Time Series and Forecasting
Atime seres is a set of observations {Yj}of some variable taken at regular time intervals (monthly, quarterly, yearly,
etc.). Forecasting refers to using past experience to predict values for the future, which requires summarizing the past
experience in some useful way, based on the behavior that is expected to continue and factoring out random variation.
The ultimate test of a forecasting method can only be carried out after the time predicted – not very helpful when the
prediction is needed – so our comparisons of various methods rely on “How well would this method (if it had been used)
have predicted what we (now) know really happened?”. Thus we use “prediction error” (exactly the same idea as residu-
als in regression) for past time periods and the mean of the squares of the errors to compare the success of different methods.
Our methods break down (broadly) into two groups
1.) Smoothing methods: the simplest description and prediction with no assumption about patterns. For forecasting,
these methods are appropriate only for forecasting one period (one month for monthly data, one year for yearly data,
etc.) into the future
2.) Classical time series: Certain patterns (seasonal variation, underlying trend) are identified and separated out (using
the methods developed for group 1 as well as regression techniques) to give a more complete representation. This requires
assumptions about the existence of patterns but then allows forecasting behavior for several time periods.
Because there is no unifying assumption of pattern (as there is with linear regression), there is no “best fit” prediction
and no tests of significance are used – these are very much descriptive techniques being used to make forecasts. Compar-
ison of results is on a case-by-case basis and based on the mean square error of the predictions of past behavior.
Accuracy of a method over a particular part of the series is measured by the mean square error MSE (smaller is
better) MSE = PYj−ˆ
Yj2
nwhere nis the number of observations used and ˆ
Yjis the value predicted by the method
for time j. [Some people call this “mean square deviation” – abbreviated MSD].
1. Smoothing:
(a) Moving averages
To summarize behavior in a time series, our first (and most basic) tool is the moving average used to smooth
out the series for forecasting. The nperiod moving average of a time series Yjis the average of nconsecutive
values.[with each new period, the oldest observation is dropped and the newest is include])
n-period moving average, through period k=
k
X
j=k−n+1
Yj
n
The forecast for period i+ 1 is the moving average for the nperiods through time i.
Fi+1 = forecast for period i+ 1 = PYj
n, where Yj= actual value at time j, sum is taken over the nperiods
ending at period i)
In using a moving average for forecasting, the main issue is the number of observations to include in the moving
average.
(b) Weighted moving averages
To treat different observations as more useful or less useful (we are likely to consider more recent observations
as more useful for forecasting) we assign a weight wjto each observation and calculate a weighted moving
average; then our forecast is
Fi+1 =PwjYj
Pwj
, with Yj= actual value at time j,wj= weight assigned to period j, sum is taken over the n
periods ending at period i.
Typically the weight is based on position in the last nobservations - so the same observation has different
weights depending on which moving average we are calculating.
(c) Exponential smoothing
This forecasting method gives a prediction based on a weighted average of the most recent observation (Yi) and
the prediction for that value (Fi) (which uses the previous observed values). The effect is a weighted average,
with more history [more periods included] in each prediction, but weighting the older data less and less heavily
as time goes on. It involves a single weighting factor αwhich is less than one.
Fi+1 =Fi+α(Yi−Fi) = αYi+ (1 −α)Fi
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