MODEL Statement

The VARMAX Procedure

MODEL Statement

MODEL dependents < = regressors >
<, dependents < = regressors > ... >
</ options > ;

The MODEL statement specifies endogenous (dependent) variables and exogenous (independent) variables for the VARMAX model. The multivariate model can have the same or different independent variables corresponding to the dependent variables. As a special case, the VARMAX procedure allows you to analyze one dependent variable with independent variables. The one MODEL statement is required.

For example, the following statements are equivalent ways of specifying the multivariate model for the vector (y1, y2, y3):

   model y1 y2 y3 </options>;
   model y1-y3 </options>;

The following statements are equivalent ways of specifying the multivariate model for the vectors (y1, y2, y3, y4) and (x1, x2, x3, x4, x5):

   model y1 y2 y3 y4 = x1 x2 x3 x4 x5 </options>;
   model y1 y2 y3 y4 = x1-x5 </options>;
   model y1 = x1-x5, y2 = x1-x5, y3 y4 = x1-x5 </options>;
   model y1-y4 = x1-x5 </options>;

When the multivariate model has different independent variables corresponding to the dependent variables, equations are separated by commas (,) and the model can be specified as illustrated by the following MODEL statement:

   model y1 = x1-x3, y2 = x3-x5, y3 y4 = x1-x5 </options>;

The following options can be used in the MODEL statement after a forward slash (/):

General Options

CENTER

centers endogenous (dependent) variables by subtracting their means. Note that centering is done after differencing when the DIF= or DIFY= option is specified. If there are exogenous (independent) variables, this option is not applicable.

DIF= (variable(number-list)<... variable(number-list)>)

specifies the degrees of differencing to be applied to the specified dependent or independent variables. The differencing can be the same for all variables, or it can vary among variables. For example, DIF=(y₁(1,4) y₃(1) x₂(2)) specifies that the y₁ series is differenced at lag 1 and at lag 4, which is (y_1t-y_1,t-1)-(y_1,t-4-y_1,t-5); y₃ at lag 1, which is (y_3t-y_3,t-1); x₂ at lag 2, which is (x_2t-x_2,t-2).

DIFX= (number-list)

specifies the degrees of differencing to be applied to all exogenous (independent) variables. For example, DIFX=(1) specifies that the series are differenced once at lag 1; DIFX=(1,4) at lag 1 and at lag 4. If exogenous variables are specified in the DIF= option, this option is ignored.

DIFY= (number-list)

specifies the degrees of differencing to be applied to all endogenous (dependent) variables. For details, see the DIFX= option. If endogenous variables are specified in the DIF= option, this option is ignored.

METHOD= value

requests the type of estimates to be computed. The possible values of the METHOD= option are

LS: specifies least-squares estimates.
ML: specifies maximum likelihood estimates.

If the PRIOR= or ECM= option or both is specified, the default is METHOD=ML; otherwise, the default is METHOD=ML.

NOCURRENTX

suppresses the current values x_t of exogenous (independent) variables. In general, the VARMAX(p,q,s) model is

$y_t={{\delta}} + \sum_{i=1}^p\Phi_i{y}_{t-i} + \sum_{i=0}^s\Theta_i^*x_{t-i} + {{\epsilon}}_t - \sum_{i=1}^q\Theta_i{{\epsilon}}_{t-i}.$

If this option is specified, it suppresses the current values x_t and starts with x_t-1.

$y_t={{\delta}} + \sum_{i=1}^p\Phi_i{y}_{t-i} + \sum_{i=0}^s\Theta_i^*x_{t-i} + {{\epsilon}}_t - \sum_{i=1}^q\Theta_i{{\epsilon}}_{t-i}.$

NOINT

suppresses the intercept parameters ${{\delta}}$ .

NSEASON= number

specifies the number of seasonal periods. When the NSEASON=number option is specified, (number-1) seasonal dummies are added to the regressors. If the NOINT option is specified, the NSEASON= option is not applicable.

SCENTER

centers seasonal dummies specified by the NSEASON= option. The centered seasonal dummies are generated by c-(1/s), where c is a seasonal dummy generated by the NSEASON=s option.

TREND= value

specifies the degree of deterministic time trend included in the model. Valid values are as follows:

LINEAR: includes a linear time trend as a regressor.
QUAD: includes linear and quadratic time trends as regressors.

The TREND=QUAD option is not applicable when the ECM= option is specified.

VARDEF= value

corrects for the degrees of freedom of the denominator. This option is used to calculate an error covariance matrix for the METHOD=LS. If the METHOD=ML is specified, the VARDEF=N is used. Valid values are as follows:

DF: specifies that the number of nonmissing observation minus the number of regressors be used.
N: specifies that the number of nonmissing observation be used.

Printing Control Options

LAGMAX= number

specifies the lag to compute and display the results obtained by the PRINT=(CORRX CORRY COVX COVY IARR IMPULSE= IMPULSX= PARCOEF PCANCORR PCORR) option. This option is also used to print cross-covariances and cross-correlations of residuals. The default is LAGMAX=min(12, T-2), where T is the number of nonmissing observations.

NOPRINT

suppresses all printed output.

PRINTALL

requests all printing control options. The options set by PRINTALL are DFTEST=, MINIC=, PRINTFORM=BOTH, and PRINT=( CORRB CORRX CORRY COVB COVPE COVX COVY DECOMPOSE IARR IMPULSE=(ALL) IMPULSX=(ALL) PARCOEF PCANCORR PCORR ROOTS YW ).

You can also specify this option as ALL.

PRINTFORM= value

requests the printing format of outputs of the PRINT= option and cross-covariances and cross-correlations of residuals. Valid values are as follows:

BOTH: prints outputs in both MATRIX and UNIVARIATE forms.
MATRIX: prints outputs in matrix form. This is the default.
UNIVARIATE: prints outputs by variables.

Printing Options

PRINT=(options)

The following options can be used in the PRINT=( ) option. The options are listed within parentheses.

CORRB

prints the estimated correlations of the parameter estimates.

CORRX

prints the cross-correlation matrices of exogenous (independent) variables using the number of lags specified by the LAGMAX=number.

CORRY

prints the cross-correlation matrices of endogenous (dependent) variables using the number of lags specified by the LAGMAX=number.

COVB

prints the estimated covariances of the parameter estimates.

COVPE

COVPE(number)

prints the covariance matrices of number-ahead prediction errors for the VARMAX(p,q,s) model. If the DIF= or DIFY= option is specified, the covariance matrices of multistep-ahead prediction errors are computed based on the differenced data. This option is not applicable when the PRIOR= option is specified. See the "Forecasting" section for details.

COVX

prints the cross-covariance matrices of exogenous (independent) variables using the number of lags specified by the LAGMAX=number.

COVY

prints the cross-covariance matrices of endogenous (dependent) variables using the number of lags specified by the LAGMAX=number.

DECOMPOSE

DECOMPOSE(number)

prints the decomposition of the prediction error covariances using the number of lags specified by number in parentheses for the VARMAX(p,q,s) model. It can be interpreted as the contribution of innovations in one variable to the mean squared error of the multistep-ahead forecast of another variable. The DECOMPOSE option also prints proportions of the forecast error variance.

If the DIF= or DIFY= option is specified, the covariance matrices of multistep-ahead prediction errors are computed based on the differenced data. This option is not applicable when the PRIOR= option is specified. See the "Forecasting" section for details.

IARR

prints the infinite order AR representation of a VARMA process. The coefficient matrices print the number of lags specified by the LAGMAX=number. If the moving-average order is zero, this option prints the maximum order of the autoregressive characteristic function. If the ECM= option is specified, the reparameterized AR coefficient matrices are printed.

IMPULSE

IMPULSE= (SIMPLE ACCUM ORTH STDERR ALL)

prints the impulse response function using the number of lags specified by the LAGMAX= number. It investigates the response of one variable to an impulse in another variable in a system that involves a number of other variables as well. It is an infinite order MA representation of a VARMA process. See the "Forecasting" section for details.

The following options can be used in the IMPULSE=( ) option. The options are listed within parentheses.


ACCUM: prints the accumulated impulse function.
ALL: equivalent to specifying all of SIMPLE, ACCUM, ORTH, and STDERR.
ORTH: prints the orthogonalized impulse function.
SIMPLE: prints the impulse response function. This is the default.
STDERR: prints the standard errors of the impulse response function, the accumulated impulse response function, or the orthogonalized impulse response function. If the exogenous variables are used to fit the model, this option is ignored.

IMPULSX

IMPULSX= (SIMPLE ACCUM ALL)

prints the impulse response function related to exogenous (independent) variables using the number of lags specified by the LAGMAX=number. See the "Forecasting" section for details.

The following options can be used in the IMPULSX=( ) option. The options are listed within parentheses.


ACCUM: prints the accumulated impulse response matrices in the transfer function.
ALL: equivalent to specifying both SIMPLE and ACCUM.
SIMPLE: prints the impulse response matrices in the transfer function. This is the default.

PARCOEF

prints the partial autoregression coefficient matrices, $\Phi_{mm}$ . With a VAR process, this option is useful for the identification of the order since the $\Phi_{mm}$ have the characteristic property that they equal zero for m>p under the hypothetical assumption of a VAR(p) model. These matrices print the number of lags specified by the LAGMAX=number. See the "Tentative Order Selection" section for details.

PCANCORR

prints the partial canonical correlations of the process at lag m and the test for testing $\Phi_{m}$ =0 for m>p. The lag m partial canonical correlations are the canonical correlations between y_t and y_t-m, after adjustment for the dependence of these variables on the intervening values y_t-1, ..., y_t-m+1. See the "Tentative Order Selection" section for details.

PCORR

prints the partial correlation matrices using the number of lags specified by the LAGMAX=number. With a VAR process, this option is useful for a tentative order selection by the same property as the partial autoregression coefficient matrices, as described in the PARCOEF option. See the "Tentative Order Selection" section for details.

ROOTS

prints the eigenvalues of the kp ×kp companion matrix associated with the AR characteristic function $\Phi(B)$ , where k is the number of endogenous (dependent) variables, and $\Phi(B)$ is the finite order matrix polynomial in the backshift operator B, such that Bⁱy_t = y_t-i. These eigenvalues indicate the stationary condition of the process since the stationary condition on the roots of $|\Phi(B)|=0$ in the VAR(p) model is equivalent to the condition in the corresponding VAR(1) representation that all eigenvalues of the companion matrix be less than one in absolute value. Similarly, you can use this option to check the invertibility of the MA process.

YW

prints Yule-Walker estimates of the preliminary autoregressive model for the endogenous (dependent) variables. The coefficient matrices are printed using the maximum order of the autoregressive process.

Lag Specification Options

P= number
P= (number-list): specifies the order of the vector autoregressive process. Subset models of vector autoregressive orders can be specified as, for example, P=(1,3,4). P=3 is equivalent to P=(1,2,3). The default is P=0.

If P=0 and there are no exogenous (independent) variables, the AR polynomial order is automatically determined by minimizing an information criterion; if P=0 and the PRIOR= or ECM= option or both is specified the AR polynomial order is determined.

If the PRIOR= or ECM= option or both is specified, subset models of vector autoregressive orders are not allowed and the AR maximum order is used.
XLAG= number
XLAG= (number-list): specifies the lags of exogenous (independent) variables. Subset models of distributed lags can be specified as, for example, XLAG=(2). The default is XLAG=0. To exclude the present values of exogenous (independent) variables from the model, the NOCURRENTX option must be used.

Tentative Order Selection Options

MINIC

MINIC= (TYPE=value P=number Q=number PERROR=number)

prints the information criterion for the appropriate AR and MA tentative order selection and for the diagnostic checks of the fitted model.

If the MINIC= option is not specified, all types of information criteria are printed for diagnostic checks of the fitted model.

The following options can be used in the MINIC=( ) option. The options are listed within parentheses.

P= number

P= (p_min:p_max)

specifies the range of AR orders. The default is P=(0:5).

PERROR= number

PERROR= ( $p_{\epsilon , min}$ : $p_{\epsilon, max}$ )

specifies the range of AR orders for obtaining the error series. The default is PERROR=(p_max:p_max+q_max).

Q= number

Q= (q_min:q_max)

specifies the range of MA orders. The default is Q=(0:5).

TYPE= value

specifies the criterion for the model order selection. Valid criteria are as follows:


AIC: specifies the Akaike Information Criterion.
AICC: specifies the Corrected Akaike Information Criterion. This is the default criterion.
FPE: specifies the Final Prediction Error criterion.
HQC: specifies the Hanna-Quinn Criterion.
SBC: specifies the Schwarz Bayesian Criterion. You can also specify this value as TYPE=BIC.

Cointegration Related Options

COINTTEST

COINTTEST= (JOHANSEN<(=options)> SW<(=options)> SIGLEVEL=number)

The following options can be used in the COINTTEST=( ) option. The options are listed within parentheses.

JOHANSEN

JOHANSEN= (TYPE=value IORDER=number NORMALIZE=variable)

prints the cointegration rank test for multivariate time series based on Johansen method. This test is provided when the number of endogenous (dependent) variables is less than or equal to 11. See the "Vector Error Correction Modeling" section for details.

The VAR(p) model can be written as the error correction model

$\Delta y_{t}=\Pi y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + A D_t + {\epsilon}_t$

where $\Pi$ , $\Phi^*_i$ , and A are coefficient parameters; D_t is a deterministic term such as a constant, a linear trend, or seasonal dummies.

The I(1) model is defined by one reduced rank condition. If the cointegration rank is r<k, then there exist k×r matrices ${\alpha}$ and ${\beta}$ of rank r such that $\Pi={\alpha} {\beta}'$ .

The I(1) model is rewritten as the I(2) model

$\Delta^2 y_{t}=\Pi y_{t-1} -\Psi \Delta y_{t-1} +\sum_{i=1}^{p-2} \Psi_i \Delta^2 y_{t-i} +A D_t + {\epsilon}_t$

where $\Psi=I_k - \sum_{i=1}^{p-1} \Phi^*_i$ and $\Psi_i=-\sum_{j=i+1}^{p-1} \Phi^*_i$ .

The I(2) model is defined by two reduced rank conditions. One is that $\Pi={\alpha} {\beta}'$ , where ${\alpha}$ and ${\beta}$ are k×r matrices of full rank r. The other is that ${\alpha}'_{\bot} \Psi {\beta}_{\bot}={\xi} {\eta}'$ where ${\xi}$ and ${\eta}$ are (k-r)×s matrices with $s\leq k-r$ .and ${\alpha}_{\bot}$ and ${\beta}_{\bot}$ are k×(k-r) matrices of full rank k-r such that ${\alpha}'{\alpha}_{\bot}=0$ and ${\beta}'{\beta}_{\bot}=0$ .

The following options can be used in the JOHANSEN=( ) option. The options are listed within parentheses.

IORDER= number

specifies the integrated order.


IORDER=1: prints the cointegration rank test for an integrated order 1 and prints the long-run parameter, ${\beta}$ ,and the adjustment coefficient, ${\alpha}$ . This is the default. If IORDER=1 is specified, the AR order should be greater than or equal to 1. When P=0, P is temporarily set to 1.
IORDER=2: prints the cointegration rank test for integrated orders 1 and 2. If IORDER=2 is specified, the AR order should be greater than or equal to 2. If P=1, the IORDER=1 is used; if P=0, P is temporarily set to 2.

NORMALIZE= variable

specifies the endogenous (dependent) variable name whose cointegration vectors are to be normalized. If the normalized variable is different from that specified in the ECM= option or the COINTEG statement, the latter is used.

TYPE= value

specifies the type of cointegration rank test to be printed. Valid values are as follows:


MAX: prints the cointegration maximum eigenvalue test.
TRACE: prints the cointegration trace test. This is the default.

If the NOINT option is not specified, the procedure prints two different cointegration rank tests in the presence of the unrestricted and restricted deterministic terms (constant or linear trend) models. If IORDER=2 is specified, the procedure automatically determines that TYPE=TRACE.

SIGLEVEL= value

sets the size of cointegration rank tests and common trends tests. The SIGLEVEL=value option must be one of 0.1, 0.05, or 0.01. The default is SIGLEVEL=0.05.

SW

SW= (TYPE=value LAG=number)

prints common trends tests for a multivariate time series based on the Stock-Watson method. This test is provided when the number of endogenous (dependent) variables is less than or equal to 6. See the "Common Trends" section for details.

The following options can be used in the SW=( ) option. The options are listed within parentheses.

LAG= number

specifies the number of lags. The default is LAG=max(1,p) for TYPE=FILTDIF or TYPE=FILTRES, where p is the AR maximum order specified by the P= option; LAG=O(T^1/4) for TYPE=KERNEL, where T is the number of nonmissing observations. If LAG= exceeds the default, it is replaced by the default.

TYPE= value

specifies the type of common trends test to be printed. Valid values are as follows:


FILTDIF: prints the common trends test based on the filtering method applied to the differenced series. This is the default.
FILTRES: prints the common trends test based on the filtering method applied to the residual series.
KERNEL: prints the common trends test based on the kernel method.

DFTEST

DFTEST= (DLAG=number)

prints the Dickey-Fuller unit root test. The DLAG=number specifies the regular or seasonal unit root test. If the number is greater than one, seasonal Dickey-Fuller tests are performed. The number should be one of 1, 2, 4, or 12. The default is DLAG=1.

Bayesian VAR Estimation Options

PRIOR

PRIOR= (MEAN=(vector) LAMBDA=value THETA=value IVAR<=(variables)> NREP=number SEED=number)

specifies the prior value of parameters for the BVAR(p) model. If the ECM= option is specified with the PRIOR option, the BVECM(p) form is fitted. The following options can be used in the PRIOR option. For the standard errors of the predictors, the bootstrap procedure is used. See the "Bayesian VAR Modeling" section for details.

The following options can be used in the PRIOR=( ) option. The options are listed within parentheses.

IVAR

IVAR= (variables)

specifies an integrated BVAR(p) model. If you use the IVAR option without variables, it sets the overall prior mean of the first lag of each variable equal to one in its own equation and sets all other coefficients to zero. If variables are specified, it sets the prior mean of the first lag of the specified variables equal to one in its own equation and sets all other coefficients to zero. When the series y_t = (y₁, y₂)' follows a bivariate BVAR(2) process, the IVAR or IVAR=(y₁ y₂) option is equivalent to specifying MEAN=( 1 0 0 0 0 1 0 0 ).

If the PRIOR=(MEAN= ) or ECM= option is specified, the IVAR= option is ignored.

LAMBDA= value

specifies the prior standard deviation of the AR coefficient parameter matrices. It should be a positive number. The default is LAMBDA=1. As the value of the LAMBDA= is larger, a BVAR(p) model is close to a VAR(p) model.

MEAN= (vector)

specifies the mean vector in the prior distribution for the AR coefficients. If the vector is not specified, the prior value is assumed to be a zero vector. See the "Bayesian VAR Modeling" section for details.

You can specify the mean vector by order of the equation. Let $B=(\delta, \Phi_1,...,\Phi_p)'$ be the parameter sets to be estimated. If $\Phi=(\Phi_1,...,\Phi_p)'$ , then MEAN= ( ${\rm vec}(\Phi)$ ).

For example, in case of a bivariate (k=2) BVAR(2) model,

${\rm MEAN}=( \phi_{1,11} \phi_{1,12} \phi_{2,11} \phi_{2,12} \phi_{1,21} \phi_{1,22} \phi_{2,21} \phi_{2,22} )$

where $\phi_{l,ij}$ is the (i,j)th element of the matrix $\Phi_l$ .

The deterministic terms are considered to shrink toward zero; you must omit prior means of deterministic terms such as a constant, seasonal dummies, or trends.

For a Bayesian error correction model, you specify a mean vector for only lagged AR coefficients, $\Phi^*_i$ , in terms of regressors $\Delta y_{t-i}$ ,for i=1,...,(p-1) in the VECM(p) representation. The diffused prior variance of ${\alpha}$ is used since ${\beta}$ is replaced by $\hat {{\beta}}$ estimated in a nonconstrained VECM(p) form.

$\Delta y_{t}={\alpha} z_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + A D_t + {\epsilon}_t$

where $z_{t}={\beta}' y_{t}$ .

For example, in case of a bivariate (k=2) BVECM(2) form,

${\rm MEAN}=( \phi^*_{1,11} \phi^*_{1,12} \phi^*_{1,21} \phi^*_{1,22} )$

where $\phi^*_{1,ij}$ is the (i,j)th element of the matrix $\Phi^*_1$ .

NREP= number

specifies the number of bootstrap replications. The default is NREP=100.

SEED= number

specifies seeds to generate uniform random numbers for resampling. By default, the system clock is used to generate the random seed.

THETA= value

specifies the prior standard deviation of the AR coefficient parameter matrices. The value is in the interval (0,1). The default is THETA=0.1. As the value of the THETA= is close to 1, a BVAR(p) model is close to a VAR(p) model.

Vector Error Correction Model Options

ECM=(RANK=number NORMALIZE=variable)

specifies a vector error correction model.

The following options can be used in the ECM=( ) option. The options are listed within parentheses.

NORMALIZE= variable: specifies a single endogenous variable name whose cointegrating vectors are normalized. If the variable name is different from that specified in the COINTEG statement, the latter is used.
RANK= number: specifies the cointegration rank. This option is required in the ECM= option. The value of the RANK= option should be greater than zero and less than or equal to the number of endogenous (dependent) variables, k. If the rank is different from that specified in the COINTEG statement, the latter is used.

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