Transcript Statistics for Business and Economics, 6/e

Statistics for
Business and Economics
6th Edition
Chapter 19
Time-Series Analysis and
Forecasting
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-1
Chapter Goals
After completing this chapter, you should be able to:
 Compute and interpret index numbers
 Weighted and unweighted price index
 Weighted quantity index
 Test for randomness in a time series
 Identify the trend, seasonality, cyclical, and irregular
components in a time series
 Use smoothing-based forecasting models, including
moving average and exponential smoothing
 Apply autoregressive models and autoregressive
integrated moving average models
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-2
Index Numbers
 Index numbers allow relative comparisons
over time
 Index numbers are reported relative to a Base
Period Index
 Base period index = 100 by definition
 Used for an individual item or measurement
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-3
Single Item Price Index
Consider observations over time on the price of a single
item
 To form a price index, one time period is chosen as a
base, and the price for every period is expressed as a
percentage of the base period price
 Let p0 denote the price in the base period
 Let p1 be the price in a second period
 The price index for this second period is
 p1 
100 
 p0 
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-4
Index Numbers: Example
 Airplane ticket prices from 1995 to 2003:
Index
Year
Price
(base year
= 2000)
1995
272
85.0
1996
288
90.0
1997
295
92.2
1998
311
97.2
1999
322
100.6
2000
320
100.0
2001
348
108.8
2002
366
114.4
2003
384
120.0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
I1996
P1996
288
 100
 (100)
 90
P2000
320
Base Year:
P2000
320
I2000  100
 (100)
 100
P2000
320
I2003
P2003
384
 100
 (100)
 120
P2000
320
Chap 19-5
Index Numbers: Interpretation
 Prices in 1996 were 90%
of base year prices
I1996
P1996
288

 100 
(100)  90
P2000
320
I2000
 Prices in 2000 were 100%
P2000
320

 100 
(100)  100
of base year prices (by
P2000
320
definition, since 2000 is the
base year)
I2003
 Prices in 2003 were 120%
P2003
384

 100 
(100)  120
of base year prices
P2000
320
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-6
Aggregate Price Indexes
 An aggregate index is used to measure the rate
of change from a base period for a group of items
Aggregate
Price Indexes
Unweighted
aggregate
price index
Weighted
aggregate
price indexes
Laspeyres Index
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-7
Unweighted
Aggregate Price Index
 Unweighted aggregate price index for period
t for a group of K items:
 K
  p ti
100 iK1

  p0i
 i1






i = item
t = time period
K = total number of items
K
p
i1
ti
K
p
i1
0i
= sum of the prices for the group of items at time t
= sum of the prices for the group of items in time period 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-8
Unweighted Aggregate Price
Index: Example
Automobile Expenses:
Monthly Amounts ($):
Index
Year
Lease payment
Fuel
Repair
Total
(2001=100)
2001
260
45
40
345
100.0
2002
280
60
40
380
110.1
2003
305
55
45
405
117.4
2004
310
50
50
410
118.8
I2004
P

 100
P
2004
2001
410
 (100)
 118.8
345
 Unweighted total expenses were 18.8%
higher in 2004 than in 2001
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-9
Weighted
Aggregate Price Indexes
 A weighted index weights the individual prices by
some measure of the quantity sold
 If the weights are based on base period quantities the
index is called a Laspeyres price index
 The Laspeyres price index for period t is the total cost of
purchasing the quantities traded in the base period at prices in
period t , expressed as a percentage of the total cost of
purchasing these same quantities in the base period
 The Laspeyres quantity index for period t is the total cost of the
quantities traded in period t , based on the base period prices,
expressed as a percentage of the total cost of the base period
quantities
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-10
Laspeyres Price Index
 Laspeyres price index for time period t:
 K

  q0ip ti 

100 iK1


  q0ip0i 
 i1

q0i = quantity of item i purchased in period 0
p 0i = price of item i in time period 0
p ti = price of item i in period t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-11
Laspeyres Quantity Index
 Laspeyres quantity index for time period t:


  qtip0i 

100 iK1


  q0ip0i 
 i1

K
p0i = price of item i in period 0
q0i = quantity of item i in time period 0
q ti = quantity of item i in period t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-12
The Runs Test for Randomness
 The runs test is used to determine whether a
pattern in time series data is random
 A run is a sequence of one or more occurrences
above or below the median
 Denote observations above the median with “+”
signs and observations below the median with
“-” signs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-13
The Runs Test for Randomness
(continued)
 Consider n time series observations
 Let R denote the number of runs in the
sequence
 The null hypothesis is that the series is random
 Appendix Table 14 gives the smallest
significance level for which the null hypothesis
can be rejected (against the alternative of
positive association between adjacent
observations) as a function of R and n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-14
The Runs Test for Randomness
(continued)
 If the alternative is a two-sided hypothesis on
nonrandomness,
 the significance level must be doubled if it is
less than 0.5
 if the significance level, , read from the table
is greater than 0.5, the appropriate
significance level for the test against the twosided alternative is 2(1 - )
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-15
Counting Runs
Sales
Median
Time
--+--++++-----++++
Runs: 1 2 3
4
5
6
n = 18 and there are R = 6 runs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-16
Runs Test Example
n = 18 and there are R = 6 runs
 Use Appendix Table 14
 n = 18 and R = 6
 the null hypothesis can be rejected (against the
alternative of positive association between adjacent
observations) at the 0.044 level of significance
 Therefore we reject that this time series is random
using  = 0.05
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-17
Runs Test: Large Samples


Given n > 20 observations
Let R be the number of sequences above or below
the median
Consider the null hypothesis H0: The series is random

If the alternative hypothesis is positive association
between adjacent observations, the decision rule is:
Reject H0 if
n
R  1
2
z
 z α
2
n  2n
4(n  1)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-18
Runs Test: Large Samples
(continued)
Consider the null hypothesis H0: The series is random

If the alternative is a two-sided hypothesis of
nonrandomness, the decision rule is:
Reject H0 if
n
R  1
2
z
 z α/2
2
n  2n
4(n  1)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
n
R  1
2
or z 
 z α/2
2
n  2n
4(n  1)
Chap 19-19
Example: Large Sample
Runs Test
 A filling process over- or under-fills packages,
compared to the median
OOO U OO U O UU OO UU OOOO UU O UU
OOO UUU OOOO UU OO UUU O U OO UUUUU
OOO U O UU OOO U OOOO UUU O UU OOO U
OO UU O U OO UUU O UU OOOO UUU OOO
n = 100 (53 overfilled, 47 underfilled)
R = 45 runs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-20
Example: Large Sample
Runs Test
(continued)
 A filling process over- or under-fills packages,
compared to the median
 n = 100 , R = 45
Z
n
100
1
45 
1
6
2
2


 1.206
2
2
n  2n
100  2(100) 4.975
4(n  1)
4(100  1)
R
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-21
Example: Large Sample
Runs Test
(continued)
H0: Fill amounts are random
H1: Fill amounts are not random
Test using  = 0.05
Rejection Region
/2 = 0.025
Rejection Region
/2 = 0.025
 1.96
0
1.96
Since z = -1.206 is not less than -z.025 = -1.96,
we do not reject H0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-22
Time-Series Data
 Numerical data ordered over time
 The time intervals can be annually, quarterly,
daily, hourly, etc.
 The sequence of the observations is important
 Example:
Year:
2001 2002 2003 2004 2005
Sales:
75.3
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74.2
78.5
79.7
80.2
Chap 19-23
Time-Series Plot
A time-series plot is a two-dimensional
plot of time series data
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
1975
 the horizontal axis
corresponds to the
time periods
U.S. Inflation Rate
Inflation Rate (%)
 the vertical axis
measures the variable
of interest
Year
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Chap 19-24
Time-Series Components
Time Series
Trend
Component
Seasonality
Component
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Cyclical
Component
Irregular
Component
Chap 19-25
Trend Component
 Long-run increase or decrease over time
(overall upward or downward movement)
 Data taken over a long period of time
Sales
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Time
Chap 19-26
Trend Component
(continued)
 Trend can be upward or downward
 Trend can be linear or non-linear
Sales
Sales
Time
Downward linear trend
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Time
Upward nonlinear trend
Chap 19-27
Seasonal Component
 Short-term regular wave-like patterns
 Observed within 1 year
 Often monthly or quarterly
Sales
Summer
Winter
Summer
Spring
Winter
Spring
Fall
Fall
Time (Quarterly)
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Chap 19-28
Cyclical Component
 Long-term wave-like patterns
 Regularly occur but may vary in length
 Often measured peak to peak or trough to
trough
1 Cycle
Sales
Year
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-29
Irregular Component
 Unpredictable, random, “residual” fluctuations
 Due to random variations of
 Nature
 Accidents or unusual events
 “Noise” in the time series
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-30
Time-Series Component Analysis
 Used primarily for forecasting
 Observed value in time series is the sum or product of
components
 Additive Model
Xt  Tt  St  Ct It
 Multiplicative model (linear in log form)
Xt  TtStCtIt
where
Tt = Trend value at period t
St = Seasonality value for period t
Ct = Cyclical value at time t
It = Irregular (random) value for period t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-31
Smoothing the Time Series
 Calculate moving averages to get an overall
impression of the pattern of movement over
time
 This smooths out the irregular component
Moving Average: averages of a designated
number of consecutive
time series values
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-32
(2m+1)-Point Moving Average
 A series of arithmetic means over time
 Result depends upon choice of m (the
number of data values in each average)
 Examples:
 For a 5 year moving average, m = 2
 For a 7 year moving average, m = 3
 Etc.
 Replace each xt with
m
1
X 
Xt  j (t  m  1,m  2,,n  m)

2m  1 jm
*
t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-33
Moving Averages
 Example: Five-year moving average
 First average:
x 5* 
x1  x 2  x 3  x 4  x 5
5
 Second average:
x *6 
x2  x3  x 4  x5  x6
5
 etc.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-34
Example: Annual Data
1
2
3
4
5
6
7
8
9
10
11
etc…
Sales
23
40
25
27
32
48
33
37
37
50
40
etc…
Annual Sales
60
50
…
40
Sales
Year
30
20
10
0
1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
2
3
4
5
6
7
8
9
10
11
…
Year
Chap 19-35
Calculating Moving Averages
 Let m = 2
Year
Sales
Average
Year
5-Year
Moving
Average
1
23
3
29.4
2
40
4
34.4
3
25
5
33.0
4
27
6
35.4
5
32
7
37.4
6
48
8
41.0
7
33
9
39.4
8
37
…
…
9
37
10
50
11
40
etc…
29.4 
23  40  25  27  32
5
 Each moving average is for a
consecutive block of (2m+1) years
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-36
Annual vs. Moving Average
Annual vs. 5-Year Moving Average
60
50
40
Sales
 The 5-year
moving average
smoothes the
data and shows
the underlying
trend
30
20
10
0
1
2
3
4
5
6
7
8
9
10
11
Year
Annual
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
5-Year Moving Average
Chap 19-37
Centered Moving Averages
(continued)
 Let the time series have period s, where s is
even number
 i.e., s = 4 for quarterly data and s = 12 for monthly data
 To obtain a centered s-point moving average series
Xt*:
 Form the s-point moving averages
x *t .5 
s/2
s s
s
s
x
(t

,

1,

2,

,
n

)

t j
2 2
2
2
j (s/2)1
 Form the centered s-point moving averages
x *t .5  x *t .5
x 
2
*
t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
(t 
s
s
s
 1,  2,,n  )
2
2
2
Chap 19-38
Centered Moving Averages
 Used when an even number of values is used in the moving
average
 Average periods of 2.5 or 3.5 don’t match the original
periods, so we average two consecutive moving averages to
get centered moving averages
Average
Period
4-Quarter
Moving
Average
Centered
Period
Centered
Moving
Average
2.5
28.75
3
29.88
3.5
31.00
4
32.00
4.5
33.00
5
34.00
5.5
6
36.25
6.5
35.00 etc…
37.50
7
38.13
7.5
38.75
8
39.00
8.5
39.25
9
40.13
9.5
41.00
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-39
Calculating the
Ratio-to-Moving Average
 Now estimate the seasonal impact
 Divide the actual sales value by the centered
moving average for that period
xt
100 *
xt
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-40
Calculating a Seasonal Index
Quarter
Sales
Centered
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
Ratio-toMoving
Average
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x3
25
100 *  (100)
 83.7
x3
29.88
Chap 19-41
Calculating Seasonal Indexes
(continued)
Fall
Fall
Fall
Quarter
Sales
Centered
Moving
Average
1
2
3
4
5
6
7
8
9
10
11
…
23
40
25
27
32
48
33
37
37
50
40
…
29.88
32.00
34.00
36.25
38.13
39.00
40.13
etc…
…
…
Ratio-toMoving
Average
83.7
84.4
94.1
132.4
86.5
94.9
92.2
etc…
…
…
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
1. Find the median
of all of the
same-season
values
2. Adjust so that
the average over
all seasons is
100
Chap 19-42
Interpreting Seasonal Indexes
 Suppose we get these
seasonal indexes:
Season
Seasonal
Index
 Interpretation:
Spring sales average 82.5% of the
annual average sales
Spring
0.825
Summer
1.310
Summer sales are 31.0% higher
than the annual average sales
Fall
0.920
etc…
Winter
0.945
 = 4.000 -- four seasons, so must sum to 4
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-43
Exponential Smoothing
 A weighted moving average
 Weights decline exponentially
 Most recent observation weighted most
 Used for smoothing and short term
forecasting (often one or two periods into
the future)
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Chap 19-44
Exponential Smoothing
(continued)
 The weight (smoothing coefficient) is 
 Subjectively chosen
 Range from 0 to 1
 Smaller  gives more smoothing, larger 
gives less smoothing
 The weight is:
 Close to 0 for smoothing out unwanted cyclical
and irregular components
 Close to 1 for forecasting
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-45
Exponential Smoothing Model
 Exponential smoothing model
xˆ 1  x1
xˆ t  α xˆ t 1  (1 α )x t (0  α  1; t  1,2,,n)
where:
xˆ t = exponentially smoothed value for period t
xˆ t -1 = exponentially smoothed value already
computed for period i - 1
xt = observed value in period t
 = weight (smoothing coefficient), 0 <  < 1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-46
Exponential Smoothing Example
 Suppose we use weight  = .2 xˆ t  0.2 xˆ t 1  (1 0.2)x t
Time
Period
(i)
1
2
3
4
5
6
7
8
9
10
etc.
Sales
(Yi)
23
40
25
27
32
48
33
37
37
50
etc.
Forecast
from prior
period (Ei-1)
Exponentially Smoothed
Value for this period (Ei)
-23
26.4
26.12
26.296
27.437
31.549
31.840
32.872
33.697
etc.
23
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
etc.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
xˆ 1 = x1
since no
prior
information
exists
Chap 19-47
Sales vs. Smoothed Sales
 Fluctuations
have been
smoothed
50
40
Sales
 NOTE: the
smoothed value in
this case is
generally a little low,
since the trend is
upward sloping and
the weighting factor
is only .2
60
30
20
10
0
1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
2
3
4
5
6
7
Time Period
Sales
8
9
10
Smoothed
Chap 19-48
Forecasting Time Period (t + 1)
 The smoothed value in the current period (t)
is used as the forecast value for next period
(t + 1)
 At time n, we obtain the forecasts of future
values, Xn+h of the series
xˆ nh  xˆ n (h  1,2,3 )
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Chap 19-49
Exponential Smoothing in Excel
 Use tools / data analysis /
exponential smoothing
 The “damping factor” is (1 - )
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-50
Forecasting with the Holt-Winters
Method: Nonseasonal Series


To perform the Holt-Winters method of forecasting:
Obtain estimates of level xˆ t and trend Tt as
xˆ 1  x 2
T2  x 2  x1
xˆ t  α(xˆ t 1  Tt 1 )  (1 α)x t (0  α  1; t  3,4,,n)
Tt  βTt 1  (1 β)(xˆ t  xˆ t 1 )


(0  β  1; t  3,4,,n)
Where  and  are smoothing constants whose
values are fixed between 0 and 1
Standing at time n , we obtain the forecasts of future
values, Xn+h of the series by
xˆ nh  xˆ n  hTn
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-51
Forecasting with the Holt-Winters
Method: Seasonal Series

Assume a seasonal time series of period s

The Holt-Winters method of forecasting uses
a set of recursive estimates from historical
series

These estimates utilize a level factor, , a
trend factor, , and a multiplicative seasonal
factor, 
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-52
Forecasting with the Holt-Winters
Method: Seasonal Series
(continued)

The recursive estimates are based on the following
equations
xt
Ft s
(0  α  1)
Tt  βTt 1  (1 β)(xˆ t  xˆ t 1 )
(0  β  1)
xˆ t  α(xˆ t 1  Tt 1 )  (1  α)
Ft  γFt s  (1  γ )
xt
xˆ t
(0  γ  1)
Where xˆ t is the smoothed level of the series, Tt is the smoothed trend
of the series, and Ft is the smoothed seasonal adjustment for the series
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-53
Forecasting with the Holt-Winters
Method: Seasonal Series
(continued)
 After the initial procedures generate the level,
trend, and seasonal factors from a historical
series we can use the results to forecast future
values h time periods ahead from the last
observation Xn in the historical series
 The forecast equation is
xˆ nh  (xˆ t  hTt )Ft hs
where the seasonal factor, Ft, is the one generated for
the most recent seasonal time period
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-54
Autoregressive Models
 Used for forecasting
 Takes advantage of autocorrelation
 1st order - correlation between consecutive values
 2nd order - correlation between values 2 periods
apart
 pth order autoregressive model:
x t  γ  φ1x t 1  φ2 x t 2    φp x t p  εt
Random
Error
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-55
Autoregressive Models
(continued)
 Let Xt (t = 1, 2, . . ., n) be a time series
 A model to represent that series is the autoregressive
model of order p:
x t  γ  φ1x t 1  φ2 x t 2    φp x t p  εt
 where
 , 1 2, . . .,p are fixed parameters
 t are random variables that have
 mean 0
 constant variance
 and are uncorrelated with one another
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-56
Autoregressive Models
(continued)
 The parameters of the autoregressive model are
estimated through a least squares algorithm, as the
values of , 1 2, . . .,p for which the sum of
squares
SS 
n
2
(x

γ

φ
x

φ
x



φ
x
)
 t
1 t 1
2 t 2
p t p
t p 1
is a minimum
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-57
Forecasting from Estimated
Autoregressive Models
 Consider time series observations x1, x2, . . . , xt
 Suppose that an autoregressive model of order p has been fitted to
these data:
x t  γˆ  φˆ1x t 1  φˆ2 x t 2    φˆp x t p  εt
 Standing at time n, we obtain forecasts of future values of the
series from
xˆ t h  γˆ  φˆ1xˆ t h1  φˆ2 xˆ t h2    φˆp xˆ t hp
(h  1,2,3, )
ˆ n j is the forecast of Xt+j standing at time n and
 Where for j > 0, x
for j  0 , x
ˆ n j is simply the observed value of Xt+j
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-58
Autoregressive Model:
Example
The Office Concept Corp. has acquired a number of office
units (in thousands of square feet) over the last eight years.
Develop the second order autoregressive model.
Year
1999
2000
2001
2002
2003
2004
2005
2006
Units
4
3
2
3
2
2
4
6
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-59
Autoregressive Model:
Example Solution
 Develop the 2nd order
table
 Use Excel to estimate a
regression model
Excel Output
Coefficients
I n te rc e p t
3.5
X V a ri a b l e 1
0.8125
X V a ri a b l e 2
-0 . 9 3 7 5
Year
99
00
01
02
03
04
05
06
xt
4
3
2
3
2
2
4
6
xt-1
-4
3
2
3
2
2
4
xt-2
--4
3
2
3
2
2
xˆ t  3.5  0.8125xt 1  0.9375xt 2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-60
Autoregressive Model
Example: Forecasting
Use the second-order equation to forecast
number of units for 2007:
xˆ t  3.5  0.8125x t 1  0.9375x t 2
xˆ 2007  3.5  0.8125(x2006 )  0.9375(x2005 )
 3.5  0.8125(6)  0.9375(4)
 4.625
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-61
Autoregressive Modeling Steps

Choose p

Form a series of “lagged predictor”
variables xt-1 , xt-2 , … ,xt-p

Run a regression model using all p
variables

Test model for significance

Use model for forecasting
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-62
Chapter Summary
 Discussed weighted and unweighted index numbers
 Used the runs test to test for randomness in time series
data
 Addressed components of the time-series model
 Addressed time series forecasting of seasonal data
using a seasonal index
 Performed smoothing of data series
 Moving averages
 Exponential smoothing
 Addressed autoregressive models for forecasting
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 19-63