case garch: modeling volatility dynamics

Case GARCH Conditionally Heteroscedastic Models

A. Conditional Heteroscedasticity

This lesson introduces you to a recent development in forecasting asset returns. ARMA forecasting model, which includes a random process as a special case, tracks the level, defined by the mean at each time period t, of the variable given the information available at time t. However, asset returns are also subject to variability around the mean, which is measured by the variance of the variable for t. ARMA models assume constancy of the variance, i.e., homoscedasticity. However, high frequency asset returns are not homoscedastic.

Figure 1A below plots 1268 daily returns of CISCO stocks from June 25, 1995 to July 5, 2001. There is no obvious upward or downward trend, i.e., the mean at t appears constant. However, the variability is not uniform, and shows chunks mixed with occasional spikes. Let (residual at time t) = return at time t sample mean. Then, the squared residual estimates the variance of the return for t. The plot of the squared residuals does not appear to follow a random process. Also, similar values of squared residuals come in chunks. See Figure 1B.

Figure 1A:

Figure 1B

The daily return, 6/25/95 to 7/5/01 The squared daily return, 6/25/95 to 7/5/01

- CSCO - CSCO

The chunkiness of squared residuals is the result of dependence of the variance of the return at time t on variances at preceding periods. We can confirm this by computing the correlogram of squared residuals. See Figure 1C. The standard error of a sample autocorrelation for 1268 observations from a random process is approximately:

Comparing ACF values in Figure 1C with the standard error, we conclude that ACF are highly significant at all lags. Q-stats are also highly significant. This dependence of the variance at time t on variances of preceding periods is called the conditional heteroscedasticity. Conditional heteroscedasticity is quite common for high frequency asset return data.

Figure 1C:

Correlogram of the Squared Daily Returns, 6/25/95 to 7/5/01

- CSCO

Sample: 6/25/1996 7/05/2001

Included observations: 1268

AutocorrelationPartial CorrelationAC PAC Q-Stat Prob

|* | |* |10.1800.18041.2680.000

|* | |* |20.1750.14780.2040.000

|* | | |30.1050.05494.1640.000

|* | | |40.0830.036103.040.000

| | | |50.0570.017107.200.000

|* | |* |60.1290.101128.280.000

|* | |* |70.1140.069144.960.000

|* | |* |80.1850.131188.820.000

|* | | |90.079-0.003196.870.000

|* | |* |100.1910.131243.770.000

We should also note that the spikes in Figure 1A or 1B most likely do not reflect conditional heteroscedasticity. They could be caused by sudden high variances or shifts of the mean for the periods.

Conditional heteroscedasticity may be present for daily returns, but it may not be so for the monthly returns. As an example, Figures 2A through 2C show the results for the monthly return of CISCO stock for the corresponding time range. The correlogram of squared residuals does not indicate serial correlation.

Figure 2A

Figure 2B

The monthly return, 96:7 to 00:6 The squared monthly returns, 96:7 to 00:6

- CSCO - CSCO

Figure 2C:

Correlogram of the squared monthly return - CSCO

Sample: 1996:07 2000:06

Included observations: 48

AutocorrelationPartial CorrelationAC PAC Q-Stat Prob

. |. | . |. |1-0.030-0.0300.04670.829

. *|. | . *|. |2-0.118-0.1190.77180.680

. *|. | . *|. |3-0.058-0.0660.94980.813

. *|. | . *| . |4-0.104-0.1251.53730.820

.*| . | **| . |5-0.159-0.1912.94360.709

. |*. | . |*. |60.1180.0683.73860.712

. | . | .*| . |7-0.009-0.0663.74350.809

. |*. | . |*. |80.1030.0934.38180.821

. | . | . | . |90.0260.0024.42380.881

.*| . | .*| . |10-0.142-0.1375.69700.840

B. Modeling Conditional Heteroscedasticity

Forecasting the conditional standard deviation is important as the volatility affects the asset price. In 1982, an Economist Engle, developed a statistical model for conditional heteroscedasticity. In below we introduce the model in the context of forecasting stock returns. Consider the following formulation for the return of a stock.

rt is the return at time, say day, t and t is an independent observation from N (0, ). Under a random walk hypothesis:

for all t, i.e., the variance of is constant for all t. As noted earlier, we say that the return rt is homoscedastic. Figure 1A through 1C, the variance of the daily return of CISCO stocks do not follow homoscedasticity, but show a dependence on the variances of preceding periods. One way to model this serial correlation of the variance is:

(1)

This formulation is called an Autoregressive Conditional Heteroscedastic (ARCH) model. The reason why it is called so is that is an unbiased estimate of , i.e., , so that defining the estimation error , we can re-write (1) for as follows:

(2)

(2) is an AR(1) specification for the squared residual. Unlike the AR(1) model for the mean, it is the error term is not constant. We could thus say that (2) is a heteroscedastic AR(1).

One can generalize (2) to include squared residuals with lags 2, 3 and so on as explanatory variables. Following the standard notation for AR models, it is convenient to denote (2) as ARCH(1). Let (L) be a q-th order polynomial in lag operator L, i.e.,

Then, ARCH (p) is:

As we have learned in ARMA modeling for the mean, it is not wise to keep increasing lagged squared residual terms for explaining the serial dependence of the variance. Instead, a useful parsimonious generalization of (1) is to include term as an explanatory variable, i.e:

(3)

The generalization is proposed by Bollerslev in 1986 and is called GARCH (Generalized ARCH). It can be shown that (3) is an heteroscedastic ARMA (1,1) specification for the squared residual. See below. From (3),

It is heteroscedastic, because is not constant. Following ARCH(1), (3) is denoted as GARCH(1,1).

Let be a q-th order polynomial in L, i.e.,

Then GARCH(p, q) is:

C. Model Fitting Using Eviews EViews offer routines that are especially suited for testing for conditional heteroscedasticity in the residual and proceeding to fit ARCH or a GARCH models. We will illustrate these features in below. We use the daily return of CSCO stocks as our data. We begin with fitting a homoscedastic model:

Select Quick/Estimate Equation and open the panel as shown below. Specify the model for the mean in the upper space. Note that the Method window indicates LS Least Squares, which is the best estimation method for homoscedastic data. Click OK and the output is generated.

Select View/Residual Tests and open the menu. See the panel below. A variety of tests for goodness of random process appears. A useful test for the presence of conditional heteroscedasticity in the residual is Correlogram Squared Residuals which we have seen before in Figure 1C. Another useful test is ARCH LM test. LM stands for Lagrange multiplier.

Select the test, the following window opens:

The LM test uses autoregression of the squared residuals. If ARCH(1) is suspected, we enter 1 in the window. Here is the result of the test.

ARCH Test:

F-statistic45.18800 Probability0.000000

Obs*R-squared43.69846 Probability0.000000

Test Equation:

Dependent Variable: RESID^2

Method: Least Squares

Sample(adjusted): 6/27/1996 5/04/2001

Included observations: 1267 after adjusting endpoints

VariableCoefficientStd. Errort-StatisticProb.

C0.0010398.70E-0511.950710.0000

RESID^2(-1)0.1860280.0276746.7222020.0000

R-squared0.034490 Mean dependent var0.001275

Adjusted R-squared0.033726 S.D. dependent var0.002880

S.E. of regression0.002831 Akaike info criterion-8.894830

Sum squared resid0.010138 Schwarz criterion-8.886710

Log likelihood5636.875 F-statistic45.18800

Durbin-Watson stat2.053261 Prob(F-statistic)0.000000

The test statistic is the number of observations, T times R2 of the autoregression, i.e., . In the absence of ARCH(1), the coefficient of is zero, the null hypothesis, and should follow a distribution with the number of degrees of freedom 1, which predicts that the exceeds 3.84 (from a table of distribution) only 5% of the time. The value of the test statistic for this example is so high that we will not hesitate rejecting the null hypothesis.

We now proceed to fit an ARCH(1) model for the data. Going back to the beginning of this section, this time, after selecting Quick/Estimate Equation, open the pull down menu for Method and select ARCH Autoregressive Conditional Heteroscedasticity.

The panel as shown below opens and indicates that it is ready to fit a GARCH(1, 1).

For fitting ARCH(1), replace 1 in GARCH window by 0 and click OK. Here are the results for ARCH(1).

Table 1: Fitting ARCH(1) for daily returns of CSCO stocks

Dependent Variable: CSCO_R

Method: ML - ARCH

Sample(adjusted): 6/26/1996 5/04/2001


Convergence achieved after 13 iterations

CoefficientStd. Errorz-StatisticProb.

C0.0018830.0008792.1418060.0322

Variance Equation

C0.0008843.90E-0522.658280.0000

ARCH(1)0.3318730.0423547.8357310.0000

R-squared-0.000129 Mean dependent var0.001478

Adjusted R-squared-0.001710 S.D. dependent var0.035717



Log likelihood2475.264 Durbin-Watson stat2.104299

The following extracts the main results for the variable.

(4)

This time, we fit GARCH(1,1) for the same data.

Sample(adjusted): 6/26/1996 5/04/2001


Convergence achieved after 28 iterations

CoefficientStd. Errorz-StatisticProb.

C0.0030380.0008213.7008910.0002

Variance Equation

C1.69E-055.69E-062.9723800.0030

ARCH(1)0.0838220.0126016.6517510.0000

GARCH(1)0.9048580.01428063.364570.0000

R-squared-0.001909 Mean dependent var0.001478

Adjusted R-squared-0.004287 S.D. dependent var0.035717



Log likelihood2565.158 Durbin-Watson stat2.100559

The estimated model for the daily return of CSCO stocks is:

with

(5)

Which model ARCH(1) or GARCH(1,1) fitts the data best? First of all, the coefficient of GARCH(1) term, i.e., is highly significant. Second, the estimation method uses the method of maximum likelihood. Comparing the value of Log likelihood, GARCH has a higher value, which means that the data are more likely from from GARCH process than ARCH. Third, GARCH uses one more parameter than ARCH. Both Akaike information criterion and Schwarz criterion adjust the likelihood for the number of parameters. The lower the values of these criteria, the better the fit. GARCH has lower values than ARCH for both criteria. From these tests, we would opt for GARCH over ARCH.

Is there rooms for improvement? To answer the question, we select View/Rresidual Tests/Correlogram Squared Residuals in the GARCH output. Eviews computes the correlogram for the standardized residual:

Here is the result:

We can also select View/Rresidual Tests/ARCH LM test and get the following result (only a part of the output is shown).

ARCH Test:

F-statistic4.324190 Probability0.037775

Obs*R-squared4.316272 Probability0.037750

Test Equation:

Dependent Variable: STD_RESID^2

Method: Least Squares

Sample(adjusted): 6/27/1996 5/04/2001


VariableCoefficientStd. Errort-StatisticProb.

C0.9429680.05529117.054590.0000

STD_RESID^2(-1)0.0584570.0281112.0794690.0378

These tests indicate that the conditional heteroscedasticity is mostly eliminated for the standardized residual. (AC are still significant but their values are small.) Therefore, we conclude that the daily stock return of CISCO stocks follow GARCH(1,1) as summarized in (4). It is instructive to compute the correlogram for the standardized squared residual. The autocorrelations are highly significant after lag 1, which indicates there is still significant conditional heteroscedasticity left in the standardized residual.

How does the estimated model track the heteroscedasticity in the data? In the GARCH(1,1) estimation output, open View/Conditional SD menu:

Then, the graph of the standard deviation as computed by equation (4) appears. See below.

D. Forecasting Using GARCH

Here is the GARCH(1,1) estimated for the daily return of CSCO stocks shown earlier.

with

(5)

How can we use this model for forecasting purposes? First, for constructing an interval forecast of the return, we can account for changing standard deviation by applying the second equation in (5) for . Second, forecasting the future value of the variance itself is important for assessing the risk of investing in CSCO stocks. These forecast advantages of the model are for short run. For a long run forecast of the variance, the MA effect will quickly disappear and the AR effect converges to the following value:

which is the sample variance of the data. That is, for a long run forecast, the sample mean and the sample variance are the best forecast of the expected return and the risk.

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case garch: modeling volatility dynamics

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