svarFit#
Purpose#
Estimate structural vector autoregressive models.
Format#
- rslt = svarFit(Y[, maxlags, const, X_exog, ctl])#
- Parameters:
Y (matrix) – NxM or Nx(M+1) data. May include date variable, which will be removed from the data matrix. The date variable is not included in the model as a regressor.
maxlags (scalar) – Optional, maximum number of lags to consider for VAR model. If user_lags is specified in the
svarControlstructure, this parameter is ignored. Default = 8.const (scalar) –
Optional, specifying deterministic components of model.
0
No constant or trend.
1
Constant. (Default)
2
Constant and trend.
X_exog (matrix) – Optional, exogenous variables. If specified, the model is estimated as a VARX model. The exogenous variables are assumed to be stationary and are included in the model as additional regressors. May include date variable, which will be removed from the data matrix. The date variable is not included in the model as a regressor.
ctl (struct) –
Optional, an instance of the
svarControlstructure containing the following members.ctl.lutStats
An indicator specifying to use the Lutkepohl (2005) versions of the information criteria be reported. Default = 1.
ctl.smallDF
An indicator specifies that a small-sample degrees-of-freedom adjustment be used when estimating sigma, the error variance–covariance matrix. Specifically, 1/(T - m) is used instead of the large-sample divisor 1/T, where m is the average number of parameters in the functional form for yt over the K equations. Default = 1.
ctl.printVAR
An indicator specifying whether to print the results to screen. Default = 1.
ctl.irf
An instance of the
irfControlstructure containing the following members.ctl.irf.nsteps
Scalar, the number of horizons for IRF computations. Default = 20.
ctl.irf.ident
String, the identification method. Options include:
”short”
Zero short-run restrictions.
”long”
Zero long-run restrictions.
”sign”
Sign restrictions.
ctl.irf.ndraws
Scalar, number of draws for bootstrapping IRFs. Default = 10000.
ctl.irf.cl
Scalar, confidence level for IRF bootstrap confidence intervals. Default = 0.95.
ctl.irf.bootMethod
String, method for bootstrapping the IRF confidence intervals. Default = “bs”.
ctl.irf.signRestrictions
Matrix, specifies restrictions for sign restricted identification. There should be a single row for each restricted shock and a column for and a single column for each endogenous variable. 0 specifies that no restrictions are placed on a variable, -1 specifies that the sign should be negative, 1 specifies that the sign should be positive.
ctl.irf.restrictionHorizon
Matrix, specifies the number of horizons over which the restrictions hold.
ctl.irf.restrictedShockNames
String array, specifies which shock has restricted impulses using variable names. Must be specified if the number of restricted shocks is less than the number of endogenous variables and ctl.irf.restrictedShock index is not specified.
ctl.irf.restrictedShock
Matrix, specifies which shock has restricted impulses using an index. Must be specified if the number of restricted shocks is less than the number of endogenous variables and ctl.irf.restrictedShockNames is not specified.
- Return:
An instance of an
svarOutstructure containing the following members.rslt.coefficients
NxM matrix, final parameter estimates for the SVAR reduced form coefficients, computed by OLS.
rslt.coefficients_se
NxM matrix, standard errors of final estimates for the SVAR reduced form coefficients.
rslt.tstats
NxM matrix, t-stats of parameter estimates for the SVAR reduced form coefficients.
rslt.yhat
TxM matrix, predicted y values.
rslt.residuals
NxM matrix, residuals.
rslt.vcb
KxK matrix, covariance matrix for the SVAR reduced form coefficients.
rslt.ll
Scalar, value of the maximized likelihood function.
rslt.aic
Scalar, Akaike Information Criterion (AIC).
rslt.sbc
Scalar, Schwarz Bayesian Criterion (SBC).
rslt.aicc
Scalar, corrected Akaike Information Criterion (AICC).
rslt.hq
Scalar, Hannan-Quinn Criterion (HQ).
rslt.nlags
Scalar, number of lags used in the model.
rslt.F
Matrix, companion matrix.
rslt.B
Matrix, short-run identification matrix. The
Bmatrix represents the contemporaneous relationships between the structural shocks and the observed variables in the SVAR model under the “oir” (orthogonalized impulse response) short-run identification scheme.rslt.C
Matrix, long-run identification matrix. The
Cmatrix represents long-run cumulative impact of structural shocks on the observed variables in the SVAR model. It is computed when long-run identification restrictions are specified.rslt.wold
TxMxM Array, the moving average (MA) form of the estimated VAR model. Each plane of the array corresponds to a different time period.
rslt.irf
MxMxh Array, the impulse response functions of the estimated VAR model. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.irf_boot_upper
MxMxh Array, the upper bound of the bootstrapped confidence intervals for the impulse response functions. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.irf_boot_median
MxMxh Array, the median of the bootstrapped confidence intervals for the impulse response functions. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.irf_boot_lower
MxMxh Array, the lower bound of the bootstrapped confidence intervals for the impulse response functions. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.fevd
MxMxh Array, the forecast error variance decompositions of the estimated VAR model. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.fevd_upper
MxMxh Array, the upper bound of the bootstrapped confidence intervals for the forecast error variance decompositions. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.fevd_lower
MxMxh Array, the lower bound of the bootstrapped confidence intervals for the forecast error variance decompositions. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.HD
MxMxT Array, the impulse response functions of the estimated VAR model. Each plane of the array corresponds to different shock variable, and each element in the plane represents the impact of that shock on the endogenous variables at different horizons.
rslt.tsmtDesc
An instance of the
tsmtModelDescstructure containing the following members:tsmtDesc.depvar
Kx1 string array, names of endogenous variables.
tsmtDesc.indvars
Mx1 string array, names of exogenous variables.
tsmtDesc.timespan
2x1 string array, range of the time series. Available if date vector is passed as part of a dataframe input.
tsmtDesc.ncases
Scalar, number of observations.
tsmtDesc.df
Scalar, degrees of freedom.
tsmtDesc.model_name
String, model name.
rslt.sumStats
An instance of the
tsmtSummaryStatsstructure containing the following members:sumStats.sse
Vector, sum of the squared errors of estimates for endogenous variables in the model.
sumStats.mse
Vector, mean squared errors of estimates for endogenous variables in the model.
sumStats.rmse
Vector, root mean squared errors of estimates for endogenous variables in the model.
sumStats.see
Vector, standard error of the estimates for endogenous variables in the model.
sumStats.rsq
Vector, r-squared of estimates for endogenous variables in the model.
sumStats.AdjRsq
Scalar, adjusted r-squared of estimates for endogenous variables in the model.
sumStats.ssy
Scalar, total sum of the squares for endogenous variables in the model.
sumStats.DW
Scalar, Durbin-Watson statistic for residuals from the estimates for endogenous variables in the model.
- Rtype:
struct
Examples#
Example One: Short-run restrictions#
This example demonstrates the use of short-run restrictions to identify the structural model. It uses the cholesky identification method to determine the structural model.
This is the default identification method so no svarControl structure is necessary.
// Load library
new;
library tsmt;
/*
** Data import
*/
lutkepohl2 = loadd(getGAUSShome("pkgs/tsmt/examples/lutkepohl2.dta"));
// Filter data
lutkepohl2 = selif(lutkepohl2, lutkepohl2[., "qtr"] .<= "1978-12-30");
// Set Y
y = packr(lutkepohl2[., "qtr" "dln_inv" "dln_inc" "dln_consump"]);
// Set up output structures
struct svarOut sout;
// Compute structural VAR model
sout = svarFit(Y);
This prints the estimates for the reduced for coefficients:
=====================================================================================================
Model: SVAR(2) Number of Eqs.: 3
Time Span: 1960-04-01: Valid cases: 73
1978-10-01
Log Likelihood: 606.307 AIC: -24.632
SBC: -24.067
=====================================================================================================
Equation R-sq DW SSE RMSE
dln_inv 0.12856 2.01020 0.14056 0.04615
dln_inc 0.11419 1.75766 0.00906 0.01172
dln_consump 0.25128 -1.84234 0.00589 0.00944
=====================================================================================================
Results for reduced form equation dln_inv
=====================================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
-----------------------------------------------------------------------------------------------------
Constant -0.01672 0.01723 -0.97073 0.33523
dln_inv L(1) -0.31963 0.12546 -2.54775 0.01318
dln_inc L(1) 0.14599 0.54567 0.26754 0.78989
dln_consump L(1) 0.96123 0.66431 1.44696 0.15264
dln_inv L(2) -0.16055 0.12491 -1.28537 0.20316
dln_inc L(2) 0.11460 0.53457 0.21438 0.83091
dln_consump L(2) 0.93440 0.66509 1.40491 0.16474
=====================================================================================================
Results for reduced form equation dln_inc
=====================================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
-----------------------------------------------------------------------------------------------------
Constant 0.01577 0.00437 3.60427 0.00060
dln_inv L(1) 0.04393 0.03186 1.37891 0.17258
dln_inc L(1) -0.15273 0.13857 -1.10219 0.27438
dln_consump L(1) 0.28850 0.16870 1.71014 0.09194
dln_inv L(2) 0.05003 0.03172 1.57726 0.11952
dln_inc L(2) 0.01916 0.13575 0.14116 0.88817
dln_consump L(2) -0.01020 0.16890 -0.06039 0.95203
=====================================================================================================
Results for reduced form equation dln_consump
=====================================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
-----------------------------------------------------------------------------------------------------
Constant 0.01293 0.00353 3.66626 0.00049
dln_inv L(1) -0.00242 0.02568 -0.09437 0.92510
dln_inc L(1) 0.22481 0.11168 2.01305 0.04819
dln_consump L(1) -0.26397 0.13596 -1.94153 0.05646
dln_inv L(2) 0.03388 0.02556 1.32534 0.18963
dln_inc L(2) 0.35491 0.10941 3.24398 0.00185
dln_consump L(2) -0.02223 0.13612 -0.16329 0.87079
=====================================================================================================
The IRFs for the model are stored in the irf member of the svarOut output structure. This member is 3-dimensional array, with each plane containing the response to shocks to a different endogenous variable. The planes house a MxH matrix of responses with each row containg the responses from different response variable, and each column representing a different horizon.
For example, let’s preview the response of our three endogenous variables to, dln_inv, dln_inc, and dln_consump, to a shock in the first variable, dln_inv.
// Index of shock variable
shk_indx = 1;
// Get matrix of responses to dln_inv
res_to_dln_inv = getMatrix(sout.irf, shk_indx);
// Print first five responses
res_to_dln_inv[., 1:3];
0.046147884 -0.011956777 -0.0009900109
0.001551898 0.002560746 0.0012599300
0.002670542 -0.000467869 0.0027831146
Example Two: Long-run restrictions#
This example demonstrates the use of long-run restrictions to identify the structural model. This is done using the ctl.irf.ident member of the svarControl structure.
// Load library
new;
library tsmt;
/*
** Data import
*/
lutkepohl2 = loadd(getGAUSShome("pkgs/tsmt/examples/lutkepohl2.dta"));
// Filter data
lutkepohl2 = selif(lutkepohl2, lutkepohl2[., "qtr"] .<= "1978-12-30");
// Set up output structures
struct svarOut sout;
// Declare controls structure
// Fill with defaults
struct svarControl ctl;
ctl = svarControlCreate();
// Use long-run restrictions for
// structural identification
ctl.irf.ident = "long";
// Set Y
y = packr(lutkepohl2[., "qtr" "dln_inv" "dln_inc" "dln_consump"]);
// Run model
maxlags = 8;
const = 1;
// Check structural VAR model
sout = svarFit(Y, maxlags, const, ctl);
The reduced for estimates for this model are the same as the first model, because identification restrictions have no impact on the reduced form estimates.
However, if we look at the IRFS using these restrictions:
// Index of shock variable
shk_indx = 1;
// Get matrix of responses to dln_inv
res_to_dln_inv = getMatrix(sout.irf, shk_indx);
// Print first five responses
res_to_dln_inv[., 1:3];
0.041667833 -0.0067978789 0.0016807041
0.0056614147 0.0026748073 0.0013125032
0.0059236730 -0.00039186770 0.0040106258
Example Three: Sign restrictions#
The sign-restrictions option implements identification based on the theoretically anticipated direction of the IRFs. For example, consider a VAR model which includes real (GDP), the personal consumption expenditure price index (PCEPI), and the federal funds rate (FFR).
We can use sign-restricted IRFs to model the theory that real GDP and the PCEPI should initially respond with negatively to a monetary policy shock.
To start we import and transform the data:
new;
rndseed 908098;
library tsmt;
// Data files
fname = getGAUSSHome("pkgs/tsmt/examples/sign_restrictions_data.csv");
// Load data from .csv file
// and take ln of GDPC1 and PCEp1
data = loadd(__FILE_DIR $+ fname, "ln(GDPC1) + ln(PCEPI) + FEDFUNDS");
// Renaming columns
data = asDF(data, "l_gdp"$|"l_pce"$|"ffr");
// Remove missing values
reg_data = packr(data);
Next we implement the sign restrictions using the svarControl structure. This requires specifying:
* The use of sign-restrictions for identification by setting the svarControl structure member ctl.irf.ident to "sign".
* Which shocks to restrict using the ctl.irf.restrictedShock control structure member.
* The horizons whose responses are restricted using the ctl.irf.restrictionHorizon control structure member.
* The direction of the restrictions using the ctl.irf.signrestrictions control structure member. This matrix should have a row for each restricted shock and a column for each response variable. A value of -1 restricts a shock to be negative, a value of 1 restricts a shock to be positive, and a value of 0 indicates no restrictions.
// Declare controls structure
// Fill with defaults
struct svarControl ctl;
ctl = svarControlCreate();
// Specify to use sign restrictions
ctl.irf.ident = "sign";
// Specify which shock variable is restricted
ctl.irf.restrictedShock = 3;
// Set up restrictions horizon
ctl.irf.restrictionHorizon = 1;
/* Specify sign restrictions
** GDP response to monetary shock must < 0 (-1)
** PCE response to monetary shock must < 0 (-1)
** FFR response to monetary shock must > 0 (1)
*/
ctl.irf.signRestrictions = { -1 -1 1 };
Finally, we run the model using svarFit().
/*
** Setup VAR estimation
*/
// Maximum lags
maxlags = 8;
// Use constant in model
const = 1;
// Check structural VAR model
struct svarOut sOut;
sout = svarFit(reg_data, maxlags, const, ctl);
Remarks#
The procedure svarFit() is designed to provide flexibility in estimating SVAR models by allowing users to specify various options for the deterministic components, number of lags, and control settings for model estimation and impulse response analysis. The inclusion of bootstrapping methods and sign restrictions further enhances the robustness and interpretability of the resulting SVAR model.
See also
Functions arimaFit(), plotIRF(), svarControlCreate(), plotFEVD(), plotHD()