void	g13fbc (Integer num, Integer nt, Integer p, Integer q, const double theta[], double gamma, double fht[], const double ht[], const double et[], NagError *fail)

The function may be called by the names: g13fbc, nag_tsa_uni_garch_asym1_forecast or nag_forecast_agarchi.

3 Description

Assume the standard

(γ = 0)

GARCH

(p, q)

process can be represented by:

ε_{t} ∣ ψ_{t - 1} \sim N (0, h_{t})

h_{t} = α_{0} + \sum_{i = 1}^{q} α_{i} ε_{t - i}^{2} + \sum_{i = 1}^{p} β_{i} h_{t - i}, t = 1, \dots, T .

or type I AGARCH

(p, q)

process with conditional variance

h_{t}

given by:

h_{t} = α_{0} + \sum_{i = 1}^{q} α_{i} {(ε_{t - i} + γ)}^{2} + \sum_{i = 1}^{p} β_{i} h_{t - i}, t = 1, \dots, T .

has been modelled by g13fac and the estimated conditional variances and residuals are contained in the arrays ht and et respectively. Then g13fbc will use the last

\max (p, q)

elements of the arrays ht and et to estimate the conditional variance forecasts,

h_{t} ∣ ψ_{T}

, where

t = T + 1, \dots, T + τ

and

τ

is the forecast horizon.

4 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

5 Arguments

1: $num$ – Integer Input: On entry: the number of terms in the arrays ht and et from the modelled sequence.

Constraint: $\max (p, q) \leq num$ .
2: $nt$ – Integer Input: On entry: $τ$ , the forecast horizon.

Constraint: $nt > 0$ .
3: $p$ – Integer Input: On entry: the GARCH $(p, q)$ argument $p$ .

Constraint: $0 < \max (p, q) \leq num, p \geq 0$ .
4: $q$ – Integer Input: On entry: the GARCH $(p, q)$ argument $q$ .

Constraint: $0 < \max (p, q) \leq num, q \geq 1$ .
5: $theta [q + p + 1]$ – const double Input: On entry: the first element must contain the coefficient $α_{o}$ and the next q elements must contain the coefficients $α_{i}$ , for $i = 1, 2, \dots, q$ . The remaining p elements must contain the coefficients $β_{j}$ , for $j = 1, 2, \dots, p$ .
6: $gamma$ – double Input: On entry: the asymmetry argument $γ$ for the GARCH $(p, q)$ sequence.
7: $fht [nt]$ – double Output: On exit: the forecast values of the conditional variance, $h_{t}$ , for $t = 1, 2, \dots, τ$ .
8: $ht [num]$ – const double Input: On entry: the sequence of past conditional variances for the GARCH $(p, q)$ process, $h_{t}$ , for $t = 1, 2, \dots, T$ .
9: $et [num]$ – const double Input: On entry: the sequence of past residuals for the GARCH $(p, q)$ process, $ε_{t}$ , for $t = 1, 2, \dots, T$ .
10: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $num = 〈value〉$ while $\max (p, q) = 〈value〉$ . These arguments must satisfy $num \geq \max (p, q)$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $nt = 〈value〉$ .
Constraint: $nt \geq 1$ .

On entry, $num = 〈value〉$ .
Constraint: $num \geq 0$ .

On entry, $p = 〈value〉$ .
Constraint: $p \geq 0$ .

On entry, $q = 〈value〉$ .
Constraint: $q \geq 1$ .