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Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version. Gloria González-Rivera University of California, Riverside and Jesús Gonzalo U. Carlos III de Madrid Forecasting II (forecasting with ARMA models) “There are two kind of forecasters: those who don´t know and those who don´t know they don´t know” John Kenneth Galbraith (1993)

Optimal forecast for ARMA models For a general ARMA process Objective: given information up to time n, want to forecast ‘l-step ahead’

Criterium: Minimize the mean square forecast error

Another interpretation of optimal forecast Consider Given a quadratic loss function, the optimal forecast is a conditional expectation, where the conditioning set is past information

Sources of forecast error Sources of forecast error When the forecast is using :

Properties of the forecast error 1.The forecast and the forecast error are uncorrelated Unbiased MA(l-1)

Properties of the forecast error (cont) 1-step ahead forecast errors,, are uncorrelated In general, l-step ahead forecast errors (l>1) are correlated n-jnn+l n-j+l

Forecast of an AR(1) process The forecast decays geometrically as l increases

Forecast of an AR(p) process You need to calculate the previous forecasts l-1,l-2,….

Forecast of a MA(1) That is the mean of the process

Forecast of a MA(q)

Forecast of an ARMA(1,1)

Forecast of an ARMA(p,q) where

Example: ARMA(2,2)

Updating forecasts Suppose you have information up to time n, such that When new information comes, can we update the previous forecasts?

Problems P1: For each of the following models: (a)Find the l-step ahead forecast of Z n+l (b)Find the variance of the l-step ahead forecast error for l=1, 2, and 3. P2: Consider the IMA(1,1) model (a)Write down the forecast equation that generates the forecasts (b)Find the 95% forecast limits produced by this model (c)Express the forecast as a weighted average of previous observations

Problems (cont) P3: With the help of the annihilation operator (defined in the appendix) write down an expression for the forecast of an AR(1) model, in terms of Z. P4: Do P3 for an MA(1) model.

Appendix I: The Annihilation operator We are looking for a compact lag operator expression to be used to express the forecasts The annihilation operator is Then if

Appendix II: Forecasting based on lagged Z´s Let Then Wiener-Kolmogorov Prediction Formula