Interpreting GARCH Return and Volatility Forecasts
Article Quant Q&A · Author: math
Summary
The discussion explains how to read multi-step forecasts from an AR/GARCH model fitted to stock returns. The forecasted mean is determined by the conditional mean equation: in a model with only a constant mean, each horizon has the same point forecast. A model with autoregressive terms can produce different mean forecasts as those terms decay or evolve across forecast steps.
The forecast output also reports conditional standard deviations from the variance model; squaring them gives conditional variance forecasts. These quantities describe different aspects of future returns: the mean is a point estimate, while volatility measures conditional dispersion. The example includes fitted coefficients and residual diagnostics for two specifications, but it does not establish that either model is reliably predictive. It also highlights that software output and model assumptions matter when interpreting forecasts; the accepted response’s long-run variance suggestion should not be confused with the separate conditional standard deviation column.
Key ideas
- A constant-only conditional mean produces the same point forecast at every future horizon.
- The forecasted mean and conditional volatility describe distinct properties of returns.
- The reported conditional standard deviation can be squared to obtain conditional variance.
- Autoregressive terms in an AR/GARCH model can make mean forecasts vary across horizons.
- Residual checks and fitted coefficients provide diagnostics, not proof of forecasting performance.
Tags
Full text
# GARCH model and prediction
# GARCH model and prediction
I have a question about the prediction of volatility and returns of a time series. Basically it is a question about `predict` in the `fGarch`package.
The following code is from the book Analysis of financial time series and it is an example of AR/GARCH models for the log returns of the SP500
```
library(fGarch)
sp5=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts/sp500.dat")#Load data
plot(sp5,type="l")
m1=garchFit(formula=~arma(3,0)+garch(1,1),data=sp5,trace=F)
summary(m1)
m2=garchFit(formula=~garch(1,1),data=sp5,trace=F,cond.dist="std")
summary(m2)
stresi=residuals(m2,standardize=T)
plot(stresi,type="l")
Box.test(stresi,10,type="Ljung")
predict(m2,5)
```
First we assume an ARMA/GARCH model. However, we see that all the coefficient from the AR model are not significant. Hence we model a pure GARCH time series. My question is about the very last command. Running this command gives the following output:
```
> predict(m2,5)
meanForecast meanError standardDeviation
1 0.008455044 0.05330089 0.05330089
2 0.008455044 0.05327886 0.05327886
3 0.008455044 0.05325781 0.05325781
4 0.008455044 0.05323769 0.05323769
5 0.008455044 0.05321847 0.05321847
```
What is here the meanForecast and why is it always the same number? Is there a way to get the n-th volatility forecast as well as the n-th return forecast, e.g. the predicted volatility for the next day as well as the return. In the case of the model `m2` the volatility forecast will be the return forecast, since we assume a pure GARCH model. But how can we extract both in the `m1`case?
Edit: as asked by user12348 here are my outputs of `summary(m1)` and `summary(m2)`.
```
> summary(m1)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(3, 0) + garch(1, 1), data = sp5, trace = F)
Mean and Variance Equation:
data ~ arma(3, 0) + garch(1, 1)
<environment: 0x6ac79b0>
[data = sp5]
Conditional Distribution:
norm
Coefficient(s):
mu ar1 ar2 ar3 omega
7.7077e-03 3.1968e-02 -3.0261e-02 -1.0649e-02 7.9746e-05
alpha1 beta1
1.2425e-01 8.5302e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 7.708e-03 1.607e-03 4.798 1.61e-06 ***
ar1 3.197e-02 3.837e-02 0.833 0.40473
ar2 -3.026e-02 3.841e-02 -0.788 0.43076
ar3 -1.065e-02 3.756e-02 -0.284 0.77677
omega 7.975e-05 2.810e-05 2.838 0.00454 **
alpha1 1.242e-01 2.247e-02 5.529 3.22e-08 ***
beta1 8.530e-01 2.183e-02 39.075 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Log Likelihood:
1272.179 normalized: 1.606287
Description:
Wed Apr 23 18:07:32 2014 by user:
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 73.04843 1.110223e-16
Shapiro-Wilk Test R W 0.9857968 5.961505e-07
Ljung-Box Test R Q(10) 11.56744 0.3150483
Ljung-Box Test R Q(15) 17.78746 0.2740041
Ljung-Box Test R Q(20) 24.11916 0.2372259
Ljung-Box Test R^2 Q(10) 10.31614 0.4132084
Ljung-Box Test R^2 Q(15) 14.22819 0.5082978
Ljung-Box Test R^2 Q(20) 16.79404 0.6663039
LM Arch Test R TR^2 13.34305 0.3446074
Information Criterion Statistics:
AIC BIC SIC HQIC
-3.194897 -3.153581 -3.195051 -3.179018
>
```
and for `summary(m2)`:
```
> summary(m2)
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = sp5, cond.dist = "std",
trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
<environment: 0x6b70f70>
[data = sp5]
Conditional Distribution:
std
Coefficient(s):
mu omega alpha1 beta1 shape
0.00845504 0.00012485 0.11302582 0.84220210 7.00318063
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 8.455e-03 1.515e-03 5.581 2.39e-08 ***
omega 1.248e-04 4.519e-05 2.763 0.00573 **
alpha1 1.130e-01 2.693e-02 4.198 2.70e-05 ***
beta1 8.422e-01 3.186e-02 26.432 < 2e-16 ***
shape 7.003e+00 1.680e+00 4.169 3.06e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Log Likelihood:
1283.417 normalized: 1.620476
Description:
Wed Apr 23 18:09:17 2014 by user:
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 99.61249 0
Shapiro-Wilk Test R W 0.9836345 9.72802e-08
Ljung-Box Test R Q(10) 11.37961 0.3287173
Ljung-Box Test R Q(15) 18.2163 0.2514649
Ljung-Box Test R Q(20) 24.91842 0.2045699
Ljung-Box Test R^2 Q(10) 10.52266 0.3958941
Ljung-Box Test R^2 Q(15) 16.14586 0.3724248
Ljung-Box Test R^2 Q(20) 18.93325 0.5261686
LM Arch Test R TR^2 14.88667 0.247693
Information Criterion Statistics:
AIC BIC SIC HQIC
-3.228325 -3.198814 -3.228404 -3.216983
>
```
## Answer by user12348 (score 5, accepted)
https://quant.stackexchange.com/a/11040
The mean could be the long run variance which is
```
sig2 = fit.Constant/(1-fit.GARCH{1}-fit.ARCH{1});
```
I hope this explains.
If not, note I ran this model through Matlab, I get different values. you can paste your m1 and m2 values and some other intermediate results so I can see why Matlab differs.
EDIT: The question refers to forecasting the returns. Using AR-GARCH model, $$r_t= μ+\epsilon_t$$ $$ z_t=\epsilon_t/σ_t $$ $z_t$ is white noise or i.i.d, and can take any distribution. $$σ_t^2=w+\alpha \epsilon_{t-1}^2+\beta σ_{t-1}^2$$ The predict function in R is forecasting $r_{t+k}$ where k is the periods into the future. It is also possible to forecast future variance, $σ_{t+k}^2$,as shown, using GARCH formula above.
## Answer by Richard Hardy (score 1)
https://quant.stackexchange.com/a/30287
> What is here the meanForecast and why is it always the same number?
It is the point forecast due to the conditional mean model. Since it is just a constant in your model `m2`, the forecast is also constant.
> Is there a way to get the n-th volatility forecast as well as the n-th return forecast, e.g. the predicted volatility for the next day as well as the return.
The documentation of the `predict` function is not very explicit, but the first column `meanForecast` gives the point forecast and I guess the third column `standardDeviation` gives the conditional standard deviation forecasts due to the GARCH model. Square them to get the conditional variance forecasts.
> In the case of the model `m2` the volatility forecast will be the return forecast, since we assume a pure GARCH model.
Volatility forecast will be the return forecast? In what sense? A return is a random variable, volatility is its property. If by return forecast you mean a point forecast, again this is not the same as volatility forecast.
> But how can we extract both in the `m1` case?
`predict` gives you the conditional mean and conditional standard deviation forecasts in the first and the last columns. Square the latter to get the conditional variance forecasts.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.