Improving GARCH modeling approach

Question

Modeling Exchange Rate Using GARCH

Let's consider the following exchange rate : USD/JPY

For each sequence, we consider changes in the daily difference between the highest price and the open price of the underlying exchange rates.

Thus, if:

• $O(t)$ is the open price of the underlying exchange rate at time $t$, and
• $H(t)$ is the highest price of the underlying at time $t$,

we transform the sequence as follows:

$$ Y(t) = \log \frac{H(t)-O(t)}{H(t-1)-O(t-1)} $$

GARCH Model is frequently used to model changes in the variance of $Y(t)$, and I suggest to investigate in this way.

Is a common known that GARCH models are appropriate for modeling time series that exhibit a heavily-tailed distribution and display some degree of serial correlation.

So as a preliminary we must verify that the sequence $Y(t)$ is in fact heavy-tailed and does indeed exhibit serial correlation

Empirical Sequence

1. I computed : skewness = 0.11 and kurtosis = 3.9. Test Ok
2. I plotted ACF & PACF : evidence of serial correlation & long term dependence among sequence

GARCH model "OK"

GARCH Modelling

1. I fit a GARCH(1,1) / GARCH(1,2) / GARCH(1,2) to sequence to obtain parameters.
2. Ljung-Box : Only GARCH(1,1) & GARCH(1,2) succeed.
3. I simulated on 1 $Y$ and compared simulated to original sequence.
4. Result does not seems to capture salient features of the empirical sequence.

Do you see any improvement in the methodology to improve my results?

Thanks.

Answer 1

I think there is some room for improvement here.

1. GARCH

GARCH models are appropriate for modeling time series that exhibit a heavily-tailed distribution and display some degree of serial correlation.

That's not the case. GARCH is used for modelling series where there is serial correlation in variance, not in actual observations. And heavy tails are just incidental, and could indicate any number of things that have nothing to do with GARCH.

As you may recall, the model for GARCH(N,M) is

$$ Y(t) = \mu + \sigma(t) \varepsilon(t) $$ where $\varepsilon(t)$ are i.i.d. (usually $N(0,1)$) and $$ \begin{aligned} \sigma^2(t) = \omega & + \alpha_1\varepsilon^2(t-1) + ...+\alpha_N\varepsilon^2(t-N) \\ & + \beta_1\sigma^2(t-1) + ...+\beta_M\sigma^2(t-M) \end{aligned} $$

So to test for appropriatness of using GARCH, you should check for variability and serial correlation of squares of residuals $(Y(t) - \mu)^2$, not of the time series itself. This can be done by e.g. comparing variances of $(Y(t) - \mu)$ on different subsets of samples and testing the hypothesis that they are different. Alternatively, this can be done by plotting ACF and PACF of $(Y(t) - \mu)^2$, though as far as I remember (and I don't remember this well), there may be some quirks there. But before you do that, check the next section:

2. Serial Correlation

I plotted ACF & PACF : evidence of serial correlation & long term dependence among sequence

Your ACF and PACF showed you serial dependence in $Y(t)$. This suggests that the first thing you do is should make sure that the series is stationary by applying one of stationarity tests and, if they they show lack of stationarity, apply a correction like differencing. Though given that you're working with daily differences between max and and opening price, I would expect that the series is stationary.

If the series is stationary and still comes up with significant ACF/PACF results, you should try one of the ARMA(N,M) models which model serial dependence in the time series itself:

$$ \begin{align} Y(t) = \mu &+ \alpha_1 \epsilon(t-1) + ... + \alpha_N \epsilon(t-N)\\ &+ \beta_1 Y(t-1) + ... + \beta_M Y(t-M) \end{align} $$