ARIMA Forecasting Lecture 7 and 8 - March 14-16, 2011 BABS 502 ARIMA Forecasting Lecture 7 and 8 - March 14-16, 2011

General Overview An ARIMA model is a mathematical model for time series data. 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 for us. 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

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

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

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

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

Random Walk (c) Martin L. Puterman

Other AR(p) models The AR(2) Model The AR(p) 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

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

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

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

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

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

Monthly Pulp Price Data (c) Martin L. Puterman

Annual Births Data (c) Martin L. Puterman

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 is a formal test for non-stationarity One such test is the Dickey-Fuller test (c) Martin L. Puterman

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

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

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

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

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

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

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

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

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

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

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