{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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 .
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern