# Can I forecast stock returns using GARCH?

I know this is a rookie question, but I have seen some comments about using GARCH to forecast stock returns.

Is it something people do? Wasn't GARCH just for volatility?

Also, can you suggest any (simple) papers that could help me better understand this technique and how to apply it in Eviews or R?

• Do you remember where you saw some comments about using GARCH to forecast stock returns? Because, like you say, GARCH is about volatility (of returns), not returns themselves. So it doesn't sound right. That some traders use both forecasts of returns and forecasts of volatility, that is more plausible. But these would be two separate things. – Alex C Jul 3 '17 at 4:59
• Stock Price Prediction Using the ARIMA Model by Adebyi et al. 2014 a paper that I found which mentions stock returns forecasting using GARCH It could be an error in the paper or the formulation might be erroneous, but it made me think :)) – Alex Jul 4 '17 at 16:54
• That Adebyi paper was published by the IEEE, unusual for a quant finance paper. The whole idea seems strange to me (but strange ideas sometime work...). – Alex C Jul 4 '17 at 19:58
• Simple paper to better understand the technique quantopian.com/lectures/arch-garch-and-gmm – Alex C Jul 6 '17 at 22:11

The GARCH(p, q) model (where p is the order of the GARCH terms ${\sigma^2}$ and q is the order of the ARCH terms ${\epsilon^2}$ ), following the notation of original paper is given by: $${\sigma_t^2}={\omega}+{\sum_{i=1}^q}{\alpha_i}{\epsilon_{t-i}^2}+{\sum_{i=1}^p}{\beta_i}{\sigma_{t-i}^2}$$

Obviously, the GARCH model is about volatility and variance of returns. It can only forecast volatility, but not returns.

Actually, It is much more difficult to forecast returns than to forecast volatility.

You could take this book to understand GARCH and apply it with R: An Introduction to Analysis of Financial Data with R.

It depends on your beliefs about the nature of stock returns. Whether or not those beliefs reflect reality is a different matter. I think it's helpful to consider what is being modeled in GARCH by breaking down the term:

Generalized - i.e., can take a number of parameters in order to fit generic data types.

Autoregressive - i.e., terms tend to revert to their means, a-la a Ornstein-Uhlenbeck process (also, Brownian Motion under friction).

Conditional - i.e., future terms are dependent on past terms and/or best estimates of future terms are based on Bayesian inference; reflects stylized beliefs on absolute return/variance clustering.

Heteroskedasticity - i.e., literally, "differing variance"; parameters change over time.

While it is technically possible to use GARCH to model the conditional expectations of stock returns, GARCH models were not intended to model returns. Implicit in the name is its intent as skedasticity (i.e., volatility) metric in which terms are squared residuals of periodic returns.

Relaxing the squared error condition, however, results in a particular form of Autoregressive Moving Averages (ARMA). ARMAs are a more generic class of econometric model which allow for negative terms and which can conform to many stylized facts of security returns.