# Documentation

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## 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);```
``` GARCH(1,1) Conditional Variance Model: ---------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 1.63e-06 4.3763e-07 3.72461 GARCH{1} 0.913708 0.00688679 132.675 ARCH{1} 0.0585264 0.00500137 11.7021 ```
`numParams(1) = sum(any(EstParamCov1)); % Number of fitted parameters`

### 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);```
``` EGARCH(1,1) Conditional Variance Model: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant -0.292511 0.0459423 -6.36693 GARCH{1} 0.969756 0.00467861 207.275 ARCH{1} 0.122918 0.0120523 10.1987 Leverage{1} -0.0132298 0.0049498 -2.6728 ```
`numParams(2) = sum(any(EstParamCov2));`

### 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);```
``` GJR(1,1) Conditional Variance Model: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 1.70832e-06 4.50809e-07 3.78946 GARCH{1} 0.911397 0.00722306 126.179 ARCH{1} 0.0588977 0.00686651 8.57754 Leverage{1} 0.00131296 0.00727963 0.180361 ```
`numParams(3) = sum(any(EstParamCov3));`

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 -3.3327 ```
```bic = 1.0e+04 * -3.3309 -3.3295 -3.3301 ```

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.