MODEL Statement

The AUTOREG Procedure

MODEL Statement

MODEL dependent = regressors / options ;

The MODEL statement specifies the dependent variable and independent regressor variables for the regression model. If no independent variables are specified in the MODEL statement, only the mean is fitted. (This is a way to obtain autocorrelations of a series.)

Models can be given labels of up to eight characters. Model labels are used in the printed output to identify the results for different models. The model label is specified as follows:

label : MODEL ... ;

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

CENTER: centers the dependent variable by subtracting its mean and suppresses the intercept parameter from the model. This option is only valid when the model does not have regressors (explanatory variables).
NOINT: suppresses the intercept parameter.

Autoregressive Error Options

NLAG= number
NLAG= ( number-list ): specifies the order of the autoregressive error process or the subset of autoregressive error lags to be fitted. Note that NLAG=3 is the same as NLAG=(1 2 3). If the NLAG= option is not specified, PROC AUTOREG does not fit an autoregressive model.

GARCH Estimation Options

GARCH= ( option-list )

Specifies a GARCH-type conditional heteroscedasticity model. The GARCH= option in the MODEL statement specifies the family of ARCH models to be estimated. The GARCH(1,1) regression model is specified in the following statement:


   model y = x1 x2 / garch=(q=1,p=1);

When you want to estimate the subset of ARCH terms, for example, ARCH(1 3), you can write the SAS statement as follows:


   model y = x1 x2 / garch=(q=(1 3));

With the TYPE= option, you can specify various GARCH models. The IGARCH(2,1) model without trend in variance is estimated as follows:


   model y = / garch=(q=2,p=1,type=integ,noint);

The following options can be used in the GARCH=( ) option. The options are listed within parentheses and separated by commas.

Q= number

Q= (number-list)

specifies the order of the process or the subset of ARCH terms to be fitted.

P= number

P= (number-list)

specifies the order of the process or the subset of GARCH terms to be fitted. If only the P= option is specified, Q=1 is assumed.

TYPE= value

specifies the type of GARCH model. The values of the TYPE= option are

EXP: specifies the exponential GARCH or EGARCH model.
INTEGRATED: specifies the integrated GARCH or IGARCH model.
NELSON | NELSONCAO: specifies the Nelson-Cao inequality constraints.
NONNEG: specifies the GARCH model with nonnegativity constraints.
STATIONARY: constrains the sum of GARCH coefficients to be less than 1.

The default is TYPE=NELSON.

MEAN= value

specifies the functional form of the GARCH-M model. The values of the MEAN= option are

LINEAR

specifies the linear function.

$y_{t} = x_{t}^{'}{\beta} + {\delta}h_{t} + {\epsilon}_{t}$

LOG

specifies the log function.

$y_{t} = x_{t}^{'}{\beta} + {\delta}\rm{ln} h_{t} + {\epsilon}_{t}$

SQRT

specifies the square root function.

$y_{t} = x_{t}^{'}{\beta} + {\delta}\sqrt{h_{t}} + {\epsilon}_{t}$

DIST= value

specifies the distribution assumed for the error term. The values of the DIST= option are

T: specifies Student's t distribution.
NORMAL: specifies the standard normal distribution. The default is DIST=NORMAL.

NOINT

suppresses the intercept parameter in the conditional variance model. This option is valid only with the TYPE=INTEG option.

STARTUP= MSE | ESTIMATE

STARTUP=ESTIMATE requests that the positive constant c for the start-up values of the GARCH conditional error variance process be estimated. By default or if STARTUP=MSE is specified, the value of the mean squared error is used as the default constant.

TR

uses the trust region method for GARCH estimation. This algorithm is numerically stable, though computation is expensive. The double quasi-Newton method is the default.

Printing Options

ALL

requests all printing options.

ARCHTEST

requests the Q and LM statistics testing for the absence of ARCH effects.

CHOW= ( obs₁ ... obs_n )

The CHOW= option computes Chow tests to evaluate the stability of the regression coefficient. The Chow test is also called the analysis-of-variance test.

COEF

prints the transformation coefficients for the first p observations. These coefficients are formed from a scalar multiplied by the inverse of the Cholesky root of the Toeplitz matrix of autocovariances.

CORRB

prints the estimated correlations of the parameter estimates.

COVB

prints the estimated covariances of the parameter estimates.

COVEST= OP | HESSIAN | QML

The COVEST= option specifies the type of covariance matrix for the GARCH or heteroscedasticity model. When COVEST=OP is specified, the outer product matrix is used to compute the covariance matrix of the parameter estimates. The COVEST=HESSIAN option produces the covariance matrix using the Hessian matrix. The quasi-maximum likelihood estimates are computed with COVEST=QML. The default is COVEST=OP.

DW= n

prints Durbin-Watson statistics up to the order n. The default is DW=1. When the LAGDEP option is specified, the Durbin-Watson statistic is not printed unless the DW= option is explicitly specified.

DWPROB

The DWPROB option now produces p-values for the generalized Durbin-Watson test statistics for large sample sizes. Previously, the Durbin-Watson probabilities were calculated only for small sample sizes. The new method of calculating Durbin-Watson probabilities is based on the algorithm of Ansley, Kohn, and Shively (1992).

GINV

prints the inverse of the Toeplitz matrix of autocovariances for the Yule-Walker solution. See "Computational Methods" later in this chapter for details.

GODFREY

GODFREY= r

The GODFREY option produces Godfrey's general Lagrange multiplier test against ARMA errors.

ITPRINT

prints the objective function and parameter estimates at each iteration. The objective function is the full log likelihood function for the maximum likelihood method, while the error sum of squares is produced as the objective function of unconditional least squares. For the ML method, the ITPRINT option prints the value of the full log likelihood function, not the concentrated likelihood.

LAGDEP

LAGDV

prints the Durbin t statistic, which is used to detect residual autocorrelation in the presence of lagged dependent variables. See "Generalized Durbin-Watson Tests" later in this chapter for details.

LAGDEP= name

LAGDV= name

prints the Durbin h statistic for testing the presence of first-order autocorrelation when regressors contain the lagged dependent variable whose name is specified as LAGDEP=name. If the Durbin h statistic cannot be computed, the asymptotically equivalent t statistic is printed instead. See "Generalized Durbin-Watson Tests" for details.

When the regression model contains several lags of the dependent variable, specify the lagged dependent variable for the smallest lag in the LAGDEP= option, for example,


   model y = x1 x2 ylag2 ylag3 / lagdep=ylag2;

LOGLIKL

The LOGLIKL option prints the log likelihood value of the regression model, assuming normally distributed errors.

NOPRINT

suppresses all printed output.

NORMAL

The NORMAL option specifies the Jarque-Bera's normality test statistic for regression residuals.

PARTIAL

prints partial autocorrelations.

PCHOW= ( obs₁ ... obs_n )

The PCHOW= option computes the predictive Chow test. The form of the PCHOW= option is the same as the CHOW= option; see the discussion of the CHOW= option earlier in this chapter.

RESET

The RESET option produces Ramsey's RESET test statistics. The RESET option tests the null model

$y_{t} = x_{t}{\beta} + u_{t}$

against the alternative

$y_{t} = x_{t}{\beta} + \sum_{j=2}^p{{\phi}_{j} \hat{y}^j_{t}} + u_{t}$

where ${ \hat{y}_{t}}$ is the predicted value from the OLS estimation of the null model. The RESET option produces three RESET test statistics for p = 2, 3, and 4.

STATIONARITY= ( PHILLIPS )

STATIONARITY= ( PHILLIPS=( value ... value ) )

The STATIONARITY= option specifies tests of stationarity or unit roots. The STATIONARITY= option provides Phillips-Perron tests.

The PHILLIPS or PHILLIPS= suboption of the STATIONARITY= option produces the Phillips-Perron unit root test when there are no regressors in the MODEL statement. When the model includes regressors, the PHILLIPS option produces the Phillips-Ouliaris cointegration test. The PHILLIPS option can be abbreviated as PP.

The PHILLIPS option performs the Phillips-Perron test for three null hypothesis cases: zero mean, single mean, and deterministic trend. For each case, the PHILLIPS option computes two test statistics, Z( $\hat{\alpha}$ ) and ${\rm{Z}(t_{\hat{{\alpha}}})}$ , and reports their p-values. These test statistics have the same limiting distributions as the corresponding Dickey-Fuller tests.

The three types of the Phillips-Perron unit root test reported by the PHILLIPS option are as follows.

Zero Mean

computes the Phillips-Perron test statistic based on the zero mean autoregressive model

$y_{t} = {\alpha}y_{t-1} + u_{t}$

Single Mean

computes the Phillips-Perron test statistic based on the autoregressive model with a constant term

$y_{t} = {\mu} + {\alpha}y_{t-1} + u_{t}$

Trend

computes the Phillips-Perron test statistic based on the autoregressive model with constant and time trend terms

$y_{t} = {\mu} + {\alpha}y_{t-1} + {\delta}t + u_{t}$

You can specify several truncation points l for weighted variance estimators using the PHILLIPS=(l₁ ... l_n) specification.

URSQ

The URSQ option prints the uncentered regression R². The uncentered regression R² is useful to compute Lagrange multiplier test statistics, since most LM test statistics are computed as T*URSQ, where T is the number of observations used in estimation.

Stepwise Selection Options

BACKSTEP: removes insignificant autoregressive parameters. The parameters are removed in order of least significance. This backward elimination is done only once on the Yule-Walker estimates computed after the initial ordinary least-squares estimation. The BACKSTEP option can be used with all estimation methods since the initial parameter values for other estimation methods are estimated using the Yule-Walker method.
SLSTAY= value: specifies the significance level criterion to be used by the BACKSTEP option. The default is SLSTAY=.05.

Estimation Control Options

CONVERGE= value

specifies the convergence criterion. If the maximum absolute value of the change in the autoregressive parameter estimates between iterations is less than this amount, then convergence is assumed. The default is CONVERGE=.001.

INITIAL= ( initial-values )

START= ( initial-values )

The INITIAL= option specifies initial values for some or all of the parameter estimates. The values specified are assigned to model parameters in the same order as the parameter estimates are printed in the AUTOREG procedure output. The order of values in the INITIAL= or START= option is: the intercept, the regressor coefficients, the autoregressive parameters, the ARCH parameters, the GARCH parameters, the inverted degrees of freedom for Student's t distribution, the start-up value for conditional variance, and the heteroscedasticity model parameters ${\eta}$ specified by the HETERO statement.

The following is an example of specifying initial values for an AR(1)-GARCH(1,1) model with regressors X1 and X2:


   model y = w x / nlag=1 garch=(p=1,q=1)
                   initial=(1 1 1 .5 .8 .1 .6);

The model specified by this MODEL statement is

$y_{t} = {\beta}_{0} + {\beta}_{1} w_{t} + {\beta}_{2} x_{t} + {\nu}_{t}$

${\nu}_{t} = {\epsilon}_{t} - {\phi}_{1}{\nu}_{t-1}$

${\epsilon}_{t} = \sqrt{ h_{t}} e_{t}$

$h_{t} = {\omega} + {\alpha}_{1} {\epsilon}_{t-1}^2 + {\gamma}_{1} h_{t-1}$

${\epsilon}_{t} \rm{N}(0, {\sigma}_{t}^2)$

The initial values for the regression parameters, INTERCEP ( ${{\beta}_{0}}$ ), X1 ( ${{\beta}_{1}}$ ), and X2 ( ${{\beta}_{2}}$ ), are specified as 1. The initial value of the AR(1) coefficient ( ${{\phi}_{1}}$ ) is specified as 0.5. The initial value of ARCH0 ( ${{\omega}}$ ) is 0.8, the initial value of ARCH1 ( ${ {\alpha}_{1}}$ ) is 0.1, and the initial value of GARCH1 ( ${ {\gamma}_{1}}$ ) is 0.6.

When you use the RESTRICT statement, the initial values specified by the INITIAL= option should satisfy the restrictions specified for the parameter estimates. If they do not, the initial values you specify are adjusted to satisfy the restrictions.

LDW

The LDW option specifies that p-values for the Durbin-Watson test be computed using a linearized approximation of the design matrix when the model is nonlinear due to the presence of an autoregressive error process. (The Durbin-Watson tests of the OLS linear regression model residuals are not affected by the LDW option.) Refer to White (1992) for Durbin-Watson testing of nonlinear models.

MAXITER= number

sets the maximum number of iterations allowed. The default is MAXITER=50.

METHOD= value

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

METHOD=ML: specifies maximum likelihood estimates
METHOD=ULS: specifies unconditional least-squares estimates
METHOD=YW: specifies Yule-Walker estimates
METHOD=ITYW: specifies iterative Yule-Walker estimates

If the GARCH= or LAGDEP option is specified, the default is METHOD=ML. Otherwise, the default is METHOD=YW.

NOMISS

requests the estimation to the first contiguous sequence of data with no missing values. Otherwise, all complete observations are used.

OPTMETHOD= QN | TR

The OPTMETHOD= option specifies the optimization technique when the GARCH or heteroscedasticity model is estimated. The OPTMETHOD=QN option specifies the quasi-Newton method. The OPTMETHOD=TR option specifies the trust region method. The default is OPTMETHOD=QN.

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