Time series forecasting

1. Time Series
Forecasting
Outline:
1. Measuring forecast error
2. The multiplicative time series model
3. Naïve extrapolation
4. The mean forecast model
5. Moving average models
6. Weighted moving average models
7. Constructing a seasonal index using a centered
moving average
8. Exponential smoothing

2. Forecast error
Forecasting Convenience Store Ice Sales
(1)
Forecasted
Month/Year
Value
(2)
Actual
Value
(3) = (2) – (1)
Error
July 2000
$390
$423
$33
Aug 2000
450
429
-21
Sept 2000
289
301
12

3. 3 measures of forecast error
• Mean absolute deviation
• Mean square error
• Root mean square error.

4. Actual
Predicted
Time
Average Absolute Error (AAE) is given by:
1
AAE =
m
m
Y
∑
t
− ˆt
Y
t=
1
Where Yt is the actual value of variable that we seek to
ˆ
forecast and Yt is the fitted or forecasted value of the
variable.

5. Actual
Predicted
Time
Mean Square Error (MSE) is given by:
1
MSE =
m
m
(Yi − ˆi ) 2
∑ Y
t=
1
Where Yt is the actual value of variable that we seek to
ˆ
forecast and Yt 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.

6. Actual
Predicted
Time
Root Mean Square Error (root MSE) is given by:
rootMSE =
1
m
m
(Yt − ˆt ) 2
∑ Y
t=
1
Root MSE is a statistic
that is typically is reported
by forecasting software
applications

7. 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:
1. The trend component (T)
2. The seasonal component (S)
3. The cyclical component (C); and
4. The irregular component (I)

8. The trend component (T)
Trend is the gradual, longrun (or secular) evolution
of the variables that we are
seeking to forecast.

9. 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 music on DVD have
slumped due to Ipods. Typewriter sales have
plumetted.
•Changes in consumer tastes and preferences.

10. Linear trends
40
20
Trend = 10 – 25t
0
-20
-40
Trend = -50 + .8t
-60
10
20
30
40
50
60
70
80
90
100

11. Non-linear, increasing trend
4000
3000
Trend = 10 + .3t + .3t2
2000
1000
0
10
20
30
40
50
60
70
80
90
100

12. Non-linear, decreasing trend
1000
Trend = 10 - .4t - .4t2
0
-1000
-2000
-3000
-4000
-5000
10
20
30
40
50
60
70
80
90
100

13. 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.

14. 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-cyclical”—for example, it flourished during
the Depression.

15. 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.

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

17. The model
Yt = Tt ×St ×Ct × It
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.

18. 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.

19. The Problem: Forecast Sales of Home
Furnishing Stores, October-December, 2007
The data:
•We have monthly data of sales of home
furniture stores January 1992 to July 2007
(187 monthly observations).
•The data are expressed in millions of
current dollars, not seasonally adjusted

20. t
1
2
3
4
5
6
7
8
9
10
11
12
"
"
187
The Data
Yr
1992
1992
1992
1992
1992
1992
1992
1992
1992
1992
1992
1992
"
"
2007
Mo
1
2
3
4
5
6
7
8
9
10
11
12
"
"
7
$
1460
1453
1556
1622
1675
1759
1789
1814
1721
1839
1925
2246
"
"
4803

21. Sales of Home Furnishing Stores, 1992-2007
(millions of dollars, NSA)
7000
6000
5000
4000
3000
2000
1000
92
94
96
98
00
02
Year/Month
Source: Economagic.com
04
06

22. Our first step is to estimate the
trend 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 home furnishing sales) from the trend
line (fitted values of monthly building materials
sales).

23. Let:
•Yt be the actual value of furniture store sales in
month t;
•Let Ŷt be the trend value of furniture store sales
in month t. The trend function we are seeking
satisfies the following condition:
187
ˆ
MIN .∑ (Y t − Yt ) 2
t =1

24. We estimate a linear
trend function with Excel.
It is displayed on the next
slide.

25. R2 = 0.83
t

26. Actual and Trend Values of Hom Furniture Sales (in millions)
7000
6000
5000
4000
3000
2000
1000
92
94
96
98
00
Year/Month
Actual
02
TREND
04
06

27. Seasonal Index
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Index
0.8799
0.8475
0.9823
0.9004
0.9939
1.0197
0.9729
1.0487
1.0042
0.9962
1.123
1.2969
•If you sum the
monthly values and
divide by 12, you
get 1.00.
•Later we show a
simple technique
for computing a
seasonal index.

28. Performing an in-sample forecast of home
furnishing sales
•An in-sample forecast means we are forecasting
home furshing 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 home furnishing 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 two business cycle contractions.

29. Let’s give an example how
we use this model to Home
furnishing sales for a
particular month, say, April
1998 . t = 76 for this month
ˆ
FApr 98 = Tt ×St ×Ct
ˆ
FApr 98 = [(17.62 × 76) +1475] × 0.900 ×1 = $2,532.71

30. In-Sample Forecast of Home Furnishing Sales Using Multiplicative Model
7000
6000
5000
4000
3000
2000
1000
94
96
Multiplcative model
98
00
Year/Month
02
04
06
Home furnishing sales (millions)

31. Residuals from In-sample Forecast of Home Furnishing Sales (in millions)
300
Recession is shaded
200
100
0
-100
-200
-300
94
96
MSE = $103.275
98
00
Year/Month
02
04
06

32. Forecasting Using the Multiplicative
Model
t
Yr/Mo
Trend
Seasonal
Cyclical
Forecast
190
2007/Oct
4822.8
0.9962
0.999
4799.669
191
2007/Nov
4840.42
1.123
0.979
5321.64
192
2007/Dec
4858.04
1.2969
0.975
6142.882