RESTRICT Statement
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The VARMAX Procedure

RESTRICT Statement

RESTRICT restriction, …, restriction ;

The RESTRICT statement restricts the specified parameters to the specified values. Only one RESTRICT statement is allowed, but multiple restrictions can be specified in one RESTRICT statement.

The syntax for restriction is parameter=value, and each restriction is separated by commas. Parameters are referred by the following keywords:

  • CONST($i$) is the intercept parameter of the $i$th time series $y_{it}$

  • AR($l,i,j$) is the autoregressive parameter of the lag $l$ value of the $j$th dependent (endogenous) variable, $y_{j,t-l}$, to the $i$th dependent variable at time $t$, $y_{it}$

  • MA($l,i,j$) is the moving-average parameter of the lag $l$ value of the $j$th error process, $\bepsilon _{j,t-l}$, to the $i$th dependent variable at time $t$, $y_{it}$

  • XL($l,i,j$) is the exogenous parameter of the lag $l$ value of the $j$th exogenous (independent) variable, $x_{j,t-l}$, to the $i$th dependent variable at time $t$, $y_{it}$

  • SDUMMY($i,j$) is the $j$th seasonal dummy of the $i$th time series at time $t$, $y_{it}$, where $j=1,\ldots ,($nseason$-1)$, where nseason is based on the NSEASON= option in the MODEL statement

  • LTREND($i$) is the linear trend parameter of the current value $i$th time series $y_{it}$

  • QTREND($i$) is the quadratic trend parameter of the current value $i$th time series $y_{it}$

The following keywords are for the fitted GARCH model. The indexes $i$ and $j$ refer to the position of the element in the coefficient matrix.

  • GCHC($i$,$j$) is the constant parameter of the covariance matrix, $H_{t}$, and ($i$,$j$) is $1\leq i=j \leq k$ for CCC representation and $1\leq i \leq j \leq k$ for BEKK representations, where $k$ is the number of dependent variables

  • ACH($l$,$i$,$j$) is the ARCH parameter of the lag $l$ value of $\bepsilon _{t} \bepsilon _{t}’$, where $i,j=1,\ldots ,k$ for BEKK representation and $i=j=1,\ldots ,k$ for CCC representation

  • GCH($l$,$i$,$j$) is the GARCH parameter of the lag $l$ value of covariance matrix, $H_{t}$, where $i,j=1,\ldots ,k$ for BEKK representation and $i=j=1,\ldots ,k$ for CCC representation

  • CCC($i$,$j$) is the constant conditional correlation parameter for only the CCC representation; ($i$,$j$) is $1\leq i < j \leq k$

To use the RESTRICT statement, you need to know the form of the model. If the P=, Q=, and XLAG= options are not specified, then the RESTRICT statement is not applicable.

Restricted parameter estimates are computed by introducing a Lagrangian parameter for each restriction (Pringle and Rayner 1971). The Lagrangian parameter measures the sensitivity of the sum of square errors to the restriction. The estimates of these Lagrangian parameters and their significance are printed in the restriction results table.

The following are examples of the RESTRICT statement. The first example shows a bivariate ($k$=2) VAR(2) model, 

   proc varmax data=one;
      model y1 y2 / p=2;
      restrict AR(1,1,2)=0, AR(2,1,2)=0.3;
   run;

The AR(1,1,2) and AR(2,1,2) parameters are fixed as AR(1,1,2)=0 and AR(2,1,2)=0.3, respectively, and other parameters are to be estimated.

The following shows a bivariate ($k$=2) VARX(1,1) model with three exogenous variables, 

   proc varmax data=two;
      model y1 = x1 x2, y2 = x2 x3 / p=1 xlag=1;
      restrict XL(0,1,1)=-1.2, XL(1,2,3)=0;
   run;

The XL(0,1,1) and XL(1,2,3) parameters are fixed as XL(0,1,1)=–1.2 and XL(1,2,3)=0, respectively, and other parameters are to be estimated.

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