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

arimaeconometricsforecastinggarchtime series

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!

Best Answer

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.

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Solved – Forecasting returns using GARCH model alone

A model for the returns $r_t$ with a GARCH structure for the conditional variance will look like this: \begin{aligned} r_t &= \mu_t + u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \varepsilon_t &\sim i.i.d.(0,1), \end{aligned} where $\mu_t$ is the conditional mean of $r_t$ which could be e.g. a constant or an ARMA process. (Here we have a GARCH(1,1) model, but the extension to GARCH(p,q) is trivial.)

An optimal (under square loss) point forecast of $r_{t+h}$ for some $h>0$, given the available information $I_t$, is \begin{aligned} \mathbb{E}(r_{t+h}|I_t) &= \mathbb{E}(\mu_{t+h}+u_{t+h}|I_t) \\ &= \mathbb{E}(\mu_{t+h}|I_t) + \mathbb{E}(u_{t+h}|I_t) \\ &= \mathbb{E}(\mu_{t+h}|I_t) + 0 \\ &= \mathbb{E}(\mu_{t+h}|I_t). \\ \end{aligned} Since we normally do not know $\mathbb{E}(\mu_{t+h}|I_t)$, we take a forecasted value from the model, $\hat\mu_{t+h|t}$. So in practice the point forecast is $\hat\mu_{t+h|t}$.

In case $\mu_t=\mu$ is a constant (could be zero, could be nonzero), the optimal (under square loss) point forecast of $r_{t+h}$ for some $h>0$ is $\mathbb{E}(\mu_{t+h}|I_t)=\mu$. In practice you would take the fitted intercept $\hat\mu$ from the conditional mean model as the point forecast.

Example with R package "fGarch":

library(fGarch)
model = garchFit(formula = ~ garch(1, 1), data = dem2gbp, cond.dist = "norm", include.mean = TRUE)
fcst=predict(model,n.ahead=5)
mean.fcst=fcst$meanForecast

The last line saves the point forecasts in an object called mean.fcst.
Check coef(model) to verify that the forecasts equal the fitted intercept mu.

Solved – Why fit ARMA before GARCH if I am interested in variance of the data, not the residuals

On the one hand, GARCH is a model for the conditional distribution of the time series $y_t$:

\begin{aligned} y_t &\sim d(\mu_t,\sigma_t^2), \\ \mu_t &= \dots \text{(e.g. some ARMA equation)} \\ \sigma_t^2 &= \omega + \alpha_1 ( y_{t-1} - \mu_{t-1} )^2 + \dotsc + \alpha_s ( y_{t-s} - \mu_{t-s} )^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &:= \frac{y_t-\mu_t}{\sigma_t} \sim i.i.d(0,1). \\ \end{aligned}

GARCH specifically characterizes the conditional variance equation, but this inevitably depends on some equation for the conditional mean; otherwise the conditional variance would be undefined. This hopefully answers your questions 1. and 3.

On the other hand, GARCH happens to characterize the conditional variance of the residuals from the conditional mean model, $u_t$, as well. The exact same model as above can be represented as follows:

\begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \dots \text{(e.g. some ARMA equation)} \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d(0,1), \\ \end{aligned}

and $\sigma_t^2$ is the conditional variance of the residual $u_t$ since $\text{Var}(u_t|I_{t-1})=\sigma_t^2$ where $I_{t-1}$ is information available at time $t-1$. This hopefully answers your question 2.

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Solved – Forecasting returns using GARCH model alone

Solved – Why fit ARMA before GARCH if I am interested in variance of the data, not the residuals

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