© 2000 John Petroff
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© 2000 John Petroff |
1)- Box-Jenkins times-series analysis
There are three patterns that are
usually investigate: autoregression, moving average and trend.
There may also be occasional erratic observations, or disturbances
that may have to be removed or corrected.The
procedure requires a sequence of three steps
- identification in which different patterns (e.g. autoregressive,
moving average or trend discussed below) are tried out,
- estimation in which estimated coefficient values are recorded
in a schedule,
- diagnostic checking in which the goodness of fit is verified
to be adequate, or if found insufficient the procedure is started
anew.
See review questions Q-5F.1 through Q-5F.7.
A stepwise multiple regression is run on as many combined lagged series as necessary until additional lagged series have no explanatory power (i.e. do not improve regression results such as R2). The equation tested is
| Yt = a1Yt-1 + + apYt-p + et |
Equation E-5F.1 |
where e is the residual or error term, and is assumed to have
a mean of zero.
The number of periods p that y needs to be lagged is determined
by when the stability of the coefficients is reached. That is,
if regressions are run with lagged Y beyond p the estimated coefficients
remain unchanged; this proves that beyond p, no additional lagged
Y has additional explanatory power.
If p turns out to be 12 for monthly data, the autoregressive model establishes a pattern of seasonal indices which are the coefficients estimates. As mentioned previously, the seasonal pattern itself can be removed to investigate if some other pattern is present over several years, or if the pattern even extends over the very long term. Naturally, the autoregressive model can also reveal cyclical variations shorter than twelve months. It will be up to the analyst to pay attention to these shorter cycles, remove them form the data or disregard them altogether.
See review question Q-5F1.1.
Box-Jenkins moving average model proposes that a time series is explained by some combination of random events going back q periods in the past.
| Yt = b1et-1 + + bpet-q + et |
Equation E-5F.2 |
where e is, as before, a random serially independent variable of mean zero.
No earthly phenomenon is protected from random events. For instance, product sales are affected by introduction of many different new products, or the stock market is bombarded by random new information all the time. The further in time is the shock the less it will have a bearing on current observations. As before, q is chosen so that the coefficients are stable and no explanatory power is gained beyond q.
See review question Q-5F2.1.
Trend is determined by running a regression where time t is the exogenous variable
Yt = a + t + et
Box-Jenkins trend is determine by differentiating the time series
dt = Yt - Yt-1
and running regressions on
| dt = c1dt-1 + . . . + cpdt-p + et |
Equation E-5F.3 |
If the regression is unstable, the time series can be differentiated a second time, i.e. differences of differences. Usually, however, this is the easiest step.
d)- Correlograms
A complete Box-Jenkins procedure is considered complex because one must obtain estimates of the coefficients a and b in equations E-5F.1 and E-5F.2 above (as well as coefficients c in E-5F.3 if needed), and decide at the same time the number of lags p and q to use. To assist in these decisions, a schedule of successive coefficients estimates is built. This schedule (or graph) is called a correlogram. If the values of the coefficient beyond a certain number K are no different from zero (i.e. be finding values of t statistic that are too low, as before), then the model is Moving Average (MA) with q=K.
In addition, a partial correlogram is constructed with estimated coefficients from regressions of the original series on a fitted series (i.e. obtained from the previous analytical work above). If the values of the coefficients beyond a certain number K are no different than zero, the model is autoregression (AR) with p=K.
A pattern can be a combination of the two if the two correlograms show convergence at q and p: it is then called autoregressive moving average (ARMA). But if convergence is not obtained, i.e. coefficients do not become stable when adding additional lagged variables, then there must be a trend present. In this case the pattern is called autoregressive integrated moving average (ARIMA). This is the most comprehensive (and most complex) pattern in times-series analysis.
See review questions Q-5F4.1.
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