Forecasting II forecasting with ARMA models There are

  • Slides: 19
Download presentation
Forecasting II (forecasting with ARMA models) “There are two kind of forecasters: those who

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) Gloria González-Rivera University of California, Riverside and Jesús Gonzalo U. Carlos III de Madrid Copyright(© MTS-2002 GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version.

Optimal forecast for ARMA models For a general ARMA process Objective: given information up

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

Criterium: Minimize the mean square forecast error

Another interpretation of optimal forecast Consider Given a quadratic loss function, the optimal forecast

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 When the forecast is using :

Sources of forecast error When the forecast is using :

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

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

Properties of the forecast error (cont) 1 -step ahead forecast errors, , are uncorrelated

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-j n n-j+l n+l

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

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 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(1) That is the mean of the process

Forecast of a MA(q)

Forecast of a MA(q)

Forecast of an ARMA(1, 1)

Forecast of an ARMA(1, 1)

Forecast of an ARMA(p, q) where

Forecast of an ARMA(p, q) where

Example: ARMA(2, 2)

Example: ARMA(2, 2)

Updating forecasts Suppose you have information up to time n, such that When new

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

Problems P 1: For each of the following models: (a) Find the l-step ahead

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

Problems (cont) P 3: With the help of the annihilation operator (defined in the

Problems (cont) P 3: 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. P 4: Do P 3 for an MA(1) model.

Appendix I: The Annihilation operator We are looking for a compact lag operator expression

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

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

Forecasting II forecasting with ARMA models There areForecasting II forecasting with ARMA models There are
Forecasting II forecasting with ARMA models There areForecasting II forecasting with ARMA models There are
Forecasting II forecasting with ARMA models There areForecasting II forecasting with ARMA models There are
Forecasting Replenishment Forecasting Replenishment Forecasting Forecasting Replenishment ForecastingForecasting Replenishment Forecasting Replenishment Forecasting Forecasting Replenishment Forecasting
ARMA CharlottePiedmont November 15 2018 ARMA International InformationARMA CharlottePiedmont November 15 2018 ARMA International Information
forecasting longterm forecasting middleterm forecasting shortterm forecasting 2forecasting longterm forecasting middleterm forecasting shortterm forecasting 2
CHAPTER 13 FORECASTING VOLATILITY I In ARMA modelsCHAPTER 13 FORECASTING VOLATILITY I In ARMA models
Forecasting Models FORECASTING Forecasting is attempting to predictForecasting Models FORECASTING Forecasting is attempting to predict
THERE IS THERE ARE THERE WAS THERE WERETHERE IS THERE ARE THERE WAS THERE WERE
There is There are There isthere are ThereThere is There are There isthere are There
THERE IS THERE ARE There is and thereTHERE IS THERE ARE There is and there
There is There are There is there areThere is There are There is there are
THERE IS THERE ARE There is There areTHERE IS THERE ARE There is There are
There is There are There is there areThere is There are There is there are
THERE IS THERE ARE There is Singular ThereTHERE IS THERE ARE There is Singular There
There was There were There was There wereThere was There were There was There were
Introduction Definition Forecasting Technology Technological Forecasting Process ForecastingIntroduction Definition Forecasting Technology Technological Forecasting Process Forecasting
Forecasting Forecasting Problems and Methods New product forecastingForecasting Forecasting Problems and Methods New product forecasting