On Fri, 16 Oct 2009, Francisco Sosa wrote: Any econometrics textbooks will do. In short, the reason why the standard error grows is that you use y_{t-1} to forecast y_{t}; if y_{t-1} is known with certainty, clearly the variance of your forecast is smaller than if y_{t-1} has to be forecast itself. Riccardo (Jack) Lucchetti Dipartimento di Economia Università Politecnica delle Marche r.lucchetti(a)univpm.it http://www.econ.univpm.it/lucchetti

On Viernes, 16 de Octubre de 2009 12:35:09 Francisco Sosa escribió: But, why are you using Prais-Winsten to estimate an AR(1)? or am I loosing something? You can estimate an AR(1) directly by OLS or using the arima command of gretl. Is your model Y_t=const+\phi Y_{t-1}+\eps_t or is it Y_t=alpha+beta X_t + E_t with E_t=rho*E_(t-1)+U_t? This last is the model that the Prais-Winsten estimator is for. -- Ignacio Diaz-Emparanza DEPARTAMENTO DE ECONOMÍA APLICADA III (ECONOMETRÍA Y ESTADÍSTICA) UPV/EHU Avda. Lehendakari Aguirre, 83 | 48015 BILBAO T.: +34 946013732 | F.: +34 946013754 www.et.bs.ehu.es

On Viernes, 16 de Octubre de 2009 13:15:56 Francisco Sosa escribió: Prais-Winsten is a type of feasible GLS estimator. I don't know the formula that gretl uses for the static forecasts but I assume it is the standard one that can be found in any econometric book. I also would like to know what gretl does when selecting "dynamic" forecast. Having a look at the gretl source code for "forecast", I think the relevant comment in forecast.c is this below, but is not of much help for me, is it for you? /* The code below generates forecasts that incorporate the predictable portion of an AR error term: u_t = r1 u_{t-1} + r2 u_{t-1} + ... + e_t where e_t is white noise. The forecasts are based on the representation of a model with such an error term as (1 - r(L)) y_t = (1 - r(L)) X_t b + e_t or y_t = r(L) y_t + (1 - r(L)) X_t b + e_t where r(L) is a polynomial in the lag operator. We also attempt to calculate forecast error variance for out-of-sample forecasts. These calculations, like those for ARMA, do not take into account parameter uncertainty. This code is used for AR and AR1 models; it is also used for dynamic forecasting with models that do not have an explicit AR error process but that have one or more lagged values of the dependent variable as regressors. */ -- Ignacio Diaz-Emparanza DEPARTAMENTO DE ECONOMÍA APLICADA III (ECONOMETRÍA Y ESTADÍSTICA) UPV/EHU Avda. Lehendakari Aguirre, 83 | 48015 BILBAO T.: +34 946013732 | F.: +34 946013754 www.et.bs.ehu.es

On Fri, 16 Oct 2009, Francisco Sosa wrote: The following script illustrates how gretl calculates the forecast standard errors for an AR(1) model estimated via Prais-Winsten. <script> open data9-7 # reserve a couple of observations smpl ; 1990:2 ar1 QNC 0 INCOME --pwe # save the Standard Error of Regression and rho-hat SER = $sigma rho = $rho # set forecast range smpl 1989:1 1990:4 fcast # save forecast standard errors series fcerr = $fcerr # calculate standard errors manually and check err1 = SER check1 = fcerr[1990:3] - err1 err2 = sqrt((1 + rho^2) * SER^2) check2 = fcerr[1990:4] - err2 </script> Allin Cottrell

On Mon, 19 Oct 2009, Francisco Sosa wrote: If you want a definitive answer to this, please send me the dataset and script that you are using, so I can see exactly where your numbers are coming from. Allin Cottrell

On Mon, 19 Oct 2009, Francisco Sosa wrote: OK, I see: your model has an AR(1) specification for the dependent variable as well as an AR(1) for the error term via Prais-Winsten. So then the formula I gave does not apply. Here is how to replicate the forecast standard errors given by gretl. <script> open Global_Sales_Beer.gdt genr y = Global_Sales ar1 y 0 y(-1) --pwe S2 = $sigma^2 beta = $coeff[2] rho = $rho psi = rho + beta addobs 1 smpl 2007 2009 fcast series fcerr = $fcerr scalar sum_psi_sq = 1 scalar k = 1 loop i=2007..2009 -q scalar err = sqrt(sum_psi_sq * S2) printf "check: %g - %g = %g\n", fcerr[i], err, fcerr[i] - err sum_psi_sq += (psi^k)^2 k++ endloop </script> See Box and Jenkins, Time Series Analysis: Forecasting and Control, 1976, p. 508, "Program 4", for the general algorithm. Allin Cottrell

Hi Francisco, Sorry for not answering, I personally couldn't answer your question, because I thought it was related to the implementation details of Prais-Winsten. Now I think maybe your question was more general, see below. Not sure if this answers your question, but: It is perfectly normal that the forecast uncertainty grows as one predicts the values of periods that are farther in the future. Intuitively, it is typically easier to predict what happens tomorrow (think of the weather) than to predict what happens next month. Slightly more formally, for a stationary variable the dynamic forecast uncertainty will converge (for h\rightarrow\infty, where h is the forecast horizon) to a constant value which is basically the variance of the unconditional mean estimate. For an integrated variable, however, the forecast uncertainty will grow without bound. As regards the detailed formulae, I don't exactly which are used in gretl. Several variants are possible, with or without small-sample corrections, with or without parameter uncertainty, etc. etc. Maybe this helps nonetheless, cheers, sven