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Guided tour on VAR innovationresponse analysis

Introduction

The theory involvedin explained in my lecturenotes (VAR.PDF) on vector time series and innovation response analysis,but here I will review the main ideas, based on the seminal papers:

• Bernanke, B.S. (1986):"Alternative Explanations of the Money-Income Correlation", Carnegie-RochesterConference Series on Public Policy 25, 49-100
• Sims, C.A. (1980):"Macroeconomics and Reality", Econometrica 48, 1-48
• Sims, C.A. (1986):"Are Forecasting Models Usable for Policy Analysis?", Federal ReserveBank of Minneapolis Quarterly Review, 1-16

where

• X_t= (X_1,t, ..... ,X_k,t)' is avector time series of macroeconomic variables,
• c₀is a k-vector of intercept parameters,
• the C_jare k´kparameter matrices, and
• U_tis the error vector, which is assumed to be i.i.d. k-variate normallydistributed with expectation the zero vector, and variance matrix S.

where

is a matrix-valued lag polynomial, with L the lag operator: LX_t= X_t-1.

The process X_tis strictly stationarity if det[C(z)] has all its roots outsidethe complex unit circle. Then C(L) is invertible, i.e., thereexist k´kparameter matrices D_j, with D₀= I_k and å_j³0D_jD_j'a finite matrix, such that

Hence, the processX_thas a stationary MA(¥)representation:

where m= (å_j³0D_j)c₀.Note that E[X_t] = mand Var[X_t] = å_j³0D_jSD_j'.

Since the U_t'sare i.i.d. with E[U_t] = 0, it follows now thatfor m ³0:

The latter isthe basis for innovation response analysis, i.e., E[X_t+m|U_t]- E[X_t+m] is the net effect of the innovationU_ton the future values X_t+m of X_t.

Non-structural VAR innovationresponse analysis

where Dis a lower triangular matrix such that

Then e_tis i.i.d. N_k[0,I_k]. The componentse_1,t,..... ,e_k,t of e_t are uniquely associatedto the corresponding components of U_t. Consequently,we can now interpret e_1,t, ..... ,e_k,tas the actual innovations, and moreover we may consider them as sequentialpolicy shocks: at time t a shock e_1,t is imposed,and after then the next shock e_2,t is imposed, etc., up to the lastshocke_k,t. Then the response of X_tto a unit shock in e_j,t is:

where d_jis column j ofD,and thus the response of X_i,t to a unit shock in e_j,tis given by

where d_i,m'is row i of D_m.

Structural VAR innovation responseanalysis

where e_tis i.i.d. N_k[0,I_k]. The matrix Brepresents the contemporaneous relations between the components of X_t.This structural VAR(p) model is related to the non-structural VAR(p)model

by

Hence, c₀= B^-1a₀,C_j = B^-1A_jfor j = 1,..,p, andU_t = B^-1e_t.The latter reads as

Therefore, effectivelythe matrix B of structural parameters links the nonstructural innovationsU_tto the structural innovations e_t.

The main differencewith the non-structural approach is the way the variance matrix Sof U_t is decomposed, i.e., instead of writing U_t= De_twith D a lower triangularmatrix we now have U_t = B^-1e_t,hence S = (B^-1)(B^-1)'= (B'B)^-1, and thus

Given S,and taking into account the symmetry of S,the equality B'B = S^-1is a system of (k + k²)/2 nonlinear equationsin thek² elements of B. Therefore, in order tosolve this system, one has to set at least (k² - k)/2off diagonal elements of B to zeros, similarly to classical simultaneousequation systems. This is where economic theory comes into the picture:The zeros in B are exclusion restrictions prescribed by economictheory.

Note thateven if we reduce the system B'B = S^-1to (k + k²)/2 equations in (k + k²)/2unknowns, there is no guarantee that there exists a solution, because theequations involved are quadratic. But assuming that we have spread thezeros in B such that a solution exists, the structural innovationresponse of X_i,t to a unit shock in e_j,tis given by

where againd_i,m'is row i of D_m, and b_jis now column j of B^-1.

Estimation and inference

EasyReg Internationalestimates a structural VAR model in three steps. First, the non-structuralVAR is estimated by maximum likelihood. Next, given the estimated variancematrixS and the specificationof the matrix B, EasyReg will try to solve the nonlinear equationssystem B'B = S^-1analytically, and if this is not possible, it will minimize the maximumabsolute value of the elements of the matrix B'B -S^-1.The latter is a form of method of moments estimation. Finally, using thenon-structural parameter estimates and the solution of B as startingvalues, EasyReg re-estimates the parameters by maximum likelihood. Giventhese maximum likelihood estimates, the innovation responses and standarderror bands are computed in the same way as in the non-structural case.

Note that directmaximum likelihood estimation of the structural VAR model (as some othereconometric software packages do) is not advisable because the likelihoodfunction is highly nonlinear in the non-zero elements of B, andtherefore you may get stuck in a local maximum.

VAR innovation response analysiswith EasyReg

The data

The data are taken from the EasyReg database, namely the following quarterly data for the US:

• federal funds rate
• M2 (= money)
• cons.price index (= consumer price index)
• nominal GDP

Since VAR innovation response analysis assumes normal errors of the VAR, and the variables involved are all positive valued, transform them by taking logs, using the 'Transform variables' option via Menu > Input:

• LN[federal funds rate]
• LN[M2]
• LN[cons.price index]
• LN[nominal GDP]

The last three variables are likely nonstationary. Therefore, take them in firstdifferences, using the option Menu > Input > Transform variables > Time series transformations.Then select the variables in X_t = (X_1,t,X_2,t,X_3,t,X_4,t)'in the following order:

• X_1,t = LN[federal funds rate]
• X_2,t = DIF1[LN[M2]]
• X_3,t = DIF1[LN[cons.price index]]
• X_4,t = DIF1[LN[nominal GDP]]

for t = 1,2,...,142, from quarter 1959.2 (t = 1) to quarter 1994.3 (t = 142).

The transformed data are also available in Excel 97 workbook formatVAR.XLS and CSV (US number setting) format VAR.CSV.

VAR model specification

OpenMenu > Multiple equations models > VAR innovation response analysis, select the variablesin the VAR in the above order, and click "Selection OK". Then the followingwindow appears.

I will not select a subset of observations. Thus click "No" and then "Continue":

This window is only for your information. Click "Continue":

In the introduction above I have discussed only a VAR model with interceptparameter vector c₀. However, if X_t (= z(t) in EasyReg) is stationary around a deterministic function of time,i.e., X_t - E[X_t] isstationary, we can still conduct VAR innovation response analysis. The VAR(p) model then takes the form:

where d(t) is a vector of deterministic functions of timet, and C₀ (= B in EasyReg) is the corresponding matrix of coefficients. Note that

The default specification of d(t) is d(t) = 1. Other options are d(t) = (1,t)', seasonaldummy variables (only in the case of seasonal data, of course), and Chebishevtime polynomials. The latter can be used to capture nonlinear time trends. TheChebishev time polynomials have been used in my paper

• Bierens, H.J. (2000), "Nonparametric Nonlinear Co-Trending Analysis, with an Application to Inflation and Interest in the U.S.", Journal of Business & Economic Statistics 18, 323-337,

Let us conduct non-structural VAR analysis first. After you are done with that, you will return to this window so that you can conduct structural VAR analysis. The sameapplies the other way around.

Non-structural VAR innovationresponse analysis

You have to choose the number of periods ahead (the innovation response horizon) for which you want to display the innovation responses. The minimum value is 10. HereI have chosen 40, so that the innovation responses are displayed over a period of 10 years.

Click "Start" to compute the innovation responses together with their standard errors.

You will have the option to write the numerical values of the innovation responses withtheir standard errors to the output file OUTPUT.TXT, but in general there is no purposein doing this. Thus, click "Continue".

The contribution of the innovation in variable i to the h-step ahead forecast error of variable j is the sum of the squared responses of variable j to a unit shock in the innovation of variable i. In this window the relative contributions of each variable i to the forecast error variance of variable j are presented. This procedure is known as "variance decomposition".

Click "Continue".

The solid curve is the response of the inflation rate to a unit shock in the innovation of the log of the federal funds rate. You see that in the first three quarters the response is significantly positive, as the two-times standard error band is above thehorizontal axis, and then dies out quickly to zero. This phenomenon is known as the price puzzle. Since the FED raises the federal funds rate in order to curtail inflation,one would expect that the response of inflation to a unit shock in the innovation of thefederal funds rate is negative rather than positive.

This picture is the response of the money growth rate to a unit shock in the innovation of the log of the federal funds rate. This pattern is what you would expect: Ifborrowing money is made more expensive, the demand for money will decrease.

When you click "Done", EasyReg will jump back to the the following window.

Structural VAR innovationresponse analysis

The matrice A in the description of the structural VAR is a matrix of structuralparameters with ones as diagonal elements, and the matrix C is a diagonal matrix. Thesematrices are related to the matrix B in the structural model

used in the previous discussion of structural VAR analysis by the equality

Recall that the structural model relates the nonstructural VAR errors U_tto the structural VAR errors e_t by the relationships

In the following four windows the non-zero elements of each row of B arespecified.

The first element on row 1 of B is always non-zero. The remaining three non-zeroelements are determined by double-clicking the correponding components of U_t. In this example I will choose the first row of B to be (b(1),0,0,0), hence I will not double-click anything, but just click "Equation OK".

I will choose (b(2),b(3),0,0) as the second row of B.The second element b(3) of row 2 is always non-zero, and b(2) is the coefficient ofthe non-structural innovation in the log of the federal funds rate. Thus, double-clickthe latter, and click "Equation OK".

When you click "Equation OK", (0,b(4),b(5),b(6)) will be chosen as the third rowof B.

Finally, when you click "Equation OK", (0,0,0,b(7)) will be chosen as the fourth rowof B. Then the matrix B is:

b(1) 0    0    0   b(2) b(3) 0    0   0    b(4) b(5) b(6)0    0    0    b(7)

Note that this specification is not intended to be a serious economic specification,but is chosen merely as an example.

EasyReg will now try to solve the equation system B'B = S_n^-1 analytically:

What you see here is equations in the systemB'B = S_n^-1. The equation indicated by (*) does not involve parameters, because the system is over-identified:there are more equations than unknown b(.).

The equation in the previous window indicated by (*) is a testable hypothesis. In this examplethe null hypothesis that the equation holds is rejected, hence we should respecifythe matrix B. However, since this is only a demonstration of structural VAR analysis, Iwill continue.

Note that the non-zero parameters in B are solved analytically.

Click "Method 2". Then the maximum absolute value of the elements of the matrixB'B - S_n^-1(called MAD: The Maximum Absolute Deviation of the elements of B'B from the corresponding elements of S_n^-1)will be minimized, using the simplex method of Nelder and Mead.

Next, switch to Option 1. Then the non-zero elements b(1),...,b(7) of B will be estimated by maximum likelihood, starting from the previous solutions.

When you click "Done with Simplex iteration" the following window appears.

From this point on structural VAR analysis is similar to the non-structural case.

Granger-causality

Introduction

and X2,t does not Granger-cause X1,t, then the matrices C_j for j = 1,...,p are lower-triangular, becausethe coefficients of the lagged X2,t in the VAR are zero.

Granger-causality testing in practice

a  0b  c

1  01  1