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lec11-2003-2010%20S2

# lec11-2003-2010%20S2 - ACTL2003 Stochastic Modelling for...

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ACTL2003 Stochastic Modelling for Actuaries Week 11 I Time Series -Forecasting I Brownian Motions I Continuous-time Martingales Readings: Chan, Chapter 6; Ross, 9th Edition, Chapter 10 (10.1) Reference: ACTED Chapter 12 and 13 CT6 1/29

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Practical time series analysis: I Description of data I Construction of a model which fits the data I Model selection I Model identification (estimation of parameters) I Model checking (diagnostic) I Forecasting future values of the process (This week) 2/29
Forecasting Box and Jenkins approach : fitting an ARIMA model and using it for forecasting purposes. Problem : Assume that I we have all observations for the process { X t } up until time n : x 1 , · · · , x n , I we have builded an ARMA model to fit the data: X t - φ 1 X t - 1 - · · · - φ p X t - p = Z t + θ 1 Z t - 1 + · · · + θ q Z t - q . I all the parameters φ j s, θ j s and σ 2 have been estimated I the process { Z t } is known for all t n . How to forecast (predict) future values X n + k for k > 0? 3/29

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Forecasting I To forecast (predict) X n + k : we use ˜ X n + k | n = E [ X n + k | X n , · · · , X 1 ] . These forecasts are best linear predictors in terms of the mean square error(see Exercise in Chapter 6 of Chan). I To forecast X n +1 : ˜ X n +1 | n = E [ X n +1 | X n , · · · , X 1 ] = E [ φ 1 X n + · · · + φ p X n +1 - p + Z n +1 + θ 1 Z n + · · · + θ q Z n +1 - q | X n , · · · , X 1 ] = φ 1 X n + · · · + φ p X n +1 - p + θ 1 Z n + · · · + θ q Z n +1 - q I To forecast X n +2 : ˜ X n +2 | n = E [ X n +2 | X n , · · · , X 1 ] = E [ φ 1 X n +1 + φ 2 X n + · · · + φ p X n +2 - p + Z n +2 + θ 1 Z n +1 + θ 2 Z n + · · · + θ q Z n +2 - q | X n , · · · , X 1 ] = φ 1 ˜ X n +1 | n + φ 2 X n · · · + φ p X n +2 - p + θ 2 Z n + · · · + θ q Z n +2 - q I This procedure is iterated to generate forecasts of X n + k , k > 0 4/29
Forecasting To sum up, the forecast value of X n + k given all observations up until time n , known as k -step ahead forecast and denoted by ˜ X n + k | n , is obtained by: I replacing the random variables X 1 , · · · , X n by their observed values x 1 , · · · , x n ; I replacing the random variables X n +1 , · · · , X n + k - 1 by their forecast values ˜ X n + k | n ; I replacing the random variables Z n +1 , · · · , Z n + k - 1 by their expectations 0 I if Z 1 , · · · , Z n are unknown, replacing Z 1 , · · · , Z n by the residuals ˜ Z 1 , ˜ Z 2 , and ˜ Z n , where ˜ Z i = E [ Z i | X n , · · · , X 1 ] i = 1 , · · · , n ; 5/29

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Forecasting Example For an AR(1) process the recursion simplifies to ˜ X n + h | n = φ h X n . Notice that ˜ X n + h 0 as h → ∞ if the process is stationary. Example Assuming that Z 1 , · · · , Z n are known, for an MA(1) process we get ˜ X n + h | n = θ Z n if h = 1 0 if h > 1 Notice there is not much information about the future in an MA(1) process. 6/29
Forecasting Example Consider the ARMA(2 , 2) process X t = 0 . 6 X t - 1 + 0 . 2 X t - 1 + Z t + 0 . 3 Z t - 1 - 0 . 4 Z t - 2 and suppose we know X n = 4 . 0, X n - 1 = 5 . 0, Z n = 1 . 0 and Z n - 1 = 0 . 5. Then ˜ X n +1 | n = 0 . 6 X n + 0 . 2 X n - 1 + 0 . 3 Z n - 0 . 4 Z n - 1 = 3 . 5 and ˜ X n +2 | n = 0 . 6 ˜ X n +1 | n + 0 . 2 X n - 0 . 4 Z n = 2 . 5 .

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