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ARIMA BOX JENKINS METHODOLOGY

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When ARIMA is to be used In many real world situations We do not know the variables determinants of the variable to be forecast Or the data on these casual variables are readily available

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Box-Jenkins methodology is Technically sophisticated way of forecasting a variable by looking only at the past pattern of the time series It uses most recent observations as a starting value Best suited for long range rather than short range forecasting.

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White noise White noise is a purely random series of numbers The numbers are normally and independently distributed There is no relationship between consecutively observed values Previous values do not help in predicting future values

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Box-Jenkins methodology (continued) We start with the observed time series itself Examine its characteristics Get an idea how to transform the series into white noise. If we get the white noise, we assume that it is the correct model

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Basic models Moving average (MA) models Autoregressive (AR) models Mixed autoregressive moving average (ARMA) models

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Moving average models Predicts Y t as a functions the past errors in predicting Y t Y t = e t + W 1 e t-1 + W 2 e t-2 +..W q e t-q MA (1) series..Y t = e t + W 1 e t-1

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To know the order Autocorrelationcorrelation between the values of the time series at different periods Partial autocorrelation measures the degree of association between the variable and that same variable in another time period after partialing out the effect of the other lags

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If the ACF abruptly stops at some point, we know the model is of MA type The number of spikes before the abrupt stop is referred to as q

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Autoregressive models Dependent variable Yt depends on its own previous values rather than white noise or residuals Y t = A 1 Y t-1 + A 2 Y t-2 ++A p Y t-p +e t Y t = A 1 Y t-1 +e t AR (1) model Y t = A 1 Y t-1 + A 2 Y t-2 +e t AR (2) model

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If the PACF stops abruptly at some point, the model is of AR type The number of spikes before the abrupt stop is equal to the order of the AR model. It is denoted by p

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Y t = A 1 Y t-1 + A 2 Y t-2 ++A p Y t-p +e t + W 1 e t-1 + W 2 e t-2 +..W q e t-q Order = ARMA (p,q)

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Any of the four frames could be patterns that could identify ARIMA (1,1) model Both ACF and PACF gradually fall to zero rather than abruptly stop.

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stationarity Two consecutive values in the series depend only on the time interval between them and not on time itself

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Non stationary data Mean value of the time series changes over time Variance of the time series changes over time Autocorrelations are usually significantly different from zero at first and then gradually fall to zero or show spurious pattern as the lags are increased

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How to remove non stationarity If caused by trend in series, differencing of the series is done When there is change in variability, log of actual series

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When differencing is used to make a time series stationary, it is common to refer the resulting model as ARIMA (p,d,q) type. The I refers to integrated or differencing term d refers to the degree of differencing

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Step I Identify If the ACF abruptly stops at some point- say, after q spikes-then the appropriate model is an MA(q) type. If the PACF abruptly stops at some point-say, after p spikes-then the model is an AR(p) type If neither function falls off abruptly, but both decline toward zero in some fashion, the appropriate model is an ARMA (p,q) type

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Step II Estimation Similar to fitting a standard regression

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Step III Diagnose Determine whether the correct model has been chosen Examine the ACF of residuals If the ACF of the residuals shows no spikes the model chosen is the correct one If you are left with only white noise in the residual series, the model chosen is likely to be the correct one

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Ljung-Box statistic test Tests whether the residual autocorrelations as a set are significantly different from zero. If the residual autocorrelations as a set are significantly different from zero, the model specification should be reformulated.

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Step IV Forecast Actually forecast using the chosen model

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SPSS