Regression-Based Trend Models

1
Example 16.4

• Regression-Based Trend Models

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REEBOK.XLS

• This file includes quarterly sales data for
  Reebok from first quarter 1986 through second
  quarter 1996.
• The following screen shows the time series plot
  of these data.
• Sales increase from 174.52 million in the first
  quarter to 817.57 million in the final quarter.
• How well does a linear trend fit these data?
• Are the residuals from this fit random?

3
Time Series Plot of Reebok Sales
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Linear Trend

• A linear trend means that the time series
  variable changes by a constant amount each time
  period.
• The relevant equation is Yt a bt Et where a
  is the intercept, b is the slope and Et is an
  error term.
• If b is positive the trend is upward, if b is
  negative then the trend is downward.
• The graph of the time series is a good place to
  start. It indicates whether a linear trend model
  is likely to provide a good fit.

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Solution

• The plot indicates an obvious upward trend with
  little or no curvature.
• Therefore, a linear trend is certainly plausible.
• We use regression to estimate the linear fit,
  where Sales is the response variable and Time is
  the single explanatory variable.
• The Time variable is coded 1-42 and is used as
  the explanatory variable in the regression.

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Solution -- continued

• The Quarter variable simply labels the quarters
  (Q1-86 to Q2-96) and is used only to label the
  horizontal axis.
• The following regression output shows that the
  estimated equation is Forecasted Sales 244.82
  16.53Time with R2 and se values of 83.8 and
  90.38 million.

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Regression Output for Linear Trend
8
Time Series Plot with Linear Trend Superimposed

• The linear trendline, superimposed on the sales
  data, appears to be a decent fit.

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Solution -- continued

• The trendline implies that sales are increasing
  by about 16.53 million per quarter during this
  period.
• The fit is far from perfect, however.
• First, the se value 90.38 million is an
  indication of the typical forecast error. This is
  substantial, approximately equal to 11 of the
  final quarters sales
• Furthermore, there is some regularity to the
  forecast errors shown in the following plot.

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Time Series Plot of Forecasted Errors
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Plot Interpretation

• They zigzag more than a random series.
• There is probably some seasonal pattern in the
  sales data, which we might be able to pick up
  with a more sophisticated forecasting method.
• However, the basic linear trend is sufficient as
  a first approximation to the behavior of sales.