guided tour on var innovation response analysis

8/2/2019 Guided Tour on VAR Innovation Response Analysis

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Guided tour on VAR innovation response analysis

Introduction

In this guided tour I will explain how to conduct vector autoregression (VAR) innovation responseanalysis, including structural vector autoregression innovation response analysis.

The theory involved in explained in my lecture notes on vector time series and innovation responseanalysis, 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-Rochester Conference Series on Public Policy25, 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 Reserve

Bank of Minneapolis Quarterly Review, 1-16

The starting point is a k-variate Gaussian VAR(p) model:

Xt = c0 + C1Xt-1 + ..... + CpXt-p + Ut ,

where

Xt = (X1,t, ..... ,Xk,t)' is a vector time series of macroeconomic variables, c0 is a k-vector of intercept parameters, the Cjare k kparameter matrices, and Ut is the error vector, which is assumed to be i.i.d. k-variate normally distributed with

expectation the zero vector, and variance matrix .

The VAR(p) model involved can be written as

C(L)Xt = c0 + Ut,

where

C(L) = Ik - C1L - ..... - CpLp

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

The processXt is strictly stationarity if det[C(z)] has all its roots outside the complex unit circle. Then

C(L) is invertible, i.e., there exist k kparameter matrices Dj, with D0 = Ikand j0 DjDj' a finite matrix,

such that

C(L)-1 = j 0DjL j.

Hence, the processXt has a stationary MA() representation:

Xt = + j 0DjUt-j,

where = (j 0Dj)c0. Note that E[Xt] = and Var[Xt] = j 0DjDj'.
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Since the Ut's are i.i.d. with E[Ut] = 0, it follows now that form 0:

E[Xt+m|Ut] - E[Xt+m] = DmUt .

The latter is the basis for innovation response analysis, i.e., E[Xt+m|Ut] - E[Xt+m] is the net effect of theinnovation Uton the future valuesXt+m ofXt.

Non-structural VAR innovation response analysis

Sims (1980) proposes to interpret the components of the innovation vectorUt = (U1,t, ..... ,Uk,t)' aspolicy shocks. The problem however is that the components U1,t, ..... ,Uk,tofUt are not independent,so that it is unrealistic to assume that a shock in one of these components does not affect the othercomponents. In order to solve this problem, Sims (1980) proposes to rewrite Ut as

Ut = et,

where is a lower triangular matrix such that

= '.

Then et is i.i.d. Nk[0,Ik]. The components e1,t, ..... ,ek,tofetare uniquely associated to thecorresponding components ofUt. Consequently, we can now interpret e1,t, ..... ,ek,tas the actualinnovations, and moreover we may consider them as sequential policy shocks: at time ta shock e1,tis imposed, and after then the next shock e2,t is imposed, etc., up to the last shock ek,t. Then theresponse ofXt to a unit shock in ej,t is:

E[Xt+m|ej,t= 1] - E[Xt+m] = Dmj form = 0,1,2,3,.....,

where j is columnjof, and thus the response ofXi,t to a unit shock in ej,t is given by

ri,j(m) = E[Xi,t+m|ej,t= 1] - E[Xi,t+m] = di,m'j form = 0,1,2,3,.....,

where di,m' is row iofDm.

Structural VAR innovation response analysis

A disadvantage of this approach is that economic theory plays a limited role. The only role ofeconomic theory is to determine the orderin which the innovation shocks are imposed. This ordercorresponds to the order in which the macroeconomic variables inXtare arranged. Therefore,Bernanke (1986) and Sims (1986) propose to set up the VAR(p) model as a system of simultaneous

equations:

B.Xt = a0 +A1Xt-1 + ..... +ApXt-p + et,

where et is i.i.d. Nk[0,Ik]. The matrix B represents the contemporaneous relations between thecomponents ofXt. This structural VAR(p) model is related to the non-structural VAR(p) model

Xt = c0 + C1Xt-1 + ..... + CpXt-p + Ut ,

by


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Xt = B-1a0 + B

-1A1Xt-1 + ..... + B-1ApXt-p + B

-1et,

.

Hence, c0 = B-1a0, Cj= B

-1Aj forj= 1,..,p, and Ut = B-1et. The latter reads as

B.Ut = et.

Therefore, effectively the matrix B of structural parameters links the nonstructural innovations Ut tothe structural innovations et.

The main difference with the non-structural approach is the way the variance matrix ofUt isdecomposed, i.e., instead of writing Ut= et with a lower triangular matrix we now have Ut= B-1et,hence = (B-1)(B-1)' = (B'B)-1, and thus

B'B = -1.

Given , and taking into account the symmetry of, the equality B'B = -1 is a system of (k+ k2)/2nonlinear equations in the k2 elements ofB. Therefore, in order to solve this system, one has to setat least (k2 - k)/2 off diagonal elements ofB to zeros, similarly to classical simultaneous equationsystems. This is where economic theory comes into the picture: The zeros in B are exclusionrestrictions prescribed by economic theory.

Note that even if we reduce the system B'B = -1 to (k+ k2)/2 equations in (k+ k2)/2 unknowns, thereis no guarantee that there exists a solution, because the equations involved are quadratic. Butassuming that we have spread the zeros in B such that a solution exists, the structural innovationresponse ofXi,t to a unit shock in ej,t is given by

ri,j(m) = E[Xi,t+m|ej,t= 1] - E[Xi,t+m] = di,m'j form = 0,1,2,3,.....,

where again di,m' is row iofDm, and j is now columnjofB-1.

Estimation and inference

The non-structural VAR(p) model can be estimated by maximum likelihood. Given the maximum

likelihood estimators of the coefficient matrices Cj forj= 1,..,p, and the variance matrix , and thejoint normal asymptotic distribution of the parameters therein, it is possible to derive asymptoticstandard error of each innovation response ri,j(m). EasyReg International endows the estimatedinnovation responses involved with one and two times standard error bands based on the asymptoticnormal distribution of each estimated innovation response around the true innovation response, in

order to determine whether the latter is significantly different from zero. The two-times standard errorband corresponds approximately to the pointwise 95% confidence interval of each innovationresponse. The one-time standard error band corresponds approximately to the pointwise 70%confidence interval.

EasyReg International estimates a structural VAR model in three steps. First, the non-structural VAR

is estimated by maximum likelihood. Next, given the estimated variance matrix and thespecification of the matrix B, EasyReg will try to solve the nonlinear equations system B'B = -1analytically, and if this is not possible, it will minimize the maximum absolute value of the elements of

the matrix B'B - -1. The latter is a form of method of moments estimation. Finally, using the non-structural parameter estimates and the solution ofB as starting values, EasyReg re-estimates the


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parameters by maximum likelihood. Given these maximum likelihood estimates, the innovationresponses and standard error bands are computed in the same way as in the non-structural case.

Note that direct maximum likelihood estimation of the structural VAR model (as some othereconometric software packages do) is not advisable because the likelihood function is highlynonlinear in the non-zero elements ofB, and therefore you may get stuck in a local maximum.

VAR innovation response analysis with 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 variablesinvolved are all positive valued, transform them by taking logs, using the 'Transform variables' optionvia 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 first differences, using theoption Menu > Input > Transform variables > Time series transformations. Then select the variablesinXt= (X1,t,X2,t,X3,t,X4,t)' in the following order:

X1,t = LN[federal funds rate] X2,t = DIF1[LN[M2]] X3,t = DIF1[LN[cons.price index]] X4,t = DIF1[LN[nominal GDP]]

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

VAR model specification

Open Menu > Multiple equations models > VAR innovation response analysis, select the variables inthe VAR in the above order, and click "Selection OK". Then the following window appears.
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I will not select a subset of observations. Thus click "No" and then "Continue":


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


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In the introduction above I have discussed only a VAR model with intercept parameter vectorc0.However, ifXt (= z(t) in EasyReg) is stationary around a deterministic function of time, i.e.,Xt - E[Xt]is stationary, we can still conduct VAR innovation response analysis. The VAR(p) model then takes

the form:

Xt = C0d(t) + C1Xt-1 + ..... + CpXt-p + Ut ,

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

E[Xt] = C(L)-1C0d(t).

The default specification ofd(t) is d(t) = 1. Other options are d(t) = (1,t)', seasonal dummy variables(only in the case of seasonal data, of course), and Chebishev time polynomials. The latter can be

used to capture nonlinear time trends. The Chebishev time polynomials have been used in my paper

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

which can be downloaded from my web site. Your have to read this paper in order to learn how andwhether to use this option. I will not discuss it here.

It is logically impossible that the (transformed) data contain a linear time trend, because that wouldimply that the expectation of some of the variables involved converge to plus or minus infinity.
http://econ.la.psu.edu/~hbierens/PAPERS.HTMhttp://econ.la.psu.edu/~hbierens/PAPERS.HTM


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Since the data are quarterly data, and because it is not clear whether the data are seasonallyadjusted, I recommend to include in first instance seasonal dummy variables, next to the constant 1.Whether seasonal dummy variables are needed can be tested. Thus click "Seasonal dummies":

Note that only three quarterly dummies are included next to the contant 1, because the four seasonadummies add up to 1 and would therefore be perfectly multicollinear with 1.

Now click "d(t) is OK":


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There are various ways to determine the orderp of the VAR(p) model

Xt= C0d(t) + C1Xt-1 + ..... + CpXt-p + Ut.

Via this window you can determinep automatically by one of three information criteria:

Akaike = ln[det()] + 2[1/(n-p)].(m+p.k2) Hannan-Quinn = ln[det()] + 2.[ln(ln((n-p))) / (n-p)].(m+p.k2) Schwarz = ln[det()] + 2 .[ln(n-p)/(n-p)].(m+p.k2)

where m is the number of parameters in the matrix C0, kis the dimension ofXt, n is the length of the

vector time series involved, and is the estimated variance matrix ofUt. The quantity ln[det()] is ameasure of the fit of the model, which is penalized by a function of the sample size n and the numbeof parameters, similarly to the adjusted R2 in OLS. Starting from an upper bound ofp (8 in this case),

the estimatedp corresponds to the minimum value of these criteria. The Akaike criterion is the mostconservative of the three. This criterion may give too large ap. The other two criteria are consistent,i.e., the estimatedp is equal to the truep with probability converging to 1 ifn converges to infinity.

Another way to determinep is through testing the joint significance of the parameters in the matricesCj. I will consider this later.

I have chosen 8 as the upper bound ofp. Now click "p OK":


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In view of these results, I have chosenp = 2.


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


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This window enables you to impose Granger-causality restrictions on the VAR. See my lecture noteson vector time series and innovation response analysis. Granger-causality will be discussed belowby a separate example. Thus, click "Continue".
http://econ.la.psu.edu/~hbierens/LECNOTES.HTMhttp://econ.la.psu.edu/~hbierens/LECNOTES.HTM


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Since each equation in the VAR model

Xt= C0d(t) + C1Xt-1 + ..... + CpXt-p + Ut, t=p+1,...,n,

has the same right-hand side variables, and there are no parameter restrictions imposed, themaximum likelihood estimators of the parameters in the matrices Cj forj= 0,1,...,p are the same asthe OLS estimators. Thus, click "OLS estimation" first. After EasyReg is done with OLS estimation,the button "FIML estimation" will be enabled. FIML stands forFull Information Maximum Likelihood.

Given the vectors Rtof OLS residuals, the maximum likelihood estimator of the variance matrix ofthe VAR error vectorUt is

n = (n-p)-1p+1 t nRtRt',

which is decomposed as

n = nn',

where n (= L in EasyReg) is a lower triangular matrix. We need FIML in order to compute thevariance matrix of the non-zero elements ofn, which in its turn is needed to compute the standarderror bands of the innovation responses. Thus, click "FIML estimation" when it becomes enabled.Then the following window appears.


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The variables L(.,.) are the non-zero elements of the lower triangular matrix n (= L).

You can now test the joint significance of any subset of parameters of the VAR. First, I have tested

the joint significance of the parameters of the seasonal dummy variables: Double-click the seasonaldummies (note that each of the four equations contains three seasonal dummy variables, so that youhave to double-click all 12 seasonal dummy variables), and then click "Test joint significance":


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The test involved is the Wald test of the null hypothesis that all the coefficients of the seasonal

dummy variables are zero. The asymptotic null distribution is 2 with 12 degrees of freedom. Clearly,the null hypothesis involved is not rejected at any conventional significance level.

In view of this result, we may now respecify and re-estimate the VAR without seasonal dummyvariables. However, I have not done that.

If you click "Again", you can conduct more tests.


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Next, I have tested whether the VAR orderp can be reduced fromp = 2 top = 1, by testing whetherthe 16 coefficients corresponding to the variables with lag 2 (i.e., the elements of the matrix C2) are

jointly zero.


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Clearly, the null hypothesis involved is rejected. Therefore I will adopt the initial choice p = 2.

Note that this procedure is an alternative way to determine the VAR orderp. Given an initial value of

p for which you are convinced that the actual VAR order does not exceed this initial value, testwhether the elements of the matrices Cj forj= q,...,p (with q 1) in the VAR model

Xt = C0d(t) + C1Xt-1 + ..... + CpXt-p + Ut

are jointly zero, and take as the newp the largest value ofq for which this hypothesis is rejected.

Now click "Continue". Then the following windows appears.


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

Non-structural VAR innovation response analysis


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

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


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You will have the option to write the numerical values of the innovation responses with their standarderrors to the output file OUTPUT.TXT, but in general there is no purpose in doing this. Thus, click"Continue".


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The contribution of the innovation in variable ito the h-step ahead forecast error of variablejis thesum of the squared responses of variablejto a unit shock in the innovation of variable i. In thiswindow the relative contributions of each variable ito the forecast error variance of variablejare

presented. This procedure is known as "variance decomposition".

Click "Continue".


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The solid curve is the response of the inflation rate to a unit shock in the innovation of the log of thefederal funds rate. You see that in the first three quarters the response is significantly positive, as thetwo-times standard error band is above the horizontal axis, and then dies out quickly to zero. This

phenomenon is known as theprice puzzle. Since the FED raises the federal funds rate in order tocurtail inflation, one would expect that the response of inflation to a unit shock in the innovation ofthe federal funds rate is negative rather than positive.


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

When you click "Done", EasyReg will jump back to the first window of module VAR (where you selectthe variables).

Structural VAR innovation response analysis

In this demonstration of structural VAR innovation response analysis I will use the same fourvariables, in a VAR(2) model. However, I have now excluded the seasonal dummy variables,because their coefficients were not jointly significant. The specification and estimation procedure issimilar to the previous case, until you reach the following window:


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The matrix A in the description of the structural VAR is a matrix of structural parameters with 1 asdiagonal elements, and the matrix C is a diagonal matrix. These matrices are related to the matrix Bin the structural model

B.Xt= a0 +A1Xt-1 + ..... +ApXt-p + et,

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

B = C-1A.

Recall that the structural model relates the nonstructural VAR errors Ut to the structural VAR errorsetby the relationships

BUt = et.

In the following four windows the non-zero elements of each row ofB are specified.


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The first element on row 1 ofB is always non-zero. The remaining three non-zero elements aredetermined by double-clicking the correponding components ofUt. In this example I will choose thefirst row ofB to be (b(1),0,0,0), hence I will not double-click anything, but just click "Equation OK".


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When you click "Equation OK", (0,b(2),b(3),b(4)) will be chosen as the second row ofB.


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


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

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

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

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


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What you see here are the equations of the system B'B = n-1. The equations indicated by (*) do notinvolve parameters, because the system is over-identified: there are more equations than unknownb(.)'s.

The non-zero parameters in B are solved analytically. If EasyReg cannot solve the systemanalytically, then you will likely have an identification problem.

The equations in the previous window indicated by (*) are testable hypotheses. In this example thenull hypothesis that these equations hold is rejected, hence we should respecify the matrix B.However, since this is only a demonstration of structural VAR analysis, I will continue.


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Click "Method 2". Then the log-likelihood will be maximized using the simplex method of Nelder andMead, starting from the non-structural parameter estimates and the solutions of the b(.)'s.


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Restart the simplex iteration until the log-likelihood does not change anymore:

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


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The rest of the structural VAR innovation response analysis is now similar to the non-structural case.

Granger-causality

Introduction

Consider a bivariate time series processXt =(X1,t,X2,t)'. As is well-known (or should be well-known),the best one-step ahead forecast of each componentXi,tofXt is the conditional expectation

E[Xi,t |Xt-1,Xt-2,Xt-3,.....],

i.e., of all functions of the past ofXt, say gi(Xt-1,Xt-2,Xt-3,.....), this conditional expectation yields thesmallest mean-square forecast error:

E{Xi,t- E[Xi,t |Xt-1,Xt-2,Xt-3,.....]}2 E{Xi,t- gi(Xt-1,Xt-2,Xt-3,.....)}2.

Now suppose that

E[X1,t |Xt-1,Xt-2,Xt-3,.....] = E[X1,t |X1,t-1,X1,t-2,X1,t-3,.....].

Then the past of the processX2,t does not contain information that can be used to improve theforecast ofX1,t. If so, it is said thatX2,tdoes not Granger-causeX1,t (called after Clive Granger atUCSD who introduced this causality concept).


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IfXt is a VAR(p) process:

Xt = c0 + C1Xt-1 + ..... + CpXt-p + Ut ,

andX2,t does not Granger-causeX1,t, then the matrices Cj forj= 1,...,p are lower-triangular, becausethe coefficients of the laggedX2,t in the VAR are zero.

Granger-causality testing in practice

Retrieve two annual time series from the EasyReg database, namely LN[nominal GDP], which is thelog of nominal GPD of the US, and LN[Income Sweden], which is the log of national income ofSweden. Then use the transformation option (via Menu > Input > Transform variables > Time seriestransformations) to transform these time series in first differences, in order to make them stationary:

1. DIF1[LN[nominal GDP]]2. DIF1[LN[Income Sweden]]

These transformed time series are now growth rates. Plot them (via Menu > Data analysis > Plottime series):

You see that these time series have quite a few similar patterns. The reason is that due to the size ofthe US economy the US GDP growth rate may be considered as a proxi for the world economicgrowth rate. Sweden is a small country and its economic performance heavily depends on exports.Therefore, the US GDP growth rate will affect the Swedish national income growth rate, but not theother way around. In other words, one may expect that DIF1[LN[Income Sweden]] does not Grangercause DIF1[LN[nominal GDP]]. To test this, select these variables in the above order in a VAR:
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It is clear from the plot that there is no time trend in these series, and you see in this window that theaverage growth rates (= sample means) are non-zero. Therefore, include only intercepts in the VAR.Thus, click "d(t) is OK":


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I have chosenp = 6 as the initial value ofp. The three information criteria all indicate that the actualvalue isp = 1. Therefore, I have chosenp = 1.


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The coefficient of one year lagged DIF1[LN[Income Sweden]] in the equation for DIF1[LN[nominalGDP]] has t-value -0.17, and is therefore not significant. Consequently, the null hypothesis thatDIF1[LN[Income Sweden]] does not Granger-cause DIF1[LN[nominal GDP]] is not rejected at any

convential significance level.

VAR innovation response analysis under Granger-causality restrictions

In order to impose this restriction on the VAR(1), click "Respecify VAR", select the two variablesagain, and choosep = 1:


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The VAR(1) involved is now of the formXt= a0 +A1Xt-1 + Ut, whereXt = (X1,t,X2,t)' with

X1,t = DIF1[LN[nominal GDP]]

X2,t = DIF1[LN[Income Sweden]]

The Granger-causality restriction now amounts to specifying the matrixA1 as

a 0 b c

say, which corresponds to the pattern

1 0

1 1

Therefore, click "Column Up", and then "Change pattern":


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Click "Continue":


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Click "OLS estimation" in order to get initial estimates. Since there are parameter restrictionsimposed on the VAR, OLS is no longer efficient. Therefore, after OLS is done, the button "SURestimation" becomes enabled. SUR stands for Seemly Unrelated Regression. SUR estimation of the

restricted VAR(1) modelXt = a0 +A1Xt-1 + Ut involves the following steps:

1. Estimate the variance matrix ofUton the basis of the vector of OLS residuals. Denote thisvariance matrix estimate by n,0.

2. Maximize the likelihood function L(a0,A1,) to a0 andA1, with replaced by n,0.3. Re-estimate on the basis of the new estimates ofa0 andA1. Denote this estimate by n,1.

The new estimates ofa0 andA1 are efficient, in the sense that the limited normal distribution of the

parameters therein is the same as for the maximum likelihood estimators, but the estimatorn,1 maynot yet be efficient. Therefore, I recommend that you repeat the steps 2 and 3, each timejreplacing

n,j-1 with n,j, until these matrices converge:


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In each SUR estimation stepjthe matrix n,j is decomposed as as n,j= Ln,jLn,j', where Ln,j is a lowertriangular matrix, and the maximum absolute deviation of the non-zero elements ofLn,j from thecorresponding elements ofLn,j-1 is computed. If the latter gets small enough, continue with full

information maximum likelihood estimation. Thus, click "FIML estimation", and then "Continue":


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You can now conduct joint significance tests.

Click "Continue", choose "Non-structural VAR", and innovation response horizon = 10:


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Since DIF1[LN[Income Sweden]] does not Granger-cause DIF1[LN[nominal GDP]], the innovationresponses involved are all zero.


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On the other hand, a unit shock in the innovation of DIF1[LN[nominal GDP]] has a significant positiveimpact on DIF1[LN[Income Sweden]], at least in the first three years after the shock in the innovationof DIF1[LN[nominal GDP]].

VAR models with exogenous variables: VARX models

In principle it is possible to include exogenous variables in a VAR model, next to determinsiticvariables such as trends, and conduct innovation response analysis via EasyReg. How to do that isexplained in PDF file VARX.PDF.

This is the end of the guided tour on VAR innovation response analysis.
http://econ.la.psu.edu/~hbierens/EasyRegTours/VAR_Tourfiles/VARX.PDFhttp://econ.la.psu.edu/~hbierens/EasyRegTours/VAR_Tourfiles/VARX.PDF

guided tour on var innovation response analysis

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