PROC MODEL: MA(1) Estimation

Example 19.4 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 statements is summarized in Output 19.4.1:

proc model outmodel=grunmod;
   var gei whi gef gec whf whc;
   parms ge_int ge_f ge_c wh_int wh_f wh_c;
   label ge_int = 'GE Intercept'
         ge_f   = 'GE Lagged Share Value Coef'
         ge_c   = 'GE Lagged Capital Stock Coef'
         wh_int = 'WH Intercept'
         wh_f   = 'WH Lagged Share Value Coef'
         wh_c   = 'WH Lagged Capital Stock Coef';
   gei = ge_int + ge_f * gef + ge_c * gec;
   whi = wh_int + wh_f * whf + wh_c * whc;
run;

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 19.4.1 PROC MODEL Results by 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 19.4.2.

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 19.4.2 PROC ARIMA Results by 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	Standard Error	t Value	Approx 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 19.3 is read in by using the MODEL= option and the moving-average terms are added using the %MA macro.

The MA(1) model using maximum likelihood is estimated by using the following statements:

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 by 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 19.4.3 and Output 19.4.4. 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 19.4.3 PROC MODEL Results by 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 19.4.4 PROC ARIMA Results by 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	Standard Error	t Value	Approx 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