# Documentation

## Compare Conditional Variance Models Using Information Criteria

This example shows how to specify and fit a GARCH, EGARCH, and GJR model to foreign exchange rate returns. Compare the fits using AIC and BIC.

Step 1. Load the data.

Load the foreign exchange rate data included with the toolbox. Convert the Swiss franc exchange rate to returns.

```load Data_FXRates y = DataTable.CHF; r = price2ret(y); T = length(r); logL = zeros(1,3); % Preallocate numParams = logL; % Preallocate figure plot(r) xlim([0,T]) title('Swiss Franc Exchange Rate Returns') ```

The returns series appears to exhibit some volatility clustering.

Step 2. Fit a GARCH(1,1) model.

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

```Mdl1 = garch(1,1); [EstMdl1,EstParamCov1,logL(1)] = estimate(Mdl1,r); numParams(1) = sum(any(EstParamCov1)); % Number of fitted parameters ```
``` GARCH(1,1) Conditional Variance Model: ---------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 1.62872e-06 4.37426e-07 3.72342 GARCH{1} 0.913754 0.00688224 132.77 ARCH{1} 0.0585042 0.00499947 11.7021 ```

Step 3. Fit an EGARCH(1,1) model.

Specify, and then fit an EGARCH(1,1) model to the returns series. Return the value of the loglikelihood objective function.

```Mdl2 = egarch(1,1); [EstMdl2,EstParamCov2,logL(2)] = estimate(Mdl2,r); numParams(2) = sum(any(EstParamCov2)); ```
``` EGARCH(1,1) Conditional Variance Model: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant -0.292496 0.0459408 -6.36681 GARCH{1} 0.969758 0.00467846 207.281 ARCH{1} 0.122915 0.0120521 10.1986 Leverage{1} -0.0132285 0.00494973 -2.67256 ```

Step 4. Fit a GJR(1,1) model.

Specify, and then fit a GJR(1,1) model to the returns series. Return the value of the loglikelihood objective function.

```Mdl3 = gjr(1,1); [EstMdl3,EstParamCov3,logL(3)] = estimate(Mdl3,r); numParams(3) = sum(any(EstParamCov3)); ```
``` GJR(1,1) Conditional Variance Model: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 3.13748e-06 6.57444e-07 4.77224 GARCH{1} 0.882518 0.0111439 79.1926 ARCH{1} 0.0733637 0.00871511 8.41799 Leverage{1} -0.0218035 0.00867496 -2.51339 ```

The leverage term in the GJR model is not statistically significant.

Step 5. Compare the model fits using AIC and BIC.

Calculate the AIC and BIC values for the GARCH, EGARCH, and GJR model fits. The GARCH model has three parameters; the EGARCH and GJR models each have four parameters.

```[aic,bic] = aicbic(logL,numParams,T) ```
```aic = 1.0e+04 * -3.3329 -3.3321 -3.3308 bic = 1.0e+04 * -3.3309 -3.3295 -3.3282 ```

The GARCH(1,1) and EGARCH(1,1) models are not nested, so you cannot compare them by conducting a likelihood ratio test. The GARCH(1,1) is nested in the GJR(1,1) model, however, so you could use a likelihood ratio test to compare these models.

Using AIC and BIC, the GARCH(1,1) model has slightly smaller (more negative) AIC and BIC values. Thus, the GARCH(1,1) model is the preferred model according to these criteria.