CHAPTER 15
Time Series Analysis page 651
15. Introduction page 651
Chapter 15 deals with the basic components of time series, time series decomposition, and
simple forecasting. At the end of this chapter, you should know the following:
 The four possible components of a time series.
 How to use smoothing techniques to remove the random variation and identify the
remaining components.
 How to use the linear and quadratic regression models to analyze the trend.
 How to measure the cyclical effect using the percentage of trend method.
 How to measure the seasonal effect by computing the seasonal indices.
 How to calculate MAD and RMSE to determine which forecasting model works best.
Definition
A time series is a collection of data obtained by observing a response variable at periodic
points in time.
Definition
If repeated observations on a variable produce a time series, the variable is called a time series
variable. We use Yi to denote the value of the variable at time i.
Four possible components:
Trend ( secular trend) -- Long term pattern or direction of the time series
Cycle ( cyclical effect) -- Wavelike pattern that varies about the long-term trend, appears over
a number of years e.g. business cycles of economic boom when the cycle lies above the trend
line and economic recession when the cycle lies below the secular trend.
Seasonal variation -- Cycles that occur over short periods of time, normally < 1 year
e.g. monthly, weekly, daily.
Random variation ( residual effect) --Random or irregular variation that a time series shows
Could be additive: Yi = Ti + Ci + Si + Ii
or
multiplicative: Yi = Ti x Ci x Si xIi
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Forecasting using smoothing techniques
The two commonly used smoothing techniques for removing random variation from a time
series are moving averages and exponential smoothing.
Moving average: ( MA)
Moving averages involve averaging the time series over a specified number of periods. We
usually choose odd number of periods so we can center the averages at particular periods for
graphing purposes. If we use an even period, we may center the averages by finding twoperiod moving averages of the moving averages. Moving averages aid in identifying the secular
trend of a time series because the averaging modifies the effect of cyclical or seasonal variation.
i.e. a plot of the moving averages yields a “smooth” time series curve that clearly shows the
long term trend and clearly shows the effect of averaging out the random variations to reveal
the trend.
Moving averages are not restricted to any periods or points. For example, you may wish to
calculate a 7-point moving average for daily data, a 12-point moving average for monthly data,
or a 5-point moving average for yearly data. Although the choice of the number of points is
arbitrary, you should search for the number N that yields a smooth series, but is not so large
that many points at the end of the series are "lost." The method of forecasting with a general
L-point moving average is outlined below where L is the length of the period.
Forecasting Using an L-Point Moving Average
1. Select L, the number of consecutive time series values Y1, Y2. . . YL
that will be averaged. (The time series values must be equally spaced.)
2. Calculate the L-point moving total, by summing the time series values over L adjacent time
periods.
3. Compute the L-point moving average, MA, by dividing the corresponding moving total by L
4. Graph the moving average MA on the vertical axis with time i on the horizontal axis. (This
plot should reveal a smooth curve that identifies the long-term trend of the time series.) Extend
the graph to a future time period to obtain the forecasted value of MA.
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Exponential smoothing:
One problem with using a moving average to forecast future values of a time series is that
values at the ends of the series are lost, thereby requiring that we subjectively extend the graph
of the moving average into the future. No exact calculation of a forecast is available since the
moving average at a future time period t requires that we know one or more future values of
the series.
Exponential smoothing is a technique that leads to forecasts that can be explicitly calculated.
Like the moving average method, exponential smoothing de-emphasizes (or smoothes) most of
the residual effects.
To obtain an exponentially smoothed time series, we first need to choose a weight W, between 0
and 1, called the exponential smoothing constant. The exponentially smoothed series, denoted
Ei, is then calculated as follows:
Ei= W Yi+(1- W)Ei-1 (for i>=2)
where Ei = exponentially smoothed time series
Yi = observed value of the time series at time i
Ei-1 = exponentially smoothed time series at time i-1
W = smoothing constant, where 0<= W <=1
Begin by setting E1=Y1
E2= W Y2+(1- W)E1
E3= W Y3+(1- W)E2
.
.
Ei= W Yi+(1- W)Ei-1
You can see that the exponentially smoothed value at time i is simply a weighted average of the
current time series value, Yi, and the exponentially smoothed value at the previous time period,
Ei-1. Smaller values of W give less weight to the current value, Yi. Whereas larger values give
more weight to Yi

The formula indicates that the smoothed time series in period i depends on all the
previous observations of the time series.

The smoothing constant W is chosen on the basis of how much smoothing is required. A
small value of W produces a great deal of smoothing. A large value of W results in very little
smoothing.
Exponential smoothing helps to remove random variation in a time series. Because it uses the
past and current values of the series, it is useful for forecasting time series. The objective of the
time series analysis is to forecast the next value of the series, Fi+1. The exponentially smoothed
forecast for Fi+1= Ei
where Ei= W Yi+(1- W)Ei-1
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is the forecast of Yi+1 since the exponential smoothing model assumed that the time series has
little or no trend or seasonal component. The forecast Fi is used to forecast not only Yi+1 but
also all future value of Yi.
i.e. Fi = W Yi+(1- W)Ei-1, i = n + 1, n + 2, . . .
This points out one disadvantage of the exponential smoothing forecasting technique. Since the exponentially
smoothed forecast is constant for all future values, any changes in trend and/or seasonality are not taken
into account. Therefore, exponentially smoothed forecasts are appropriate only when the trend and seasonal
components of the time series are relatively insignificant.
Forecasting: The Regression Approach
Many firms use past sales to forecast future sales. Suppose a wholesale distributor of sporting goods is
interested in forecasting its sales revenue for each of the
next 5 years. Since an inaccurate forecast
may have dire consequences to the distributor, some measure of the forecast’s reliability is required. To
make such forecasts and assess their reliability, an inferential time series forecasting model must be
constructed. The familiar general linear regression model represents one type of inferential model since it
allows us to calculate prediction intervals for the forecasts.
YEAR SALES
t
y
1
4.8
2
4.0
3
5.5
4
15.6
5
23.1
6
23.3
7
31.4
8
46.0
9
46.1
10 41.9
11 45.5
12 53 5
YEAR SALES
t
y
13 48.4
14 61.6
15 65.6
16 71.4
17 83.4
18 93.6
19 94.2
20 85.4
21 86.2
22 89.9
23 89.2
24 99.1
YEAR SALES
t
y
25 100.3
26 111.7
27 108.2
28 115.5
29 119.2
30 125.2
31 136.3
32 146.8
33 146.1
34 151.4
35 150.9
To illustrate the technique of forecasting with regression, consider the data in the Table
above. The data are annual sales (in thousands of dollars) for a firm (say, the sporting goods
distributor) in each of its 35 years of operation.
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A scatter plot of the data is shown below and reveals a linearly increasing trend ,
so the first-order (straight-line) model
E(Yi) = βo + β1i
seems plausible for describing the secular trend. The regression analysis printout for the model
gives R2 = .98, F = 1,615.724 (p-value < .0001), and s = 6.38524. The least squares prediction
equation is
Yi = Bo + B1i = .401513 + 4.295630i
The prediction intervals for i = 36, 37, . . ., 40 widen as we attempt to forecast farther into the future.
Intuitively, we know that the farther into the future we forecast, the less certain we are of the accuracy of
the forecast since some unexpected change in business and economic conditions may make the
model inappropriate. Since we have less confidence in the forecast for, say, i = 40 than for t
=-36, it follows that the prediction interval for i = 40 must be wider to attain a 95% level of
confidence. For this reason, time series forecasting (regardless of the forecasting method) is
generally confined to the short term.
Multiple regression models can also be used to forecast future values of a time series with
seasonal variation. We illustrate with an example.
EXAMPLE
Refer to the 1991-1994 quarterly power loads listed in the attached Table .
a. Propose a model for quarterly power load, y`, that will account for both the secular
trend and seasonal variation present in the series.
b. Fit the model to the data, and use the least squares prediction equation to forecast the
utility company's quarterly power loads in 1995. Construct 95% confidence intervals
for the forecasts.
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Solution
a. A common way to describe seasonal differences in a time series is with dummy variables.
For quarterly data, a model that includes both trend and seasonal components is
E(Yi) = Bo + Bli
+ B2X1 + B3X2 + B4X3
Trend
Seasonal component
where
where
i = Time period, ranging from i = 1 for quarter I of 1991 to i = 16 for quarter IV of 1994
Yi = Power load (megawatts) in time i
X1_= 1 if quarter I
O if not
X3 =1 if quarter III
X2 =1 if quarter II
O if not
Base level = quarter IV
O if not
The  coefficients associated with the seasonal dummy variables determine the mean increase (or decrease)
in power load for each quarter, relative to the base level quarter, quarter IV.
b. The model is fitted to the data using the SAS multiple regression routine. The resulting SAS printout is
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shown below. Note that the model appears to fit the data quite well: R = .9972, indicating that the model
accounts for 99.7% of the sample variation in power loads over the 4-year period; F = 968.962 strongly
supports the hypothesis that the model has predictive utility (p-value = .0001); and the standard deviation,
Root MSE = 1.53242, implies that the model predictions will usually be accurate to within approximately
+2(1.53), or about +3.06 megawatts.
Forecasts and corresponding 95% prediction intervals for the 1995 power loads are reported in the bottom
portion of the printout . For example, the forecast for power load in quarter I of 1995 is 184.7 megawatts
with the 95% prediction interval (180.5, 188.9). Therefore, using a 95% prediction interval, we expect the
power load in quarter I of 1995 to fall between 180.5 and 188.9 megawatts. Recall from the Table that the
actual 1995 quarterly power loads are 181.5, 175.2, 195.0 and189.3, respectively. Note that each of these falls
within its respective 95% prediction interval shown.
Many descriptive forecasting techniques have proved their merit by providing good forecasts for particular
applications. Nevertheless, the advantage of forecasting using the regression approach is clear: Regression
analysis provides us with a measure of reliability for each forecast through prediction intervals. However,
there are two problems associated with forecasting time series using a multiple regression model.
PROBLEM 1 We are using the least squares prediction equation to forecast values outside the region of
observation of the independent variable, t. For example, suppose we are forecasting for values of t between
17 and 20 (the four quarters of 1995), even though the observed power loads are for t values between 1 and
6
16. As noted earlier, it is risky to use a least squares regression model for prediction outside the range of the
observed data because some unusual change—economic, political, etc.—may make the model inappropriate
for predicting future events. Because forecasting always involves predictions about future values of a time
series, this problem obviously cannot be avoided. However, it is important that the forecaster recognize the
dangers of this type of prediction.
PROBLEM 2 Recall the standard assumptions made about the random error component of a multiple
regression model . We assume that the errors have mean 0, constant variance, normal probability
distributions, and are independent. The latter assumption is often violated in time series that exhibit short term trends. As an illustration, refer to the plot of the sales revenue data. Notice that the observed sales tend
to deviate about the least squares line in positive and negative runs. That is, if the difference between the
observed sales and predicted sales in year t is positive (or negative), the difference in year t + 1 tends to be
positive (or negative). Since the variation in the yearly sales is systematic, the implication is that the errors
are correlated. Violation of this standard regression assumption could lead to unreliable forecasts.
Measuring forecast accuracy (MAD)
Forecast error is defined as the actual value of the series at time i minus the forecast
value,
i.e. Yi - Ŷ i. This can be used to evaluate the accuracy of the forecast. Two of the
procedures for the evaluation are to find the mean absolute deviation
(1)
MAD= | Yi - Ŷ i | /N and (2) RMSE=  ( Yi - Ŷ i)2 /N
Descriptive Analysis: Index numbers
Time series data like other sets of data are subject to two kinds of analyses: Descriptive
and inferential analysis
The most common method to describe a business or economic time series is to compute index
numbers.
INDEX NUMBERS
Definition: An index number is a number that measures the change in a variable over time
relative to the value of the variable during a specific base period. The two most important are
price index and quantity index. Price indexes measure changes in price of a commodity or a
group of commodities over time. The Consumer Price Index (CPI) is a price index, which
measures price changes of a group of commodities over time. On the other hand, an index
constructed to measure the change in the total number of commodities produced annually is a
good example of a quantity index, computations here could be complicated.
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Simple Index Numbers:
Definition: When an index number is based on the price or quantity of a single commodity, it
is called a simple index number.
e.g. to construct a simple index # for the price of silver between 1975-1995, we might choose a
base period, say 1975. To calculate the simple index number for any year, we divide that year’s
price by the price during the base year and multiply by 100
e.g. for 1980, silver price index =(20.64/4.42)*100=466.97
The index # for the base year is always 100. The increase 466.97 – 100= 366.97 gives the
increase in the price of silver over the year.
Measuring the cyclical effect:
Differences between cyclical and seasonal variation is the length of time period. Also,
seasonal effects are predictable while cyclical effects are quite unpredictable.
We use percentage of trend to identify cyclical variation. The calculation is carried out as
follows:
(1) Determine the trend line (by regression)
(2) For each time period, compute the trend value Ŷ
(3) The % of trend is (Y/ Ŷ)*100
See study guide pages 294- 296
Whether graphed or not, we should look for cycle whose peaks and valleys and time periods
are approximately equal. However, in practice, random variation is likely to make it difficult
to see cycles.
Measuring the Seasonal Effect:
Seasonal effect may occur within a year, month, week or day. To measure seasonal effect,
we construct seasonal indexes, which attempt to gauge the degree to which the seasons differ
from one another. One requirement for this method is that we have a time series sufficiently
long to allow us to observe the variable over several seasons. The seasonal indexes are
computed as follows:
(1) Remove the effect of seasonal and random variation. This can be accomplished in one of
two ways:
First:
Calculate moving average by setting the number of periods in the moving average equal to
the number of types of seasons, e.g. if the seasons represent quarters, then a 4-period moving
average will be most suitable. The effect of the Mi is seen as follows:
Assuming a multiplicative model,
Yi = Ti x Ci x Si xIi
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MA removes Si and Ri leaving MAi = Ti x Ci
By taking the ratio of the time series and the MA, we obtain
Yi / MAi = Ti x Ci x Si xIi / Ti x Ci = Si xIi
Second Method:
If the time series is not cyclical, we have Yi = Ti x Si xIi
So, Yi/Ŷi= Si xIi
Since Ŷi represents the trend = Ti
The two methods thus yield the same results.
(2) For each type of seasons, calculate the average of the ratios in step 1. This procedure
removes most of the random variation. The average is a measure of seasonal
differences.
The seasonal indexes are the average ratio from step2 adjusted to ensure that the average
seasonal index is 1.
Detecting Residual Correlation: The Durbin—Watson Test
Many types of business data are observed at regular time intervals thus giving rise to time
series data. We often use regression models to estimate the Trend component of a time series
variable. However, regression models of time series may pose a special problem. Because
business time series tend to follow economic trends and seasonal cycles, the value of a time
series at time t is often indicative of its value at time (i + 1). That is, the value of a time series at
time t is correlated with its value at time (i + 1). If such a series is used as the dependent
variable in a regression analysis, the result is that the random errors are correlated, and this
violates one of the assumptions basic to the least squares inferential procedures. Consequently,
we cannot apply the standard least squares inference-making tools and have confidence in
their validity. In this section, we present a method of testing for the presence of residual
correlation.
Consider the time series data in the Table below, which gives sales data for the 35 year history
of a company. The computer printout shown below gives the regression analysis for the
first-order linear model
Yi = B0 + B1i + e
This model seems to fit the data very well, since R2 = .98 and the F value (1,615.72) that tests
the adequacy of the model is significant. The hypothesis that the B1 coefficient is positive is
accepted at almost any alpha level (i = 40.2 with 33 df)
The residuals e-hat = Y - (B0 + B1i) are plotted in . Note that there is a distinct tendency for
the residuals to have long positive and negative runs. That is, if the residual for year
i is positive, there is a tendency for the residual for year
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(i+ 1) to be positive. These cycles are indicative of possible positive correlation
between residuals.
T A B L E A Finn's Annual Sales Revenue
(thousands of dollars)
YEAR SALES
t
y
1
4.8
2
4.0
3
5.5
4
15.6
5
23.1
6
23.3
7
31.4
8
46.0
9
46.1
10 41.9
11 45.5
12 53 5
YEAR SALES
t
Y
13 48.4
14 61.6
15 65.6
16 71.4
17 83.4
18 93.6
19 94.2
20 85.4
21 86.2
22 89.9
23 89.2
24 99.1
YEAR SALES
t
y
25 100.3
26 111.7
27 108.2
28 115.5
29 119.2
30 125.2
31 136.3
32 146.8
33 146.1
34 151.4
35 150.9
For most economic time series models, we want to test the null hypothesis
Ho: No residual correlation against the alternative
Ha: Positive residual correlation exists
since the hypothesis of positive residual correlation is consistent with economic trends and
seasonal cycles.
The Durbin-Watson d statistic is used to test for the presence of residual correlation. This
statistic is given by the formula at the bottom of page 714 and has values that range from 0-4
with interpretations as follows:
Summary of the interpretations of the Durbin-Watson d statistic
1. If the residuals are uncorrelated, d is approx. = 2
2. If the residuals are positively correlated, d < 2, and if the correlation is very strong, d is
approx. = 0
3. If the residuals are negatively correlated, d > 2, and if the correlation is very strong, d is
approx. = 4
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As indicated in the printout for the sales example, the computed value of d, .821, is less than
the tabulated value of dL, 1.40. Thus, we conclude that the residuals of the straight-line model
for sales are positively correlated.
Moving averages can also be used to provide some measure of seasonal effects in a time series.
The ratio between the observed Yi and the moving average measures the seasonal effect for
that period.
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