time series - 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 fGarchpackage.

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 m1case?

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)

 [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)

 [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

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.