### Spline-GARCH Model

#### Definition

Consider a return time series ${r}_{t}=\mu +{\epsilon }_{t}$, where $\mu$ is the expected return and ${\epsilon }_{t}$ is a zero-mean white noise. Despite of being serially uncorrelated, the series ${\epsilon }_{t}$ does not need to be serially independent. For instance, it can present conditional heteroskedasticity. The $\mathrm{Spline-GARCH}$ model assumes a specific parametric form for this conditional heteroskedasticity. More specifically, we say that ${\epsilon }_{t}~\mathrm{Spline-GARCH}$ if we can write ${\epsilon }_{t}=\sqrt{{\sigma }_{t}^{2}{\tau }_{t}}{z}_{t}$, where ${z}_{t}$ is standard Gaussian:

$σt2 = ω + αεt-12 + βσt-12$

is a usual $\mathrm{GARCH}$ specification, and:

$τt = exp ∑ i=1 k ϕi t-ti 2$

is the exponential of a quadratic Spline with $k$ knots ${t}_{1},{t}_{2},...,{t}_{k}$. The idea of adding this ${\tau }_{t}$ is to capture lower frequency variations on the volatility, like seasonalities and trends. The number of knots, $k$, is chosen by minimizing the Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC), while the knots are placed equidistantly.

#### Estimation

V-Lab estimates all the parameters simultaneously, by maximizing the log likelihood. The assumption that ${z}_{t}$ is Gaussian does not imply the the returns are Gaussian. Even though their conditional distribution is Gaussian, it can be proved that their unconditional distribution presents excess kurtosis (fat tails). In fact, assuming that the conditional distribution is Gaussian is not as restrictive as it seems: even if the true distribution is different, the so-called Quasi-Maximum Likelihood (QML) estimator is still consistent, under fairly mild regularity conditions.

Besides leptokurtic returns, the $\mathrm{Spline-GARCH}$ model, as the $\mathrm{GARCH}$ model, captures other stylized facts in financial time series, like volatility clustering. The volatility is more likely to be high at time $t$ if it was also high at time $t-1$. Another way of seeing this is noting that a shock at time $t-1$ also impacts the variance at time $t$.

Usual restrictions on the parameters are $\omega ,\alpha ,\beta >0$. There are no restrictions on the parameters of the spline, ${\varphi }_{1},...,{\varphi }_{k}$. The $\mathrm{GARCH}$ model is in fact a restricted version of the $\mathrm{Spline-GARCH}$, with ${\varphi }_{1}=...={\varphi }_{k}=0$.

#### Prediction

Let ${r}_{T}$ be the last observation in the sample, and let $\stackrel{^}{\omega }$, $\stackrel{^}{\alpha }$, $\stackrel{^}{\beta }$ and ${\stackrel{^}{\varphi }}_{1},...,{\stackrel{^}{\varphi }}_{k}$ be the QML estimators of the parameters $\omega$, $\alpha$, $\beta$ and ${\varphi }_{1},...,{\varphi }_{k}$, respectively.

For out-of-sample forecasts, we need to extrapolate the spline. We have good results when the extrapolation simply uses the last in-sample value. That is, V-Lab assumes that, $\forall h\ge 0$:

$τ^t+h = τ^T = exp ∑ i=1 k ϕ^i T-ti 2$

Hence, the $\mathrm{Spline-GARCH}$ model implies that the forecast of the conditional variance at time $T+h$ is:

$σ^T+h2 τ^T+h = ω^ + α^ + β^ σ^ T+h-1 2 exp ∑ i=1 k ϕ^i T-ti 2$

and so, by applying the above formula iteratively, we can forecast the conditional variance for any horizon $h$. Then, the forecast of the compound volatility at time $T+h$ is:

$σ^ T+1:T+h τ^ T+1:T+h = ∑ i=1 h σ^ T+i 2 τ^ T+i$

Notice that, for large $h$, this forecast of the compound volatility converges to:

$h ω^ 1-α^-β^ exp ∑ i=1 k ϕ^i T-ti 2$

scaling over the forecast horizon with the well known square-root law.

#### Zero Slope Spline-GARCH

The Zero Slope $\mathrm{Spline-GARCH}$ model requires that the low-frequency component (i.e. the exponential of the spline) has zero slope in the end of the sample. That is, the coefficients are estimated by QML with the additional restriction that:

$2 ∑ i=1 k ϕi T-ti exp ∑ i=1 k ϕi T-ti 2 = 0$

or:

$∑ i=1 k ϕi T-ti = 0$

Note that we can solve the last equation for, say, ${\varphi }_{1}$. That is, from the zero-slope constraint, we can write ${\varphi }_{1}$ as a function of ${\varphi }_{2},...,{\varphi }_{k}$. But then we can substitute this expression wherever ${\varphi }_{1}$ appears in the maximum likelihood estimation. In this way, we can transform a constrained optimization in an unconstrained optimization with one reduced dimension. This leads to a more reliable, fast optimization.

#### Spline-GARCH(p,q)

The specific model just described can be generalized to account for more lags in the conditional variance. A $\mathrm{Spline-GARCH}\left(p,q\right)$ model assumes that:

$σ t 2 = ω + ∑ i=1 p αi ε t-i 2 + ∑ j=1 q βj σ t-j 2$

The best model ($p$ and $q$) can be chosen, for instance, by Bayesian Information Criterion (BIC), or by Akaike Information Criterion (AIC). The former tends to be more parsimonious than the latter. V-Lab uses $p=1$ and $q=1$ though, because this is usually the option that best fits financial time series.

#### Bibliography

Engle, R. F. and J. G. Rangel, 2008. The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes. Review of Financial Studies 21(3): 1187-1222.