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 The MODEL Procedure

## 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

 The MODEL Procedure

 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

 The ARIMA Procedure

 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 3.68608 Variance Estimate 80.9754 Std Error Estimate 8.99863 AIC 149.004 SBC 152.987 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

 The MODEL Procedure

 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

 The ARIMA Procedure

 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 2.95645 Variance Estimate 81.2964 Std Error Estimate 9.01646 AIC 148.911 SBC 152.894 Number of Residuals 20

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