SAS/ETS 9.22 User''''s Guide 220 pot

the BY variables NAME, a character variable that contains the name of endogenous dependent variables SIGMA_i, numeric variables that contain the estimate of the innovation covariance m

2182 F Chapter 32: The VARMAX Procedure

Consider the following example:

proc varmax data=simul2 outest=est;

model y1 y2 / p=2 noint

ecm=(rank=1 normalize=y1) noprint;

run;

proc print data=est;

run;

The output inFigure 32.67shows the results of the OUTEST= data set

Figure 32.67 OUTEST= Data Set

Obs NAME TYPE AR1_1 AR1_2 AR2_1 AR2_2

1 y1 EST -0.46680 0.91295 -0.74332 -0.74621

2 STD 0.04786 0.09359 0.04526 0.04769

3 y2 EST 0.10667 -0.20862 0.40493 -0.57157

4 STD 0.05146 0.10064 0.04867 0.05128

OUTHT= Data Set

The OUTHT= data set contains prediction of the fitted GARCH model produced by the GARCH statement The following output variables can be created

 the BY variables

 Hi_j , numeric variables that contain the prediction of covariance, where 1  i < j  k, where k is the number of dependent variables

The OUTHT= data set contains the values shown inTable 32.6for a bivariate case

Table 32.6 OUTHT= Data Set

Obs H1_1 H1_2 H2_2

1 h111 h121 h221

2 h112 h122 h222

Consider the following example of the OUTHT= option:

proc varmax data=garch;

model y1 y2 / p=1

print=(roots estimates diagnose);

garch q=1 outht=ht;

run;

proc print data=ht(firstobs=495);

run;

The output inFigure 32.68shows the part of the OUTHT= data set

Figure 32.68 OUTHT= Data Set

Obs h1_1 h1_2 h2_2

495 9.36568 -1.10406 2.44644

496 8.46807 -0.17464 1.60330

497 9.19686 0.09762 1.69639

498 8.40787 -0.33463 2.07687

499 8.88429 0.03646 1.69401

500 8.60844 -0.40260 1.79703

OUTSTAT= Data Set

The OUTSTAT= data set contains estimation results of the fitted model produced by the VARMAX statement The following output variables can be created The subindex i is 1; : : : ; k, where k is the number of endogenous variables

 the BY variables

 NAME, a character variable that contains the name of endogenous (dependent) variables

 SIGMA_i, numeric variables that contain the estimate of the innovation covariance matrix

 AICC, a numeric variable that contains the corrected Akaike’s information criterion value

 HQC, a numeric variable that contains the Hannan-Quinn’s information criterion value

 AIC, a numeric variable that contains the Akaike’s information criterion value

 SBC, a numeric variable that contains the Schwarz Bayesian’s information criterion value

 FPEC, a numeric variable that contains the final prediction error criterion value

 FValue, a numeric variable that contains the F statistics

 PValue, a numeric variable that contains p-value for the F statistics

If the JOHANSEN= option is specified, the following items are added:

 Eigenvalue, a numeric variable that contains eigenvalues for the cointegration rank test of integrated order 1

2184 F Chapter 32: The VARMAX Procedure

 RestrictedEigenvalue, a numeric variable that contains eigenvalues for the cointegration rank test of integrated order 1 when the NOINT option is not specified

 Beta_i, numeric variables that contain long-run effect parameter estimates, ˇ

 Alpha_i, numeric variables that contain adjustment parameter estimates, ˛

If the JOHANSEN=(IORDER=2) option is specified, the following items are added:

 EValueI2_i, numeric variables that contain eigenvalues for the cointegration rank test of integrated order 2

 EValueI1, a numeric variable that contains eigenvalues for the cointegration rank test of integrated order 1

 Eta_i, numeric variables that contain the parameter estimates in integrated order 2, 

 Xi_i, numeric variables that contain the parameter estimates in integrated order 2, 

The OUTSTAT= data set contains the values shownTable 32.7for a bivariate case

Table 32.7 OUTSTAT= Data Set

Obs NAME SIGMA_1 SIGMA_2 AICC RSquare FValue PValue

Obs EValueI2_1 EValueI2_2 EValueI1 Beta_1 Beta_2

Obs Alpha_1 Alpha_2 Eta_1 Eta_2 Xi_1 Xi_2

Consider the following example:

proc varmax data=simul2 outstat=stat;

model y1 y2 / p=2 noint

cointtest=(johansen=(iorder=2)) ecm=(rank=1 normalize=y1)

noprint;

run;

proc print data=stat;

run;

The output inFigure 32.69shows the results of the OUTSTAT= data set

Figure 32.69 OUTSTAT= Data Set

Obs NAME SIGMA_1 SIGMA_2 AICC HQC AIC SBC FPEC

1 y1 94.7557 4.527 9.37221 9.43236 9.36834 9.52661 11712.14

EValue EValue EValue Obs RSquare FValue PValue I2_1 I2_2 I1 Beta_1 Beta_2

1 0.93900 482.308 6.1637E-57 0.98486 0.95079 0.50864 1.00000 1.00000

2 0.93912 483.334 5.6124E-57 0.81451 0.01108 -1.95575 -1.33622

Obs Alpha_1 Alpha_2 Eta_1 Eta_2 Xi_1 Xi_2

1 -0.46680 0.007937 -0.012307 0.027030 54.1606 -52.3144

2 0.10667 0.033530 0.015555 0.023086 -79.4240 -18.3308

Printed Output

The default printed output produced by the VARMAX procedure is described in the following list:

 descriptive statistics, which include the number of observations used, the names of the variables, their means and standard deviations (STD), their minimums and maximums, the differencing operations used, and the labels of the variables

 a type of model to fit the data and an estimation method

 a table of parameter estimates that shows the following for each parameter: the variable name for the left-hand side of equation, the parameter name, the parameter estimate, the approximate standard error, t value, the approximate probability (P r >jtj), and the variable name for the right-hand side of equations in terms of each parameter

 the innovation covariance matrix

 the information criteria

If PRINT=ESTIMATES is specified, the VARMAX procedure prints the following list with the default printed output:

 the estimates of the constant vector (or seasonal constant matrix), the trend vector, the coef-ficient matrices of the distributed lags, the AR coefcoef-ficient matrices, and the MA coefcoef-ficient matrices

 the ALPHA and BETA parameter estimates for the error correction model

 the schematic representation of parameter estimates

2186 F Chapter 32: The VARMAX Procedure

If PRINT=DIAGNOSE is specified, the VARMAX procedure prints the following list with the default printed output:

 the cross-covariance and cross-correlation matrices of the residuals

 the tables of test statistics for the hypothesis that the residuals of the model are white noise: – Durbin-Watson (DW) statistics

– F test for autoregressive conditional heteroscedastic (ARCH) disturbances

– F test for AR disturbance

– Jarque-Bera normality test

– Portmanteau test

ODS Table Names

The VARMAX procedure assigns a name to each table it creates You can use these names to reference the table when using the Output Delivery System (ODS) to select tables and create output data sets These names are listed in the following table:

Table 32.8 ODS Tables Produced in the VARMAX Procedure

ODS Tables Created by the MODEL Statement

AccumImpulse Accumulated impulse response matrices IMPULSE=(ACCUM)

IMPULSE=(ALL) AccumImpulsebyVar Accumulated impulse response by

vari-able

IMPULSE=(ACCUM) IMPULSE=(ALL) AccumImpulseX Accumulated transfer function matrices IMPULSX=(ACCUM)

IMPULSX=(ALL) AccumImpulseXbyVar Accumulated transfer function by

vari-able

IMPULSX=(ACCUM) IMPULSX=(ALL)

AlphaInECM ˛ coefficients when rank=r ECM=

AlphaOnDrift ˛ coefficients under the restriction of a

deterministic term

JOHANSEN=

AlphaBetaInECM …D ˛ˇ0coefficients when rank=r ECM=

ANOVA Univariate model diagnostic checks for

the residuals

PRINT=DIAGNOSE

ARRoots Roots of AR characteristic polynomial ROOTS with P=

BetaInECM ˇ coefficients when rank=r ECM=

BetaOnDrift ˇ coefficients under the restriction of a

deterministic term

JOHANSEN=

Table 32.8 continued

CorrB Correlations of parameter estimates CORRB

CorrResiduals Correlations of residuals PRINT=DIAGNOSE

CorrResidualsbyVar Correlations of residuals by variable PRINT=DIAGNOSE

CorrResidualsGraph Schematic representation of correlations

of residuals

PRINT=DIAGNOSE

CorrXGraph Schematic representation of sample

cor-relations of independent series

CORRX

CorrYGraph Schematic representation of sample

cor-relations of dependent series

CORRY CorrXLags Correlations of independent series CORRX

CorrXbyVar Correlations of independent series by

variable

CORRX CorrYLags Correlations of dependent series CORRY

CorrYbyVar Correlations of dependent series by

vari-able

CORRY

CovB Covariances of parameter estimates COVB

CovInnovation Covariances of the innovations default

CovPredictError Covariance matrices of the prediction

er-ror

COVPE CovPredictErrorbyVar Covariances of the prediction error by

variable

COVPE

CovResiduals Covariances of residuals PRINT=DIAGNOSE

CovResidualsbyVar Covariances of residuals by variable PRINT=DIAGNOSE

CovXLags Covariances of independent series COVX

CovXbyVar Covariances of independent series by

variable

COVX

CovYLags Covariances of dependent series COVY

CovYbyVar Covariances of dependent series by

vari-able

COVY

DecomposeCov-

Pre-dictError

Decomposition of the prediction error co-variances

DECOMPOSE DecomposeCov-

Pre-dictErrorbyVar

Decomposition of the prediction error co-variances by variable

DECOMPOSE

DiagnostAR Test the AR disturbance for the residuals PRINT=DIAGNOSE

DiagnostWN Test the ARCH disturbance and

normal-ity for the residuals

PRINT=DIAGNOSE DynamicARCoef AR coefficients of the dynamic model DYNAMIC

DynamicConstant Constant estimates of the dynamic model DYNAMIC

DynamicCov-

Inno-vation

Covariances of the innovations of the dy-namic model

DYNAMIC

DynamicLinearTrend Linear trend estimates of the dynamic

model

DYNAMIC DynamicMACoef MA coefficients of the dynamic model DYNAMIC

2188 F Chapter 32: The VARMAX Procedure

Table 32.8 continued

DynamicSConstant Seasonal constant estimates of the

dy-namic model

DYNAMIC

DynamicParameter-Estimates

Parameter estimates table of the dynamic model

DYNAMIC

DynamicParameter-Graph

Schematic representation of the parame-ters of the dynamic model

DYNAMIC

DynamicQuadTrend Quadratic trend estimates of the dynamic

model

DYNAMIC

DynamicSeasonGraph Schematic representation of the seasonal

dummies of the dynamic model

DYNAMIC DynamicXLagCoef Dependent coefficients of the dynamic

model

DYNAMIC

Hypothesis Hypothesis of different deterministic

terms in cointegration rank test

JOHANSEN=

HypothesisTest Test hypothesis of different deterministic

terms in cointegration rank test

JOHANSEN=

EigenvalueI2 Eigenvalues in integrated order 2 JOHANSEN=

(IORDER=2)

(IORDER=2) InfiniteARRepresent Infinite order ar representation IARR

InfoCriteria Information criteria default

MARoots Roots of MA characteristic polynomial ROOTS with Q= MaxTest Cointegration rank test using the

maxi-mum eigenvalue

JOHANSEN=

(TYPE=MAX)

OrthoImpulse Orthogonalized impulse response

matri-ces

IMPULSE=(ORTH) IM-PULSE=(ALL)

OrthoImpulsebyVar Orthogonalized impulse response by

vari-able

IMPULSE=(ORTH) IM-PULSE=(ALL)

ParameterEstimates Parameter estimates table default

ParameterGraph Schematic representation of the

parame-ters

PRINT=ESTIMATES

PartialAR Partial autoregression matrices PARCOEF

PartialARGraph Schematic representation of partial

au-toregression

PARCOEF

PartialCanCorr Partial canonical correlation analysis PCANCORR

PartialCorr Partial cross-correlation matrices PCORR

PartialCorrbyVar Partial cross-correlations by variable PCORR

PartialCorrGraph Schematic representation of partial

cross-correlations

PCORR

Table 32.8 continued

PortmanteauTest Chi-square test table for residual

cross-correlations

PRINT=DIAGNOSE

ProportionCov-

Pre-dictError

Proportions of prediction error covari-ance decomposition

DECOMPOSE ProportionCov-

Pre-dictErrorbyVar

Proportions of prediction error covari-ance decomposition by variable

DECOMPOSE

RankTestI2 Cointegration rank test in integrated order

2

JOHANSEN=

(IORDER=2) RestrictMaxTest Cointegration rank test using the

maxi-mum eigenvalue under the restriction of

a deterministic term

JOHANSEN=

(TYPE=MAX) without NOINT RestrictTraceTest Cointegration rank test using the trace

under the restriction of a deterministic term

JOHANSEN=

(TYPE=TRACE) without NOINT QuadTrend Quadratic trend estimates TREND=QUAD

SeasonGraph Schematic representation of the seasonal

dummies

PRINT=ESTIMATES SConstant Seasonal constant estimates NSEASON=

SimpleImpulse Impulse response matrices IMPULSE=(SIMPLE)

IMPULSE=(ALL) SimpleImpulsebyVar Impulse response by variable IMPULSE=(SIMPLE)

IMPULSE=(ALL) SimpleImpulseX Impulse response matrices of transfer

function

IMPULSX=(SIMPLE) IMPULSX=(ALL) SimpleImpulseXbyVar Impulse response of transfer function by

variable

IMPULSX=(SIMPLE) IMPULSX=(ALL) Summary Simple summary statistics default

TraceTest Cointegration rank test using the trace JOHANSEN=

(TYPE=TRACE)

(IORDER=2)

ODS Tables Created by the GARCH Statement

GARCHConstant GARCH constant estimates PRINT=ESTIMATES

GARCHParameter-Estimates

GARCH parameter estimates table default

GARCHParameter-Graph

Schematic representation of the garch pa-rameters

PRINT=ESTIMATES

2190 F Chapter 32: The VARMAX Procedure

Table 32.8 continued

GARCHRoots Roots of GARCH characteristic

polyno-mial

ROOTS

ODS Tables Created by the COINTEG Statement or the ECM option

AlphaInECM ˛ coefficients when rank=r PRINT=ESTIMATES AlphaBetaInECM …D ˛ˇ0coefficients when rank=r PRINT=ESTIMATES AlphaOnAlpha ˛ coefficients under the restriction of ˛ J=

AlphaOnBeta ˛ coefficients under the restriction of ˇ H=

AlphaTestResults Hypothesis testing of ˇ J=

BetaInECM ˇ coefficients when rank=r PRINT=ESTIMATES BetaOnBeta ˇ coefficients under the restriction of ˇ H=

BetaOnAlpha ˇ coefficients under the restriction of ˛ J=

BetaTestResults Hypothesis testing of ˇ H=

GrangerRepresent Coefficient of Granger representation PRINT=ESTIMATES

WeakExogeneity Testing weak exogeneity of each

depen-dent variable with respect to BETA

EXOGENEITY

ODS Tables Created by the CAUSAL Statement

CausalityTest Granger causality test default

GroupVars Two groups of variables default

ODS Tables Created by the RESTRICT Statement

ODS Tables Created by the TEST Statement

ODS Tables Created by the OUTPUT Statement

Note that the ODS table names suffixed by “byVar” can be obtained with the PRINT-FORM=UNIVARIATE option

ODS Graphics

This section describes the use of ODS for creating statistical graphs with the VARMAX procedure

To request these graphs, you must specify the ODS GRAPHICS ON statement

When ODS GRAPHICS are in effect, the VARMAX procedure produces a variety of plots for each dependent variable

The plots available are as follows:

 The procedure displays the following plots for each dependent variable in the MODEL statement with the PLOT= option in the VARMAX statement:

– impulse response function

– impulse response of the transfer function

– time series and predicted series

– prediction errors

– distribution of the prediction errors

– normal quantile of the prediction errors

– ACF of the prediction errors

– PACF of the prediction errors

– IACF of the prediction errors

– log scaled white noise test of the prediction errors

 The procedure displays forecast plots for each dependent variable in the OUTPUT statement with the PLOT= option in the VARMAX statement

ODS Graph Names

The VARMAX procedure assigns a name to each graph it creates by using ODS You can use these names to reference the graphs when using ODS The names are listed inTable 32.9

Table 32.9 ODS Graphics Produced in the VARMAX Procedure

ErrorACFPlot Autocorrelation function of prediction

er-rors

MODEL

ErrorIACFPlot Inverse autocorrelation function of

pre-diction errors

MODEL

ErrorPACFPlot Partial autocorrelation function of

predic-tion errors

MODEL ErrorDiagnosticsPanel Diagnostics of prediction errors MODEL

ErrorNormalityPanel Histogram and Q-Q plot of prediction

er-rors

MODEL