ARIMA models use past patterns in time series data to forecast future values of a variable when causal factors are unknown. The Box-Jenkins methodology identifies the best ARIMA model by examining a time series' characteristics to transform it into white noise. ARIMA models include moving average (MA), autoregressive (AR), and mixed (ARMA) components. Stationarity is important and differencing can make data stationary. The steps are to identify the model type, estimate parameters, diagnose the model fit, and forecast.

1. 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