naginterfaces.library.tsa.uni_​garch_​asym2_​forecast

naginterfaces.library.tsa.uni_garch_asym2_forecast(nt, ip, iq, theta, gamma, ht, et)

uni_garch_asym2_forecast forecasts the conditional variances, $h_t,t=T+1,\dots ,T+\xi$ from a type II $\text{AGARCH}(p,q)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Engle and Ng (1993)).

For full information please refer to the NAG Library document for g13fd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13fdf.html

Parameters

ntint

$\xi$, the forecast horizon.

ipint

The number of coefficients, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$.

iqint

The number of coefficients, ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

thetafloat, array-like, shape $(iq+ip+1)$

The first element must contain the coefficient ${\alpha }_o$ and the next $iq$ elements must contain the coefficients ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The remaining $ip$ elements must contain the coefficients ${\beta }_{\text{j}}$, for $\text{j}=1,2,\dots ,p$.

gammafloat

The asymmetry parameter $\gamma$ for the $\text{GARCH}(p,q)$ sequence.

htfloat, array-like, shape $\left(\text{num}\right)$

The sequence of past conditional variances for the $\text{GARCH}(p,q)$ process, $h_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

etfloat, array-like, shape $\left(\text{num}\right)$

The sequence of past residuals for the $\text{GARCH}(p,q)$ process, ${ϵ}_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

Returns

fhtfloat, ndarray, shape $\left(nt\right)$

The forecast values of the conditional variance, $h_t$, for $\text{t}=T+1,\dots ,T+\xi$.

Raises

NagValueError

(errno $1$)

On entry, $max(ip-1,iq-1)=⟨value⟩$.

Constraint: $max(ip-1,iq-1)\le 20$.

(errno $1$)

On entry, $\text{num}=⟨value⟩$.

Constraint: $max(ip,iq)\le \text{num}$.

(errno $1$)

On entry, $nt=⟨value⟩$.

Constraint: $nt>0$.

(errno $1$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 1$.

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 0$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Assume the $\text{GARCH}(p,q)$ process can be represented by:

$$h_t={\alpha }_0+\sum _{i=1}^q{\alpha }_i{(\left|{ϵ}_{t-i}\right|+\gamma {ϵ}_{t-i})}^2+\sum _{i=1}^p{\beta }_i{h}_{t-i}\text{,}t=1,2,\dots ,T\text{.}$$

where ${ϵ}_t|{\psi }_{t-1}=N(0,h_t)$ or ${ϵ}_t|{\psi }_{t-1}=S_t(\text{df},h_t)$, has been modelled by uni_garch_asym2_estim() and the estimated conditional variances and residuals are contained in the arrays $ht$ and $et$ respectively.

uni_garch_asym2_forecast will then use the last $max(p,q)$ elements of the arrays $ht$ and $et$ to estimate the conditional variance forecasts, $h_t|{\psi }_T$, where $t=T+1,\dots ,T+\xi$ and $\xi$ is the forecast horizon.

References

Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307–327

Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987–1008

Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749–1777

Hamilton, J, 1994, Time Series Analysis, Princeton University Press