Estimate Conditional Mean and Variance Models

This example shows how to estimate a composite conditional mean and variance model using estimate.

Load the Data and Specify the Model.

Load the NASDAQ data included with the toolbox. Convert the daily close composite index series to a return series. For numerical stability, convert the returns to percentage returns. Specify an AR(1) and GARCH(1,1) composite model. This is a model of the form

$${r_t} = c + {\phi _1}{r_{t - 1}} + {\varepsilon _t},$$

where ${\varepsilon _t} = {\sigma _t}{z_t}$,

$$\sigma _t^2 = \kappa  + {\gamma _1}\sigma _{t - 1}^2 + {\alpha _1}\varepsilon _{t - 1}^2,$$

and $z_t$ is an independent and identically distributed standardized Gaussian process.

load Data_EquityIdx
nasdaq = DataTable.NASDAQ;
r = 100*price2ret(nasdaq);
T = length(r);

Mdl = arima('ARLags',1,'Variance',garch(1,1))
Mdl = 

    ARIMA(1,0,0) Model:
    --------------------
    Distribution: Name = 'Gaussian'
               P: 1
               D: 0
               Q: 0
        Constant: NaN
              AR: {NaN} at Lags [1]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: [GARCH(1,1) Model]

Estimate the Model Parameters Without Using Presample Data.

Fit the model, Mdl, to the return series, r, using estimate. Use the presample observations that estimate automatically generates.

EstMdl = estimate(Mdl,r);
 
    ARIMA(1,0,0) Model:
    --------------------
    Conditional Probability Distribution: Gaussian

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant      0.0726323     0.0180473        4.02456
        AR{1}       0.138157     0.0198931        6.94499
 
 
    GARCH(1,1) Conditional Variance Model:
    ----------------------------------------
    Conditional Probability Distribution: Gaussian

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant      0.0223769    0.00332008        6.73988
     GARCH{1}        0.87312    0.00910186        95.9276
      ARCH{1}       0.118649    0.00871698        13.6112

The estimation display shows the five estimated parameters and their corresponding standard errors (the AR(1) conditional mean model has two parameters, and the GARCH(1,1) conditional variance model has three parameters).

The fitted model (EstMdl) is

$${r_t} = 7.1 \times {10^{ - 4}} + 0.14{r_{t - 1}} + {\varepsilon _t},$$

where ${\varepsilon _t} = {\sigma _t}{z_t}$ and

$$\sigma _t^2 = 2.3 \times {10^{ - 6}} + 0.87\sigma _{t - 1}^2 + 0.12\varepsilon _{t - 1}^2.$$

All t statistics are greater than two, suggesting all parameters are statistically significant.

Infer the Conditional Variances and Residuals.

Infer and plot the conditional variances and standardized residuals. Also output the loglikelihood objective function value.

[res,v,logL] = infer(EstMdl,r);

figure
subplot(2,1,1)
plot(v)
xlim([0,T])
title('Conditional Variance')

subplot(2,1,2)
plot(res./sqrt(v))
xlim([0,T])
title('Standardized Residuals')

The conditional variances increase after observation 2000. This corresponds to the increased volatility seen in the original return series.

The standardized residuals have more large values (larger than 2 or 3 in absolute value) than expected under a standard normal distribution. This suggests a Student's t distribution might be more appropriate for the innovation distribution.

Fit a Model With a t-Innovation Distribution.

Modify the model so that it has a Student's t-innovation distribution. Fit the modified model to the NASDAQ return series. Specify an initial value for the variance model constant term.

MdlT = Mdl;
MdlT.Distribution = 't';
EstMdlT = estimate(MdlT,r,'Variance0',{'Constant0',0.001});
 
    ARIMA(1,0,0) Model:
    --------------------
    Conditional Probability Distribution: t

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant       0.093488     0.0166938        5.60018
        AR{1}       0.139107     0.0188565        7.37711
          DoF         7.4775      0.882615        8.47198
 
 
    GARCH(1,1) Conditional Variance Model:
    ----------------------------------------
    Conditional Probability Distribution: t

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant      0.0112457    0.00363047        3.09758
     GARCH{1}       0.907662     0.0105156        86.3156
      ARCH{1}      0.0898968     0.0108354        8.29662
          DoF         7.4775      0.882615        8.47198

The coefficient estimates change slightly when the t distribution is used for the innovations. The second model fit (EstMdlT) has one additional parameter estimate, the t distribution degrees of freedom. The estimated degrees of freedom are relatively small (about 8), indicating significant departure from normality.

Compare the Model Fits.

Compare the two model fits (Gaussian and t-innovation distribution) using the Akaike information criterion (AIC) and Bayesian information criterion (BIC). First, obtain the loglikelihood objective function value for the second fit.

[resT,vT,logLT] = infer(EstMdlT,r);
[aic,bic] = aicbic([logL,logLT],[5,6],T)
aic =

   1.0e+03 *

    9.4929    9.3807


bic =

   1.0e+03 *

    9.5230    9.4168

The second model has six parameters compared to five in the first model (because of the t distribution degrees of freedom). Despite this, both information criteria favor the model with the Student's t distribution. The AIC and BIC values are smaller for the t innovation distribution.

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