An Introduction to Time Series Analysis by moti


									Treasury Board of Canada Secretariat

An Introduction to Time Series Analysis

“With the past, we can see trajectories into the future - both catastrophic and creative
John Ralston Saul

There are two major categories of statistical information: cross section and time series. To
illustrate, econometricians estimating how U.S. consumer expenditure is related to national
income ("the consumption function") sometimes use a detailed breakdown of individuals’
consumption at various income levels at one point in time (cross section). At other times, they
examine how total consumption is related to national income over a number of time periods
(time series) and sometimes they use a combination of the two. We shall demonstrate the use of
some familiar techniques (linear regression) and develop some new methods to analyse time
series. Although our examples will use quarterly data, the techniques are also applicable to
monthly data, weekly data, etc.
The main characteristic of a time series (which distinguishes it from a simple random sample) is
that its observations have some form of dependence on time. The problem is that there are any
number of patterns that this dependence may take. The major ones are illustrated in Figure 1-1.
Panel (a) shows a time series with only a trend. Panel (b) shows a time series with only a
quarterly pattern, repeated identically

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           FIGURE 1-1 Three possible patterns of time dependence in a time series:
           (a) Trend; (b) Seasonal; (c) Random tracking.
every year; thus, for example, the fourth quarter of 1955 is the same as the fourth quarter of 1956
or any other year. Panel (c) displays a random tracking time series of auto-correlated or serially
correlated terms; that is, each value is related to the preceding values, with a random disturbance
added. (There are many examples of "series that follow themselves" outside of business and the
social sciences: for example, a garden hose leaves a random-tracking path of water along a wall.)
If a time series followed only one of these patterns, there would be no problem. In practice,
however, it is typically a mixture of all three that is very difficult to unscramble. Consider, for
example, the quarterly data on plant and equipment expenditures shown in Figure 1-2. What
combination of these three patterns can be perceived? There appears to be some

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       FIGURE 1-2 U.S. new plant and equipment expenditures, in durable manufacturing (in
       billions of dollars). Source: Survey of Current Business; U.S. Department of Commerce.
trend, some seasonal influence and some random element, but how much of each is a mystery.
Time series analysis may be viewed as simply the attempt to break a time series up into these
various components. We therefore shall consider each pattern in Figure 1-1 in turn.

Trend is often the most important element in a time series. It may be linear, as shown in Figure
1-1 (a), displaying a constant increase in each time period, or it may be exponential (a geometric
series), displaying a constant percentage increase in each time period. Examples might be
population growth in a developing country or the growth of a trust fund over a long period. Then
the observations’ logarithms will display a linear trend. The trend also may be a polynomial, or
an even more complicated function.
The simplest way to deal with trend is with regression, simultaneously making seasonal
adjustment, as we discuss in the section below. We assume that the trend is linear, but if it is not,
non-linear regression techniques exist to tackle more complex relationships.

There may be seasonal fluctuation in a time series, for several reasons. For example, a religious
holiday, in particular Christmas, results in completely different economic and purchasing
patterns. Or the seasons may affect economic activity: in the summer, agricultural production is

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high, while the sale of ski equipment is low.
As an example of seasonal fluctuation, we often note both the possibility of a very slight upward
trend and an obvious seasonal pattern marked by the sharp rise in sales every fourth quarter
because of Christmas. If we made the mistake of trying to estimate only trend, say by a simple
linear regression of sales S on time T, the result would be a substantial bias.
Note: The upward bias in slope largely is caused by the fact that the fourth quarter observations
would be high. To avoid this, both trend and seasonal should be put into the regression model, in
order to estimate their separate effects.
The fourth-quarter observations may be treated as a dummy variable. Let Q4 be the fourth
quarter dummy. Hence, the model becomes:
S = a + b1T + b4Q4 + e

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                       Canadian Department Store Sales - Jewellery
                               Sales and Seasonal Dummies

                Time,      T           Sales, S ($100,000's)   Q4 Seasonal
                (YYYY - Q)                                     Dummy

                1957           1                 36                 0

                               2                 44                 0

                               3                 45                 0

                               4                106                 1

                1958           5                 38                 0

                               6                 46                 0

                               7                 47                 0

                               8                112                 1

                1959           9                 42                 0

                              10                 49                 0

                              11                 48                 0

                              12                118                 1

                1960          13                 42                 0

                              14                 50                 0

                              15                 51                 0

                              16                118                 1

                             Source: Statistics Canada, 63-002.
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Even this model may not be adequate. If allowance should be made for shifts in the other
quarters, dummies Q2 and Q3 should be added.
Whether or not to include various regressors such as Q4 can be decided on statistical grounds, by
testing for statistical discernibility.
Our equation has allowed us to break down the total jewellery sales series into its three
   1. Trend (the a + b1T component of the equation) ;
   2. Seasonal (the b4Q4 term); and
   3. Random Fluctuations (the error term e).

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