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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_uni_garch_exp_forecast (g13fh)

## Purpose

nag_tsa_uni_garch_exp_forecast (g13fh) forecasts the conditional variances, ht,t = T + 1,,T + ξ${h}_{t},t=T+1,\dots ,T+\xi$ from an exponential GARCH(p,q)$\text{GARCH}\left(p,q\right)$ sequence, where ξ$\xi$ is the forecast horizon and T$T$ is the current time (see Engle and Ng (1993)).

## Syntax

[fht, ifail] = g13fh(nt, ip, iq, theta, ht, et, 'num', num)
[fht, ifail] = nag_tsa_uni_garch_exp_forecast(nt, ip, iq, theta, ht, et, 'num', num)

## Description

Assume the GARCH(p,q)$\text{GARCH}\left(p,q\right)$ process represented by:
 q q p ln(ht) = α0 + ∑ αizt − i + ∑ φi(|zt − j| − E[|zt − i|]) + ∑ βiln(ht − j),  t = 1,2, … ,T. i = 1 j = 1 j = 1
$ln(ht)=α0+∑i=1qαizt-i+∑j=1qϕi(|zt-j|-E[|zt-i|])+∑j=1pβiln(ht-j), t=1,2,…,T.$
where εtψt1 = N(0,ht)${\epsilon }_{t}\mid {\psi }_{t-1}=N\left(0,{h}_{t}\right)$ or εtψt1 = St(df,ht)${\epsilon }_{t}\mid {\psi }_{t-1}={S}_{t}\left(\mathit{df},{h}_{t}\right)$, and zt = (εt)/(sqrt(ht)) ${z}_{t}=\frac{{\epsilon }_{t}}{\sqrt{{h}_{t}}}$, E[|zti|]$E\left[|{z}_{t-i}|\right]$ denotes the expected value of |zti|$|{z}_{t-i}|$, has been modelled by nag_tsa_uni_garch_exp_estim (g13fg), and the estimated conditional variances and residuals are contained in the arrays ht and et respectively.
nag_tsa_uni_garch_exp_forecast (g13fh) will then use the last max (p,q)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(p,q\right)$ elements of the arrays ht and et to estimate the conditional variance forecasts, htψT${h}_{t}\mid {\psi }_{T}$, where t = T + 1,,T + ξ$t=T+1,\dots ,T+\xi$ and ξ$\xi$ 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

## Parameters

### Compulsory Input Parameters

1:     nt – int64int32nag_int scalar
ξ$\xi$, the forecast horizon.
Constraint: nt > 0${\mathbf{nt}}>0$.
2:     ip – int64int32nag_int scalar
The number of coefficients, βi${\beta }_{\mathit{i}}$, for i = 1,2,,p$\mathit{i}=1,2,\dots ,p$.
Constraints:
• max (ip,iq)20$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le 20$;
• ip0${\mathbf{ip}}\ge 0$.
3:     iq – int64int32nag_int scalar
The number of coefficients, αi${\alpha }_{\mathit{i}}$, for i = 1,2,,q$\mathit{i}=1,2,\dots ,q$.
Constraints:
• max (ip,iq)20$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le 20$;
• iq1${\mathbf{iq}}\ge 1$.
4:     theta(2 × iq + ip + 1$2×{\mathbf{iq}}+{\mathbf{ip}}+1$) – double array
The initial parameter estimates for the vector θ$\theta$. The first element must contain the coefficient αo${\alpha }_{o}$ and the next iq elements must contain the autoregressive coefficients αi${\alpha }_{\mathit{i}}$, for i = 1,2,,q$\mathit{i}=1,2,\dots ,q$. The next iq elements must contain the coefficients φi${\varphi }_{\mathit{i}}$, for i = 1,2,,q$\mathit{i}=1,2,\dots ,q$. The next ip elements must contain the moving average coefficients βj${\beta }_{\mathit{j}}$, for j = 1,2,,p$\mathit{j}=1,2,\dots ,p$.
5:     ht(num) – double array
num, the dimension of the array, must satisfy the constraint max (ip,iq)num$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le {\mathbf{num}}$.
The sequence of past conditional variances for the GARCH(p,q)$\text{GARCH}\left(p,q\right)$ process, ht${h}_{\mathit{t}}$, for t = 1,2,,T$\mathit{t}=1,2,\dots ,T$.
6:     et(num) – double array
num, the dimension of the array, must satisfy the constraint max (ip,iq)num$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le {\mathbf{num}}$.
The sequence of past residuals for the GARCH(p,q)$\text{GARCH}\left(p,q\right)$ process, εt${\epsilon }_{\mathit{t}}$, for t = 1,2,,T$\mathit{t}=1,2,\dots ,T$.

### Optional Input Parameters

1:     num – int64int32nag_int scalar
Default: The dimension of the arrays ht, et. (An error is raised if these dimensions are not equal.)
The number of terms in the arrays ht and et from the modelled sequence.
Constraint: max (ip,iq)num$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)\le {\mathbf{num}}$.

None.

### Output Parameters

1:     fht(nt) – double array
The forecast values of the conditional variance, ht${h}_{t}$, for t = T + 1,,T + ξ$\mathit{t}=T+1,\dots ,T+\xi$.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, num < max (ip,iq)${\mathbf{num}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)$, or iq < 1${\mathbf{iq}}<1$, or ip < 0${\mathbf{ip}}<0$, or max (ip,iq) > 20$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}\right)>20$, or nt ≤ 0${\mathbf{nt}}\le 0$.

Not applicable.

None.

## Example

```function nag_tsa_uni_garch_exp_forecast_example
num = 100;
mn = 1;
nreg = 2;

% The series
yt = [7.53; 6.64; 7.39; 7.15; 6.42; 6.32; 6.98; 7.09; 6.63; 6.93;
7.01; 5.30; 7.86; 6.73; 7.39; 5.61; 7.02; 6.04; 7.46; 4.33;
6.02; 6.37; 3.93; 7.24; 8.58; 5.70; 9.13; 7.99; 7.79; 6.13;
8.78; 6.52; 6.79; 7.77; 7.31; 7.58; 8.78; 7.39; 8.00; 7.07;
7.65; 9.15; 8.32; 7.32; 7.58; 9.78; 8.17; 9.26; 7.79; 7.03;
7.45; 7.09; 8.06; 7.06; 9.91; 7.01; 8.32; 6.41; 8.59; 8.55;
7.77; 8.04; 9.54; 8.28; 7.97; 8.42; 8.30; 7.98; 7.60; 8.77;
7.54; 7.40; 9.26; 7.30; 9.33; 9.54; 8.08; 6.93; 4.27; 2.65;
5.03; 0.91;12.63;10.87; 9.26; 8.30; 6.85; 7.48; 9.67; 9.54;
7.33; 8.84; 7.75; 8.12; 7.29; 8.58; 7.80; 3.07; 9.33;16.91];

% The exogenous variables
x = zeros(num, nreg+mn);
x(:, 1:nreg) = [2.40, 0.12; 2.40, 0.12; 2.40, 0.13; 2.40, 0.14;
2.40, 0.14; 2.40, 0.15; 2.40, 0.16; 2.40, 0.16;
2.40, 0.17; 2.41, 0.18; 2.41, 0.19; 2.41, 0.19;
2.41, 0.20; 2.41, 0.21; 2.41, 0.21; 2.41, 0.22;
2.41, 0.23; 2.41, 0.23; 2.41, 0.24; 2.42, 0.25;
2.42, 0.25; 2.42, 0.26; 2.42, 0.26; 2.42, 0.27;
2.42, 0.28; 2.42, 0.28; 2.42, 0.29; 2.42, 0.30;
2.42, 0.30; 2.43, 0.31; 2.43, 0.32; 2.43, 0.32;
2.43, 0.33; 2.43, 0.33; 2.43, 0.34; 2.43, 0.35;
2.43, 0.35; 2.43, 0.36; 2.43, 0.37; 2.44, 0.37;
2.44, 0.38; 2.44, 0.38; 2.44, 0.39; 2.44, 0.39;
2.44, 0.40; 2.44, 0.41; 2.44, 0.41; 2.44, 0.42;
2.44, 0.42; 2.45, 0.43; 2.45, 0.43; 2.45, 0.44;
2.45, 0.45; 2.45, 0.45; 2.45, 0.46; 2.45, 0.46;
2.45, 0.47; 2.45, 0.47; 2.45, 0.48; 2.46, 0.48;
2.46, 0.49; 2.46, 0.49; 2.46, 0.50; 2.46, 0.50;
2.46, 0.51; 2.46, 0.51; 2.46, 0.52; 2.46, 0.52;
2.46, 0.53; 2.47, 0.53; 2.47, 0.54; 2.47, 0.54;
2.47, 0.54; 2.47, 0.55; 2.47, 0.55; 2.47, 0.56;
2.47, 0.56; 2.47, 0.57; 2.47, 0.57; 2.48, 0.57;
2.48, 0.58; 2.48, 0.58; 2.48, 0.59; 2.48, 0.59;
2.48, 0.59; 2.48, 0.60; 2.48, 0.60; 2.48, 0.61;
2.48, 0.61; 2.49, 0.61; 2.49, 0.62; 2.49, 0.62;
2.49, 0.62; 2.49, 0.63; 2.49, 0.63; 2.49, 0.63;
2.49, 0.64; 2.49, 0.64; 2.49, 0.64; 2.50, 0.64];

% Details of the model to fit
dist = 't';
ip = 1;
iq = 2;

% Control parameters
copts = true;
maxit = int64(200);
tol = 0.00001;

% Initial parameter estimates
theta = [0.05; -0.15; -0.05; 0.05; 0.15; 0.35; 3.25; 1.5; 0; 0];

% Forecast horizon
nt = 4;

% Fit the GARCH model
[theta, se, sc, covar, hp, et, ht, lgf, ifail] = ...
nag_tsa_uni_garch_exp_estim(dist, yt, x, int64(ip), ...
int64(iq), int64(nreg), int64(mn), theta, 0, copts, ...
maxit, tol, 'ifail', int64(-1));

% Calculate the volatility forecast
[fht, ifail] = nag_tsa_uni_garch_exp_forecast(...
int64(nt), int64(ip), int64(iq), theta, ht, et);

% Output the results
fprintf('\n               Parameter        Standard\n');
fprintf('               estimates         errors\n');

% Output the coefficient alpha_0
fprintf('Alpha0 %16.2f%16.2f\n', theta(1), se(1));
l = 2;

% Output the coefficients alpha_i
for i = l:l+iq-1
fprintf('Alpha%d %16.2f%16.2f\n', i-1, theta(i), se(i));
end
l = l+iq;

% Output the coefficients psi_i
for i = l:l+iq-1
fprintf('  Psi%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end
l = l+iq;

% Output the coefficients beta_j
fprintf('\n');
for i = l:l+ip-1
fprintf(' Beta%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end
l = l+ip;

% Output the estimated degrees of freedom, df
if (dist == 't')
fprintf('\n    DF %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated mean term, b_0
if (mn == 1)
fprintf('\n    B0 %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated linear regression coefficients, b_i
for i = l:l+nreg-1
fprintf('    B%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end

% Display the volatility forecast
fprintf('\nVolatility forecast = %12.4f\n', fht(nt));
```
```

Parameter        Standard
estimates         errors
Alpha0             0.17            0.20
Alpha1            -0.64            0.29
Alpha2            -0.44            0.25
Psi1            -0.07            0.23
Psi2             0.35            0.24

Beta1             0.42            0.17

DF             5.65            3.69

B0           129.22           50.19
B1           -51.94           21.18
B2            13.06            3.83

Volatility forecast =       1.3373

```
```function g13fh_example
num = 100;
mn = 1;
nreg = 2;

% The series
yt = [7.53; 6.64; 7.39; 7.15; 6.42; 6.32; 6.98; 7.09; 6.63; 6.93;
7.01; 5.30; 7.86; 6.73; 7.39; 5.61; 7.02; 6.04; 7.46; 4.33;
6.02; 6.37; 3.93; 7.24; 8.58; 5.70; 9.13; 7.99; 7.79; 6.13;
8.78; 6.52; 6.79; 7.77; 7.31; 7.58; 8.78; 7.39; 8.00; 7.07;
7.65; 9.15; 8.32; 7.32; 7.58; 9.78; 8.17; 9.26; 7.79; 7.03;
7.45; 7.09; 8.06; 7.06; 9.91; 7.01; 8.32; 6.41; 8.59; 8.55;
7.77; 8.04; 9.54; 8.28; 7.97; 8.42; 8.30; 7.98; 7.60; 8.77;
7.54; 7.40; 9.26; 7.30; 9.33; 9.54; 8.08; 6.93; 4.27; 2.65;
5.03; 0.91;12.63;10.87; 9.26; 8.30; 6.85; 7.48; 9.67; 9.54;
7.33; 8.84; 7.75; 8.12; 7.29; 8.58; 7.80; 3.07; 9.33;16.91];

% The exogenous variables
x = zeros(num, nreg+mn);
x(:, 1:nreg) = [2.40, 0.12; 2.40, 0.12; 2.40, 0.13; 2.40, 0.14;
2.40, 0.14; 2.40, 0.15; 2.40, 0.16; 2.40, 0.16;
2.40, 0.17; 2.41, 0.18; 2.41, 0.19; 2.41, 0.19;
2.41, 0.20; 2.41, 0.21; 2.41, 0.21; 2.41, 0.22;
2.41, 0.23; 2.41, 0.23; 2.41, 0.24; 2.42, 0.25;
2.42, 0.25; 2.42, 0.26; 2.42, 0.26; 2.42, 0.27;
2.42, 0.28; 2.42, 0.28; 2.42, 0.29; 2.42, 0.30;
2.42, 0.30; 2.43, 0.31; 2.43, 0.32; 2.43, 0.32;
2.43, 0.33; 2.43, 0.33; 2.43, 0.34; 2.43, 0.35;
2.43, 0.35; 2.43, 0.36; 2.43, 0.37; 2.44, 0.37;
2.44, 0.38; 2.44, 0.38; 2.44, 0.39; 2.44, 0.39;
2.44, 0.40; 2.44, 0.41; 2.44, 0.41; 2.44, 0.42;
2.44, 0.42; 2.45, 0.43; 2.45, 0.43; 2.45, 0.44;
2.45, 0.45; 2.45, 0.45; 2.45, 0.46; 2.45, 0.46;
2.45, 0.47; 2.45, 0.47; 2.45, 0.48; 2.46, 0.48;
2.46, 0.49; 2.46, 0.49; 2.46, 0.50; 2.46, 0.50;
2.46, 0.51; 2.46, 0.51; 2.46, 0.52; 2.46, 0.52;
2.46, 0.53; 2.47, 0.53; 2.47, 0.54; 2.47, 0.54;
2.47, 0.54; 2.47, 0.55; 2.47, 0.55; 2.47, 0.56;
2.47, 0.56; 2.47, 0.57; 2.47, 0.57; 2.48, 0.57;
2.48, 0.58; 2.48, 0.58; 2.48, 0.59; 2.48, 0.59;
2.48, 0.59; 2.48, 0.60; 2.48, 0.60; 2.48, 0.61;
2.48, 0.61; 2.49, 0.61; 2.49, 0.62; 2.49, 0.62;
2.49, 0.62; 2.49, 0.63; 2.49, 0.63; 2.49, 0.63;
2.49, 0.64; 2.49, 0.64; 2.49, 0.64; 2.50, 0.64];

% Details of the model to fit
dist = 't';
ip = 1;
iq = 2;

% Control parameters
copts = true;
maxit = int64(200);
tol = 0.00001;

% Initial parameter estimates
theta = [0.05; -0.15; -0.05; 0.05; 0.15; 0.35; 3.25; 1.5; 0; 0];

% Forecast horizon
nt = 4;

% Fit the GARCH model
[theta, se, sc, covar, hp, et, ht, lgf, ifail] = ...
g13fg(dist, yt, x, int64(ip), int64(iq), int64(nreg), ...
int64(mn), theta, 0, copts, maxit, tol, 'ifail', int64(-1));

% Calculate the volatility forecast
[fht, ifail] = g13fh(int64(nt), int64(ip), int64(iq), theta, ht, et);

% Output the results
fprintf('\n               Parameter        Standard\n');
fprintf('               estimates         errors\n');

% Output the coefficient alpha_0
fprintf('Alpha0 %16.2f%16.2f\n', theta(1), se(1));
l = 2;

% Output the coefficients alpha_i
for i = l:l+iq-1
fprintf('Alpha%d %16.2f%16.2f\n', i-1, theta(i), se(i));
end
l = l+iq;

% Output the coefficients psi_i
for i = l:l+iq-1
fprintf('  Psi%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end
l = l+iq;

% Output the coefficients beta_j
fprintf('\n');
for i = l:l+ip-1
fprintf(' Beta%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end
l = l+ip;

% Output the estimated degrees of freedom, df
if (dist == 't')
fprintf('\n    DF %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated mean term, b_0
if (mn == 1)
fprintf('\n    B0 %16.2f%16.2f\n', theta(l), se(l));
l = l + 1;
end

% Output the estimated linear regression coefficients, b_i
for i = l:l+nreg-1
fprintf('    B%d %16.2f%16.2f\n', i-l+1, theta(i), se(i));
end

% Display the volatility forecast
fprintf('\nVolatility forecast = %12.4f\n', fht(nt));
```
```

Parameter        Standard
estimates         errors
Alpha0             0.17            0.20
Alpha1            -0.64            0.29
Alpha2            -0.44            0.25
Psi1            -0.07            0.23
Psi2             0.35            0.24

Beta1             0.42            0.17

DF             5.65            3.69

B0           129.22           50.19
B1           -51.94           21.18
B2            13.06            3.83

Volatility forecast =       1.3373

```