The AUTOREG Procedure

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.

specifies the distribution assumed for the error term in GARCH-type estimation. If no GARCH= option is specified, the option is ignored. If EGARCH is specified, the distribution is always the normal distribution. The values of the DIST= option are as follows:

T

specifies Student’s t distribution.

NORMAL

specifies the standard normal distribution. The default is DIST=NORMAL.

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 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, such as ARCH, 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 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.

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

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

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

EXP | EGARCH

specifies the exponential GARCH or EGARCH model.

INTEGRATED | IGARCH

specifies the integrated GARCH or IGARCH model.

NELSON | NELSONCAO

specifies the Nelson-Cao inequality constraints.

NONNEG

specifies the GARCH model with nonnegativity constraints.

POWER | PGARCH

specifies the power GARCH or PGARCH model.

QUADR | QUADRATIC | QGARCH

specifies the quadratic GARCH or QGARCH model.

STATIONARY

constrains the sum of GARCH coefficients to be less than 1.

THRES | THRESHOLD | TGARCH

specifies the threshold GARCH or TGARCH model.

The default is TYPE=NELSON.

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

LINEAR

specifies the linear function:

LOG

specifies the log function:

SQRT

specifies the square root function:

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

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.

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.

requests all printing options.

model return = x1 x2 / archtest;

is equivalent to the statement

model return = x1 x2 / archtest=(qlm);

The following statement requests Lee and King’s tests and Wong and Li’s tests:

model return = / archtest=(lk,wl);

If BDS is defined without additional suboptions, all suboptions are set as default values. That is, the statement

model return = x1 x2 / nlag=1 BDS;

is equivalent to the statement

model return = x1 x2 / nlag=1 BDS=(M=20, D=1.5, PVALUE=DIST, Z=Y);

To do the specification check of a GARCH(1,1) model, you can write the SAS statement as follows:

model return = / garch=(p=1,q=1) BDS=(Z=SR);

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

Each value listed on the CHOW= option specifies a break point of the sample. The sample is divided into parts at the specified break point, with observations before in the first part and and later observations in the second part, and the fits of the model in the two parts are compared to whether both parts of the sample are consistent with the same model.

The break points refer to observations within the time range of the dependent variable, ignoring missing values before the start of the dependent series. Thus, CHOW=20 specifies the 20th observation after the first nonmissing observation for the dependent variable. For example, if the dependent variable Y contains 10 missing values before the first observation with a nonmissing Y value, then CHOW=20 actually refers to the 30th observation in the data set.

When you specify the break point, you should note the number of presample missing values.

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.

prints the estimated correlations of the parameter estimates.

prints the estimated covariances of the parameter estimates.

specifies the type of covariance matrix. 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 by using the Hessian matrix. The quasi-maximum likelihood estimates are computed with COVEST=QML, which is equivalent to COVEST=HC0. When the final model is an OLS or AR error model, COVEST=OP, HESSIAN, or QML, is ignored, and the method to calculate the estimate of covariance matrix is illustrated in the section Variance Estimates and Standard Errors. When COVEST=HC, where is specified, the corresponding heteroscedasticity consistent covariance matrix estimator (HCCME) is calculated. See the section Heteroscedasticity Consistent Covariance Matrix Estimator for more details. The default is COVEST=OP.

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.

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).

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

produces Godfrey’s general Lagrange multiplier test against ARMA errors.

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.

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

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 the section 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;

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

suppresses all printed output.

specifies the Jarque-Bera’s normality test statistic for regression residuals.

prints partial autocorrelations.

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.

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

against the alternative

where is the predicted value from the OLS estimation of the null model. The RESET option produces three RESET test statistics for , 3, and 4.

specifies the runs test for independence. The Z= suboption specifies the type of the time series or residuals to be tested. The values of the Z= suboption are as follows:

Y

specifies the regressand. The default is Z=Y.

RO

specifies the OLS residuals.

R

specifies the residuals of the final model.

RM

specifies the structural residuals of the final model.

SR

specifies the standardized residuals of the final model, defined by residuals over the square root of the conditional variance.

specifies tests of stationarity or unit roots. The STATIONARITY= option provides Phillips-Perron, Phillips-Ouliaris, augmented Dickey-Fuller, Engle-Granger, KPSS, ERS, and NP 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, and (in the original paper they are referred to as and ) , 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:

Single mean

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

Trend

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

You can specify several truncation points for weighted variance estimators by using the PHILLIPS=() specification. The statistic for each truncation point is computed as

where and are OLS residuals. If you specify the PHILLIPS option without specifying truncation points, the default truncation point is , where is the number of observations.

The Phillips-Perron test can be used in general time series models since its limiting distribution is derived in the context of a class of weakly dependent and heterogeneously distributed data. The marginal probability for the Phillips-Perron test is computed assuming that error disturbances are normally distributed.

When there are regressors in the MODEL statement, the PHILLIPS option computes the Phillips-Ouliaris cointegration test statistic by using the least squares residuals. The normalized cointegrating vector is estimated using OLS regression. Therefore, the cointegrating vector estimates might vary with the regressand (normalized element) unless the regression R-square is 1.

The marginal probabilities for cointegration testing are not produced. You can refer to Phillips and Ouliaris (1990) tables Ia–Ic for the test and tables IIa–IIc for the test. The standard residual-based cointegration test can be obtained using the NOINT option in the MODEL statement, while the demeaned test is computed by including the intercept term. To obtain the demeaned and detrended cointegration tests, you should include the time trend variable in the regressors. Refer to Phillips and Ouliaris (1990) or Hamilton (1994, Tbl. 19.1) for information about the Phillips-Ouliaris cointegration test. Note that Hamilton (1994, Tbl. 19.1) uses and instead of the original Phillips and Ouliaris (1990) notation. We adopt the notation introduced in Hamilton. To distinguish from Student’s distribution, these two statistics are named accordingly as (rho) and (tau).

The ADF or ADF= suboption produces the augmented Dickey-Fuller unit root test (Dickey and Fuller; 1979). As in the Phillips-Perron test, three regression models can be specified for the null hypothesis for the augmented Dickey-Fuller test (zero mean, single mean, and trend). These models assume that the disturbances are distributed as white noise. The augmented Dickey-Fuller test can account for the serial correlation between the disturbances in some way. The model, with the time trend specification for example, is

This formulation has the advantage that it can accommodate higher-order autoregressive processes in . The test statistic follows the same distribution as the Dickey-Fuller test statistic. For more information, see the section PROBDF Function for Dickey-Fuller Tests.

In the presence of regressors, the ADF option tests the cointegration relation between the dependent variable and the regressors. Following Engle and Granger (1987), a two-step estimation and testing procedure is carried out, in a fashion similar to the Phillips-Ouliaris test. The OLS residuals of the regression in the MODEL statement are used to compute the t statistic of the augmented Dickey-Fuller regression in a second step. Three cases arise based on which type of deterministic terms are included in the first step of regression. Only the constant term and linear trend cases are practically useful (Davidson and MacKinnon; 1993, page 721), and therefore are computed and reported. The test statistic, as shown in Phillips and Ouliaris (1990), follows the same distribution as the statistic in the Phillips-Ouliaris cointegration test. The asymptotic distribution is tabulated in tables IIa–IIc of Phillips and Ouliaris (1990), and the finite sample distribution is obtained in Table 2 and Table 3 in Engle and Yoo (1987) by Monte Carlo simulation.

The experimental ERS or ERS= suboption and NP or NP= suboption provide a class of efficient unit root tests, in the sense that they reduce the size distortion and improve the power compared with traditional unit root tests such as augmented Dickey-Fuller and Phillips-Perron tests. Two test statistics are provided by the ERS test: the point optimal test and the DF-GLS test, which are originally proposed in Elliott, Rothenberg, and Stock (1996). Four different tests, discussed in Ng and Perron (2001), are reported by NP test. These four tests include the two in the ERS test and two other tests, the modified PP test and the modified point optimal test, discussed in Ng and Perron (2001). The authors suggest using the modified AIC to select the optimal lag length in the augmented Dickey-Fuller type regression. The maximum lag length can be provided by using the NP= suboption. The default maximum lag length is 8.

The KPSS, KPSS=(KERNEL=TYPE), or KPSS=(KERNEL=TYPE TRUNCPOINTMETHOD) specifications of the STATIONARITY= option produce the Kwiatkowski, Phillips, Schmidt, and Shin (1992) (KPSS) unit root test.

Unlike the null hypothesis of the Dickey-Fuller and Phillips-Perron tests, the null hypothesis of the KPSS states that the time series is stationary. As a result, it tends to reject a random walk more often. If the model does not have an intercept, the KPSS option performs the KPSS test for three null hypothesis cases: zero mean, single mean, and deterministic trend. Otherwise, it reports the single mean and deterministic trend only. It computes a test statistic and provides tabulated critical values (see Hobijn, Franses, and Ooms (2004)) for the hypothesis that the random walk component of the time series is equal to zero in the following cases (for details, see Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) Unit Root Test):

Zero mean

computes the KPSS test statistic based on the zero mean autoregressive model. The p-value reported is used from Hobijn, Franses, and Ooms (2004).

Single mean

computes the KPSS test statistic based on the autoregressive model with a constant term. The p-value reported is used from Kwiatkowski et al. (1992).

Trend

computes the KPSS test statistic based on the autoregressive model with constant and time trend terms. The p-value reported is from Kwiatkowski et al. (1992).

This test depends on the long-run variance of the series being defined as

where is a kernel, is a maximum lag (truncation point), and are OLS residuals or original data series. You can specify two types of the kernel:

KERNEL=NW | BART

Newey-West (or Bartlett) kernel

KERNEL=QS

Quadratic spectral kernel

You can set the truncation point by using three different methods:

SCHW=c

Schwert maximum lag formula

LAG=

LAG= manually defined number of lags.

AUTO

Automatic bandwidth selection (Hobijn, Franses, and Ooms; 2004) (for details, see Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) Unit Root Test).

If STATIONARITY=KPSS is defined without additional parameters, the Newey-West kernel is used. For the Newey-West kernel the default is the Schwert truncation point method with . For the quadratic spectral kernel the default is AUTO.

The KPSS test can be used in general time series models since its limiting distribution is derived in the context of a class of weakly dependent and heterogeneously distributed data. The limiting probability for the KPSS test is computed assuming that error disturbances are normally distributed.

The marginal probabilities for the test are reported. They are copied from Kwiatkowski et al. (1992) and Hobijn, Franses, and Ooms (2004). When there is an intercept in the model, results for mean and trend tests statistics are provided.

Examples: To test for stationarity of a variable, y, using default KERNEL= NW and SCHW= 4, you can use the following code:

/*-- test for stationarity of regression residuals --*/
proc autoreg data=a;
   model y=  / stationarity = (KPSS);
run;

To test for stationarity of a variable, y, using quadratic spectral kernel and automatic bandwidth selection, you can use:

/*-- test for stationarity using quadratic
     spectral kernel and automatic bandwidth selection --*/
proc autoreg data=a;
   model y= /
   stationarity = (KPSS=(KERNEL=QS AUTO));
run;

specifies the turning point test for independence. The Z= suboption specifies the type of the time series or residuals to be tested. The values of the Z= suboption are as follows:

Y

specifies the regressand. The default is Z=Y.

RO

specifies the OLS residuals.

R

specifies the residuals of the final model.

RM

specifies the structural residuals of the final model.

SR

specifies the standardized residuals of the final model, defined by residuals over the square root of the conditional variance.

prints the uncentered regression . The uncentered regression 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.

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.

specifies the significance level criterion to be used by the BACKSTEP option. The default is SLSTAY=.05.

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.

If the GARCH= option and/or the HETERO statement is specified, convergence is assumed when the absolute maximum gradient is smaller than the value specified by the CONVERGE= option or when the relative gradient is smaller than 1E–8. By default, CONVERGE=1E–5.

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 as follows: 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 specified by the HETERO statement.

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

/*-- specifying initial values --*/
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

The initial values for the regression parameters, INTERCEPT (), X1 (), and X2 (), are specified as 1. The initial value of the AR(1) coefficient () is specified as 0.5. The initial value of ARCH0 () is 0.8, the initial value of ARCH1 () is 0.1, and the initial value of GARCH1 () 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.

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.

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

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

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.

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

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.