GARCH Forecasting in R – How to Perform GARCH Forecasting with Constant Mean in R?

arimaforecastinggarchr

I'm trying to forecast a time series of a stock option using ARMA-GARCH modelling in R. First I determine the ARMA order using AIC and I found (0,1) to be the best one.

But when I run

garchFit(formula=~arma(0,1)+garch(1,1),data=XX,trace=FALSE,include.mean=TRUE)

I get constant mean forecasts.

Whereas when I use ARMA(1,1):

garchFit(formula=~arma(1,1)+garch(1,1),data=brentlog1,trace=FALSE,include.mean=TRUE)

I get variables mean forecasts. Does anyone know why is that?

How do I forecast the actual prices of the stock (not its variance)?

Best Answer

ARMA(0,1) (also known as MA(1)) will give a constant forecast beyond the time period $t+1$. More generally, MA($q$) will give a constant forecast beyond the time period $t+q$. This is nothing new; you may check any introductory time series textbook or lecture notes where forecasts from an MA(1) or MA($q$) model are discussed. Alternatively, you may check this post or search through other posts tagged with arima and forecasting.

On the other hand, ARMA(1,1) will give variable mean forecasts. Again, you may find that in any introductory time series textbook.

Edit: the following paragraph applies to stock returns and may be irrelevant with regards to stock options.

A side note regarding forecasts for stock returns: there is very little chance that an ARMA($p$,$q$) model will beat a naive no-change forecast. It would be more realistic to assume an ARMA(0,0) model for the conditional mean of the stock returns. I am not so sure about the option include.mean. In the medium run, the stock price could be going up together with inflation, then include.mean=TRUE could make sense (assuming that the inflation is constant). In the long run, the assumption of constant inflation may fail, so you could try disinflating the stock price before modelling it using the ARMA-GARCH model.

Related Solutions

R – How to Estimate GARCH with Exogenous Variables in the Mean Equation?

The solution is to use another package called rugarch.

install.packages("rugarch")
require(rugarch)

Let's construct a (Tx2) matrix called data, where column 1 is $Y_t$ and column 2 is $X_t$.

data <- cbind(rnorm(1000),rnorm(1000))

We can then compute the GARCH(1,1) model that I described in my question:

spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1), 
submodel = NULL, external.regressors = NULL, variance.targeting = FALSE), 
mean.model = list(armaOrder = c(0, 0), external.regressors = matrix(data[,2]), 
distribution.model = "norm", start.pars = list(), fixed.pars = list()))

garch <- ugarchfit(spec=spec,data=data[,1],solver.control=list(trace=0))

Comparing this to EViews: The R coefficient estimates that are greater than $0.01$ differ to the EViews estimates by approximately $1-2\%$, which is acceptable in my mind. $t$-statistics differ by more but it doesn't effect inference.

GARCH Model – How to Choose the Order of a GARCH Model?

You should determine both the ARMA and the GARCH orders simultaneously.

If the process is indeed well approximated by an ARMA-GARCH model, considering the conditional mean model (ARMA) while neglecting the conditional variance model (GARCH) -- and this way (implicitly) assuming the conditional variance to be constant -- will lead to trouble. Similarly, when considering the conditional variance model you should not neglect the model for the conditional mean.

This is because neither the conditional mean model nor the conditional variance model can be estimated consistently if taken separately, unless under special conditions (e.g. if the MA part of the ARMA model is empty, the AR part can be estimated consistently even if the GARCH model is neglected). However, joint estimation will typically be more efficient and that is why it is preferred.

Unfortunately, the task of jointly determining the ARMA-GARCH orders is difficult, as I understand it. You may experiment with a few different models and compare their AICs or BICs. The larger the pool of candidate models, the more likely you are to overfit. If you have a large enough sample, you may try cross validation. That is,
(1) define a pool of candidate models,
(2) estimate the models on part of the sample,
(3) use the estimated models to predict the remainder of the sample,
(4) pick the model that has the lowest prediction error.
Still, this will not prevent overfitting if the pool of candidate models is quite large as compared with the sample size.

You can also examine the properties of the residuals of the different models. You would prefer a model with well-behaved residuals (no remaining autocorrelations, no remaining ARCH patterns, etc.). Again, this is subject to overfitting. It is easy to imagine data dredging or model mining when one is not stopping until one gets all the model coefficients to be significant. But can this result be trusted if achieved via data dredging? (Of course, not.)

I am sorry I do not have a better solution.

Best Answer

Related Solutions

R – How to Estimate GARCH with Exogenous Variables in the Mean Equation?

GARCH Model – How to Choose the Order of a GARCH Model?

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