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

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

uni_garch_exp_forecast forecasts the conditional variances, $h_t,t=T+1,\dots ,T+\xi$ from an exponential $\text{GARCH}(p,q)$ 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 g13fh

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g13/g13fhf.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 $(2\times iq+ip+1)$

The initial parameter estimates for the vector $\theta$. The first element must contain the coefficient ${\alpha }_o$ and the next $iq$ elements must contain the autoregressive coefficients ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The next $iq$ elements must contain the coefficients ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The next $ip$ elements must contain the moving average coefficients ${\beta }_{\text{j}}$, for $\text{j}=1,2,\dots ,p$.

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

No equivalent traditional C interface for this routine exists in the NAG Library.

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

$$ln\left(h_t\right)={\alpha }_0+\sum _{i=1}^q{\alpha }_i{z}_{t-i}+\sum _{j=1}^q{\varphi }_i(\left|z_{t-j}\right|-E\left[\left|z_{t-i}\right|\right])+\sum _{j=1}^p{\beta }_{i}ln\left(h_{t-j}\right)\text{,}t=1,2,\dots ,T\text{.}$$

where ${ϵ}_t|{\psi }_{t-1}=N(0,h_t)$ or ${ϵ}_t|{\psi }_{t-1}=S_t(\text{df},h_t)$, and $z_t=\frac{{ϵ}_t}{\sqrt{h_t}}$, $E\left[\left|z_{t-i}\right|\right]$ denotes the expected value of $\left|z_{t-i}\right|$, has been modelled by uni_garch_exp_estim(), and the estimated conditional variances and residuals are contained in the arrays $ht$ and $et$ respectively.

uni_garch_exp_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

Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779–1801

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