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Consider the time series

$${y}_{t}=\mu +{\epsilon}_{t},$$

A *conditional variance model* specifies
the dynamic evolution of the innovation variance,

$${\sigma}_{t}^{2}=Var\left({\epsilon}_{t}|{H}_{t-1}\right),$$

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*) = 0._{t}*E*(*ε*_{t}*ε*) = 0 for all_{t–h}*t*and $$h\ne 0.$$

However, if $${\sigma}_{t}^{2}$$ depends on $${\sigma}_{t-1}^{2}$$, for example, then *ε _{t}* depends
on

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

$${y}_{t}=\mu +{\epsilon}_{t},$$

$${\sigma}_{t}^{2}=\kappa +{\gamma}_{1}{\sigma}_{t-1}^{2}+\dots +{\gamma}_{P}{\sigma}_{t-P}^{2}+{\alpha}_{1}{\epsilon}_{t-1}^{2}+\dots +{\alpha}_{Q}{\epsilon}_{t-Q}^{2}.$$

The `Constant`

property of a `garch`

model
corresponds to *κ*, and the `Offset`

property
corresponds to *μ*.

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

$${y}_{t}=\mu +{\epsilon}_{t},$$

$$\mathrm{log}{\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}\mathrm{log}}{\sigma}_{t-i}^{2}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}\left[\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}-E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}\right]}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}\left(\frac{{\epsilon}_{t-j}}{{\sigma}_{t-j}}\right).$$

The `Constant`

property of an `egarch`

model
corresponds to *κ*, and the `Offset`

property
corresponds to *μ*.

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

$${y}_{t}=\mu +{\epsilon}_{t},$$

$${\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}{\epsilon}_{t-j}^{2}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}I\left[{\epsilon}_{t-j}<0\right]{\epsilon}_{t-j}^{2}.$$

The `Constant`

property of a `gjr`

model
corresponds to *κ*, and the `Offset`

property
corresponds to *μ*.

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.

- Specify GARCH Models Using garch
- Specify EGARCH Models Using egarch
- Specify GJR Models Using gjr
- Specify Conditional Mean and Variance Models
- Assess EGARCH Forecast Bias Using Simulations

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