Time Series
Forecasting
Outline:
1. Measuring forecast error
2. The multiplicative time series model
3. Naïve extrapolatio...
Forecast error
Forecasting Convenience Store Ice Sales
(1)
Forecasted
Month/Year
Value

(2)
Actual
Value

(3) = (2) – (1)
...
3 measures of forecast error
• Mean absolute deviation
• Mean square error
• Root mean square error.
Actual
Predicted

Time

Average Absolute Error (AAE) is given by:
1
AAE =
m

m

Y
∑

t

− ˆt
Y

t=
1

Where Yt is the actu...
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 v...
Actual

Predicted

Time

Root Mean Square Error (root MSE) is given by:
rootMSE =

1
m

m

(Yt − ˆt ) 2
∑ Y
t=
1

Root MSE...
The time path of a variable (such as monthly sales of
building materials by supply stores) is produced by the
interaction ...
The trend component (T)
Trend is the gradual, longrun (or secular) evolution
of the variables that we are
seeking to forec...
Factors affecting the trend component of a
time series
•Population changes
•Demographic changes. For example, spending for...
Linear trends
40

20

Trend = 10 – 25t

0

-20

-40

Trend = -50 + .8t
-60
10

20

30

40

50

60

70

80

90

100
Non-linear, increasing trend
4000

3000

Trend = 10 + .3t + .3t2
2000

1000

0
10

20

30

40

50

60

70

80

90

100
Non-linear, decreasing trend
1000

Trend = 10 - .4t - .4t2

0
-1000
-2000
-3000
-4000
-5000
10

20

30

40

50

60

70

80...
The seasonal component (S)

•Many series display a regular pattern of
variability depending on the time of year.
•For exam...
The cyclical component (C)
•The time path of a series can be influenced by business
cycle fluctuations.
•For example, we e...
The irregular component (I)
•The irregular component of the series, sometimes
called white noise, is the remaining variabi...
If you have a well-designed
forecasting model, then forecasting
errors should be mainly accounted
for by irregular factors
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....
The trend component (T) is measured in
the units in which the time series itself is
measured. So, for example, the trend
c...
The Problem: Forecast Sales of Home
Furnishing Stores, October-December, 2007
The data:
•We have monthly data of sales of ...
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...
Sales of Home Furnishing Stores, 1992-2007
(millions of dollars, NSA)

7000
6000
5000
4000
3000
2000
1000
92

94

96

98

...
Our first step is to estimate the
trend component of our series.
This is accomplished using a
ordinary least squares, or O...
Let:
•Yt be the actual value of furniture store sales in
month t;
•Let Ŷt be the trend value of furniture store sales
in m...
We estimate a linear
trend function with Excel.
It is displayed on the next
slide.
R2 = 0.83

t
Actual and Trend Values of Hom Furniture Sales (in millions)

7000
6000
5000
4000
3000
2000
1000
92

94

96

98
00
Year/Mo...
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
0.97...
Performing an in-sample forecast of home
furnishing sales
•An in-sample forecast means we are forecasting
home furshing sa...
Let’s give an example how
we use this model to Home
furnishing sales for a
particular month, say, April
1998 . t = 76 for ...
In-Sample Forecast of Home Furnishing Sales Using Multiplicative Model

7000
6000
5000
4000
3000
2000
1000
94

96

Multipl...
Residuals from In-sample Forecast of Home Furnishing Sales (in millions)

300

Recession is shaded

200
100
0
-100
-200
-3...
Forecasting Using the Multiplicative
Model

t

Yr/Mo

Trend

Seasonal

Cyclical

Forecast

190

2007/Oct

4822.8

0.9962

...
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Timeseries forecasting

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Timeseries forecasting

  1. 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. 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. 3 measures of forecast error • Mean absolute deviation • Mean square error • Root mean square error.
  4. 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. 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. 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. 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. 8. The trend component (T) Trend is the gradual, longrun (or secular) evolution of the variables that we are seeking to forecast.
  9. 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. 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. 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. 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. 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. 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. 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. 16. If you have a well-designed forecasting model, then forecasting errors should be mainly accounted for by irregular factors
  17. 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. 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. 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. 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. 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. 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. 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. 24. We estimate a linear trend function with Excel. It is displayed on the next slide.
  25. 25. R2 = 0.83 t
  26. 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. 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. 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. 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. 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. 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. 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

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