One-Step AR(1) Forecast Timing and a Higher-Order AR Bug

# One-step ahead forecast of a AR(1) process (GARCH context)


# One-step ahead forecast of a AR(1) process (GARCH context)












I am using a Matlab toolbox for obtaining one-step ahead forecasts of the conditional mean from the ARMA(1,0)-GARCH(1,1) process and I have encountered a piece of code that contains, in my opinion, a mistake. The full code of the forecasting function is available for viewing at: http://uk.mathworks.com/matlabcentral/fileexchange/32882-armax-garch-k-toolbox--estimation--forecasting--simulation-and-value-at-risk-applications-/content/garchfor.m

The fragment that I was referring to is:

```
% Forecasting the Mean
MF = parameters(1:1+z)'*[1; data(end-(1:ar)); resids(end-(0:ma-1))]; % 1-period ahead forecast
 for i = 2:max_forecast
     MF(i,1) = sum([parameters(1); ones(1,ar)*parameters(2:2+ar)*MF(i-1,1); ones(1,ma)*parameters(3+ar:2+ar+ma)*resids(end-(0:ma-1-i))]);
 end
 clear i
```

From this code it seems that when I am considering ARMA(1,0) the function takes the one before last observation for the forecast. In other words, when the data spans time points $1,...,t$ and I want to obtain a forecast for period $t+1$ I multiply the AR(1) coefficient by the $t-1$ observation.In my opinion, for time $t+1$ AR(1) forecast I should be taking the last observation ($t$) from the data-set and multiply it by the AR(1) coefficient.

Could you please confirm my suspicions about this piece of code?

## Answer by Malick (score 1, accepted)

https://quant.stackexchange.com/a/22506

The code is correct regarding your question (and only for an AR(1) ), you made a mistake because the last observation of the data set is $t-1$ and not $t$ since you are forecasting the point at time $t$.

In the code : `MF(i,1)` is the current point forecast ($t$) and lag one observation ( `MF(i-1,1)` which is $t-1$ ) is correctly related to the AR part.

However it seems to me that there is an error in the following part :

```
ones(1,ar)*parameters(2:2+ar)*MF(i-1,1)
```

It is only correct if you are estimating an AR(1) , for a higher order the `MF(i-1,1)` part is wrong because you apply different coefficients to the same observation ( ex : $ \alpha_{1} y_{t-1} + \alpha_{2} y_{t-1} $ instead of $\alpha_{1} y_{t-1} + \alpha_{2} y_{t-2} $ ). I would recommend you to use more reliable codes such as functions you can find in the MFE Toolbox.