EGARCH(1,1) asymmetric volatility model

$$\log \sigma_t^2 = \omega + \alpha (|z_{t-1}| - E|z_{t-1}|) + \gamma z_{t-1} + \beta \log \sigma_{t-1}^2$$ \(\alpha\) captures the size effect and \(\gamma\) the sign (leverage) effect. Note that this naming is not universal – rugarch uses the two letters the other way round – so compare coefficients by which term they multiply, not by name. \(E|z| = \sqrt{2/\pi}\) under the Gaussian likelihood used here. Modelling the log variance means no positivity constraints are needed; stationarity requires \(|\beta| < 1\).

Usage

morie_egarch_model(x)

Arguments

x

Numeric return series.

Value

Named list with \(omega, alpha, gamma, beta, loglik, conditional_variance, n, method\).

Details

Native Gaussian quasi-maximum-likelihood fit; no GARCH package is loaded or called.

References

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59(2), 347-370.

Examples

set.seed(1)
morie_egarch_model(x = rnorm(50))
#> $omega
#> [1] -0.05480421
#> 
#> $alpha
#> [1] -1
#> 
#> $gamma
#> [1] 0.2652565
#> 
#> $beta
#> [1] 0.8945284
#> 
#> $loglik
#> [1] -53.01563
#> 
#> $conditional_variance
#>  [1] 0.6912159 0.4998199 1.0369213 0.6786919 0.3918728 0.6950435 0.3753246
#>  [8] 0.5501104 0.6548030 0.9349346 1.1639775 0.9210457 1.5651034 1.5127008
#> [15] 0.2812973 0.1635340 0.2640701 0.4794071 0.4450554 0.4607405 0.6161617
#> [22] 0.6336511 0.7450788 1.5556783 0.3747275 0.4684427 0.7987312 1.1963581
#> [29] 0.4008899 0.2920567 0.4540415 0.2631078 0.3857355 0.6383678 1.1021970
#> [36] 0.3865369 0.3146907 0.2448747 0.3969216 0.2867302 0.2770014 0.3526595
#> [43] 0.3894307 0.4480487 0.6213443 0.3869808 0.1738679 0.2760189 0.2611244
#> [50] 0.3734619
#> 
#> $n
#> [1] 50
#> 
#> $method
#> [1] "EGARCH(1,1) Gaussian QMLE"
#>