1. BOX JENKINS METHODOLOGY
ARIMA
BOX JENKINS METHODOLOGY

2. 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

3. 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.

4. 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

5. 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

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

7. Moving average models
Predicts Yt as a functions the past errors in predicting Yt
Yt = et + W1 et-1 + W2 et-2 +…..Wq et-q
MA (1) series……………..Yt = et + W1 et-1

8. To know the order
Autocorrelation—correlation 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

10. 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

11. Autoregressive models
Dependent variable Yt depends on its own previous values rather than white noise or residuals
Yt = A1 Yt-1 + A2 Yt-2 +……+ApYt-p +et
Yt = A1 Yt-1 +et…………AR (1) model
Yt = A1 Yt-1 + A2 Yt-2 +et…………AR (2) model

12. 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

14. Yt = A1 Yt-1 + A2 Yt-2 +……+ApYt-p +et + W1 et-1 + W2 et-2 +…..Wq et-q
Order = ARMA (p,q)

16. 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.

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

18. 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

19. 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

20. 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

21. 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

22. Step II
Estimation
Similar to fitting a standard regression

23. 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

24. 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.

25. Step IV
Forecast
Actually forecast using the chosen model

26. SPSS