ARIMA
BOX JENKINS METHODOLOGY
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
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.
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
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
Basic models
• Moving average (MA) models
• Autoregressive (AR) models
• Mixed autoregressive moving average (ARMA)
models
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
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
• 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
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
• 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
• Yt = A1 Yt-1 + A2 Yt-2 +……+ApYt-p +et + W1 et-1 +
W2 et-2 +…..Wq et-q
• Order = ARMA (p,q)
• 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.
stationarity
• Two consecutive values in the series depend
only on the time interval between them and
not on time itself
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
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
• 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
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
Step II
• Estimation
• Similar to fitting a standard regression
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
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.
Step IV
• Forecast
• Actually forecast using the chosen model
SPSS