Conditional Variance Models

General Conditional Variance Model Definition

Consider the time series

$y_{t} = μ + ε_{t},$

where $ε_{t} = σ_{t} z_{t}$ . Here,z_tis an independent and identically distributed series of standardized random variables. Econometrics Toolbox™ supports standardized Gaussian and standardized Student’stinnovation distributions. The constant term, $μ$ , is a mean offset.

Aconditional variance modelspecifies the dynamic evolution of the innovation variance,

$σ_{t}^{2} = V a r (ε_{t} | H_{t - 1}),$

whereH_t–1is the history of the process. The history includes:

Past variances, $σ_{1}^{2}, σ_{2}^{2}, \dots, σ_{t - 1}^{2}$
Past innovations, $ε_{1}, ε_{2}, \dots, ε_{t - 1}$

Conditional variance models are appropriate for time series that do not exhibit significant autocorrelation, but are serially dependent. The innovation series $ε_{t} = σ_{t} z_{t}$ is uncorrelated, because:

E(ε_t) = 0.
E(ε_tε_t–h) = 0 for alltand $h \neq 0.$

However, if $σ_{t}^{2}$ depends on $σ_{t - 1}^{2}$ , for example, thenε_tdepends onε_t–1, even though they are uncorrelated. This kind of dependence exhibits itself as autocorrelation in the squared innovation series, $ε_{t}^{2} .$

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.

GARCH Model

Thegeneralized autoregressive conditional heteroscedastic(GARCH)模型是一种扩展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, withPGARCH coefficients associated with lagged variances, andQARCH coefficients associated with lagged squared innovations. The form of the GARCH(P,Q) model in Econometrics Toolbox is

$y_{t} = μ + ε_{t},$

where $ε_{t} = σ_{t} z_{t}$ and

$σ_{t}^{2} = κ + γ_{1} σ_{t - 1}^{2} + \dots + γ_{P} σ_{t - P}^{2} + α_{1} ε_{t - 1}^{2} + \dots + α_{Q} ε_{t - Q}^{2} .$

Note

TheConstant财产的garchmodel corresponds toκ, and theOffsetproperty corresponds toμ.

For stationarity and positivity, the GARCH model has the following constraints:

$κ > 0$
$γ_{i} \geq 0, α_{j} \geq 0$
$\sum_{i = 1}^{P} γ_{i} + \sum_{j = 1}^{Q} α_{j} < 1$

To specify Engle’s original ARCH(Q) model, use the equivalent GARCH(0,Q) specification.

EGARCH Model

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 hasPGARCH coefficients associated with lagged log variance terms,QARCH coefficients associated with the magnitude of lagged standardized innovations, andQleverage coefficients associated with signed, lagged standardized innovations. The form of the EGARCH(P,Q) model in Econometrics Toolbox is

$y_{t} = μ + ε_{t},$

where $ε_{t} = σ_{t} z_{t}$ and

$\log σ_{t}^{2} = κ + \sum_{i = 1}^{P} γ_{i} \log σ_{t - i}^{2} + \sum_{j = 1}^{Q} α_{j} [\frac{| ε_{t - j} |}{σ_{t - j}} - E {\frac{| ε_{t - j} |}{σ_{t - j}}}] + \sum_{j = 1}^{Q} ξ_{j} (\frac{ε_{t - j}}{σ_{t - j}}) .$

Note

TheConstant财产的negarchmodel corresponds toκ, and theOffsetproperty corresponds toμ.

The form of the expected value terms associated with ARCH coefficients in the EGARCH equation depends on the distribution ofz_t:

If the innovation distribution is Gaussian, then

$E {\frac{| ε_{t - j} |}{σ_{t - j}}} = E {| z_{t - j} |} = \sqrt{\frac{2}{π}} .$
If the innovation distribution is Student’stwithν> 2 degrees of freedom, then

$E {\frac{| ε_{t - j} |}{σ_{t - j}}} = E {| z_{t - j} |} = \sqrt{\frac{ν - 2}{π}} \frac{Γ (\frac{ν - 1}{2})}{Γ (\frac{ν}{2})} .$

The toolbox treats the EGARCH(P,Q) model as an ARMA model for $\log σ_{t}^{2} .$ Thus, to ensure stationarity, all roots of the GARCH coefficient polynomial, $(1 - γ_{1} L - \dots - γ_{P} L^{P})$ , 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,

$E (σ_{t}^{2}) \geq \exp {E (\log σ_{t}^{2})} .$

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.

GJR Model

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 hasPGARCH coefficients associated with lagged variances,QARCH coefficients associated with lagged squared innovations, andQleverage coefficients associated with the square of negative lagged innovations. The form of the GJR(P,Q) model in Econometrics Toolbox is

$y_{t} = μ + ε_{t},$

where $ε_{t} = σ_{t} z_{t}$ and

$σ_{t}^{2} = κ + \sum_{i = 1}^{P} γ_{i} σ_{t - i}^{2} + \sum_{j = 1}^{Q} α_{j} ε_{t - j}^{2} + \sum_{j = 1}^{Q} ξ_{j} I [ε_{t - j} < 0] ε_{t - j}^{2} .$

The indicator function $I [ε_{t - j} < 0]$ equals 1 if $ε_{t - j} < 0$ , and 0 otherwise. Thus, the leverage coefficients are applied to negative innovations, giving negative changes additional weight.

Note

TheConstant财产的gjrmodel corresponds toκ, and theOffsetproperty corresponds toμ.

For stationarity and positivity, the GJR model has the following constraints:

$κ > 0$
$γ_{i} \geq 0, α_{j} \geq 0$
$α_{j} + ξ_{j} \geq 0$
$\sum_{i = 1}^{P} γ_{i} + \sum_{j = 1}^{Q} α_{j} + \frac{1}{2} \sum_{j = 1}^{Q} ξ_{j} < 1$

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.

References

[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.

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