be observed in y 1t alone. {y 2t } provides marginal predictive power for {y 1t } Can you improve on MSE by adding ỹ 2t? If y 1t can be predicted more efficiently when the information in the y 2t process is taken into account then y 2t is Granger-causal for y 1t.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) St. Louis Regression (60s) St. Louis Regression (60s): Regress output growth on lagged money growth Y t = α + β M t 1 + ε t, They find β > 0 when money growth is high today, output growth will be high tomorrow. St. Louis Fed said it is causal relationship.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) Tobin (1970, AER) Tobin (1970, AER) β > 0 can reflect: 1 M causes Y 2 output, Y, causes money, M (monetary authority just respond to economic conditions) and output, Y t 1, forecasts future output, Y t, i.e. cov( Y t, Y t 1) > 0, cov( Y t 1, M t 1) > 0. Purely passive rule with real causality of income causing money.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) Sims (1972) Sims (1972) 1 If Y causes M, and Y predicts Y, then output growth Granger causes M money. 2 If Y does not causes M, then Y does not Granger causes M. Estimate VAR: Sims finds: Y t = c Y + φ 11 Y t 1 + φ 12 M t 1 + e 1t, M t = c M + φ 21 Y t 1 + φ 22 M t 1 + e 2t. φ 12 0 M Granger causes Y (Note φ 12 β.) φ 21 = 0 Y does not Granger causes M. Money is not predicted by income.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Granger Causality Granger(AER) St. Louis Regression (60s) Tobin (1970, AER) Catch Catch: Sims paper rejects Tobin s story. With updated data you don t get the second result: money Granger cause income and income Granger cause money. If both GC each other it might be that both causes each other if we have this simultaneity we have to do the identification restriction. Hamilton: there are pitfalls in expectations: stock market can be found to GC a lot of variables: does it mean they cause them? stock market prices can reflect expectations. You have to find unidirectional causality. Have to convince that this unidirectional causality is not because of expectations. Also, failure to reject GC might be due to low power. If data are not stationary and you perform Granger causality, sizes of the tests are different than usual. It is relevant to know if there is a unit root.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary VAR A VAR is an n-equation, n-variable model in which each variable is in turn explained by its own lagged values, plus current and past values of the remaining n 1 variables. In data description and forecasting, VARs have proven to be powerful and reliable tools. Structural inference and policy analysis are, however, inherently more difficult because they require differentiating between correlation and causation; the identification problem.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Reduced-form VAR A reduced form VAR expresses each variables as a linear function of its own past values, the past values of all other variables being considered and a serially uncorrelated forecast error term. Each equations is estimated by ordinary least squares regression. The error terms are the surprise movements in the variables. If the different variables are correlated with each other then the eror terms in the reduced form model will also be correlated across equations.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Recursive VAR A recursive VAR constructs the error terms in each regression equation to be uncorrelated with the error in the preceding equations. This is done by judiciously including some contemporaneous values as regressors. Estimation of each equation by ordinary least squares produces residuals that are uncorrelated across equations. The results depend on the order of the variables: changing the order changes the VAR equations, coefficients, and residuals.
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Structural VAR A structural VAR uses economic theory to sort out the contemporaneous links among variables. Structural VARs require identyfing assumptions that allow correlations to be interpreted causally.