Orthogonalized VARs A. Recursively orthogonalized VAR B. Variance decomposition C. Historical decomposition D. Structural interpretation E. Generalized IRFs 1
A. Recursively orthogonalized Nonorthogonal IRF: VARs s Ey ts y t,y t1,...,y tp y nn t Column 1 Ey ts y 1t,y 2t,...,y nt,y t1,...,y tp y 1t e.g., already have data on y 2t,...,y nt and ask how a 1-unit change in y 1t affects forecast. 2
Could instead ask Ey ts y 1t,y t1,...,y tp y 1t e.g., don t have any data from period t except for y 1t and ask how 1-unit change in y 1t affects forecast. Knowing y 1t gives us information about y 2t,...,y nt if VAR forecast errors are correlated. 3
How calculate Method 1: local projection h s1 Ey ts y 1t,y t1,...,y tp y 1t? n1 Estimate by n OLS equations y ts c s h s1 y 1t H s2 y t1 H sp y tp1 u ts 4
Method 2: calculate answer implied by VAR y t y 1t,y 2t,...,y nt n1 x t 1,y t1 k1 k np 1 y t x t t E t t,y t2,...,y tp 5
Given parameters, observation of y 1t,y t1,...,y tp allows us to observe 1t y 1t 1 x t 6
Can calculate optimal forecast of it given 1t as E it 1t i1 11 1t 1 E t 1t 21 / 11 n1 / 11 1t a 1 1t 7
Ey t y 1t,y t1,...,y tp y 1t a 1 Ey ts y 1t,y t1,...,y tp y 1t Ey ts y t,y t1,...,y tp y t Ey t y 1t,y t1,...,y tp y 1t s a 1 8
T t1 nk y t x t T t1 x t x t 1 s t y t x t n1 T 1 T t1 nn t t â 1 1 21 / 11 n1 / 11 9
Could also do this using Cholesky factor: P P â 1 p 1 /p 11 p 1 column 1 of P (P lower triangular) p 11 row 1 col 1 element of P 10
s a 1 Ey ts y 1t,y t1,...,y tp y 1t is effect of one-unit increase in y 1t or 1t on forecast s p 1 is effect of one-standard-deviation increase in 1t on forecast. 11
s a 1 shows IRF to shock in observed units s p 1 shows IRF to shock of typical size Plots will look identical just with different units on vertical axis s p 1 s a 1 p 11 12
Recursively orthgonalized VAR estimated 1954-2007 1.0 0.8 0.6 0.4 0.2 0.0-0.2 0.12 0.10 0.08 0.06 0.04 0.02 0.00-0.02-0.04 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 Response of GDP to GDP alone (VAR) 0 5 10 15 Response of inflation to GDP alone (VAR) 0 5 10 15 Response of fed funds to GDP alone (VAR) 0 5 10 15 13
Recursively orthgonalized local projection estimated 1954-2007 1.0 Response of GDP to GDP alone (local projection) 0.8 0.6 0.4 0.2 0.0-0.2 0.125 0.100 0.075 0.050 0.025 0.000-0.025-0.050 0.250 0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 5 10 15 20 Response of inflation to GDP alone (local projection) 5 10 15 20 Response of fed funds to GDP alone (local projection) 5 10 15 20 14
Suppose next that we ve observed y 1t and y 2t but not y 3t,y 4t,...,y nt. What is effect on forecast of changing y 2t? Ey t y 1t,y 2t,y t1,...,y tp y 2t a 2 p 2 /p 22 p 2 column 2 of P p 22 row 2 col 2 element of P first element of a 2 is zero 15
Ey ts y 1t,y 2t,y t1,...,y tp y 2t s a 2 Ey ts y t,y t1,...,y tp y t Ey t y 1t,y 2t,y t1,...,y tp y 2t 16
-0.00-0.05-0.10-0.15-0.20-0.25-0.30 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 Response of GDP to inflation given GDP (VAR) 0 5 10 15 Response of inflation to inflation given GDP (VAR) 0 5 10 15 Response of fed funds to inflation given GDP (VAR) 0 5 10 15 17
n n matrix of recursively orthogonalized shocks: s P or s A 1 0 0 A a 21 1 0 a n1 a n2 1 PdiagP 1 Note last col of s A is identical to last col of s 18
We have broken down the news arriving in period t into n separate uncorrelated components 1t y 1t 1 x t news about y 1t u 2t 2t a 21 1t news about y 2t not already revealed by y 1t u nt nt E nt 1t,..., n1,t news about y nt not already revealed by y 1t,...,y n1,t 19
Simple way to summarize these components: v t P 1 t n1 p 11 0 0 p 21 p 22 0 p n1 p n2 p nn 1t 2t nt Ev t v t P 1 E t t P 1 P 1 P 1 P 1 PP P 1 I n 20
v t is a linear combination of t whose elements are uncorrelated with each other v 1t p 11 1t rescaled error forecasting y 1t v 2t p 21 1t p 22 2t rescaled error forecasting 2t from 1t v nt p n1 1t p n2 2t p nn nt rescaled error forecasting nt from 1t,..., n1,t 21
B. Variance decomposition y ts ŷ ts t 0 ts 1 ts1 2 ts2 s1 t1 Ey ts ŷ ts t y ts ŷ ts t s1 m0 m m t Pv t p 1 v 1t p 2 v 2t p n v nt Contribution of v i,t1,v i,t2,...,v i,ts to forecast error: 0 p i v i,ts 1 p i v i,ts1 s1 p i v i,t1 22
Ey ts ŷ ts t y ts ŷ ts t s1 m0 m p 1 p 1 m s1 m0 m p n p n m First term: amount by which could reduce MSE if we knew the values of 1,t1,..., 1,ts Second term: amount by which we could reduce MSE if we knew the values of u 2,t1,...,u 2,ts 23
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C. Historical decomposition y ts ŷ ts t s1 m0 m tsm ŷ ts t s1 m0 m p 1 v 1,tsm p n v n,tsm Can decompose the observed value for any variable at any date into component that could have been predicted as of some earlier date plus innovations in individual v i,tm since then. 27
Historical Decomposition of GDP 20 15 10 5 0-5 -10 20 15 10 5 0-5 -10 20 15 10 5 0-5 -10 Effect of GDP 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Effect of inflation 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Effect of fed funds 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 28
D. Structural interpretation Suppose we hypothesized the following structural model for the behavior of the Fed: i t 3 y y t t b 31 y t1 b 3p y tp u 3t i t fed funds rate y t GDP growth rate t inflation rate y, coefficients in Taylor Rule b 3m allow for inertia in monetary policy u 3t serially uncorrelated shock to monetary policy deviation from Fed s usual rule, uncorrelated with y t1,...,y tp by definition Would like to know y ts /u 3t 29
Suppose I also thought there was a Phillips Curve of the form t 2 y t b 21 slope of Phillips Curve b 2m allow for inertia in PC y t1 b 2p y tp u 2t u 2t unpredictable shock to PC u 2t uncorrelated with y t1,...,y tp by definition u 2t also assumed to be uncorrelated with u 3t (assumption that monetary policy shocks take more than one period to affect inflation) 30
Model equilibrium output as y t 1 b 11 y t1 b 1p y tp u 1t u 1t error forecasting GDP one period ahead u 1t uncorrelated with y t1,...,y tp by definition u 1t also assumed to be uncorrelated with u 2t, u 3t (assumption that PC and monetary shocks take more than one period to affect output) 31
i t 3 y y t t b 31 y t1 b 3p y tp u 3t Above assumptions mean u 3t uncorrelated with y t and t. could estimate by OLS y and are same as step 0 Jordá projection y and are same as â 31 and â 32 32
t 2 y t b 21 y t1 b 2p y tp u 2t Above assumptions mean u 2t uncorrelated with y t. could estimate by OLS is same as step 0 Jordá projection is same as â 21 33
Conclusion: under above assumptions with A PdiagP 1 u t A 1 t u 1t 1t The error I make forecasting y 1t given y t1, y t2,...,y tp is the shock to equilibrium output. 34
The error I make forecasting y 2t given y 1t, y t1, y t2,...,y tp is the shock to PC. The error I make forecasting y 3t given y 1t, y 2t, y t1,y t2,...,y tp is the shock to monetary policy. 35
Ey ts y 1t,y 2t,y 3t,y t1,...,y tp y 3t y ts u 3t Recursively orthogonalized VAR gives the dynamic effects of monetary policy. 36
Orthogonal Cholesky IRF with 95% confidence intervals (54-07) GDP inflation 4 4 4 fed funds 3 3 3 2 2 2 GDP 1 1 1 0 0 0-1 -1-1 -2 0 5 10 15-2 0 5 10 15-2 0 5 10 15 Responses of inflation 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4 0 5 10 15 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4 0 5 10 15 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4 0 5 10 15 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 fed funds 0.4 0.2 0.4 0.2 0.4 0.2 0.0 0.0 0.0-0.2-0.2-0.2-0.4 0 5 10 15 GDP -0.4 0 5 10 15 inflation -0.4 0 5 10 15 fed funds 37
A monetary contraction (higher fed funds rate) is followed by slower GDP growth 2-3 quarters later But unanticipated monetary policy shocks account for only 10% of variance of output Most of variation in fed funds rate comes from predictable response of monetary policy to output and inflation A monetary contraction is followed by higher inflation (known as price puzzle ) 38
Assumption-free statement of price puzzle: if you tell me that fed funds rate is higher than I would have predicted given current output, inflation, and lags, then I will revise my expectation of future inflation up. Natural interpretation: Fed raised funds rate because it anticipated future inflation. Our 3-variable equation is too simplistic a description of Fed 39
Popular fix for price puzzle: Add other variables that better capture information about future inflation (such as commodity prices) to Fed policy equation 40
Christiano, Eichenbaum, Evans (1996) y 1t log of real GDP y 2t log of GDP deflator y 3t index of sensitive commodity prices y 4t fed funds rate y 5t nonborrowed reserves y 6t total reserves y 7t one of a set of macro variables 41
Structural model: B 0 y t Bx t u t x t 1,y t1,y t2,...,y tp u t vector of structural shocks Eu t u t D (diagonal) 42
1 0 0 0 0 0 0 x 1 0 0 0 0 0 B 0 x x 1 0 0 0 0 x x x 1 0 0 0 x x x x 1 0 0 x x x x x 1 0 x x x x x x 1 Variable 4 is fed funds rate, equation 4 is monetary policy equation. 43
Note that Ey ts y 1t,y 2t,y 3t,y 4t,y t1,y t2,...,y tp y 4t Ey ts y 2t,y 1t,y 3t,y 4t,y t1,y t2,...,y tp y 4t Will have the identical answer for effect of variable 4 any way we order variables 1-3 and any way we order variables 5-7. Jordá estimate identical if reorder (keeping 4 in place). 44
If all we care about is effect of monetary policy, we only need to assume block-recursive x x x 0 0 0 0 x x x 0 0 0 0 B 0 x x x 0 0 0 0 x x x 1 0 0 0 x x x x x x x x x x x x x x x x x x x x x 45
67% confidence bands 46
E. Generalized IRFs If we put fed funds fourth, estimated effect of monetary policy does not depend on how we order variables 1-3. But if we switch fed funds from 4 to 3, results could change 47
Pesaran and Shin (1998): generalized impulse-response function Put variable #1 first to find effect of variable 1 Put variable #2 first to find effect of variable 2 Put variable #n first to find effect of variable n 48
GIRF: for every i, calculate Ey ts y it,y t1,...,y tp y it 49
Conclusion: any IRF or GIRF is giving answer to a forecasting question. Best practice: describe forecasting question explicitly and explain the reason that question is interesting. 50