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Notes to Accompany MBS Section 2.10   -- Time Series Plots
These notes Copyright ©   2001     Stanley Louis Sclove

Against All Odds: Inside Statistics, program #6 (Time Series)

What is a ``time series''?

Observations on one or more variables over time are called time-ordered, or temporal data.

A time series is a set of time-ordered data obtained at regular, equally spaced time points.

Example. Stock prices. The series of prices of a stock every time it is traded is time-ordered data. The series of closing prices of that stock, consisting of the price of the stock at the close of each trading day, is a  time series.

Typical notation for a time series is
 
 

y₁, y₂,  . . . ,  y_n.

Sometimes we write this as y_t,  t = 1, 2,  . . .  , n.

The importance of forecasting in business

Many economic and business datasets are time series.  Areas of application of time series analysis include Marketing, Banking, Finance and Investment.  In Operations Management, time series analysis plays an important role in forecasting demand, planning production and controlling inventory.

Example.  Demand-driven Production/Inventory system.  Manufacturing firms produce goods and maintain inventory to meet demand. Production, Demand, and Inventory are closely related.  Production and Inventory must be balanced with Demand.

Here the role of statistics is to predict ("forecast'') demand.  Demand may be growing, shrinking, or seasonal, and we need to forecast it.  Statistical methods are used to organize and summarize the demand data, to develop forecasts and thus help set ideal production and inventory levels.

How might you forecast Demand?   First, you need an accounting system which keeps track not only of Sales, but also of Demand. (Demand equals Sales plus unfilled orders).  The symbol  t  denotes the current time period.  Then  t+1  is the next time period.   One forecast would be to guess that demand next period,  D_t+1,  will be equal to demand this period, D_t .   Using  F_t+1  to denote the forecast of the value at time t+1, we write  F_t+1 = D_t.  This simplest of all predictions, namely, that the value next period will be the same as this period, is called the 'naive'' prediction.   Another forecast would be to guess that the demand next period,   D_t+1,  will be equal to demand this period, plus a change which was equal to the change just preceding: F_t+1 = D_t + (D_t - D_t-1)  =  2D_t - D_t-1.   You can see that there could be many reasonable schemes for predicting next period's demand, based on the stream of past demands. We need some theory to suggest what might be best. And after we see what might be best, we need to be able to do the required computation. Except in the simplest cases, the computations are heavy, as with regression analysis.  Excel has only limited capability for time series analysis, so statistical computing packages (such as Minitab) are usually used for all but the simplest time series calculations.
 

Time series notation and concepts

We need to discuss some general ideas and notation for time series.

Example.   Income statements.   An income statement showing comparisons with preceding years can be viewed as an example of multiple time series.  Each item (row) on the income statement can be considered as a  variable. There is a  time series across the years shown for each of these variables.
 

SMOOTHING AND FORECASTING BY MOVING AVERAGES

Each observation in a time series is considered to consist of a mean value for that time point, plus error.

The mean values are the "true'' values that would still be present if we could observe the variable more than once at each time point.  The error is random and would differ on each occasion of observation.

We would like to estimate the series of "true'' mean values.  They contain the true pattern of the series .  Smoothing  means the averaging of adjacent observations. One reasonable way of estimating the mean values is by smoothing.  The reason smoothing works is that the trend, the underlying true values, usually moves slowly, so that its adjacent values, and consequently those of the observed series, are not far apart. When we average adjacent values, the errors tend to cancel out and the trend is well estimated. The  smoothed series consists of these averages. In the smoothed series, the smoothed value at time t  replaces the observed value at time t;  S_t replaces y_t.   The resulting series is called "smoothed'' because usually the sizes of any short-term ups and downs are decreased by the averaging.  It is usually easier to spot a trend in the smoothed series than in the original series.

Forming a  moving average is a way of smoothing.  A moving average of  width three (three-point moving average) is the result of averaging the observations in a moving window of width three.  First there is the average of observations at times 1, 2 and 3, then the average of observations at times 2, 3 and 4, then the average of the observations at times 3, 4 and 5,  etc. The analogous definition applies for windows of widths other than three.  Moving averages are also sometimes called  running averages.
 

Example.   In  AAO Pgm 6, the running median of Boston Marathon winning times is computed.
 

Moving Averages

Of course, in forming a moving average we could use a mean instead of a median. The resulting moving mean is usually called simply a  moving average. The centered moving mean of three is the smoothed value

S_t = mean of y_t-1, y_t,  y_t+1   =   (y_t-1+ y_t + y_t+1)/3.    For example, with the winning Marathon times,
S₁₉₇₃  =  mean of  y₁₉₇₂, y₁₉₇₃, y₁₉₇₄
            =  mean of  190, 186, 167
            = (190 + 186 + 167)/3 = 181.0.

A three-point moving average is one choice; another common choice is a five-point moving average.

Forecasting

In considering forecasting in general terms, we denote by F_t+1  the forecast of  y_t+1  at time t. The ``naive'' forecast is F<sub>t+1</sub> = y<sub>t</sub> .   We would like also to combine earlier values such as y<sub>t-1</sub>  and  y<sub>t-2</sub>  into the forecast, since they also give some hint of where the series is going.  Since the smoothed values estimate the series of ``true'' mean values, it is reasonable is to take F_t+1   to be some sort of smoothed value, that is,  F_t+1 = S_t,   where  S_t   is a smoothed value of the series up to time  t.  An example is the moving average of the three most recent values,  F_t+1 = (y_t +  y_t-1 +  y_t-2)/3.
 

Weights

The ordinary moving average uses equal weights. Although equal weights are reasonable, the average would be more sensitive to change if we weighted recent observations more heavily.  We want the weights to sum to 1.
This is because we don't want the result to be too large or too small. If the true values were all equal to a constant  c,  we  would want the weighted average to be equal to one times c,  not 0.5 or 2 times c.   An example of a weighted moving average would be a 'sum-of-digits'' moving average of the three most recent values.  Since 1 + 2 + 3 = 6, we can use weights 3/6, 2/6 and 1/6 to form the forecast  F_t+1  =  (3y_t  + 2y_t-1 +  1y_t-2)/6.

Suppose we want to forecast using just a two-point average.
We could take F_t+1 = S_t,   where S_t = .5y_t + .5y_t-1.   The next value of the smoothed series would be  S_t+1,  or
.5y_t+1 + .5 y_t.   It would be more proactive if we could take   F_t+1    to be  S_t+1  rather than  S_t.   We can't do this, because we haven't yet observed  y_t+1.   But we could predict it, using a preliminary predictor  F'_t+1 = .5y_t + .5y_t-1   and then take our final predictor  F_t+1  to be  F_t+1 =  .5F'_t+1 + .5y_t-1.   This is  F_t+1 = .5(.5y_t + .5y_t-1) + .5y_t  =  .75y_t + .25y_t-1.

This ``look ahead'' method is called a predictor-corrector method because we first predict the missing future observations and then use the predictions to correct our initial forecast.  Note that, although we started with equal weights, the final weights are not equal, and the more recent observation is weighted more heavily.

Exercises

Form a two point predictor-corrector forecast using weights 0.8, 0.2 rather than 0.5, 0.5. 2.

Form a two-point predictor-corrector forecast using weights 0.9, 0.1.

Form a three-point predictor-corrector forecast using initial equal weights.

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