Chapter 16 - Times series forecasting and index numbers. This chapter includes contents: Time series components and models, time series regression, multiplicative decomposition, simple exponential smoothing, Holt-Winter’s Models, the Box Jenkins methodology (optional advanced section), forecast error comparisons, index numbers.
Time Series Forecasting
16.1 Time Series Components and Models
16.2 Time Series Regression
16.3 Multiplicative Decomposition
16.4 Simple Exponential Smoothing
16.5 HoltWinter’s Models
16.6 The Box Jenkins Methodology (Optional
Advanced Section)
16.7 Forecast Error Comparisons
16.8 Index Numbers
162
LO16-1: Identify the
components of a times
series.
16.1 Time Series Components and
Models
Trend Longrun growth or decline
Cycle Longrun up and down fluctuation
around the trend level
Seasonal Regular periodic up and down
movements that repeat within the
calendar year
Irregular Erratic very shortrun movements
that follow no regular pattern
163
LO16-1
Time Series Exhibiting Trend, Seasonal, and
Cyclical Components
Figure 16.1 164
LO16-1
Seasonality
Some products have demand that varies a
great deal by period
◦Coats, bathing suits, bicycles
This periodic variation is called seasonality
◦Constant seasonality: the magnitude of the swing
does not depend on the level of the time series
◦Increasing seasonality: the magnitude of the
swing increases as the level of the time series
increases
Seasonality alters the linear relationship
between time and demand
165
LO16-2: Use time series
regression to forecast
time series having
16.2 Time Series Regression
linear, quadratic, and
certain types of
seasonal patterns.
Within regression, seasonality can be modeled
using dummy variables
Consider the model:
yt = 0 + 1t + Q2Q2 + Q3 Q3 + Q4 Q4 + t
◦For Quarter 1, Q2 = 0, Q3 = 0 and Q4 = 0
◦For Quarter 2, Q2 = 1, Q3 = 0 and Q4 = 0
◦For Quarter 3, Q2 = 0, Q3 = 1 and Q4 = 0
◦For Quarter 4, Q2 = 0, Q3 = 0 and Q4 = 1
The coefficient will then give us the seasonal
impact of that quarter relative to Quarter 1
◦Negative means lower sales, positive lower sales
166
LO16-3: Use data
transformations to
forecast time series
Transformations
having increasing
seasonal variation.
Sometimes, transforming the sales data
makes it easier to forecast
◦Square root
◦Quartic roots
◦Natural logarithms
While these transformations can make the
forecasting easier, they make it harder to
understand the resulting model
167
LO 4: Use multiplicative
decomposition and
moving averages to
16.3 Multiplicative Decomposition
forecast time series
having increasing
seasonal variation.
We can use the multiplicative decomposition
method to decompose a time series into its
components:
Trend
Seasonal
Cyclical
Irregular
168
LO 16-5: Use simple
exponential smoothing
to forecast a time
16.4 Simple Exponential Smoothing
series.
Earlier, we saw that when there is no trend, the least
squares point estimate b0 of β0 is just the average y
value
◦yt = β0 + t
That gave us a horizontal line that crosses the y axis
at its average value
Since we estimate b0 using regression, each period
is weighted the same
If β0 is slowly changing over time, we want to
weight more recent periods heavier
Exponential smoothing does just this
169
LO16-6: Use double
exponential smoothing
to forecast a time
16.5 Holt–Winters’ Models
series.
Simple exponential smoothing cannot handle trend
or seasonality
Holt–Winters’ double exponential smoothing can
handle trended data of the form
yt = β0 + β1t + t
◦Assumes β0 and β1 changing slowly over time
◦We first find initial estimates of β0 and β1
◦ Then use updating equations to track changes over time
Requires smoothing constants called alpha and gamma
1610
LO16-7: Use
multiplicative Winters’
method to forecast a
Multiplicative Winters’ Method
time series.
Double exponential smoothing cannot handle
seasonality
Multiplicative Winters’ method can handle trended
data of the form
yt = (β0 + β1t) ∙ SNt + t
◦Assumes β0, β1, and SNt changing slowly over time
◦We first find initial estimates of β0 and β1 and seasonal
factors
◦ Then use updating equations to track over time
Requires smoothing constants alpha, gamma and delta
1611
LO16-8: Use the
Box–Jenkins
methodology to
forecast a time
series.
16.6 The Box–Jenkins Methodology
(Optional Advanced Section)
Uses a quite different approach
Begins by determining if the time series is
stationary
◦The statistical properties of the time series are
constant through time
Plots can help
If nonstationary, will transform series
1612
LO16-9: Compare time
series models by using
forecast errors.
16.7 Forecast Error Comparison
Forecast errors: et = yt ŷt
Error comparison criteria
◦Mean absolute deviation (MAD)
n n
et yt yˆ t
MAD t 1 t 1
n n
◦Mean squared deviation (MSD)
n n
2
e t ( yt yˆ t ) 2
MSD t 1 t 1
n n
1613
LO16-10: Use index
numbers to compare
economic data over
16.8 Index Numbers
time.
Index numbers allow us to compare changes
in time series over time
We begin by selecting a base period
Every period is converted to an index by
dividing its value by the base period and then
multiplying the results by 100
◦Simple (quantity) index
yt
it 100
y0
1614
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