PROC MODEL to estimate ARMA - GARCH

data garch;

lu = 0;
lh = 0;
ly = 0;
do i = 1 to 1000;
h = 0.3 + 0.4 * lu ** 2 + 0.5 * lh;
u = sqrt(h) * rannor(1);
y = 1 + 0.6 * (ly - 1) + u - 0.7 * lu;
lu = u;
lh = h;
ly = y;
output;
end;
run;

proc model data = garch;
parms ar1 ma1 mu;
y = mu + ar1 * zlag1(y - mu) + ma1 * zlag1(resid.y);
fit y / method = marquardt fiml white;
/* Nonlinear FIML Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
ar1 0.777183 0.0702 11.07 <.0001
ma1 0.841133 0.0639 13.16 <.0001
mu 1.020479 0.0392 26.01 <.0001
RESULT: ESTIMATION OF ARMA(1, 1) WITHOUT GARCH COMPONENT.

Heteroscedasticity Test
Equation Test Statistic DF Pr > ChiSq
y White's Test 89.85 9 <.0001
RESULT: WHITE'S TEST SUGGESTS HETEROSCEKASTICITY.
*/
test "AR1 = 0.6" ar1 = 0.6;
test "MA1 = 0.7" ma1 = 0.7;
/* Test Results
Test Type Statistic Pr > ChiSq
AR1 = 0.6 Wald 6.37 0.0116
MA1 = 0.7 Wald 4.88 0.0272
RESULT: ESTIMATES ARE DIFFERENT FROM SIMULATION PARAMETERS.
*/
run;
quit;

proc model data = garch;
parms ar1 ma1 mu arch0 arch1 garch1;
y = mu + ar1 * zlag1(y - mu) + ma1 * zlag1(resid.y);
h.y = arch0 + arch1 * xlag(resid.y ** 2, mse.y) +
garch1 * xlag(h.y, mse.y);
fit y / method = marquardt fiml out = forecast outall;
/* Nonlinear FIML Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
ar1 0.51709 0.1036 4.99 <.0001
ma1 0.658866 0.0866 7.61 <.0001
mu 0.989957 0.0257 38.54 <.0001
arch0 0.391227 0.0702 5.58 <.0001
arch1 0.398557 0.0539 7.39 <.0001
garch1 0.464724 0.0530 8.76 <.0001
RESULT: ESTIMATION OF ARMA(1, 1) WITH GARCH(1, 1).
*/
test "AR1 = 0.6" ar1 = 0.6;
test "MA1 = 0.7" ma1 = 0.7;
test "ARCH1 = 0.4" arch1 = 0.4;
test "GARCH1 = 0.5" garch1 = 0.5;
/* Test Results
Test Type Statistic Pr > ChiSq
AR1 = 0.6 Wald 0.64 0.4237
MA1 = 0.7 Wald 0.23 0.6347
ARCH1 = 0.4 Wald 0.00 0.9786
GARCH1 = 0.5 Wald 0.44 0.5059
RESULT: ESTIMATES ARE IN LINE WITH SIMULATION PARAMETERS.
*/
run;
quit;