Documentation Center

  • Trial Software
  • Product Updates

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 = Dataset.CHF;
r = price2ret(y);
n = length(r);

logL = zeros(1,3); % Preallocate
numParams = logL;  % Preallocate

figure
plot(r)
xlim([0,n])
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    1.64696e-06   4.40319e-07        3.74037
     GARCH{1}       0.913111    0.00694616        131.456
      ARCH{1}      0.0588077    0.00502597        11.7008

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      -0.292505      0.045942       -6.36684
     GARCH{1}       0.969757    0.00467858        207.276
      ARCH{1}       0.122917     0.0120523        10.1986
  Leverage{1}     -0.0132288     0.0049498       -2.67259

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    1.70879e-06   4.50886e-07        3.78986
     GARCH{1}       0.911385    0.00722458        126.151
      ARCH{1}      0.0589008    0.00686726        8.57703
  Leverage{1}     0.00131901    0.00728059       0.181168

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,n)
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.

See Also

| | | | | |

Related Examples

More About

Was this topic helpful?