Example 14.4: MA(1) Estimation
This example estimates parameters for an MA(1) error process for the
Grunfeld model, using both the unconditional
least-squares and the maximum-likelihood methods.
The ARIMA procedure estimates for Westinghouse equation
are shown for comparison.
The output of the following code is summarized in Output 14.4.1:
title1 'Example of MA(1) Error Process Using Grunfeld''s Model';
title2 'MA(1) Error Process Using Unconditional Least Squares';
proc model data=grunfeld model=grunmod;
%ma(gei,1, m=uls);
%ma(whi,1, m=uls);
fit whi gei start=( gei_m1 0.8 -0.8) / startiter=2;
run;
Output 14.4.1: PROC MODEL Results Using ULS Estimation
|
| Example of MA(1) Error Process Using Grunfeld's Model |
| MA(1) Error Process Using Unconditional Least Squares |
| Nonlinear OLS Summary of Residual Errors |
| Equation |
DF Model |
DF Error |
SSE |
MSE |
Root MSE |
R-Square |
Adj R-Sq |
Label |
| whi |
4 |
16 |
1874.0 |
117.1 |
10.8224 |
0.7299 |
0.6793 |
Gross Investment WH |
| resid.whi |
|
16 |
1295.6 |
80.9754 |
8.9986 |
|
|
Gross Investment WH |
| gei |
4 |
16 |
13835.0 |
864.7 |
29.4055 |
0.6915 |
0.6337 |
Gross Investment GE |
| resid.gei |
|
16 |
7646.2 |
477.9 |
21.8607 |
|
|
Gross Investment GE |
| Nonlinear OLS Parameter Estimates |
| Parameter |
Estimate |
Approx Std Err |
t Value |
Approx
Pr > |t| |
Label |
| ge_int |
-26.839 |
32.0908 |
-0.84 |
0.4153 |
GE Intercept |
| ge_f |
0.038226 |
0.0150 |
2.54 |
0.0217 |
GE Lagged Share Value Coef |
| ge_c |
0.137099 |
0.0352 |
3.90 |
0.0013 |
GE Lagged Capital Stock Coef |
| wh_int |
3.680835 |
9.5448 |
0.39 |
0.7048 |
WH Intercept |
| wh_f |
0.049156 |
0.0172 |
2.85 |
0.0115 |
WH Lagged Share Value Coef |
| wh_c |
0.067271 |
0.0708 |
0.95 |
0.3559 |
WH Lagged Capital Stock Coef |
| gei_m1 |
-0.87615 |
0.1614 |
-5.43 |
<.0001 |
MA(gei) gei lag1 parameter |
| whi_m1 |
-0.75001 |
0.2368 |
-3.17 |
0.0060 |
MA(whi) whi lag1 parameter |
|
The estimation summary from the following PROC ARIMA statements is
shown in Output 14.4.2.
title2 'PROC ARIMA Using Unconditional Least Squares';
proc arima data=grunfeld;
identify var=whi cross=(whf whc ) noprint;
estimate q=1 input=(whf whc) method=uls maxiter=40;
run;
Output 14.4.2: PROC ARIMA Results Using ULS Estimation
|
| Example of MA(1) Error Process Using Grunfeld's Model |
| PROC ARIMA Using Unconditional Least Squares |
| Unconditional Least Squares Estimation |
| Parameter |
Estimate |
Approx Std Error |
t Value |
Pr > |t| |
Lag |
Variable |
Shift |
| MU |
3.68608 |
9.54425 |
0.39 |
0.7044 |
0 |
whi |
0 |
| MA1,1 |
-0.75005 |
0.23704 |
-3.16 |
0.0060 |
1 |
whi |
0 |
| NUM1 |
0.04914 |
0.01723 |
2.85 |
0.0115 |
0 |
whf |
0 |
| NUM2 |
0.06731 |
0.07077 |
0.95 |
0.3557 |
0 |
whc |
0 |
| Constant Estimate |
3.686077 |
| Variance Estimate |
80.97535 |
| Std Error Estimate |
8.998631 |
| AIC |
149.0044 |
| SBC |
152.9873 |
| Number of Residuals |
20 |
|
The model stored in Example 14.3 is read in using the MODEL= option
and the moving average terms are added using the %MA macro.
The MA(1) model using maximum likelihood is estimated using the following:
title2 'MA(1) Error Process Using Maximum Likelihood ';
proc model data=grunfeld model=grunmod;
%ma(gei,1, m=ml);
%ma(whi,1, m=ml);
fit whi gei;
run;
For comparison, the model is estimated using PROC ARIMA as follows:
title2 'PROC ARIMA Using Maximum Likelihood ';
proc arima data=grunfeld;
identify var=whi cross=(whf whc) noprint;
estimate q=1 input=(whf whc) method=ml;
run;
PROC ARIMA does not estimate systems so only one equation is
evaluated.
The estimation results are shown in Output 14.4.3 and Output 14.4.4.
The small differences in the parameter values between
PROC MODEL and PROC ARIMA can be eliminated
by tightening the convergence criteria for both procedures.
Output 14.4.3: PROC MODEL Results Using ML Estimation
|
| Example of MA(1) Error Process Using Grunfeld's Model |
| MA(1) Error Process Using Maximum Likelihood |
| Nonlinear OLS Summary of Residual Errors |
| Equation |
DF Model |
DF Error |
SSE |
MSE |
Root MSE |
R-Square |
Adj R-Sq |
Label |
| whi |
4 |
16 |
1857.5 |
116.1 |
10.7746 |
0.7323 |
0.6821 |
Gross Investment WH |
| resid.whi |
|
16 |
1344.0 |
84.0012 |
9.1652 |
|
|
Gross Investment WH |
| gei |
4 |
16 |
13742.5 |
858.9 |
29.3071 |
0.6936 |
0.6361 |
Gross Investment GE |
| resid.gei |
|
16 |
8095.3 |
506.0 |
22.4935 |
|
|
Gross Investment GE |
| Nonlinear OLS Parameter Estimates |
| Parameter |
Estimate |
Approx Std Err |
t Value |
Approx
Pr > |t| |
Label |
| ge_int |
-25.002 |
34.2933 |
-0.73 |
0.4765 |
GE Intercept |
| ge_f |
0.03712 |
0.0161 |
2.30 |
0.0351 |
GE Lagged Share Value Coef |
| ge_c |
0.137788 |
0.0380 |
3.63 |
0.0023 |
GE Lagged Capital Stock Coef |
| wh_int |
2.946761 |
9.5638 |
0.31 |
0.7620 |
WH Intercept |
| wh_f |
0.050395 |
0.0174 |
2.89 |
0.0106 |
WH Lagged Share Value Coef |
| wh_c |
0.066531 |
0.0729 |
0.91 |
0.3749 |
WH Lagged Capital Stock Coef |
| gei_m1 |
-0.78516 |
0.1942 |
-4.04 |
0.0009 |
MA(gei) gei lag1 parameter |
| whi_m1 |
-0.69389 |
0.2540 |
-2.73 |
0.0148 |
MA(whi) whi lag1 parameter |
|
Output 14.4.4: PROC ARIMA Results Using ML Estimation
|
| Example of MA(1) Error Process Using Grunfeld's Model |
| PROC ARIMA Using Maximum Likelihood |
| Maximum Likelihood Estimation |
| Parameter |
Estimate |
Approx Std Error |
t Value |
Pr > |t| |
Lag |
Variable |
Shift |
| MU |
2.95645 |
9.20752 |
0.32 |
0.7481 |
0 |
whi |
0 |
| MA1,1 |
-0.69305 |
0.25307 |
-2.74 |
0.0062 |
1 |
whi |
0 |
| NUM1 |
0.05036 |
0.01686 |
2.99 |
0.0028 |
0 |
whf |
0 |
| NUM2 |
0.06672 |
0.06939 |
0.96 |
0.3363 |
0 |
whc |
0 |
| Constant Estimate |
2.956449 |
| Variance Estimate |
81.29645 |
| Std Error Estimate |
9.016455 |
| AIC |
148.9113 |
| SBC |
152.8942 |
| Number of Residuals |
20 |
|
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
