IRFCONSTRAIN.RPF

IRFCONSTRAIN.RPF is an example of the use of the @IRFRESTRICT procedure to put restrictions at steps other than just impact and long-run. Note that this is different from what is done in SHUTDOWN.RPF, as the latter computes a sequence of shocks (not just impact shocks) to zero out responses, while @IRFRESTRICT computes a factorization so the shocks themselves produce the desired effects at steps higher than impact. As a result, you are limited to only a modest number of constraints: for an $N$ variable VAR, the model is just identified with $N(N-1)/2$ constraints.

This example is based upon the four variable short- and long-run model from Peersman(2005) with the long-run zero restrictions replaced with a one step out zero restriction. The impact shape is

\begin{equation} \begin{array}{*{20}c} \bullet & 0 & 0 & 0 \\ \bullet & \bullet & \bullet & 0 \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \end{array} \label{eq:irfconstrain_impact} \end{equation}

and one step out is

\begin{equation} \begin{array}{*{20}c} \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & 0 & 0 \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \end{array} \end{equation}

That satisfies the RRWZ conditions for a B model as it has 0-1-2-3 zeros in the columns (in some order).

After estimating the VAR, the following computes the responses to unit shocks in each variable.

impulse(model=peersman,factor=%identity(%nvar),steps=3,results=irfs)

and this pulls out from IRFS the matrix of impact and one-step-out responses:

compute step0=%xt(irfs,1)

compute step1=%xt(irfs,2)

The following sequence builds up the matrix of restrictions on the vec of the $\bf{B}$ matrix:

dec rect rr(0,0)

@IRFRestrict(irf=step0,variable=1,shock=2) rr

@IRFRestrict(irf=step0,variable=1,shock=3) rr

@IRFRestrict(irf=step0,variable=1,shock=4) rr

@IRFRestrict(irf=step0,variable=2,shock=4) rr

@IRFRestrict(irf=step1,variable=2,shock=3) rr

@IRFRestrict(irf=step1,variable=2,shock=4) rr

The first four of those will do the obvious: directly setting specific elements of B to zero as shown in $\eqref{irfconstrain_impact}$. The last two add columns to RR that will make the one step out response zero for shock 3 on variable 2 (output) and for shock 4, also on output. We can then use the %PERP function to convert

\begin{equation} {\bf{R}}vec({\bf{B}}) = 0 \end{equation}

into the form

\begin{equation} vec({\bf{B}}) = {{\bf{R}}^ \bot }\Theta \end{equation}

where $\Theta$ are the free parameters that we can use in a CVMODEL instruction.

compute rp=%perp(rr)

dec vect theta(%cols(rp))

This creates a FRML which maps the THETA vector back to the implied $\bf{B}$ matrix:

dec frml[rect] lrfrml

frml lrfrml = %vectorect(rp*theta,%nvar)

The THETA vector has no obvious interpretation, so it's not clear what a good set of guess values would be. What this does is to pick the guess values which comes closest to a Cholesky factor of the covariance matrix:

compute [vect] theta=%ginv(rp)*%vec(%decomp(%sigma))

and this estimates the model:

nonlin theta

cvmodel(b=lrfrml,factor=f,dmatrix=identity) %sigma

and this checks that the factor has the desired properties:

?"Impact responses with multi-step restrictions" ###.### f

?"Step one responses" ###.### step1*f

Full Program

open data peersman_data.dat

calendar(q) 1970:1

data(format=prn,nolabels,org=columns) 1970:01 2002:02 oilprice outputemu $

cpiemu rateemu outputus cpius rateus

set groil = 100.0*(log(oilprice/oilprice{1}))

set grout = 100.0*(log(outputus/outputus{1}))

set grp = 100.0*(log(cpius/cpius{1}))

set r = rateus

set trend = t

system(model=peersman)

variables groil grout grp r

lags 1 2 3

det constant trend

end(system)

estimate(sigma) 1979:5 *

* These need to be responses to unit shocks, hence the FACTOR=%IDENTITY(%NVAR)

impulse(model=peersman,factor=%identity(%nvar),steps=3,results=irfs)

* This will do constraints of the form

* Impact:

* . 0 0 0

* . . . 0

* . . . .

* First step:

* . . . .

* . . 0 0

* . . . .

compute step0=%xt(irfs,1)

compute step1=%xt(irfs,2)

***************************************************************************

dec rect rr(0,0)

@IRFRestrict(irf=step0,variable=1,shock=2) rr

@IRFRestrict(irf=step0,variable=1,shock=3) rr

@IRFRestrict(irf=step0,variable=1,shock=4) rr

@IRFRestrict(irf=step0,variable=2,shock=4) rr

@IRFRestrict(irf=step1,variable=2,shock=3) rr

@IRFRestrict(irf=step1,variable=2,shock=4) rr

compute rp=%perp(rr)

dec vect theta(%cols(rp))

dec frml[rect] lrfrml

frml lrfrml = %vectorect(rp*theta,%nvar)

* Get guess values as closest to a Cholesky factor

compute [vect] theta=%ginv(rp)*%vec(%chol(%sigma))

nonlin theta

cvmodel(b=lrfrml,factor=f,dmatrix=identity) %sigma

?"Impact responses with multi-step restrictions" ###.### f

?"Step one responses" ###.### step1*f

Output

VAR/System - Estimation by Least Squares

Quarterly Data From 1980:01 To 2002:02

Usable Observations 90

Dependent Variable GROIL

Mean of Dependent Variable -0.52882624

Std Error of Dependent Variable 14.63158931

Standard Error of Estimate 14.11004908

Sum of Squared Residuals 15131.104867

Durbin-Watson Statistic 2.0055

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. GROIL{1} 0.22534895 0.12337533 1.82653 0.07169704

2. GROIL{2} -0.43423891 0.12557276 -3.45807 0.00089464

3. GROIL{3} 0.15593820 0.12191368 1.27909 0.20475811

4. GROUT{1} 3.06894295 2.81275016 1.09108 0.27868356

5. GROUT{2} 1.23549417 2.82479878 0.43737 0.66308035

6. GROUT{3} -0.26920860 2.16748670 -0.12420 0.90148237

7. GRP{1} 1.96759913 7.16940892 0.27444 0.78448828

8. GRP{2} 9.91409078 7.76761918 1.27634 0.20572338

9. GRP{3} -6.72635085 7.27073613 -0.92513 0.35782835

10. R{1} -0.37745577 1.74372783 -0.21646 0.82920541

11. R{2} -1.01071543 2.20012084 -0.45939 0.64726485

12. R{3} 1.80175458 1.77402131 1.01563 0.31302662

13. Constant -25.39422369 18.73585331 -1.35538 0.17931064

14. TREND 0.16576962 0.12246401 1.35362 0.17987005

F-Tests, Dependent Variable GROIL

Variable F-Statistic Signif

*******************************************************

GROIL 4.8777 0.0037250

GROUT 0.5027 0.6815454

GRP 0.8185 0.4876489

R 0.3736 0.7722730

Dependent Variable GROUT

Mean of Dependent Variable 0.7130123583

Std Error of Dependent Variable 0.7723798141

Standard Error of Estimate 0.5772537367

Sum of Squared Residuals 25.324862614

Durbin-Watson Statistic 1.9712

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. GROIL{1} -0.000681960 0.005047386 -0.13511 0.89288086

2. GROIL{2} 0.012234253 0.005137285 2.38146 0.01974751

3. GROIL{3} -0.004696148 0.004987589 -0.94157 0.34939783

4. GROUT{1} 0.298096997 0.115071927 2.59053 0.01148271

5. GROUT{2} 0.393804582 0.115564846 3.40765 0.00105066

6. GROUT{3} -0.033488259 0.088673667 -0.37766 0.70673783

7. GRP{1} -0.753534228 0.293306427 -2.56910 0.01215545

8. GRP{2} 0.254907073 0.317779702 0.80215 0.42496721

9. GRP{3} 0.537294293 0.297451807 1.80632 0.07482658

10. R{1} -0.015051005 0.071337343 -0.21098 0.83346481

11. R{2} -0.407076868 0.090008757 -4.52264 0.00002212

12. R{3} 0.323482595 0.072576674 4.45712 0.00002821

13. Constant 1.647074297 0.766499200 2.14883 0.03483234

14. TREND -0.008802387 0.005010104 -1.75693 0.08295783

F-Tests, Dependent Variable GROUT

Variable F-Statistic Signif

*******************************************************

GROIL 1.9954 0.1217798

GROUT 6.9731 0.0003311

GRP 2.7335 0.0494781

R 11.7950 0.0000020

Dependent Variable GRP

Mean of Dependent Variable 0.8480124029

Std Error of Dependent Variable 0.5459623590

Standard Error of Estimate 0.2357486026

Sum of Squared Residuals 4.2238826762

Durbin-Watson Statistic 1.8792

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. GROIL{1} 0.006469164 0.002061337 3.13833 0.00241867

2. GROIL{2} -0.004793438 0.002098051 -2.28471 0.02512095

3. GROIL{3} 0.002264989 0.002036916 1.11197 0.26965622

4. GROUT{1} -0.016278566 0.046995012 -0.34639 0.73000739

5. GROUT{2} 0.002669069 0.047196318 0.05655 0.95505010

6. GROUT{3} -0.052445708 0.036214046 -1.44821 0.15167016

7. GRP{1} 0.359217167 0.119785418 2.99884 0.00366185

8. GRP{2} 0.201840033 0.129780227 1.55524 0.12404244

9. GRP{3} 0.182583963 0.121478379 1.50302 0.13697900

10. R{1} 0.050091575 0.029133945 1.71935 0.08962040

11. R{2} -0.050631770 0.036759292 -1.37739 0.17243568

12. R{3} -0.011385255 0.029640084 -0.38412 0.70196448

13. Constant 0.652570858 0.313035852 2.08465 0.04046008

14. TREND -0.003760847 0.002046111 -1.83805 0.06996315

F-Tests, Dependent Variable GRP

Variable F-Statistic Signif

*******************************************************

GROIL 4.7200 0.0044906

GROUT 0.7928 0.5016731

GRP 14.9563 0.0000001

R 1.5906 0.1986110

Dependent Variable R

Mean of Dependent Variable 7.2516666667

Std Error of Dependent Variable 3.3681179521

Standard Error of Estimate 0.9505662131

Sum of Squared Residuals 68.671785533

Durbin-Watson Statistic 1.9800

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. GROIL{1} 0.005195422 0.008311553 0.62508 0.53378807

2. GROIL{2} 0.004713180 0.008459589 0.55714 0.57906861

3. GROIL{3} -0.006775541 0.008213085 -0.82497 0.41197102

4. GROUT{1} 0.431784017 0.189489438 2.27867 0.02549564

5. GROUT{2} 0.620819990 0.190301129 3.26230 0.00165652

6. GROUT{3} 0.105351767 0.146019309 0.72149 0.47282068

7. GRP{1} 0.100960167 0.482988957 0.20903 0.83498249

8. GRP{2} 0.890088701 0.523289204 1.70095 0.09304089

9. GRP{3} 1.048473330 0.489815172 2.14055 0.03551774

10. R{1} 0.717687246 0.117471509 6.10946 0.00000004

11. R{2} -0.408239906 0.148217807 -2.75432 0.00735532

12. R{3} 0.457223146 0.119512321 3.82574 0.00026558

13. Constant -1.730636608 1.262197532 -1.37113 0.17436976

14. TREND 0.007237484 0.008250159 0.87725 0.38311346

F-Tests, Dependent Variable R

Variable F-Statistic Signif

*******************************************************

GROIL 0.5196 0.6700529

GROUT 6.8713 0.0003714

GRP 7.1476 0.0002722

R 49.6658 0.0000000

Covariance\Correlation Matrix of Residuals

GROIL GROUT GRP R

GROIL 168.12338741 0.02513615 0.42361138 0.17084446

GROUT 0.17288813 0.28138736 0.11012051 0.37325540

GRP 1.18991613 0.01265480 0.04693203 0.13606964

R 1.93500847 0.17295229 0.02574920 0.76301984

Responses to Shock in GROIL

Entry GROIL GROUT GRP R

1 1.0000000 0.0000000 0.0000000 0.00000000

2 0.2253490 -0.0006820 0.0064692 0.00519542

3 -0.3747820 0.0069244 -0.0007404 0.00997132

Responses to Shock in GROUT

Entry GROIL GROUT GRP R

1 0.0000000 1.0000000 0.0000000 0.00000000

2 3.0689429 0.2980970 -0.0162786 0.43178402

3 2.6469109 0.4863412 0.0334512 1.07372036

Responses to Shock in GRP

Entry GROIL GROUT GRP R

1 0.0000000 0.0000000 1.0000000 0.00000000

2 1.9675991 -0.7535342 0.3592172 0.10096017

3 8.7136210 -0.2432630 0.3609294 0.68367162

Responses to Shock in R

Entry GROIL GROUT GRP R

1 0.0000000 0.0000000 0.0000000 1.00000000

2 -0.3774558 -0.0150510 0.0500916 0.71768725

3 -1.3143004 -0.4598537 0.0011153 0.10343251

Covariance Model-Likelihood - Estimation by BFGS

Convergence in 22 Iterations. Final criterion was 0.0000063 <= 0.0000100

Observations 90

Log Likelihood -516.8800

Log Likelihood Unrestricted -516.8800

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. THETA(1) 12.96624057 0.99976819 12.96925 0.00000000

2. THETA(2) 0.15246586 0.09315225 1.63674 0.10168515

3. THETA(3) -0.09177035 0.02249968 -4.07874 0.00004528

4. THETA(4) 0.44389748 0.04503593 9.85652 0.00000000

5. THETA(5) 0.00918847 0.08785408 0.10459 0.91670285

6. THETA(6) 0.29546063 0.08750216 3.37661 0.00073385

7. THETA(7) 0.12829000 0.01867063 6.87122 0.00000000

8. THETA(8) 0.01333377 0.05832839 0.22860 0.81918112

9. THETA(9) 0.34436656 0.04973888 6.92349 0.00000000

10. THETA(10) 0.80001274 0.05982697 13.37211 0.00000000

Impact responses with multi-step restrictions

12.966 0.000 0.000 -0.000

0.013 0.360 0.389 -0.000

0.092 -0.128 0.148 -0.016

0.149 0.152 0.298 0.793

Step one responses

3.087 0.797 1.372 -0.330

-0.076 0.202 -0.000 -0.000

0.124 -0.044 0.062 0.034

0.189 0.252 0.397 0.567