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Specify Conditional Mean and Variance Models

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

Load the Data

Load the NASDAQ data included with the toolbox. Convert the daily close composite index series to a return series.

load Data_EquityIdx
ReturnsTbl = price2ret(DataTable);
T = height(ReturnsTbl);

figure
plot(ReturnsTbl.NASDAQ)
axis tight
title("NASDAQ Daily Returns")

Figure contains an axes object. The axes object with title NASDAQ Daily Returns contains an object of type line.

The returns appear to fluctuate around a constant level, but 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.

Scale the series by 100 and center the percentage returns.

ReturnsTbl.Residuals_NASDAQ = 100*(ReturnsTbl.NASDAQ - mean(ReturnsTbl.NASDAQ));

Check for Autocorrelation

Plot the sample autocorrelation function (ACF) and partial autocorrelation function (PACF) for the residual series. Adjust the y-axes of the plots to remedy the plot saturation of the lag 0 correlations.

figure
tiledlayout(2,1)
nexttile
autocorr(ReturnsTbl,DataVariable="Residuals_NASDAQ")
h = gca;
h.YLim(end) = 0.15;
nexttile
parcorr(ReturnsTbl,DataVariable="Residuals_NASDAQ")
h = gca;
h.YLim(end) = 0.15;

Figure contains 2 axes objects. Axes object 1 with title Sample Autocorrelation Function contains 4 objects of type stem, line. Axes object 2 with title Sample Partial Autocorrelation Function contains 4 objects of type stem, line.

The autocorrelation functions suggests there is significant autocorrelation at lag one and several higher lags.

Test the Significance of Autocorrelations

Conduct a Ljung-Box Q-test at lag 5.

StatTbl = lbqtest(ReturnsTbl,DataVariable="Residuals_NASDAQ",Lags=5)
StatTbl=1×7 table
                h       pValue      stat     cValue    Lags    Alpha    DoF
              _____    ________    ______    ______    ____    _____    ___

    Test 1    true     0.011956    14.652    11.07      5      0.05      5 

The null hypothesis that all autocorrelations are 0 up to lag 5 is rejected (h = 1).

Check for Conditional Heteroscedasticity.

Plot the sample ACF and PACF of the squared return series.

figure
tiledlayout(2,1)
nexttile
autocorr(ReturnsTbl.Residuals_NASDAQ.^2)
nexttile
parcorr(ReturnsTbl.Residuals_NASDAQ.^2)

Figure contains 2 axes objects. Axes object 1 with title Sample Autocorrelation Function contains 4 objects of type stem, line. Axes object 2 with title Sample Partial Autocorrelation Function contains 4 objects of type stem, line.

The autocorrelation functions show significant serial dependence, which suggests that the series is conditionally heteroscedastic.

Test for Significant ARCH Effects

Conduct an Engle's ARCH test. Test the null hypothesis of no conditional heteroscedasticity against the alternative hypothesis of an ARCH model with two lags (which is locally equivalent to a GARCH(1,1) model).

StatTbl = archtest(ReturnsTbl,DataVariable="Residuals_NASDAQ",Lags=2)
StatTbl=1×6 table
                h      pValue     stat     cValue    Lags    Alpha
              _____    ______    ______    ______    ____    _____

    Test 1    true       0       399.97    5.9915     2      0.05 

The null hypothesis is rejected in favor of the alternative hypothesis (h = 1).

Specify a Conditional Mean and Variance Model.

Specify an AR(1) model for the conditional mean of the centered NASDAQ percentage returns, and a GARCH(1,1) model for the conditional variance. This is a model of the form

rt=c+ϕ1rt-1+εt,

where εt=σtzt,

σt2=κ+γ1σt-12+α1εt-12,

and zt is an independent and identically distributed standardized Gaussian process.

CondVarMdl = garch(1,1);
Mdl = arima(ARLags=1,Variance=CondVarMdl)
Mdl = 
  arima with properties:

     Description: "ARIMA(1,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 1
               D: 0
               Q: 0
        Constant: NaN
              AR: {NaN} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: [GARCH(1,1) Model]

The model output shows that a garch model is stored in the Variance property of the arima model, Mdl.

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