# Estimation

We use the Deutsche Mark British Pound dataset for this demonstration throughout the document.

Code
library(tsgarch)
suppressMessages(library(data.table))
suppressMessages(library(xts))
data(dmbp)
dmbp <- xts(dmbp, as.Date(1:nrow(dmbp), origin = '1970-01-01'))
spec <- garch_modelspec(dmbp[1:1500,1], model = 'fgarch', constant = TRUE,
init = 'unconditional', distribution = 'jsu')
mod <- estimate(spec)
as_flextable(summary(mod))
FGARCH Model Summary

Estimate

Std. Error

t value

Pr(>|t|)

$\mu$

-0.0154

0.0099

-1.5574

0.1194

$\omega$

0.0050

0.0032

1.5543

0.1201

$\alpha_1$

0.1621

0.0373

4.3419

0.0000

***

$\gamma_1$

-0.3522

0.1108

-3.1776

0.0015

**

$\eta_1$

0.7829

0.1594

4.9108

0.0000

***

$\beta_1$

0.8727

0.0278

31.4083

0.0000

***

$\delta$

1.4659

0.3593

4.0799

0.0000

***

$\zeta$

-0.2595

0.0835

-3.1076

0.0019

**

$\nu$

1.4762

0.1046

14.1174

0.0000

***

$P$

0.9903

0.0104

95.0454

0.0000

***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

variance targeting: FALSE

initialization value: 0.2405

LogLik: -829.1715

AIC: 1678 | BIC: 1731

Model Equation

$z_t = \frac{\varepsilon_t}{\sigma_t} \sim JSU\left(0,1,\zeta, \nu\right)$

$\sigma^{\delta}_t = \omega + \alpha_1 \sigma^{\delta}_{t-j}\left(\left|z_{t-1} - \eta_1\right| - \gamma_1 \left(z_{t-1} - \eta_1\right)\right)^{\delta} + \beta_1 \sigma^{\delta}_{t-1}$

Persistence (P) and Unconditional Variance Equations

$P = \sum_{j=1}^p \beta_j + \sum_{j=1}^q \alpha_j \kappa_j,\quad \kappa_j = E\left[\left(\left|z_{t-j} - \eta_j\right| - \gamma_j \left(z_{t-j} - \eta_j\right)\right)^{\delta}\right]$

$E\left[\varepsilon^2_t\right] = \left(\frac{\omega}{1 - P}\right)^{2/\delta}$

Notice that the as_flextable method provides for a publication ready option for printing out the model summary.

We next take a look at a summary plot of the estimated model:

Code
oldpar <- par(mfrow = c(1,1))
par(mar = c(2,2,2,2))
plot(mod)

Code
par(oldpar)

Notice the news impact curve which is both shifted and rotated, a particularly appealing feature of the Family GARCH model.

# Simulation and Prediction

The code below shows how to predict from an estimated model and how to simulate using a specification with fixed parameters.

Code
delta <- coef(mod)["delta"]
new_spec <- spec
new_spec$parmatrix <- copy(mod$parmatrix)
sim <- simulate(new_spec, nsim = 500, h = 10000, seed = 100, burn = 1000)
mean_sim <- mean(rowMeans(sim$sigma^delta))^(2/delta) pred <- predict(mod, h = 1000, nsim = 0) oldpar <- par(mfrow = c(1,1)) par(mar = c(2,2,2,2)) plot(as.numeric(pred$sigma^2), type = "l", xlab = "horizon",
ylab = expression(sigma^2), ylim = c(0.25, 0.41), main = "Family GARCH - JSU Prediction")
abline(h = as.numeric(unconditional(mod)), col = 2)
abline(h = mean_sim, col = 3)
legend("bottomright", c("h-step prediction","unconditional variance","simulated unconditional variance"), col = c(1,2,3), lty = 1, bty = "n")

Code
par(oldpar)

# Conclusion

There are other methods available such as one for profiling the parameter distribution (tsprofile), a backtest method (tsbacktest) as well as other methods for extracting information from the estimated object such as vcov, pit (probability integral transform), confint etc.

The package does not include any tests, as most of these have been moved to the tstests package and there are ample packages which include testing methods.