# Which econometric models can be used to forecast security returns + ARIMA/GARCH questions

I'm trying to write an undergraduate thesis wherein I test the predictive power of a given econometric model on a given financial time series. I need some advice on how I should go about doing this. To put matters into context, I have mostly self-studied econometrics; the only course I took on the subject stopped short of delving into time series models, so I am by no means an expert on the subject.

To my dismay, I recently read that ARIMA models are very poor at predicting stock (and other security) returns. A professor in my school's economics department also confirmed this. All this time I was hoping they could perhaps be even remotely useful for forecasting some financial time series... Are there any other models I could look at? My goal is simply to learn some econometric modeling of time series in R or MATLAB and hopefully find statistically significant predictive results. Also, is there a particular market you would look at (energy, rates, equities)?

Lastly, is GARCH only used for forecasting volatility? The professor I mentioned seemed to suggest I should turn toward GARCH or ARIMA-GARCH models to model stock returns. I read some papers that seemed to imply it could also be used for actual returns... Perhaps I misunderstood. Would the AR and MA components in an ARIMA-GARCH model differ from those in an ARMA model? From what I vaguely understood, ARIMA and GARCH are two completely separate things (with the former being used to predict the actual time series and the other to predict its volatility).

I hope that's not too many questions, but I just don't know where to turn to anymore, I've been researching this on my own for so long. Thanks a lot!

• If you believe in efficient markets, then it should come as no surprise to you that it is very difficult to predict stock returns. If you could, you'd have found a way of printing money. There may be predictable patterns that one can exploit to make money, but from what I've heard, you need to be able to react (trade) in milliseconds these days - so it's not for you and me. – Christoph Hanck Apr 14 '15 at 7:41
• And yes, GARCH is used for volatility modelling, not levels. – Christoph Hanck Apr 14 '15 at 7:42
• Long term returns are predictable, somewhat. For undergrad it's more fun to look at modern behavior finance stuff. Lookup paper on this subject, there's a ton. It's usually on portfolio, not individual stocks. – Aksakal Apr 14 '15 at 11:36
• Thanks a lot for the info! I'll see if there are other behavioral-type topics I could also include in my paper (maybe price action on each day of the week). – Ninja7777 Apr 14 '15 at 18:09

My goal is simply to ... find statistically significant predictive results. Also, is there a particular market you would look at (energy, rates, equities)?

Most if not all the established and liquid financial markets will be very hard to predict whatever model you will use. If markets were relatively easy to predict, market participant would try to exploit that to make money. By doing that they would eliminate the predictability. This brings a contradiction, and thus the markets are not easy to predict.

Lastly, is GARCH only used for forecasting volatility? The professor I mentioned seemed to suggest I should turn toward GARCH or ARIMA-GARCH models to model stock returns. I read some papers that seemed to imply it could also be used for actual returns...

GARCH model is used for modelling the conditional variance of the disturbance term of the conditional mean model for a dependent variable $y_t$. E.g. if you have a conditional mean model $y_t=\alpha+\epsilon_t$, the GARCH model will describe the conditional variance of $\epsilon_t$. Sometimes the conditional mean model is "empty" ($y_t=\epsilon_t$), then GARCH model is used to model the conditional variance of $y_t$ itself.

Even if you are primarily interested in the conditional mean model (e.g. you want to predict stock returns using an ARMA model), a GARCH model combined with a model for the conditional mean can be useful. If the conditional variance of the dependent variable is time-varying, that should be accounted for, and a GARCH model does exactly that. If a time-varying conditional variance is neglected, the conditional mean model may (and likely will) be invalid.

Would the AR and MA components in an ARIMA-GARCH model differ from those in an ARMA model?

Yes. That also illustrates my last remark above.

From what I vaguely understood, ARIMA and GARCH are two completely separate things (with the former being used to predict the actual time series and the other to predict its volatility).

This is true. But as I have already explained, the two models can work together nicely.

• Thank you very much, now I get it. So the GARCH coefficients depend on your presumed conditional mean model, which you could model as an ARIMA process (which will make it so that you no longer assume a mean of 0). – Ninja7777 Apr 15 '15 at 21:41
• Yes. Also, the coefficients of the conditional mean model depend on GARCH because GARCH yields that different observations get different weights in estimation of the cond. mean model (the ones with high cond. variance get low weights, the ones with low cond. variance get high weights -- similarly to weighted least squares (WLS) as opposed to OLS). – Richard Hardy Apr 16 '15 at 5:54
• Hi Richard, I just wanted to follow up on this point if that's alright, as I've encountered a small issue in my report. Could you please elaborate on how a GARCH-modeled variance causes different weights in our time series observations? I initially thought the conditional mean was impacted somehow because of how GARCH affects the error term and MA component in the ARMA model, but I suspect I'm completely off. Do you know of anywhere where I can find a brief mathematical explanation perhaps? I still can't find any documentation anywhere. Thank you very much for all your help. – Ninja7777 Apr 27 '15 at 7:37
• Essentially, yes. What I said was looking at the same problem from a different angle. Suppose you have a simple regression $$y=\beta_0+\beta_1 x+\varepsilon$$ and you know the true underlying variance of each error term; that is, you know $\sigma^2_1$,...,$\sigma^2_T$. Then the efficient estimator of $\beta$'s will be weighted least squares, not ordinary least squares. With a GARCH model, you do not quite know the true underlying variances, but you have their estimates, and you use those in a way similar to the use of the true variances in weighted least squares estimation. – Richard Hardy Apr 27 '15 at 7:58
• So what you wrote above is focusing on the mechanics of the data generating process (if you believe that ARMA-GARCH is the exact, true underlying process). Meanwhile, what I said is about the estimation of that process. But as far as I can tell, you got the idea right anyway. – Richard Hardy Apr 27 '15 at 8:01

I applaud your enthusiasm for the subject. There is a lot of applications and methods to help with prediction but it is clear that there is no silver bullet. Just like there is no one weather model that predict all weather in all locations with equal accuracy, there is not model that can predict financial time series.

I would encourage you to look at a single sub-behavior of markets to see if you can understand it. Some quick examples are

• Month end price action
• Price movements around earnings-releases / economic data
• influences of winter storms on US natural gas markets
• futures contract rolls

As for techniques, one new-classic method is Cointegration:

I am in no way endorsing the analysis and results in the above links. They are simply some top google results to get you on the path to learn more about co-integration.

• Thank you, this is what I was looking for. Cointegration seems very interesting, I'll see if I can feasibly write something up on it in the next few weeks alongside the ARIMA-GARCH stuff. – Ninja7777 Apr 14 '15 at 18:08