# GARCH modelling

Is this simplistic way of describing GARCH processes correct:

Future prices do not depend on previous prices per se, but rather on the previous variance.

If this is incorrect, is there a model that does this?

• The GARCH model states that the variance of the current price depends on the variance of the previous price(s). If you want a model where the levels of the current price depend on the previous variance, you need to look at ARCH-in-mean models. Commented Feb 15, 2015 at 8:39

Suppose you have some univariate time series $y_{t}$ $$y_{t}=f\left(x_{t}\right)+\epsilon_{t}$$ with $$\epsilon_{t}\equiv z_{t}\sigma_{t}$$ where $f(x_{t})$ is some arbitrary function and $z_{t}$ is iid with zero mean. If $y_{t}$ is in levels, then it will likely have implications for how you model $f(x_{t})$. It is important that $\epsilon_{t}$ is stationary. This might involve fitting an ARMA model or setting $f(x)\equiv y_{t-1}$, effectively differencing.

If you model $\sigma_{t}$ as $$\sigma_{t}^{2}=\alpha_{0}+\sum_{i=1}^{p}\alpha_{i}\epsilon_{t-i}^{2}+\sum_{j=1}^{q}\beta_{j}\sigma_{t-j}^{2}$$ then you would say that the variance is Garch(p,q).

The above means that today's variance is a linear function of the previous period's squared error and variance. The expected value of $y_{t}$, however, does not necessarily depend on the variance. It is driven by whatever $f(x_{t})$ is (since $z_{t}$ has mean of zero). If $f(x_{t})$ contains a term for variances, such as the Garch-in-mean model, then the expected value of $y_{t}$ may depend on variances. Whether that is true is an empirical question, depending on what prices you're looking at.

Nevertheless, it is more common to perform econometric analysis on log prices (or returns) and then convert any forecasts back in terms of prices (if you need them that way). If $y_{t}$ is log prices and you try to convert $E(y_{t+k})$ to $E(\exp(y_{t+k}))$ (which would be the expected prices in $t+k$ instead of the expected log prices), then the expected price is influenced by the variance of the log price.

To that extent, prices often depend on past prices and past variances. For some types of prices, like stock returns for liquid stocks, you might be able to say that future returns depend more on past variances than past returns, at least in short forecast horizons. But do not mix up prices and returns.

• What is $x$ and $x_t$ ? (I assume $y_t$ is the asset price we want to model for example?)
– Basj
Commented Feb 26, 2016 at 16:17