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Compare GARCH Models Using Likelihood Ratio Test

This example shows how to conduct a likelihood ratio test to choose the number of lags in a GARCH model.

Load the Deutschmark/British pound foreign-exchange rate data included with the toolbox. Convert the daily rates to returns.

Y = Data;
r = price2ret(Y);
N = length(r);

figure
plot(r)
xlim([0,N])
title('Mark-Pound Exchange Rate Returns') The daily returns exhibit volatility clustering. Large changes in the returns tend to cluster together, and small changes tend to cluster together. That is, the series exhibits conditional heteroscedasticity.

The returns are of relatively high frequency. Therefore, the daily changes can be small. For numerical stability, it is good practice to scale such data. In this case, scale the returns to percentage returns.

r = 100*r;

Specify and Fit a GARCH(1,1) Model.

Specify and fit a GARCH(1,1) model (with a mean offset) to the returns series. Return the value of the loglikelihood objective function.

model1 = garch('Offset',NaN,'GARCHLags',1,'ARCHLags',1);
[fit1,~,LogL1] = estimate(model1,r);

GARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution):

Value       StandardError    TStatistic      PValue
__________    _____________    __________    __________

Constant      0.010761       0.001323         8.1342     4.1455e-16
GARCH{1}       0.80597        0.01656         48.669              0
ARCH{1}        0.15313       0.013974         10.959     6.0378e-28
Offset      -0.0061904      0.0084336       -0.73402        0.46294

Specify and Fit a GARCH(2,1) Model.

Specify and fit a GARCH(2,1) model with a mean offset.

model2 = garch(2,1);
model2.Offset = NaN;
[fit2,~,LogL2] = estimate(model2,r);

GARCH(2,1) Conditional Variance Model with Offset (Gaussian Distribution):

Value       StandardError    TStatistic      PValue
__________    _____________    __________    __________

Constant      0.011226       0.001538         7.2992     2.8948e-13
GARCH{1}       0.48965        0.11159         4.3878     1.1452e-05
GARCH{2}       0.29769        0.10218         2.9133      0.0035762
ARCH{1}        0.16842       0.016583         10.156      3.116e-24
Offset      -0.0049837      0.0084764       -0.58795        0.55657

Conduct a Likelihood Ratio Test.

Conduct a likelihood ratio test to compare the restricted GARCH(1,1) model fit to the unrestricted GARCH(2,1) model fit. The degree of freedom for this test is one (the number of restrictions).

[h,p] = lratiotest(LogL2,LogL1,1)
h = logical
1

p = 0.0218

At the 0.05 significance level, the null GARCH(1,1) model is rejected (h = 1) in favor of the unrestricted GARCH(2,1) alternative.