SAS/ETS 9.22 User''''s Guide 116 ppt

The %AR macro can be used for the following types of autoregression: univariate autoregression unrestricted vector autoregression restricted vector autoregression Univariate Autoregres

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MA Initial Conditions

The initial lags of the error terms of MA(q ) models can also be modeled in different ways The following moving-average error start-up paradigms are supported by the ARIMA and MODEL procedures:

ULS unconditional least squares

CLS conditional least squares

ML maximum likelihood

The conditional least squares method of estimating moving-average error terms is not optimal because it ignores the start-up problem This reduces the efficiency of the estimates, although they remain unbiased The initial lagged residuals, extending before the start of the data, are assumed

to be 0, their unconditional expected value This introduces a difference between these residuals and the generalized least squares residuals for the moving-average covariance, which, unlike the autoregressive model, persists through the data set Usually this difference converges quickly to 0, but for nearly noninvertible moving-average processes the convergence is quite slow To minimize this problem, you should have plenty of data, and the moving-average parameter estimates should be well within the invertible range

This problem can be corrected at the expense of writing a more complex program Unconditional least squares estimates for the MA(1) process can be produced by specifying the model as follows:

yhat = compute structural predicted value here ;

if _obs_ = 1 then do;

h = sqrt( 1 + ma1 ** 2 );

y = yhat;

resid.y = ( y - yhat ) / h;

end;

else do;

g = ma1 / zlag1( h );

h = sqrt( 1 + ma1 ** 2 - g ** 2 );

y = yhat + g * zlag1( resid.y );

resid.y = ( ( y - yhat) - g * zlag1( resid.y ) ) / h;

end;

Moving-average errors can be difficult to estimate You should consider using an AR(p ) approxima-tion to the moving-average process A moving-average process can usually be well-approximated by

an autoregressive process if the data have not been smoothed or differenced

The %AR Macro

The SAS macro %AR generates programming statements for PROC MODEL for autoregressive models The %AR macro is part of SAS/ETS software, and no special options need to be set to use the macro The autoregressive process can be applied to the structural equation errors or to the endogenous series themselves

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The %AR macro can be used for the following types of autoregression:

univariate autoregression

unrestricted vector autoregression

restricted vector autoregression

Univariate Autoregression

To model the error term of an equation as an autoregressive process, use the following statement after the equation:

%ar( varname, nlags )

For example, suppose that Y is a linear function of X1, X2, and an AR(2) error You would write this model as follows:

proc model data=in;

parms a b c;

y = a + b * x1 + c * x2;

%ar( y, 2 )

fit y / list;

run;

The calls to %AR must come after all of the equations that the process applies to

The preceding macro invocation, %AR(y,2), produces the statements shown in the LIST output in Figure 18.58

Figure 18.58 LIST Option Output for an AR(2) Model

The MODEL Procedure

Listing of Compiled Program Code Stmt Line:Col Statement as Parsed

1 2148:4 PRED.y = a + b * x1 + c * x2;

1 2148:4 RESID.y = PRED.y - ACTUAL.y;

1 2148:4 ERROR.y = PRED.y - y;

2 2149:14 _PRED y = PRED.y;

3 2149:15 _OLD_PRED.y = PRED.y + y_l1

* ZLAG1( y - _PRED y ) + y_l2

* ZLAG2( y - _PRED y );

3 2149:15 PRED.y = _OLD_PRED.y;

3 2149:15 ERROR.y = PRED.y - y;

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The _PRED prefixed variables are temporary program variables used so that the lags of the residuals are the correct residuals and not the ones redefined by this equation Note that this is equivalent to the statements explicitly written in the section “General Form for ARMA Models” on page 1140

You can also restrict the autoregressive parameters to zero at selected lags For example, if you wanted autoregressive parameters at lags 1, 12, and 13, you can use the following statements:

parms a b c;

y = a + b * x1 + c * x2;

%ar( y, 13, , 1 12 13 )

fit y / list;

run;

These statements generate the output shown inFigure 18.59

Figure 18.59 LIST Option Output for an AR Model with Lags at 1, 12, and 13

1 2157:4 PRED.y = a + b * x1 + c * x2;

1 2157:4 ERROR.y = PRED.y - y;

2 2158:14 _PRED y = PRED.y;

3 2158:15 _OLD_PRED.y = PRED.y + y_l1 * ZLAG1( y

_PRED y ) + y_l12 * ZLAG12( y -_PRED y ) + y_l13 * ZLAG13(

y - _PRED y );

3 2158:15 PRED.y = _OLD_PRED.y;

3 2158:15 ERROR.y = PRED.y - y;

There are variations on the conditional least squares method, depending on whether observations

at the start of the series are used to “warm up” the AR process By default, the %AR conditional least squares method uses all the observations and assumes zeros for the initial lags of autoregressive terms By using the M= option, you can request that %AR use the unconditional least squares (ULS)

or maximum-likelihood (ML) method instead For example,

y = a + b * x1 + c * x2;

%ar( y, 2, m=uls )

fit y;

run;

Discussions of these methods is provided in the section “AR Initial Conditions” on page 1141

By using the M=CLSn option, you can request that the first n observations be used to compute estimates of the initial autoregressive lags In this case, the analysis starts with observation n +1 For example:

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y = a + b * x1 + c * x2;

%ar( y, 2, m=cls2 )

fit y;

run;

You can use the %AR macro to apply an autoregressive model to the endogenous variable, instead of

to the error term, by using the TYPE=V option For example, if you want to add the five past lags of

Y to the equation in the previous example, you could use %AR to generate the parameters and lags

by using the following statements:

parms a b c;

y = a + b * x1 + c * x2;

%ar( y, 5, type=v )

fit y / list;

run;

The preceding statements generate the output shown inFigure 18.60

Figure 18.60 LIST Option Output for an AR model of Y

1 2180:4 PRED.y = a + b * x1 + c * x2;

1 2180:4 ERROR.y = PRED.y - y;

2 2181:15 _OLD_PRED.y = PRED.y + y_l1 * ZLAG1( y )

+ y_l2 * ZLAG2( y ) + y_l3 * ZLAG3( y ) + y_l4 * ZLAG4( y ) + y_l5 * ZLAG5( y );

2 2181:15 PRED.y = _OLD_PRED.y;

2 2181:15 ERROR.y = PRED.y - y;

This model predicts Y as a linear combination of X1, X2, an intercept, and the values of Y in the most recent five periods

Unrestricted Vector Autoregression

To model the error terms of a set of equations as a vector autoregressive process, use the following form of the %AR macro after the equations:

%ar( process_name, nlags, variable_list )

The process_name value is any name that you supply for %AR to use in making names for the autoregressive parameters You can use the %AR macro to model several different AR processes for

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different sets of equations by using different process names for each set The process name ensures that the variable names used are unique Use a short process_name value for the process if parameter estimates are to be written to an output data set The %AR macro tries to construct parameter names less than or equal to eight characters, but this is limited by the length of process_name, which is used

as a prefix for the AR parameter names

The variable_list value is the list of endogenous variables for the equations

For example, suppose that errors for equations Y1, Y2, and Y3 are generated by a second-order vector autoregressive process You can use the following statements:

y1 = equation for y1 ;

%ar( name, 2, y1 y2 y3 )

fit y1 y2 y3;

run;

which generate the following for Y1 and similar code for Y2 and Y3:

y1 = pred.y1 + name1_1_1*zlag1(y1-name_y1) +

name1_1_2*zlag1(y2-name_y2) +

name2_1_3*zlag2(y3-name_y3) ;

Only the conditional least squares (M=CLS or M=CLSn ) method can be used for vector processes You can also use the same form with restrictions that the coefficient matrix be 0 at selected lags For example, the following statements apply a third-order vector process to the equation errors with all the coefficients at lag 2 restricted to 0 and with the coefficients at lags 1 and 3 unrestricted:

%ar( name, 3, y1 y2 y3, 1 3 )

fit y1 y2 y3;

You can model the three series Y1–Y3 as a vector autoregressive process in the variables instead of

in the errors by using the TYPE=V option If you want to model Y1–Y3 as a function of past values

of Y1–Y3 and some exogenous variables or constants, you can use %AR to generate the statements for the lag terms Write an equation for each variable for the nonautoregressive part of the model, and then call %AR with the TYPE=V option For example,

parms a1-a3 b1-b3;

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y1 = a1 + b1 * x;

y2 = a2 + b2 * x;

y3 = a3 + b3 * x;

%ar( name, 2, y1 y2 y3, type=v )

fit y1 y2 y3;

run;

The nonautoregressive part of the model can be a function of exogenous variables, or it can be intercept parameters If there are no exogenous components to the vector autoregression model, including no intercepts, then assign zero to each of the variables There must be an assignment to each of the variables before %AR is called

y1=0;

y2=0;

y3=0;

%ar( name, 2, y1 y2 y3, type=v )

fit y1 y2 y3;

run;

This example models the vector Y=(Y1 Y2 Y3)0as a linear function only of its value in the previous two periods and a white noise error vector The model has 18=(3 3 + 3 3) parameters

Syntax of the %AR Macro

There are two cases of the syntax of the %AR macro When restrictions on a vector AR process are not needed, the syntax of the %AR macro has the general form

%AR ( name , nlag < ,endolist < , laglist > > < ,M= method > < ,TYPE= V > ) ;

where

name specifies a prefix for %AR to use in constructing names of variables needed to

define the AR process If the endolist is not specified, the endogenous list defaults

to name, which must be the name of the equation to which the AR error process

is to be applied The name value cannot exceed 32 characters

nlag is the order of the AR process

endolist specifies the list of equations to which the AR process is to be applied If more

than one name is given, an unrestricted vector process is created with the structural residuals of all the equations included as regressors in each of the equations If not specified, endolist defaults to name

laglist specifies the list of lags at which the AR terms are to be added The coefficients

of the terms at lags not listed are set to 0 All of the listed lags must be less than

or equal to nlag, and there must be no duplicates If not specified, the laglist defaults to all lags 1 through nlag

M=method specifies the estimation method to implement Valid values of M= are CLS

(conditional least squares estimates), ULS (unconditional least squares estimates), and ML (maximum likelihood estimates) M=CLS is the default Only M=CLS

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is allowed when more than one equation is specified The ULS and ML methods are not supported for vector AR models by %AR

TYPE=V specifies that the AR process is to be applied to the endogenous variables

them-selves instead of to the structural residuals of the equations

Restricted Vector Autoregression

You can control which parameters are included in the process, restricting to 0 those parameters that you do not include First, use %AR with the DEFER option to declare the variable list and define the dimension of the process Then, use additional %AR calls to generate terms for selected equations with selected variables at selected lags For example,

proc model data=d;

%ar( name, 2, y1 y2 y3, defer )

%ar( name, y1, y1 y2 )

%ar( name, y2 y3, , 1 )

fit y1 y2 y3;

run;

The error equations produced are as follows:

name1_1_2*zlag1(y2-name_y2) + name2_1_1*zlag2(y1-name_y1) +

name2_1_2*zlag2(y2-name_y2) ;

name1_2_2*zlag1(y2-name_y2) + name1_2_3*zlag1(y3-name_y3) ;

name1_3_2*zlag1(y2-name_y2) + name1_3_3*zlag1(y3-name_y3) ;

This model states that the errors for Y1 depend on the errors of both Y1 and Y2 (but not Y3) at both lags 1 and 2, and that the errors for Y2 and Y3 depend on the previous errors for all three variables, but only at lag 1

%AR Macro Syntax for Restricted Vector AR

An alternative use of %AR is allowed to impose restrictions on a vector AR process by calling %AR several times to specify different AR terms and lags for different equations

The first call has the general form

%AR( name, nlag, endolist , DEFER ) ;

where

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name specifies a prefix for %AR to use in constructing names of variables needed to

define the vector AR process

nlag specifies the order of the AR process

endolist specifies the list of equations to which the AR process is to be applied

DEFER specifies that %AR is not to generate the AR process but is to wait for further

information specified in later %AR calls for the same name value

The subsequent calls have the general form

%AR( name, eqlist, varlist, laglist,TYPE= )

where

name is the same as in the first call

eqlist specifies the list of equations to which the specifications in this %AR call are

to be applied Only names specified in the endolist value of the first call for the namevalue can appear in the list of equations in eqlist

varlist specifies the list of equations whose lagged structural residuals are to be included

as regressors in the equations in eqlist Only names in the endolist of the first call for the name value can appear in varlist If not specified, varlist defaults to endolist

laglist specifies the list of lags at which the AR terms are to be added The coefficients

of the terms at lags not listed are set to 0 All of the listed lags must be less than

or equal to the value of nlag, and there must be no duplicates If not specified, laglistdefaults to all lags 1 through nlag

The %MA Macro

The SAS macro %MA generates programming statements for PROC MODEL for moving-average models The %MA macro is part of SAS/ETS software, and no special options are needed to use the macro The moving-average error process can be applied to the structural equation errors The syntax of the %MA macro is the same as the %AR macro except there is no TYPE= argument

When you are using the %MA and %AR macros combined, the %MA macro must follow the %AR macro The following SAS/IML statements produce an ARMA(1, (1 3)) error process and save it in the data set MADAT2

/* use IML module to simulate a MA process */

proc iml;

phi = { 1 2 };

theta = { 1 3 0 5 };

y = armasim( phi, theta, 0, 1, 200, 32565 );

create madat2 from y[colname='y'];

append from y;

quit;

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The following PROC MODEL statements are used to estimate the parameters of this model by using maximum likelihood error structure:

title 'Maximum Likelihood ARMA(1, (1 3))';

proc model data=madat2;

y=0;

%ar( y, 1, , M=ml )

%ma( y, 3, , 1 3, M=ml ) /* %MA always after %AR */

fit y;

run;

title;

The estimates of the parameters produced by this run are shown inFigure 18.61

Figure 18.61 Estimates from an ARMA(1, (1 3)) Process

Maximum Likelihood ARMA(1, (1 3))

Nonlinear OLS Summary of Residual Errors

Nonlinear OLS Parameter Estimates

Parameter Estimate Std Err t Value Pr > |t| Label

parameter

parameter y_m3 -0.59384 0.0601 -9.88 <.0001 MA(y) y lag3

parameter

Syntax of the %MA Macro

There are two cases of the syntax for the %MA macro When restrictions on a vector MA process are not needed, the syntax of the %MA macro has the general form

%MA ( name , nlag < , endolist < , laglist > > < ,M= method > ) ;

where

name specifies a prefix for %MA to use in constructing names of variables needed to

define the MA process and is the default endolist

nlag is the order of the MA process

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endolist specifies the equations to which the MA process is to be applied If more than

one name is given, CLS estimation is used for the vector process

laglist specifies the lags at which the MA terms are to be added All of the listed lags

must be less than or equal to nlag, and there must be no duplicates If not specified, the laglist defaults to all lags 1 through nlag

M=method specifies the estimation method to implement Valid values of M= are CLS

(conditional least squares estimates), ULS (unconditional least squares estimates), and ML (maximum likelihood estimates) M=CLS is the default Only M=CLS

is allowed when more than one equation is specified in the endolist

%MA Macro Syntax for Restricted Vector Moving-Average

An alternative use of %MA is allowed to impose restrictions on a vector MA process by calling

%MA several times to specify different MA terms and lags for different equations

The first call has the general form

%MA( name , nlag , endolist , DEFER ) ;

where

name specifies a prefix for %MA to use in constructing names of variables needed to

define the vector MA process

nlag specifies the order of the MA process

endolist specifies the list of equations to which the MA process is to be applied

DEFER specifies that %MA is not to generate the MA process but is to wait for further

information specified in later %MA calls for the same name value

The subsequent calls have the general form

%MA( name, eqlist, varlist, laglist )

where

name is the same as in the first call

eqlist specifies the list of equations to which the specifications in this %MA call are to

be applied

varlist specifies the list of equations whose lagged structural residuals are to be included

as regressors in the equations in eqlist

laglist specifies the list of lags at which the MA terms are to be added

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