NAG Library Function Document

nag_forecast_agarchI (g13fbc)

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

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 nag_estimate_agarchI (g13fac) and the estimated conditional variances and residuals are contained in the arrays ht and et respectively. Then nag_forecast_agarchI (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: $num$ – IntegerInput: On entry: the number of terms in the arrays ht and et from the modelled sequence.

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

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

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

Constraint: $0 < \max (p, q) \leq num, q \geq 1$ .
5: $theta [q + p + 1]$ – const doubleInput: 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: $gamma$ – doubleInput: On entry: the asymmetry argument $γ$ for the GARCH $(p, q)$ sequence.
7: $fht [nt]$ – doubleOutput: On exit: the forecast values of the conditional variance, $h_{t}$ , for $t = 1, 2, \dots, τ$ .
8: $ht [num]$ – const doubleInput: 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 doubleInput: 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 3.6 in the Essential Introduction).

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: $q \geq 1$ .