Time Series Forecasting

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Time Series Forecasting

• Outline
• Measuring forecast error
• The multiplicative time series model
• Naïve extrapolation
• The mean forecast model
• Moving average models
• Weighted moving average models
• Constructing a seasonal index using a centered
  moving average
• Exponential smoothing

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Forecast error
Forecasting Convenience Store Ice Sales
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3 measures of forecast error

• Mean absolute deviation
• Mean square error
• Root mean square error.

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Actual
Predicted
Time
Average Absolute Error (AAE) is given by

Where Yt is the actual value of variable that we
seek to forecast and is the fitted or
forecasted value of the variable.
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Actual
Predicted
Time
Mean Square Error (MSE) is given by

Where Yt is the actual value of variable that we
seek to forecast and is the fitted or
forecasted value of the variable.

• You can think of MSE as the average forecast
  error.
• If we have a perfect forecast, then MSE 0.

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Actual
Predicted
Time
Root Mean Square Error (root MSE) is given by

Root MSE is a statistic that is typically is
reported by forecasting software applications
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The Multiplicative Time Series Model

• The time path of a variable (such as monthly
  sales of building materials by supply stores) is
  produced by the interaction of 4 factors or
  components. These components are
• The trend component (T)
• The seasonal component (S)
• The cyclical component (C) and
• The irregular component (I)

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The trend component (T)
Trend is the gradual, long-run (or secular)
evolution of the variables that we are seeking to
forecast.
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Factors affecting the trend component of a time
series

• Population changes
• Demographic changes. For example, spending for
  healthcare services is likely to rise due to the
  aging of the population. Sales of fast food are
  up due to the secular increase in the female
  labor force participation rate.
• Technological change. Sales of typewriter and
  vinyl records have trended downward due product
  innovation.
• Changes in consumer tastes and preferences.

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Linear trends
Trend 10 25t
Trend -50 .8t
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Non-linear, increasing trend
Trend 10 .3t .3t2
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Non-linear, decreasing trend
Trend 10 - .4t - .4t2
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The seasonal component (S)

• Many series display a regular pattern of
  variability depending on the time of year.
• For example, sales of toys and scotch whiskey
  peak in December each year.
• Ice cream sales are higher in summer months than
  in winter months.
• Car sales tend typically to be strong in May and
  June and weaker in November and December.

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The cyclical component (C)

• The time path of a series can be influenced by
  business cycle fluctuations.
• For example, we expect housing starts to decline
  in the contractionary phase of the business
  cycle.
• The same holds true for federal or state tax
  receipts
• The time path of spending for consumer durable
  goods is also shaped by cyclical forces.
• Spending for capital goods is likewise cyclical.
• The movie industry has the reputation for being
  counter-cyclicalfor example, it flourished
  during the Depression.

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The irregular component (I)

• The irregular component of the series, sometimes
  called white noise, is the remaining variability
  (relative to trend) that cannot be explained by
  seasonal or cyclical factors. The irregular
  component is an unexpected, non-recurring factor
  that affects the series.
• For example, hamburger sales plunge due to panic
  about E-Coli bacteria.
• Production of trucks slumps because of a strike
  at a GM parts plant in Ohio.
• Airline slump after 9/11.
• A cold snap affects July ice cream sales in
  upstate NY.

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If you have a well-designed forecasting model,
then forecasting errors should be mainly
accounted for by irregular factors
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The model

• Where
• Yt is the value of the time series variable in
  period t (month t, quarter t, etc.)
• Tt trend component of the series in period t
• St is the seasonal component of the series in
  period t
• Ct is the cylical component of the series at
  period t and
• It is the irregular component of the series in
  period t.

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The trend component (T) is measured in the units
in which the time series itself is measured. So,
for example, the trend component for state
revenues would be measured in dollars whereas
the trend component for steel production might be
measured in tons.
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The problem forecast sales of building materials
through supply stores for 20008 to 20017

• The data
• We have monthly data of building material sales
  through supply stores for the period January
  1967 to July 2000 (402 monthly observations).
• The data are expressed in millions of current
  dollars.

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All data in millions
www.economagic.com
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www.economagic.com
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Our first step is to estimate thetrend component
of our series.This is accomplished using a
ordinary least squares, or OLS for short.

• OLS is a method of finding the line, or curve, of
  best fit.
• The trend function of best fit is the one that
  minimizes the squared sum of the vertical
  distances of the sample points (the actual
  monthly values of building materials sales) from
  the trend line (fitted values of monthly building
  materials sales).

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OLS

• Let
• Yt be the actual value of building materials
  sales in month t
• Let Yt be the trend value of building materials
  sales in month t. The trend function we are
  seeking satisfies the following condition

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• Professor Brown has estimated two trend
  functionsone linear and one non-linear. They are
  displayed on the the following two slides.
• The trend of of building materials sales since
  1967 is positive and increasing (non-linear).

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Trend -771.28 25.79t
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Trend 957.77 0.11t .063t2
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Trend 957.77 0.11t .063t2
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Example Trend value of building material sales
for March 81
Note that, for March 1981 t 169
Trend 957.77 0.11t .063t2
Thus we have TrendMar 81 957.77 (.11)(169)
(.063)(1692
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The Seasonal Index

• If you sum the indices for each month, and divide
  by 12, you get 1.00.
• Notice that, on average for the period 1967-2000,
  July has been the best month for sales of
  building materials, and February the worst month.
• Later, we will show you a simple technique for
  constructional a seasonal indexa centered moving
  average.

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Performing an in-sample forecast of building
materials sales

• An in-sample forecast means we are forecasting
  building material sales for those months for
  which we already have data that have been used to
  estimate the trend, seasonal, and other
  components. Comparing forecasted, or fitted
  values of building material sales with actual
  time series data gives us an idea of how well
  this performs.
• We will assume that the cyclical index is equal
  to 1 (Ct 1). This is a poor assumption since
  our period contains several business cycle
  episodes.

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Lets give an example how we we use this model to
forecast building material sales for a particular
month, say, March 1981 again.Recall that t 169
for this month
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In-sample forecasts using the multiplicative time
series model
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Residuals for in-sample forecast
MSE 179,288root MSE 423 million
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Assumption that Ct 1 results in substantial
in-sample forecast errors
Recessionary periods are shaded
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Forecasting Sales of Building Materials Using the
Multiplicative Time Series Model
All data in millions