Consider the time series
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$. Here, z_{t} is an independent and identically distributed series of standardized random variables. Econometrics Toolbox™ supports standardized Gaussian and standardized Student's t innovation distributions. The constant term, $$\mu $$, is a mean offset.
A conditional variance model specifies the dynamic evolution of the innovation variance,
$${\sigma}_{t}^{2}=Var\left({\epsilon}_{t}|{H}_{t-1}\right),$$
where H_{t–1} is the history of the process. The history includes:
Past variances, $${\sigma}_{1}^{2},{\sigma}_{2}^{2},\dots ,{\sigma}_{t-1}^{2}$$
Past innovations, $${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{t-1}$$
Conditional variance models are appropriate for time series that do not exhibit significant autocorrelation, but are serially dependent. The innovation series $${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ is uncorrelated, because:
E(ε_{t}) = 0.
E(ε_{t}ε_{t–h}) = 0 for all t and $$h\ne 0.$$
However, if $${\sigma}_{t}^{2}$$ depends on $${\sigma}_{t-1}^{2}$$, for example, then ε_{t} depends on ε_{t–1}, even though they are uncorrelated. This kind of dependence exhibits itself as autocorrelation in the squared innovation series, $${\epsilon}_{t}^{2}.$$
Tip For modeling time series that are both autocorrelated and serially dependent, you can consider using a composite conditional mean and variance model. |
Two characteristics of financial time series that conditional variance models address are:
Volatility clustering. Volatility is the conditional standard deviation of a time series. Autocorrelation in the conditional variance process results in volatility clustering. The GARCH model and its variants model autoregression in the variance series.
Leverage effects. The volatility of some time series responds more to large decreases than to large increases. This asymmetric clustering behavior is known as the leverage effect. The EGARCH and GJR models have leverage terms to model this asymmetry.
The generalized autoregressive conditional heteroscedastic (GARCH) model is an extension of Engle's ARCH model for variance heteroscedasticity [1]. If a series exhibits volatility clustering, this suggests that past variances might be predictive of the current variance.
The GARCH(P,Q) model is an autoregressive moving average model for conditional variances, with P GARCH coefficients associated with lagged variances, and Q ARCH coefficients associated with lagged squared innovations. The form of the GARCH(P,Q) model in Econometrics Toolbox is
$${y}_{t}=\mu +{\epsilon}_{t},$$
where$${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ and
$${\sigma}_{t}^{2}=\kappa +{\gamma}_{1}{\sigma}_{t-1}^{2}+\dots +{\gamma}_{P}{\sigma}_{t-P}^{2}+{\alpha}_{1}{\epsilon}_{t-1}^{2}+\dots +{\alpha}_{Q}{\epsilon}_{t-Q}^{2}.$$
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For stationarity and positivity, the GARCH model has the following constraints:
$$\kappa >0$$
$${\gamma}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha}_{j}\ge 0$$
$${\sum}_{i=1}^{P}{\gamma}_{i}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}}<1$$
To specify Engle's original ARCH(Q) model, use the equivalent GARCH(0,Q) specification.
The exponential GARCH (EGARCH) model is a GARCH variant that models the logarithm of the conditional variance process. In addition to modeling the logarithm, the EGARCH model has additional leverage terms to capture asymmetry in volatility clustering.
The EGARCH(P,Q) model has P GARCH coefficients associated with lagged log variance terms, Q ARCH coefficients associated with the magnitude of lagged standardized innovations, and Q leverage coefficients associated with signed, lagged standardized innovations. The form of the EGARCH(P,Q) model in Econometrics Toolbox is
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ and
$$\mathrm{log}{\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}\mathrm{log}}{\sigma}_{t-i}^{2}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}\left[\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}-E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}\right]}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}\left(\frac{{\epsilon}_{t-j}}{{\sigma}_{t-j}}\right).$$
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The form of the expected value terms associated with ARCH coefficients in the EGARCH equation depends on the distribution of z_{t}:
If the innovation distribution is Gaussian, then
$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{2}{\pi}}.$$
If the innovation distribution is Student's t with ν > 2 degrees of freedom, then
$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{\nu -2}{\pi}}\frac{\Gamma \left(\frac{\nu -1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}.$$
The toolbox treats the EGARCH(P,Q) model as an ARMA model for$$\mathrm{log}{\sigma}_{t}^{2}.$$ Thus, to ensure stationarity, all roots of the GARCH coefficient polynomial,$$(1-{\gamma}_{1}L-\dots -{\gamma}_{P}{L}^{P})$$, must lie outside the unit circle.
The EGARCH model is unique from the GARCH and GJR models because it models the logarithm of the variance. By modeling the logarithm, positivity constraints on the model parameters are relaxed. However, forecasts of conditional variances from an EGARCH model are biased, because by Jensen's inequality,
$$E({\sigma}_{t}^{2})\ge \mathrm{exp}\{E(\mathrm{log}{\sigma}_{t}^{2})\}.$$
An EGARCH(1,1) specification will be complex enough for most applications. For an EGARCH(1,1) model, the GARCH and ARCH coefficients are expected to be positive, and the leverage coefficient is expected to be negative; large unanticipated downward shocks should increase the variance. If you get signs opposite to those expected, you might encounter difficulties inferring volatility sequences and forecasting (a negative ARCH coefficient can be particularly problematic). In this case, an EGARCH model might not be the best choice for your application.
The GJR model is a GARCH variant that includes leverage terms for modeling asymmetric volatility clustering. In the GJR formulation, large negative changes are more likely to be clustered than positive changes. The GJR model is named for Glosten, Jagannathan, and Runkle [2]. Close similarities exist between the GJR model and the threshold GARCH (TGARCH) model—a GJR model is a recursive equation for the variance process, and a TGARCH is the same recursion applied to the standard deviation process.
The GJR(P,Q) model has P GARCH coefficients associated with lagged variances, Q ARCH coefficients associated with lagged squared innovations, and Q leverage coefficients associated with the square of negative lagged innovations. The form of the GJR(P,Q) model in Econometrics Toolbox is
$${y}_{t}=\mu +{\epsilon}_{t},$$
where$${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ and
$${\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}{\epsilon}_{t-j}^{2}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}I\left[{\epsilon}_{t-j}<0\right]{\epsilon}_{t-j}^{2}.$$
The indicator function $$I\left[{\epsilon}_{t-j}<0\right]$$ equals 1 if $${\epsilon}_{t-j}<0$$, and 0 otherwise. Thus, the leverage coefficients are applied to negative innovations, giving negative changes additional weight.
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For stationarity and positivity, the GJR model has the following constraints:
$$\kappa >0$$
$${\gamma}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha}_{j}\ge 0$$
$${\alpha}_{j}+{\xi}_{j}\ge 0$$
$${\sum}_{i=1}^{P}{\gamma}_{i}}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}+\frac{1}{2}}{\displaystyle {\sum}_{j=1}^{Q}{\xi}_{j}}<1$$
The GARCH model is nested in the GJR model. If all leverage coefficients are zero, then the GJR model reduces to the GARCH model. This means you can test a GARCH model against a GJR model using the likelihood ratio test.
[1] Engle, Robert F. "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation." Econometrica. Vol. 50, 1982, pp. 987–1007.
[2] Glosten, L. R., R. Jagannathan, and D. E. Runkle. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks." The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.