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BABS 502

BABS 502. ARIMA Forecasting March 18, 2014. General Overview. An ARIMA model is a mathematical model for time series data. Statisticians George Box and Gwilym Jenkins developed a systematic approach for fitting these models to data so these models are often called Box-Jenkins models.

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BABS 502

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  1. BABS 502 ARIMA Forecasting March 18, 2014

  2. General Overview • An ARIMA model is a mathematical model for time series data. • Statisticians George Box and Gwilym Jenkins developed a systematic approach for fitting these models to data so these models are often called Box-Jenkins models. • We always use statistical or forecasting programs to fit these models • The programs fit models and produce forecasts. • Some choose best model automatically. • But it is beneficial to understand the basic model to know that what the software is doing makes sense • Especially if we use an automatic forecasting program. (c) Martin L. Puterman

  3. ARIMA Models • ARIMA Stands for AutoRegressive Integrated Moving Average • We speak also of AR models, MA models, ARMA models, IMA models which are special cases of this general class. • Models generalize regression but “independent” variables are past values of the series itself and unobservable random disturbances. • Estimation is based on maximum likelihood; not least squares. • We distinguish between seasonal and non-seasonal models. (c) Martin L. Puterman

  4. Notation • Y1, Y2, …, Yt denotes a series of values for a time series. • These are observable. • e1, e2, …, et denotes a series of random disturbances. • These are not observable. • They may be thought of as a series of random shocks. • Usually they are assumed to be generated from a Normal distribution with mean 0 and standard deviation  and to be uncorrelated with each other. • They are often called “white noise”. (c) Martin L. Puterman

  5. An Autoregressive (AR(1)) Model • AR(1) Model: Yt = A1Yt-1 + et • A1 is an unknown parameter with values between -1 and +1 which is to be estimated from data • As a first approximation we can estimate A1 by linear regression (with intercept set equal to 0) (How?) • When A1 = 1, the model is called a random walk. • In this case, Yt = Yt-1 + et • or alternatively Yt - Yt-1 = et • We can show (by back substitution and assuming Y0 = 0) that for a random walk • E(Yt ) = 0 and Var(Yt) = t2 • Hence the values get more variable as you move out in the series. • This means that when data follows a random walk the best prediction of the future is the present (a naïve forecast) and the prediction gets less accurate the further into the future we forecast. (c) Martin L. Puterman

  6. The ACF for a random walk • If Yt is a random walk, it can be represented by Yt = et + et-1 + … + e1 • Consequently • cov(Yt,Yt-1) = (t-1)σ2 • Var(Yt) = tσ2 • So that • Corr(Yt,Yt-1) = (t-1)/t • Corr(Yt,Yt-k) = (t-k)/t • This gives the ACF this shape -> (c) Martin L. Puterman

  7. Random Walk (c) Martin L. Puterman

  8. Other AR(p) models • The AR(2) Model • Yt = A1Yt-1 +A2 Yt-2 + et • Here, A1 and A2 are unknown parameters • The AR(p) Model • Yt = A1Yt-1 +A2 Yt-2 + … + Ap Yt-p+ et • Here, A1, … Ap are unknown parameters • To apply these in practice, we estimate the parameters and then use the model for forecasting by substituting past observed values. • These models are called ARIMA(p,0,0) models. (c) Martin L. Puterman

  9. Models with constants • Above models assume 0 mean. • An AR model with a constant c has form Yt = c + A1Yt-1 +A2 Yt-2 + … + Ap Yt-p+ et (c) Martin L. Puterman

  10. Which Model to Fit? • The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) give some insight into what model to fit to data. • We work backwards here. • Given a theoretical model, we can determine theoretically what its ACF and PACF should be. • So if the ACF and PACF from the data have a recognizable pattern then we try fitting a model that could generate that pattern to the data. • What is a PACF? • The pth partial autocorrelation is the coefficient of Yt-p in a regression of Yt on Yt-1, Yt-2, …, Yt-p. • Thus, if the data was generated by an AR(2) model, in theory the first two PACFs would be non-zero and all PACF’s higher than two would be zero. (c) Martin L. Puterman

  11. Some further comments on ACFs and PACFs • Computing autocorrelations (ACs) is similar to performing a series of simple regressions of Yt on Yt-1, then on Yt-2, then on Yt-3, …. • The AC coefficients reflect only the relationship between the two quantities included in the regression. • Computing partial autocorrelations (PACs) is more in the spirit of multiple regression. The PACs remove the effects of all lower order lags before computing the autocorrelation. • For example the 2nd order PAC is the effect of observations two periods ago on the current observation, given that the effect of the observation one period ago has been removed. • This can be viewed as multiple regression. (c) Martin L. Puterman

  12. Example: AR(1) model A1 = .8 (c) Martin L. Puterman

  13. Example: AR(1) Model; A1 =-.7 (c) Martin L. Puterman

  14. Example: AR(2) Model (c) Martin L. Puterman

  15. Monthly Pulp Price Data (c) Martin L. Puterman

  16. Annual Births Data (c) Martin L. Puterman

  17. Stationarity • A time series is stationary if: • It’s mean is the same at every time • It’s variance is the same every time • It’s autocorrelations are the same at every time • A series of outcomes from independent identical trials is stationary. • A series with a trend is not stationary. • A random walk is not stationary. (Why?) • If a time series is non-stationary, its ACF dies off slowly and the first partial autocorrelation is near 1. • In such cases we can sometimes create a stationary series by differencing the original series. • If Yt is a random walk, then its differences are white noise which is stationary • A unit root test (Section 8.1) is a formal test for non-stationarity • One such test is the Dickey-Fuller test (adf.test in R) • See also the KPSS test (c) Martin L. Puterman

  18. Differenced Births Data The PACF suggests that the differences of the birth data may follow an AR(1) or AR(2) or AR(5) model. (c) Martin L. Puterman

  19. Differenced Pulp Price Data The story is less clear here. Perhaps the differences follow an AR(1), the lag 1 PAC is .346, the lag 2 PAC is .184. (c) Martin L. Puterman

  20. Differenced Models • We let Zt = Yt – Yt-1. • When the differenced model is stationary, we can write a model in terms of Zt . • If Zt follows an AR(p) model, then Yt follows and ARIMA(p,1,0) model. • In practice ARIMA(1,1,0) and ARIMA(2,1,0) are quite common. (c) Martin L. Puterman

  21. Pulp Data • The fit from an ARIMA(1,1,0) model is • A1 =.346 (t-value 5.46) • So fitted model is • Zt = .346 Zt-1 + et • The residuals appear to have no remaining autocorrelation • Forecasts seem pretty flat; 561.7, 562.3, 562.6, 562.6, 562.6 (c) Martin L. Puterman

  22. MA(q) Models • These are less plausible but fit many series well. • MA(1) model: • Yt = et + W1 et-1 • MA(2) model: • Yt = et + W1 et-1 + W2 et-2 • MA(q) model • Yt = et + W1 et-1 + W2 et-2 +…+ Wq et-q • This is referred to as an ARIMA(0,0,q) model. • Rationale for MA models is that effects of disturbances are short lived (q periods) as opposed to an AR model where they persist forever. • Note that the disturbances are not observable. (c) Martin L. Puterman

  23. Differenced models with constants If Zt = c + et Then • Yt = c + Yt-1 + et which is a random walk with “drift” (c) Martin L. Puterman

  24. An MA(1) Model: W1 = .7 (c) Martin L. Puterman

  25. An MA(1) Model: W1 = -.7 (c) Martin L. Puterman

  26. Births Data • Clearly differencing is required • Consider fitting an MA(1) model to the differenced data • Find that estimated coefficient is -.42 with a T-value of -3.87 • But autocorrelation of residuals contains information • Note lag 2 AC = .349 (c) Martin L. Puterman

  27. Births Data • Try an ARIMA(0,1,2) model • Parameters are -.37 (t =-3.47 ), -.59 (t=-5.76) • Residuals appear to be white noise. • Forecasts are 338311, 340936, 340936,…. (c) Martin L. Puterman

  28. The ARIMA(0,1,1) Model Revisited • This model can be written as (letting w = -W1) Yt –Yt-1 = et - w et-1 • The forecast from this model is Ft = Yt-1 - w(Yt-1 - Ft-1) = (1-w) Yt-1 + w Ft-1 • This is simple exponential smoothing • The new concept here is that the ARIMA(0,1,1) model is a formal statistical model while simple exponential is an ad hoc approach to forecasting. This means that there is an error term and hence forecast errors and hypothesis tests are part of the model. (c) Martin L. Puterman

  29. Relationship between MA and AR Models • Any finite AR model can be written as an infinite MA model • Any finite MA model can be written as an infinite AR model. • These results can be shown by backward substitution (as we did previously for the AR models) • Two consequences of these observations • Model Selection • If your best fit is an AR model with several terms (i.e., 4 or more); try an MA model with a few terms and conversely • Identification • AR models have ACF with several terms and short PACFs • MA models have short ACF’s and long PACFs (c) Martin L. Puterman

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