The software MUST calculate forecasts for you, so you don't have to think about how you are gong to predict using the model you've chosen.

What is the exact model you are using?

Ar(1) is x[t]=x[t-1]+e.

But where do you use your regressor?

With autocorrelation the OLS estimators are not the best among the linear and the correct ones (it is possible to obtain better estimators: GLS, for example); the Eviews software eliminates this problem with the AR terms, thus you must refer to the coefficient estimates with these terms. The forecast must be made mantaining the AR terms.

@anyone, please be specific of how I calculate outputs using ar(1) models with a quadratic regressor...

Is yor model a purely autorregressive model (i.e., Y[t] = f(Y[t-1]) or are there other regressors (for example, Yt = f(X1[t], X2[t], ... , Xk[t], Y[t-1]) ?

It is not clear yet what you are asking about.

Please formulate your model precisely.

As Davydenko said in his first answer, Eviews calculates forecast for you. You have to do nothing else than the following steps:

1) open the estimated equation

2) press the FORECAST button

3) select the proposed options in the forecast function (I suggest "dynamic forecast" and "coefficient uncertainty in standard error calculation")

4) evaluate results

hi: I don't use eviews but, in general, without other assumtons,  AR(1) error terms allow you to only forecast one step ahead. To make the next forecast at time t, you need the error term from the previous time period t-1 and you only have that for time t. After time t, you have to assume the error term is zero, which, IMHO, is unrealistic. But, if you're only doing one step ahead forecasts and you express your equation exactly, someone here including myself can write down the expression for the forecast. The quadratic regressor doesn't complicate anything.

Dear Mark, the problem that you highlight is the reason why I suggested the option "dynamic forecast" : in Eviews this option updates the AR term and allows to forecast more than one  step ahead.

Thanks Quirino. Like I said, I have no clue about Eviews but, most likely, the only thing I can think of  is that the option just sets future epsilon's to zero and updates the AR term that way. I'm definitely opened to other methodologies though because I've run into this issue in what I work on. All the best.

@Javier and Andrey, my model is Yt=Yt-1*rho+a*Xt+rho*Xt-1+bXt^2-rho*Xt-1

a) can I use this model for long term forecasting ? some papers say ar(1) are good for long term forecasting

b) my model involve a quadratic regressor. Are there any specific issues regarding to ar(1) models with a quadratic regressor?

I don't have time to study it but that model is somewhat related to the koyck disributed but it's got some kind of error correction attempt in it. Is there supposed to be an opened parenthesis after the b and a closing parenthesis after the last rho*X_t-1. I think this must be the case because otherwise the rho*X_t-1 terms cancel out. It will be much clearer if you send a latex document with the equation in it.

in above, I meant "koyck distributed lag". my apologies for the typo and missing word. I am rushing out.

Dear Muhamad: X[t-1] and (X[t])^2 have no problems, they can be treated as exogenous regressors. The problem is Y[t-1]: The OLS estimators are biased and not consistent if you include Y[t-1] as an explanatory variable. Are you using E-views? If you are using E-views, and you estimate directly the model you have writen (Yt=Yt-1*rho+a*Xt+rho*Xt-1+bXt^2-rho*Xt-1), you are going to get a biased result. But if you type AR(1) instead of Y(t-1), E-views automatically switches to an iterative estimation method and you get consistent estimates.

The problem for long-term forecasting with this model is that you need to know the value of the X variable for the periods you want to predict. This is: let's suppose you have monthly data until October of 2015 and you want to predict November and December of 2015. To do it, you need to know the value of the X variable for November and December of 2015. If you have these values or a good prediction of them, you can predict these two observations (simply, when estimating the model, set the sample until October of 2015, and when you select the "Forecast" option, change the sample to "First observation - December of 2015" and you'll get the predictions you need. The problem is obtaining the values of the X variable for two months that are still in the future!

If your model is a "pure" autorregressive model (this is: Y[t] = a*Y[t-1] + u[t] ), then this problem does not arise, as you can feed the model with the previously predicted values of Y (this is, first, you predict Y[t+1] using Y[t]; then, you predict Y[t+2] using the predicted value of Y[t+1] and so on -remember the variance of the predictions increases as you go further into the future-),  but in the model you have, there is no way of knowing or estimating the future values of X, so you cannot use it for prediction.

You might try to specify an auxiliary model for X, which should be a "pure" ARIMA model (this is, X[t] = f( X[t-1], X[t-2], X[t-3], ...) to predict future values of X. Then, for future observations, you can "feed" the model with the predicted values of X and get predicted values for Y. I'm afraid the variance of this predictions would be quite large, though.

a) can I use this model for long term forecasting ? some papers say ar(1) are good for long term forecasting

> As was previously noted, your estimates will be biased. But this effect tends to be negligible as the number of observations grows. If you want to calculate forecasts, you will need to do it recursively to find Y[T] and you will need to know X[t].

b) my model involve a quadratic regressor. Are there any specific issues regarding to ar(1) models with a quadratic regressor?

> I am not aware of such issues

> P.S. I would suggest using ARIMA with regressors for your case.

I think that there is a mistake in your equation. Could you have a look at it and correct. You almost have a standard regression with AR(1) errors. This is not an AR modeli in the standard meaning (Box-Jenkins) of the tern.

Hi John: I mentioned earlier that two terms seem to cancel out so there must be paren  mistakenly left out somewhere. But I didn't see anything after that. There are many model names for the first part of the model such as lagged dependent variable model, ARDL etc. Second part seems to be some kind of error correction term but I'm confused. Thanks for letting me know that someone else was also :).