Chapter 15: Time Series Forecasting and Index Numbers
Chapter 15
Time-Series Forecasting and Index Numbers
LEARNING OBJECTIVES
This chapter discusses the general use of forecasting in business, several tools that are
available for making business forecasts, and the nature of time series data, thereby
enabling you to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Gain a general understanding time series forecasting techniques.
Understand the four possible components of time-series data.
Understand stationary forecasting techniques.
Understand how to use regression models for trend analysis.
Understand how can we establish the validity of forecasts?
Understand the different forms of smoothing techniques?
Learn how to decompose time-series data into their various elements and how to
forecast by using decomposition techniques
Understand the nature of autocorrelation and how to test for it.
Understand autoregression
Understand index numbers
CHAPTER TEACHING STRATEGY
Time series analysis attempts to determine if there is something inherent in the
history of a variable that can be captured in a way that will help business analysts forecast
the future values for the variable.
The first section of the chapter contains a general discussion about the various
possible components of time-series data. It creates the setting against which the chapter
later proceeds into trend analysis and seasonal effects. In addition, two measurements of
forecasting error are presented so that students can measure the error of forecasts
produced by the various techniques and begin to compare the merits of each.
A full gamut of time series forecasting techniques has been presented beginning
with the most naïve models and progressing through averaging models and exponential
smoothing. An attempt is made in the section on exponential smoothing to show the
student, through algebra, why it is called by that name. Using the derived equations and
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
a few selected values for alpha, the student is shown how past values and forecasts are
smoothed in the prediction of future values. The more advanced smoothing techniques
are briefly introduced in later sections but are explained in much greater detail on
WileyPLUS.
Trend is solved for next using the time periods as the predictor variable. In this
chapter both linear and quadratic trends are explored and compared. There is a brief
introduction to Holt’s two-parameter exponential smoothing method that includes trend.
A more detailed explanation of Holt’s method is available on WileyPLUS. The trend
analysis section is placed earlier in the chapter than seasonal effects because finding
seasonal effects makes more sense when there are no trend effects in the data or the trend
effect has been removed.
Section 15.4 includes a rather classic presentation of time series decomposition
only it is done on a smaller set of data so as not to lose the reader. It was felt that there
may be a significant number of instructors who want to show how a time series of data
can be broken down into the components of trend, cycle, and seasonality. This text
assumes a multiplicative model rather than an additive model. The main example used
throughout this section is a database of 20 quarters of actual data on Household
Appliances. A graph of these data is presented both before and after deseasonalization so
that the student can visualize what happens when the seasonal effects are removed. First,
4-quarter centered moving averages are computed which dampen out the seasonal and
irregular effects leaving trend and cycle. By dividing the original data by these 4-quarter
centered moving averages (trendcycle), the researcher is left with seasonal effects and
irregular effects. By casting out the high and low values and averaging the seasonal
effects for each quarter, the irregular effects are removed.
In regression analysis involving data over time, autocorrelation can be a problem.
Because of this, section 15.5 contains a discussion on autocorrelation and autoregression.
The Durbin-Watson test is presented as a mechanism for testing for the presence of
autocorrelation. Several possible ways of overcoming the autocorrelation problem are
presented such as the addition of independent variables, transforming variables, and
autoregressive models.
The last section in this chapter is a classic presentation of Index Numbers. This
section is essentially a shortened version of an entire chapter on Index Numbers. It
includes most of the traditional topics of simple index numbers, unweighted aggregate
price index numbers, weighted price index numbers, Laspeyres price indexes, and
Paasche price indexes.
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
CHAPTER OUTLINE
15.1
Introduction to Forecasting
Time Series Components
The Measurement of Forecasting Error
Error
Mean Absolute Deviation (MAD)
Mean Square Error (MSE)
15.2
Smoothing Techniques
Naïve Forecasting Models
Averaging Models
Simple Averages
Moving Averages
Weighted Moving Averages
Exponential Smoothing
15.3
Trend Analysis
Linear Regression Trend Analysis
Regression Trend Analysis Using Quadratic Models
Holt’s Two-Parameter Exponential Smoothing Method
15.4
Seasonal Effects
Decomposition
Winters’ Three-Parameter Exponential Smoothing Method
15.5
Autocorrelation and Autoregression
Autocorrelation
Ways to Overcome the Autocorrelation Problem
Addition of Independent Variables
Transforming Variables
Autoregression
15.6
Index Numbers
Simple Index Numbers and Unweighted Aggregate Price Indexes
Unweighted Aggregate Price Indexes
Weighted Aggregate Price Index Numbers
Laspeyres Price Index
Paasche Price Index
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Chapter 15: Time Series Forecasting and Index Numbers
KEY TERMS
Autocorrelation
Autoregression
Averaging Models
Cycles
Cyclical Effects
Decomposition
Deseasonalized Data
Durbin-Watson Test
Error of an Individual Forecast
Exponential Smoothing
First-Difference Approach
Forecasting
Forecasting Error
Index Number
Irregular Fluctuations
Laspeyres Price Index
Mean Absolute Deviation (MAD)
Mean Squared Error (MSE)
Moving Average
Naïve Forecasting Methods
Paasche Price Index
Seasonal Effects
Serial Correlation
Simple Average
Simple Average Model
Simple Index Number
Smoothing Techniques
Stationary
Time-Series Data
Trend
Unweighted Aggregate Price
Index Number
Weighted Aggregate Price
Index Number
Weighted Moving Average
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Chapter 15: Time Series Forecasting and Index Numbers
SOLUTIONS TO PROBLEMS IN CHAPTER 15
15.1
Period
1
2
3
4
5
6
7
8
9
Total
MAD =
MSE =
15.2
e
2.30
1.60
-1.40
1.10
0.30
-0.90
-1.90
-2.10
0.70
-0.30
e
2.30
1.60
1.40
1.10
0.30
0.90
1.90
2.10
0.70
12.30
e
no. forecasts
e
Period Value F
1
202
2
191 202
3
173 192
4
169 181
5
171 174
6
175 172
7
182 174
8
196 179
9
204 189
10
219 198
11
227 211
Total
MAD =
MSE =
12.30
= 1.367
9

20.43
= 2.27
9
e
e2
-11
11
-19
19
-12
12
-3
3
3
3
8
8
17
17
15
15
21
21
16
16
35 125
121
361
144
9
9
64
289
225
441
256
1919
e
e
no. forecasts
e

2
no. forecasts

2
no. forecasts
e2
5.29
2.56
1.96
1.21
0.09
0.81
3.61
4.41
0.49
20.43

125.00
= 12.5
10
1,919
= 191.9
10
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Chapter 15: Time Series Forecasting and Index Numbers
15.3
Period Value F
1
2
3
4
5
6
19.4 16.6 2.8
23.6 19.1 4.5
24.0 22.0 2.0
26.8 24.8 2.0
29.2 25.9 3.3
35.5 28.6 6.9
Total
21.5
MAD =
Year
1
2
3
4
5
6
7
8
9
10
11

215
.
= 3.583
6

94.59
= 15.765
6
2
No.Forecasts
MAD =
MSE =
2.8
7.84
4.5 20.25
2.0
4.00
2.0
4.00
3.3 10.89
6.9 47.61
21.5 94.59
No.Forecasts
Acres
140,000
141,730
134,590
131,710
131,910
134,250
135,220
131,020
120,640
115,190
114,510
Total
e2
e
e
e
MSE =
15.4
e
Forecast
140,000
141,038
137,169
133,894
132,704
133,632
134,585
132,446
125,362
119,259
e
1730
-6448
-5459
-1984
1546
1588
-3565
-11806
-10172
-4749
-39,319
e
49,047
= 4,904.7
10

361,331,847
= 36,133,184.7
10
2
No.Forecasts
e2
2,992,900
41,576,704
29,800,681
3,936,256
2,390,116
2,521,744
12,709,225
139,381,636
103,469,584
22,553,001
361,331,847

No.Forecasts
e
e
1730
6448
5459
1984
1546
1588
3565
11806
10172
4749
49047
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Chapter 15: Time Series Forecasting and Index Numbers
15.5
a)
4-mo. mov. avg.
44.75
52.75
61.50
64.75
70.50
81.00
error
14.25
13.25
9.50
21.25
30.50
16.00
b)
4-mo. wt. mov. avg. error
53.25
5.75
56.375
9.625
62.875
8.125
67.25
18.75
76.375
24.625
89.125
7.875
c)
difference in errors
14.25 - 5.75 = 8.5
3.626
1.375
2.5
5.875
8.125
In each time period, the four-month moving average produces greater errors of
forecast than the four-month weighted moving average.
15.6
Period
1
2
3
4
5
6
7
8
Value
211
228
236
241
242
227
217
203
F( =.1)
Error
F( =.8)
Error
Difference
211
213
215
218
220
221
221
23
26
24
7
-4
-18
211
225
234
240
242
230
220
11
7
2
-15
-13
-17
12
19
22
22
9
-1
Using alpha of .1 produced forecasting errors that were larger than those using
alpha = .8 for the first three forecasts. For the next two forecasts (periods 6
and 7), the forecasts using alpha = .1 produced smaller errors. Each exponential
smoothing model produced nearly the same amount of error in forecasting the
value for period 8. There is no strong argument in favour of either model.
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
15.7
Period
1
2
3
4
5
6
7
8
9
Value
9.4
8.2
7.9
9.0
9.8
11.0
10.3
9.5
9.1
 =.3
Error
 =.7
Error 3-mo.avg. Error
9.4
9.0
8.7
8.8
9.1
9.7
9.9
9.8
-1.2
-1.1
0.3
1.0
1.9
0.6
-0.4
-0.7
9.4
8.6
8.1
8.7
9.5
10.6
10.4
9.8
-1.2
-0.7
0.9
1.1
1.5
-0.3
-0.9
-0.7
8.5
8.4
8.9
9.9
10.4
10.3
0.5
1.4
1.1
0.4
-0.9
-1.2
15.8
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
Actual
52
52
51.2
51.5
51.6
51.9
52.9
54.7
56.3
58.1
60.4
60.4
59.9
Forecast
Wtd.
Forecast
51.66
51.64
51.82
52.52
53.48
54.78
56.48
57.98
59.02
51.57
51.71
52.21
53.34
54.71
56.29
58.21
59.37
59.73
Forecast error
Wtd.
Forecast
errors
-0.24
-1.26
-2.88
-3.78
-4.62
-5.62
-3.92
-1.92
-0.33
-1.19
-2.49
-2.96
-3.39
-4.11
-2.19
-0.53
Note that the weighted rather than the unweighted forecasts in this example are
closer to actual after-tax income (except for 1996).
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
15.9
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
No.Issues
332
694
518
222
209
172
366
512
667
571
575
865
609
F(=.2)
332.0
404.4
427.1
386.1
350.7
315.0
325.2
362.6
423.5
453.0
477.4
554.9
e
F(=.9)
e
362.0
113.6
205.1
177.1
178.7
51.0
186.8
304.4
147.5
122.0
387.6
54.1
332.0
657.8
532.0
253.0
213.4
176.1
347.0
495.5
649.9
578.9
575.4
836.0
362.0
139.8
310.0
44.0
41.4
189.9
165.0
171.5
78.9
3.9
289.6
227.0
 e = 2289.9
For  = .2, MAD =
2289.9
= 190.8
12
For  = .9, MAD =
2023.0
= 168.6
12
 e =2023.0
 = .9 produces a smaller mean average error.
15.10 Simple Regression Trend Model:
ŷ = 37,969.6 + 9899.0 Period
F = 1603 (p = .000), R2 = .988, adjusted R2 = .988,
se = 6,861, t = 40.04 (p = .000)
Quadratic Regression Trend Model:
ŷ = 35,767.3 + 10,473.5 Period - 26.1 Period2
F = 772.68 (p = .000), R2 = .988, adjusted R2 = .987
se = 6,988, tperiod = 9.91 (p = .000), tperiodsq = -0.56 (p = .583)
The simple linear regression trend model is superior; the period2 variable is not a
significant addition to the model.
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
15.11 Simple regression model:
R2 = 0.97
se = 2.43
Consumer Price Index = – 4409.2 + 2.2623 Year
F = 668.56
Quadratic Model:
Consumer Price Index = – 162020 + 160.43Year – 0.03968 Year2
R2 = 0.98
se = 2.06
F = 467
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
The graph indicates a quadratic fit rather than a linear fit. The quadratic model
produced an R2 = 0.98 compared to R2 = 0.97 for linear trend indicating a better
fit for the quadratic model. In addition, the standard error of the estimate drops
from 2.43 to 2.06 with the quadratic model. The t values for the quadratic model
are significant.
15.12 Simple Regression Model:
Part-Time Employment = 98.60 – 0.04 Year
R2 = 0.092
t = – 0.84(p = .43)
F = 0.71 (p = .43)
Quadratic Model:
Part-Time Employment = 89365.1 – 89.26208 Year + 0.0222944 Year2
R2 = 0. 24
tyear = – 1.08 (p = .32)
tyearsq = 1.08 (p = .32)
F = .94 (p = .4407)
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
Both regression models show quite poor predictability.
15.13
12-Month
Orange
Juice
Price ($)
Year
1
January
February
March
April
May
June
12-Month
Moving
Total
2-Year
Moving
Total
T*C
S*I
44.537
1.856
98.94
44.644
1.860
101.44
44.658
1.861
102.06
44.756
1.865
105.75
1.847
1.881
1.808
1.785
1.731
1.825
22.213
July
1.836
22.324
August
1.887
September
1.899
22.32
22.338
October
1.972
22.418
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Chapter 15: Time Series Forecasting and Index Numbers
November
1.906
44.921
1.872
101.83
45.098
1.879
97.71
45.233
1.885
103.89
45.263
1.886
99.52
45.169
1.882
97.02
45.055
1.877
99.35
45.044
1.877
96.76
45.199
1.883
101.79
22.503
December
1.836
22.595
Year
2
January
1.958
22.638
February
1.877
22.625
March
1.826
22.544
April
1.865
22.511
May
1.816
22.533
June
1.917
22.666
July
August
September
October
November
December
1.879
1.874
1.818
1.939
1.928
1.969
15.14
Month
Ship
12m tot
2yr tot
TC
SI
TCI
T
Jan(Yr1) 1891
1952.50
2042.72
Feb
1986
1975.73
2049.87
Mar
1987
1973.78
2057.02
Apr
1987
1972.40
2064.17
May
2000
1976.87
2071.32
June
2082
1982.67
2078.46
C
23822
July
1878
Aug
2074
47689
1987.04
94.51 1970.62
2085.61
94.49
47852
1993.83 104.02 2011.83
2092.76
96.13
23867
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Chapter 15: Time Series Forecasting and Index Numbers
23985
Sept
2086
48109
2004.54 104.06 2008.47
2099.91
95.65
48392
2016.33 101.42 1969.76
2107.06
93.48
48699
2029.13
95.85 2024.57
2114.20
95.76
49126
2046.92
90.92 2002.80
2121.35
94.41
49621
2067.54
93.64 1998.97
2128.50
93.91
49989
2082.88 101.01 2093.12
2135.65
98.01
50308
2096.17 101.42 2111.85
2142.80
98.56
50730
2113.75 100.82 2115.35
2149.94
98.39
51132
2130.50 101.53 2137.99
2157.09
99.11
51510
2146.25 109.31 2234.07
2164.24 103.23
51973
2165.54
2171.39 101.92
52346
2181.08 101.37 2144.73
2178.54
98.45
52568
2190.33 103.55 2183.71
2185.68
99.91
52852
2202.17 103.76 2200.93
2192.83
100.37
53246
2218.58
94.97 2193.19
2199.98
99.69
53635
2234.79
92.94 2235.26
2207.13
101.27
53976
2249.00
97.07 2254.00
2214.28
101.79
54380
2265.83
98.42 2218.46
2221.42
99.87
54882
2286.75
97.17 2207.21
2228.57
99.04
55355
2306.46 100.54 2301.97
2235.72
102.96
55779
2324.13 101.93 2341.60
2242.87
104.40
56186
2341.08 108.03 2408.34
2250.02
107.04
56539
2355.79
2257.17
105.39
24124
Oct
2045
24268
Nov
1945
Dec
1861
24431
24695
Jan(Yr2) 1936
24926
Feb
2104
Mar
2126
Apr
2131
25063
25245
25485
May
2163
25647
June
2346
25863
July
2109
97.39 2213.01
26110
Aug
2211
26236
Sept
2268
Oct
2285
Nov
2107
Dec
2077
26332
26520
26726
26909
Jan(Yr3) 2183
27067
Feb
2230
27313
Mar
2222
27569
Apr
2319
27786
May
2369
27993
June
2529
28193
July
2267
96.23 2378.80
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Chapter 15: Time Series Forecasting and Index Numbers
28346
Aug
2457
56936
2372.33 103.57 2383.35
2264.31
105.26
57504
2396.00 105.34 2430.19
2271.46
106.99
58075
2419.79 103.40 2409.94
2278.61
105.76
58426
2434.42
95.05 2408.66
2285.76
105.38
58573
2440.54
93.30 2450.50
2292.91
106.87
58685
2445.21
95.53 2411.98
2300.05
104.87
58815
2450.63 100.95 2461.20
2307.20
106.67
58806
2450.25 103.91 2529.06
2314.35
109.28
58793
2449.71 104.75 2547.15
2321.50
109.72
58920
2455.00 100.73 2444.40
2328.65
104.97
59018
2459.08 104.59 2449.29
2335.79
104.86
59099
2462.46
94.86 2451.21
2342.94
104.62
59141
2464.21 102.18 2442.53
2350.09
103.93
59106
2462.75
99.64 2362.80
2357.24
100.24
58933
2455.54 104.21 2464.84
2364.39
104.25
58779
2449.13
97.34 2481.52
2371.53
104.64
58694
2445.58
94.25 2480.63
2378.68
104.29
58582
2440.92
97.87 2466.70
2385.83
103.39
58543
2439.29 100.97 2450.26
2392.98
102.39
58576
2440.67 103.33 2505.22
2400.13
104.38
58587
2441.13
99.01 2399.25
2407.27
99.67
58555
2439.79 101.16 2439.46
2414.42
101.04
58458
2435.75 102.31 2373.11
2421.57
98.00
28590
Sept
2524
28914
Oct
2502
Nov
2314
Dec
2277
29161
29265
29308
Jan(Yr4) 2336
29377
Feb
2474
Mar
2546
29438
29368
Apr
2566
29425
May
2473
29495
June
2572
29523
July
2336
29576
Aug
2518
Sept
2454
Oct
2559
Nov
2384
Dec
2305
29565
29541
29392
29387
29307
Jan(Yr5) 2389
29275
Feb
2463
29268
Mar
2522
29308
Apr
2417
29279
May
2468
29276
June
2492
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Chapter 15: Time Series Forecasting and Index Numbers
29182
July
2304
58352
2431.33
94.76 2417.63
2428.72
99.54
58258
2427.42 103.44 2435.74
2435.87
99.99
57922
2413.42 103.34 2401.31
2443.01
98.29
57658
2402.42 105.31 2436.91
2450.16
99.46
57547
2397.79
99.30 2478.40
2457.31
100.86
57400
2391.67
92.45 2379.47
2464.46
96.55
57391
2391.29
99.40 2454.31
2471.61
99.30
57408
2392.00
99.54 2368.68
2478.76
95.56
57346
2389.42
94.92 2252.91
2485.90
90.63
57335
2388.96 100.76 2389.32
2493.05
95.84
57362
2390.08
99.03 2339.63
2500.20
93.58
57424
2392.67 102.23 2329.30
2507.35
92.90
29170
Aug
2511
29088
Sept
2494
Oct
2530
Nov
2381
28834
28824
28723
Dec
2211
28677
Jan(Yr6) 2377
28714
Feb
2381
28694
Mar
2268
28652
Apr
2407
28683
May
2367
28679
June
2446
28745
July
Aug
Sept
Oct
Nov
Dec
2341
2491
2452
2561
2377
2277
Seasonal Indexing:
Month Year1 Year2
Jan
93.64
Feb
101.01
Mar
101.42
Apr
100.82
May
101.53
June
109.31
July
94.51
97.39
Aug 104.02
101.37
Sept 104.60
103.55
Oct
101.42
103.76
Nov
95.85
94.97
Dec
90.92
92.94
Year3
97.07
98.42
97.17
100.54
101.93
108.03
96.23
103.57
105.34
103.40
95.05
93.30
Year4
95.53
100.95
103.91
104.75
100.73
104.59
94.86
102.18
99.64
104.21
97.24
94.25
Year5
97.87
100.97
103.33
99.01
101.16
102.31
94.76
103.44
103.34
105.31
99.30
92.45
Year6
99.40
99.54
94.92
100.76
99.03
102.23
© 2010 John Wiley & Sons Canada, Ltd.
Index
96.82
100.49
100.64
100.71
101.14
104.98
95.28
103.06
103.83
103.79
96.05
92.90
477
Chapter 15: Time Series Forecasting and Index Numbers
Total
1199.69
Adjust each seasonal index by 1.0002584
Final Seasonal Indexes:
Month Index
Jan
96.85
Feb
100.52
Mar
100.67
Apr
100.74
May
101.17
June
105.01
July
95.30
Aug
103.09
Sept
103.86
Oct
103.82
Nov
96.07
Dec
92.92
Yˆ = 2035.58 + 7.1481 X
R2 = .682, se = 102.9
Note: Trend Line was determined after seasonal effects were removed (based on TCI
column).
Regression Output for Trend Line:
15.15 Regression Analysis
The regression equation is:
Predictor
Coef
Constant
0.427
U.S. Rate
1.638
se = 1.261
Year
1980
1981
1982
1983
1984
1985
U.S.
10.3
11.2
11.5
9.2
11.2
9.3
Canada
16
17.9
20.7
17.2
17
16
Canadian Rate = 0.427 + 1.638 U.S. Rate
t-ratio
p
0.82
0.418
21.73
0.0000
R-sq = 0.952
Ŷ
17.298
18.773
19.264
15.497
18.773
15.660
et
-1.298
-0.873
1.436
1.703
-1.773
0.340
R-sq(adj) = 0.950
et2
1.686
0.761
2.062
2.902
3.142
0.115
© 2010 John Wiley & Sons Canada, Ltd.
et - et-1
(et - et-1)2
0.426
2.309
0.267
-3.476
2.112
0.181
5.330
0.072
12.083
4.461
478
Chapter 15: Time Series Forecasting and Index Numbers
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
8.5
7.2
7.5
7.4
7.2
7.5
7.6
6.1
5.3
5.1
4.2
3.7
4.5
2.5
2.4
1.8
2.5
2.4
1.9
-0.4
D =
13.7
12.1
12.6
13.3
13.3
13.6
13.3
12.1
9.6
9.4
7.2
5
4.9
4.1
4.8
5.3
3.5
2.8
2.7
1.2
14.350
12.221
12.712
12.548
12.221
12.712
12.876
10.419
9.108
8.781
7.307
6.488
7.798
4.522
4.358
3.375
4.522
4.358
3.539
-0.228
 (e  e
e
t
t 1
2
t
)2

-0.650
-0.121
-0.112
0.752
1.079
0.888
0.424
1.681
0.492
0.619
-0.107
-1.488
-2.898
-0.422
0.442
1.925
-1.022
-1.558
-0.839
1.428
Total
0.422
0.015
0.013
0.565
1.165
0.789
0.180
2.826
0.242
0.383
0.011
2.213
8.398
0.178
0.195
3.704
1.044
2.428
0.704
2.040
38.185
-0.990
0.529
0.009
0.864
0.328
-0.191
-0.464
1.257
-1.190
0.128
-0.726
-1.381
-1.410
2.476
0.864
1.483
-2.947
-0.536
0.719
2.267
0.979
0.280
0.000
0.746
0.107
0.037
0.215
1.580
1.415
0.016
0.527
1.907
1.989
6.131
0.746
2.199
8.682
0.288
0.517
5.141
55.629
55.629
= 1.457
38185
.
Critical values of D: Using 1 independent variable, n = 26, and  = .05,
dL = 1.30 and dU = 1.46
Since D = 1.457 falls between dL = 1.30 and dU = 1.46, the Durbin-Watson test is
inconclusive.
15.16 Regression Analysis
The regression equation is:
First Diff. in Canadian Rate = – 0.270 + 0.753 First Diff. in U.S. Rate
Predictor
Coef
t-ratio
p
Constant
– 0.270
–1.020
0.318
First Diff
0.753
3.193
0.004
se = 1.222
R-sq = 0.307
R-sq(adj) = 0.277
© 2010 John Wiley & Sons Canada, Ltd.
479
Chapter 15: Time Series Forecasting and Index Numbers
1-Diff. in
1-Diff. in
U.S.
Rate
Canadian Rate
Ŷ
et
et2
e t - et-1
(e t - et-1)2
0.9
0.3
-2.3
2
-1.9
-0.8
-1.3
0.3
-0.1
-0.2
0.3
0.1
-1.5
-0.8
-0.2
-0.9
-0.5
0.8
-2
-0.1
-0.6
0.7
-0.1
-0.5
-2.3
1.9
2.8
-3.5
-0.2
-1
-2.3
-1.6
0.5
0.7
0
0.3
-0.3
-1.2
-2.5
-0.2
-2.2
-2.2
-0.1
-0.8
0.7
0.5
-1.8
-0.7
-0.1
-1.5
0.408
-0.044
-2.002
1.236
-1.701
-0.872
-1.249
-0.044
-0.345
-0.421
-0.044
-0.195
-1.400
-0.872
-0.421
-0.948
-0.647
0.332
-1.776
-0.345
-0.722
0.257
-0.345
-0.647
-2.002
1.492
2.844
-1.498
-1.436
0.701
-1.428
-0.351
0.544
1.045
0.421
0.344
-0.105
0.199
-1.628
0.221
-1.252
-1.554
-0.432
0.976
1.045
1.222
-2.057
-0.355
0.547
0.502
Total
2.227
8.089
2.244
2.062
0.491
2.038
0.123
0.296
1.093
0.177
0.118
0.011
0.040
2.649
0.049
1.568
2.413
0.187
0.953
1.093
1.493
4.232
0.126
0.299
0.252
34.322
1.352
-4.342
0.062
2.137
-2.128
1.077
0.895
0.501
-0.625
-0.077
-0.449
0.305
-1.827
1.848
-1.473
-0.301
1.121
1.408
0.069
0.177
-3.279
1.702
0.901
-0.045
1.827
18.855
0.004
4.565
4.530
1.159
0.801
0.251
0.390
0.006
0.202
0.093
3.338
3.416
2.169
0.091
1.257
1.984
0.005
0.031
10.751
2.898
0.812
0.002
59.438
D =
 (e  e
e
t
t 1
2
t
)2

59.438
34.322
= 1.732
Critical values of D: Using 1 independent variable, n = 25, and  = .05,
dL = 1.29 and dU = 1.45
Since D = 1.732 is above dU, we fail to reject the null hypothesis.
There is no significant autocorrelation.
© 2010 John Wiley & Sons Canada, Ltd.
480
Chapter 15: Time Series Forecasting and Index Numbers
15.17 The regression equation is:
CPI = 3.277 + 1.001 PPI
R2 = 0.867
adjusted R2 = 0.859
Ŷ
80.654
82.156
82.456
81.555
81.956
84.859
89.763
96.170
96.570
97.271
97.571
99.273
103.377
104.378
104.378
103.077
106.180
107.781
PPI
77.3
78.8
79.1
78.2
78.6
81.5
86.4
92.8
93.2
93.9
94.2
95.9
100
101
101
99.7
102.8
104.4
CPI
74.7
78.4
82.1
86.8
88.1
89.7
89.8
91.8
93.2
94.7
95.7
97.3
100
102.5
104.8
107.7
109.7
112.2
D =
 (e  e
e
t 1
t
2
t
)2

et
-5.954
-3.756
-0.356
5.245
6.144
4.842
0.037
-4.370
-3.370
-2.571
-1.871
-1.973
-3.377
-1.878
0.422
4.623
3.520
4.419
Total
se = 3.963
et2
35.454
14.106
0.127
27.508
37.754
23.440
0.001
19.095
11.358
6.610
3.501
3.892
11.404
3.527
0.178
21.375
12.392
19.524
251.246
F = 104.20, p = .000
e t - et-1
(e t - et-1)2
2.199
3.400
5.601
0.900
-1.303
-4.805
-4.406
1.000
0.799
0.700
-0.102
-1.404
1.499
2.300
4.201
-1.103
0.898
4.833
11.558
31.370
0.809
1.698
23.087
19.416
0.999
0.639
0.490
0.010
1.971
2.247
5.290
17.651
1.217
0.807
124.093
124.093
= 0.494
251246
.
The critical table values for k = 1 and n = 18 are dL = 1.16 and dU = 1.39. Since
the observed value of D = 0.494 is below dL, the we reject the null
hypothesis. There is a significant autocorrelation.
© 2010 John Wiley & Sons Canada, Ltd.
481
Chapter 15: Time Series Forecasting and Index Numbers
15.18
The regression equation is:
First Diff. in CPI = 2.537 – 0.207 First Diff. in PPI
R2 = 0.146
1-Diff. in
PPI
1.5
0.3
-0.9
0.4
2.9
4.9
6.4
0.4
0.7
0.3
1.7
4.1
1
0
-1.3
3.1
1.6
adjusted R2 = 0.089
1-Diff. in
CPI
3.7
3.7
4.7
1.3
1.6
0.1
2
1.4
1.5
1
1.6
2.7
2.5
2.3
2.9
2
2.5
Ŷ
2.227
2.475
2.723
2.454
1.937
1.523
1.212
2.454
2.392
2.475
2.185
1.688
2.330
2.537
2.806
1.895
2.206
se = 1.074
et
et2
1.474
2.171
1.225
1.501
1.977
3.907
-1.154 1.332
-0.337 0.113
-1.423 2.024
0.788
0.621
-1.054 1.111
-0.892 0.796
-1.475 2.175
-0.585 0.342
1.012
1.024
0.170
0.029
-0.237 0.056
0.094
0.009
0.105
0.011
0.294
0.087
Total 17.309
F = 2.565, p = .130
e t - et-1
(e t - et-1)2
-0.248
0.752
-3.131
0.818
-1.086
2.211
-1.842
0.162
-0.583
0.890
1.597
-0.842
-0.407
0.331
0.011
0.190
0.062
0.565
9.803
0.668
1.179
4.886
3.393
0.026
0.340
0.792
2.550
0.708
0.166
0.109
0.000
0.036
25.283
The Durbin Watson statistic for this model is:
D =
 (e  e
e
t
t 1
2
t
)2

25.283
= 1.461
17.309
The critical table values for k = 1 and n = 17 are dL = 1.13 and dU = 1.38. Since
the observed value of D = 1.461 is above dU, the we fail to reject the null
hypothesis. There is no significant autocorrelation.
© 2010 John Wiley & Sons Canada, Ltd.
482
Chapter 15: Time Series Forecasting and Index Numbers
15.19
Crude Oil
Production Yt
84.4
81.4
73.7
73.5
77.3
83.3
84.1
85.5
90.3
93.9
91.8
91.6
92
96.4
101.3
105.3
110.3
113.5
119
124.7
119.9
124.8
126.6
132.9
140.4
145.8
143.4
One Period
Lagged Yt-1 (X1)
Two Periods
Lagged Yt-2 (X2)
84.4
81.4
73.7
73.5
77.3
83.3
84.1
85.5
90.3
93.9
91.8
91.6
92
96.4
101.3
105.3
110.3
113.5
119
124.7
119.9
124.8
126.6
132.9
140.4
145.8
84.4
81.4
73.7
73.5
77.3
83.3
84.1
85.5
90.3
93.9
91.8
91.6
92
96.4
101.3
105.3
110.3
113.5
119
124.7
119.9
124.8
126.6
132.9
140.4
The model with 1 lagged variable:
Crude Oil Production = – 1.604 + 1.037 Lag 1
F = 829.67
p = .000 R2 = 97.2% adjusted R2 = 97.1% se = 3.82
The model with 2 lagged variables:
Housing Starts = 0.983 + 1.242 Lag1 – 0.233 Lag2
© 2010 John Wiley & Sons Canada, Ltd.
483
Chapter 15: Time Series Forecasting and Index Numbers
F = 405.94
p = .000 R2 = 97.4% adjusted R2 = 97.1% se = 3.78
Both models are very strong.
15.20 The autoregression model is:
Energy Supply = 3.837 + 0.848 Lag1 – 0.083 Lag2
Energy Supply
15.2
15.2
15.4
15.9
14.2
14.3
14.3
14.9
14.7
15.2
16
16.2
17.1
17.2
17.9
18.1
17.8
16.7
15.4
16.1
16.6
16.6
16.3
16.6
16.7
17.1
16.7
16.3
16.7
16.9
One Period
Lagged Yt-1
(X1)
Two Periods
Lagged Yt-2 (X2)
15.2
15.2
15.4
15.9
14.2
14.3
14.3
14.9
14.7
15.2
16
16.2
17.1
17.2
17.9
18.1
17.8
16.7
15.4
16.1
16.6
16.6
16.3
16.6
16.7
17.1
16.7
16.3
16.7
15.2
15.2
15.4
15.9
14.2
14.3
14.3
14.9
14.7
15.2
16
16.2
17.1
17.2
17.9
18.1
17.8
16.7
15.4
16.1
16.6
16.6
16.3
16.6
16.7
17.1
16.7
16.3
© 2010 John Wiley & Sons Canada, Ltd.
484
Chapter 15: Time Series Forecasting and Index Numbers
16.1
16.7
15.5
15.4
16.4
16.9
16.1
16.7
15.5
15.4
16.7
16.9
16.1
16.7
15.5
The F value for this model is 25.25 with p = .000.
The value of R2 is 62.7% which denotes relatively strong predictability. The
adjusted R2 is 60.3%. The standard error of the estimate is 0.637.
15.21 Year
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Price
22.45
31.40
32.33
36.50
44.90
61.24
69.75
73.44
80.05
84.61
87.28
89.56
a.) Index1950
100.0
139.9
144.0
162.6
200.0
272.8
310.7
327.1
356.6
376.9
388.8
398.9
b.) Index1980
32.2
45.0
46.4
52.3
64.4
87.8
100.0
105.3
114.8
121.3
125.1
128.4
15.22
Year
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Triadic Patent
Families for Canadian R&D
290
275
270
285
353
380
436
543
532
595
643
661
685
710
© 2010 John Wiley & Sons Canada, Ltd.
Simple
Index
100.0
94.8
93.1
98.3
121.7
131.0
150.3
187.2
183.4
205.2
221.7
227.9
236.2
244.8
485
Chapter 15: Time Series Forecasting and Index Numbers
15.23
1995
3.37
4.86
4.22
7.44
19.89
Total
Year
2002
3.08
4.73
5.9
6.82
20.53
2009
4.77
5.52
5.72
8.80
24.81
Index2002 =
20.53
(100) = 103.2
19.89
Index2009 =
24.81
(100) = 124.7
19.89
15.24
Total
2001
1.10
1.58
1.80
7.95
12.43
2002
1.16
1.61
1.82
7.96
12.55
2003
1.23
1.78
1.98
8.24
13.23
2004
1.23
1.77
1.96
8.21
13.17
Index2001 =
12.43
(100) = 94.0
13.23
Index2002 =
12.55
(100) = 94.9
13.23
Index2004 =
1317
.
(100) = 99.5
13.23
Index2005 =
12.82
(100) = 96.9
13.23
Index2006 =
13.22
(100) = 99.9
13.23
Year
2005
1.08
1.61
1.94
8.19
12.82
2006
1.56
1.71
1.90
8.05
13.22
© 2010 John Wiley & Sons Canada, Ltd.
2007
1.85
1.90
1.92
8.12
13.79
2008
2.59
2.05
1.94
8.10
14.68
2009
2.89
2.08
1.96
8.24
15.17
486
Chapter 15: Time Series Forecasting and Index Numbers
Index2007 =
13.79
(100) = 104.2
13.23
Index2008 =
14.68
(100) = 111.0
13.23
Index2009 =
1517
.
(100) = 114.7
13.23
15.25
Item
Quantity
2000
1
2
3
4
21
6
17
43
2000
Price
2007 2008
2009
0.50
1.23
0.84
0.15
0.67
1.85
0.75
0.21
0.71
1.91
0.80
0.25
0.68
1.90
0.75
0.25
P2000Q2000 P2007Q2000 P2008Q2000
Totals
P2009Q2000
10.50
7.38
14.28
6.45
14.07
11.10
12.75
9.03
14.28
11.40
12.75
10.75
14.91
11.46
13.60
10.75
38.61
46.95
49.18
50.72
Index2007 =
Index2008 =
Index2009 =
P
P
2007
Q2000
2000
Q2000
P
P
2008
Q2000
2000
Q2000
P
P
2009
Q2000
2000
Q2000
=
46.95
(100) = 121.6
38.61
=
49.18
(100) = 127.4
38.61
=
50.72
(100) = 131.4
38.61
© 2010 John Wiley & Sons Canada, Ltd.
487
Chapter 15: Time Series Forecasting and Index Numbers
15.26
Item
Price
2000
Price Quantity Price Quantity
2008
2008
2009
2009
1
2
3
22.50
10.90
1.85
27.80
13.10
2.25
P2000Q2008 P2000Q2009
Totals
28.11
13.25
2.35
270.00
87.20
81.40
361.40
65.50
92.25
337.32
106.00
103.40
422.85
438.60
519.15
546.72
Index2008 =
P
P
2008
Q2008
2000
Q2008
P
P
12
8
44
P2008Q2008 P2009Q2009
292.50
54.50
75.85
Index2006 =
15.27 a)
13
5
41
2009
Q2009
2000
Q2009
(100) =
519.15
(100) = 122.8
422.85
(100) =
546.72
(100) = 124.7
438.60
The linear model:
Yield = 9.96 - 0.14 Month
F = 219.24 p = .000
R2 = 90.9
se = .3212
The quadratic model:
Yield = 10.4 - 0.252 Month + .00445 Month2
F = 176.21 p = .000
R2 = 94.4%
se = .2582
In the quadratic model, both t ratios are significant,
for Month: t = - 7.93, p = .000 and for Month2: t = 3.61, p = .002
The linear model is a strong model. The quadratic term adds some
predictability but has a smaller t ratio than does the linear term.
© 2010 John Wiley & Sons Canada, Ltd.
488
Chapter 15: Time Series Forecasting and Index Numbers
b)
x
10.08
10.05
9.24
9.23
9.69
9.55
9.37
8.55
8.36
8.59
7.99
8.12
7.91
7.73
7.39
7.48
7.52
7.48
7.35
7.04
6.88
6.88
7.17
7.22
F
9.65
9.55
9.43
9.46
9.29
8.96
8.72
8.37
8.27
8.15
7.94
7.79
7.63
7.53
7.47
7.46
7.35
7.19
7.04
6.99
MAD =
c)
│e │
.04
.00
.06
.91
.93
.37
.73
.25
.36
.42
.55
.31
.11
.05
.12
.42
.47
.31
.13
.23
 e = 6.77
6.77
= .3385
20
 = .3
 = .7
x
F
e
10.08
10.05 10.08 .03
9.24 10.07 .83
9.23 9.82 .59
9.69 9.64 .05
9.55 9.66 .11
9.37 9.63 .26
8.55 9.55 1.00
8.36 9.25 .89
8.59 8.98 .39
F
10.08
10.06
9.49
9.31
9.58
9.56
9.43
8.81
8.50
e
.03
.82
.26
.38
.03
.19
.88
.45
.09
© 2010 John Wiley & Sons Canada, Ltd.
489
Chapter 15: Time Series Forecasting and Index Numbers
7.99
8.12
7.91
7.73
7.39
7.48
7.52
7.48
7.35
7.04
6.88
6.88
7.17
7.22
8.86 .87
8.60 .48
8.46 .55
8.30 .57
8.13 .74
7.91 .43
7.78 .26
7.70 .22
7.63 .28
7.55 .51
7.40 .52
7.24 .36
7.13 .04
7.14 .08
 e = 10.06
MAD=.3 =
8.56
8.16
8.13
7.98
7.81
7.52
7.49
7.51
7.49
7.39
7.15
6.96
6.90
7.09
e =
10.06
= .4374
23
.57
.04
.22
.25
.42
.04
.03
.03
.14
.35
.27
.08
.27
.13
5.97
MAD=.7 =
5.97
= .2596
23
 = .7 produces better forecasts based on MAD.
d)
MAD for b) .3385, c) .4374 and .2596. Exponential smoothing with  = .7
produces the lowest error (.2596 from part c).
e)
TCSI
10.08
10.05
4 period
moving tots
8 period
moving tots
TC
SI
76.81
9.60
96.25
75.92
9.49
97.26
75.55
9.44
102.65
75.00
9.38
101.81
72.99
9.12
102.74
70.70
8.84
96.72
68.36
8.55
97.78
66.55
8.32
103.25
65.67
8.21
97.32
38.60
9.24
38.21
9.23
37.71
9.69
37.84
9.55
37.16
9.37
35.83
8.55
34.87
8.36
33.49
8.59
33.06
7.99
© 2010 John Wiley & Sons Canada, Ltd.
490
Chapter 15: Time Series Forecasting and Index Numbers
32.61
8.12
64.36
8.05
100.87
62.90
7.86
100.64
61.66
7.71
100.26
60.63
7.58
97.49
59.99
7.50
99.73
59.70
7.46
100.80
59.22
7.40
101.08
58.14
7.27
101.10
56.90
7.11
99.02
56.12
7.02
98.01
56.12
7.02
98.01
31.75
7.91
31.15
7.73
30.51
7.39
30.12
7.48
29.87
7.52
29.83
7.48
29.39
7.35
28.75
7.04
28.15
6.88
27.97
6.88
28.15
7.17
7.22
1st Period
2nd Period
3rd Period
4th Period
102.65 97.78 100.64
101.81 103.25 100.26
96.25 102.74 97.32
97.26 96.72 100.87
100.80 98.01
101.08 98.01
97.49 101.10
99.73 99.02
The highs and lows of each period (underlined) are eliminated and the others are
averaged resulting in:
Seasonal Indexes:
1st 99.82
2nd 101.05
3rd 98.64
4th 98.67
total 398.18
400
Since the total is not 400, adjust each seasonal index by multiplying by
=
398.18
1.004571 resulting in the final seasonal indexes of:
1st 100.28
2nd 101.51
3rd 99.09
4th 99.12
© 2010 John Wiley & Sons Canada, Ltd.
491
Chapter 15: Time Series Forecasting and Index Numbers
15.28
Year
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
15.29
Item
1
2
3
4
5
6
Totals
Index2005 =
Quantity
2073
2290
2349
2313
2456
2508
2463
2499
2520
2529
2483
2467
2397
2351
2308
2005
3.21
0.51
0.83
1.30
1.67
0.62
8.14
Index Number
100.0
110.5
113.3
111.6
118.5
121.0
118.8
120.5
121.6
122.0
119.8
119.0
115.6
113.4
111.3
2006
3.37
0.55
0.90
1.32
1.72
0.67
8.53
(100) 
814
.
(100) = 100.0
814
.
P
P
(100) 
8.53
(100) = 104.8
814
.
P
P
(100) 
9.32
(100) = 114.5
814
.
P
P
(100) 
9.40
(100) = 115.5
814
.
2005
2006
2005
Index2007 =
2007
2005
Index2008 =
2008
3.73
0.62
1.02
1.32
1.99
0.72
9.40
P
P
2005
Index2006 =
2007
3.80
0.68
0.91
1.33
1.90
0.70
9.32
2008
2005
© 2010 John Wiley & Sons Canada, Ltd.
2009
3.65
0.59
1.06
1.30
1.98
0.71
9.29
492
Chapter 15: Time Series Forecasting and Index Numbers
Index2009 =
P
P
2009
9.29
(100) = 114.1
814
.
(100) 
2005
15.30
Item
1
2
3
2006
P
Q
2.75 12
0.85 47
1.33 20
Laspeyres:
2007
P Q
2.98 9
0.89 52
1.32 28
2008
P
Q
3.10 9
0.95 61
1.36 25
P2006Q2006
P2009Q2006
33.00
39.95
26.60
99.55
Totals
Laspeyres Index2009 =
Paasche2005:
2009
P
Q
3.21 11
0.98 66
1.40 32
38.52
46.06
28.00
112.58
P
P
2009
Q2006
2006
Q2006
(100) =
112.58
(100) = 113.1
99.55
P2006Q2008 P2008Q2008
24.75
51.85
33.25
Totals 109.85
Paasche Index2008 =
P
P
2008
Q2008
2006
Q2008
27.90
57.95
34.00
119.85
(100) =
119.85
(100) = 109.1
109.85
© 2010 John Wiley & Sons Canada, Ltd.
493
Chapter 15: Time Series Forecasting and Index Numbers
15.31 a) and b)
Year
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
Emissions
340
358
376
386
378
393
406
409
423
428
411
393
385
402
403
394
406
437
453
429
423
435
435
450
461
476
493
498
508
530
523
531
556
551
3-year
Moving Ave
358.0
373.3
380.0
385.7
392.3
402.7
412.7
420.0
420.7
410.7
396.3
393.3
396.7
399.7
401.0
412.3
432.0
439.7
435.0
429.0
431.0
440.0
448.7
462.3
476.7
489.0
499.7
512.0
520.3
528.0
536.7
Total
│e│
28.0
4.7
13.0
20.3
16.7
20.3
15.3
9.0
27.7
25.7
5.7
9.7
2.7
6.3
36.0
40.7
3.0
16.7
0.0
6.0
19.0
21.0
27.3
30.7
21.3
19.0
30.3
11.0
10.7
28.0
14.3
540.0
© 2010 John Wiley & Sons Canada, Ltd.
α =0.2
F
340.0
343.6
350.1
357.3
361.4
367.7
375.4
382.1
390.3
397.8
400.5
399.0
396.2
397.3
398.5
397.6
399.3
406.8
416.0
418.6
419.5
422.6
425.1
430.1
436.3
444.2
454.0
462.8
471.8
483.5
491.4
499.3
510.6
│e│
18.0
32.4
35.9
20.7
31.6
38.3
33.6
40.9
37.7
13.2
7.5
14.0
5.8
5.7
4.5
8.4
37.7
46.2
13.0
4.4
15.5
12.4
24.9
30.9
39.7
48.8
44.0
45.2
58.2
39.5
39.6
56.7
40.4
945.3
494
Chapter 15: Time Series Forecasting and Index Numbers
MADmoving average =
MAD=.2 =
c)
e
numberforecasts
e
numberforecasts
=
=
540.0
= 17.4
31
945.3
= 28.6
33
The three-year moving average produced a smaller MAD (17.4) than did
exponential smoothing with  = .2 (MAD = 28.6). Using MAD as the criterion,
the three-year moving average was a better forecasting tool than the exponential
smoothing with  = .2.
15.32
Actual
values
(T.C.S.I)
12. mo.
12-mo. 2yr.
moving moving
total
total
Year
2005
2006
January
February
March
April
May
June
July
August
September
October
November
December
January
February
March
April
May
June
July
1591
1337
2122
2781
2216
1518
1167
1998
2565
2702
2224
2477
1478
2031
2220
3436
3917
2913
2415
24698
24585
25279
25377
26032
27733
29128
30376
31543
31482
31774
33282
33692
33929
49283
49864
50656
51409
53765
56861
59504
61919
63025
63256
65056
66974
67621
© 2010 John Wiley & Sons Canada, Ltd.
Ratios
of
actual
centred
moving
average
(T.C)
Values
to
moving
averages
(S.I*100)
2053
2078
2111
2142
2240
2369
2479
2580
2626
2636
2711
2791
2818
57
96
122
126
99
105
60
79
84.54
130
145
104
86
495
Chapter 15: Time Series Forecasting and Index Numbers
2007
2008
2009
August
September
October
November
December
January
February
March
April
May
June
July
August
September
October
November
December
January
February
March
April
May
June
July
August
September
October
November
December
January
February
March
April
May
June
July
August
September
October
November
December
3165
2504
2994
3732
2887
1715
2862
2324
4191
2500
2488
4344
3004
3632
4121
3626
2963
3044
2128
2726
3760
3805
3829
2209
4482
3021
3698
3888
3215
4097
2511
3064
3879
3555
3505
4715
4088
3179
4210
4226
2776
34760
34864
35619
34202
33777
35706
35545
36673
37800
37694
37770
39099
38365
38767
38336
39641
40982
38847
40325
39714
39291
39553
39805
40858
41241
41579
41698
41448
41124
43630
43236
43394
43906
44244
43805
68689
69624
70483
69821
67979
69483
71251
72218
74473
75494
75464
76869
77464
77132
77103
77977
80623
79829
79172
80039
79005
78844
79358
80663
82099
82820
83277
83146
82572
84754
86866
86630
87300
88150
88049
© 2010 John Wiley & Sons Canada, Ltd.
2862
2901
2937
2909
2832
2895
2969
3009
3103
3146
3144
3203
3228
3214
3213
3249
3359
3326
3299
3335
3292
3285
3307
3361
3421
3451
3470
3464
3441
3531
3619
3610
3638
3673
3669
111
86
102
128
102
59
96
77
135
79
79
136
93
113
128
112
88
92
65
82
114
116
116
66
131
88
107
112
93
116
69
84.89
107
97
96
496
Chapter 15: Time Series Forecasting and Index Numbers
15.33
Month
Year
2005
2006
2007
Actual
values
Seasonal Deseasonalized
(T.C.S.I) Index S data T.C.I
January
1591
76
2105
February
1337
74
1806
March
2122
83
2552
April
2781
122
2274
May
2216
88
2514
June
1518
100
1519
July
1167
76
1541
August
1998
103
1933
September
2565
100
2558
October
2702
116
2322
November
2224
112
1987
December
2477
98
2536
January
1478
76
1956
February
2031
74
2743
March
2220
83
2670
April
3436
122
2810
May
3917
88
4444
June
2913
100
2914
July
2415
76
3189
August
3165
103
3062
September
2504
100
2497
October
2994
116
2573
November
3732
112
3335
December
2887
98
2955
January
1715
76
2270
February
2862
74
3865
March
2324
83
2795
April
4191
122
3427
May
2500
88
2837
June
2488
100
2489
July
4344
76
5737
August
3004
103
2906
September
3632
100
3622
October
4121
116
3542
November
3626
112
3240
December
2963
98
3033
© 2010 John Wiley & Sons Canada, Ltd.
497
Chapter 15: Time Series Forecasting and Index Numbers
2008
2009
January
February
March
April
May
June
July
August
September
October
November
December
January
February
March
April
May
June
July
August
September
October
November
December
3044
2128
2726
3760
3805
3829
2209
4482
3021
3698
3888
3215
4097
2511
3064
3879
3555
3505
4715
4088
3179
4210
4226
2776
76
74
83
122
88
100
76
103
100
116
112
98
76
74
83
122
88
100
76
103
100
116
112
98
4028
2874
3279
3075
4317
3830
2917
4336
3013
3178
3474
3291
5422
3391
3685
3172
4034
3506
6227
3955
3170
3618
3776
2842
15.34
Linear model:
Sales = 1999.8988700565+32.6645179216449*Month
Adjusted R-squared = 0.419
Quadratic model:
Sales = 1676.699+63.942*Month -0.513*MonthSquared
Adjusted R-squared = 0.435
The quadratic model contributes very little to the model. At a t-value of 1.6 Month
Squared is not statistically significant. Therefore the linear model is a better
choice.
© 2010 John Wiley & Sons Canada, Ltd.
498
Chapter 15: Time Series Forecasting and Index Numbers
15.35
2007
Price Quantity
1.26
21
0.94
5
1.43
70
1.05
12
2.81
27
7.49
Item
Margarine (500 g)
Shortening (500 g)
Milk (2 L)
Cola (2 litres)
Potato Chips (750 g)
Total
Index2007 =
P
P
(100) 
7.49
(100) = 100.0
7.49
P
P
(100) 
7.73
(100) = 103.2
7.49
P
P
(100) 
8.37
(100) = 111.7
7.49
2007
2007
Index2008 =
2008
2007
Index2009 =
2009
2007
P2007Q2007
P2008Q2007
P2009Q2007
26.46
4.70
100.10
12.60
75.87
219.73
27.72
4.85
109.20
12.24
77.22
231.23
29.19
5.60
113.40
15.00
80.73
243.92
Totals
IndexLaspeyres2008 =
IndexLaspeyres2009 =
Total
2008
Price Quantity
1.32
23
0.97
3
1.56
68
1.02
13
2.86
29
7.73
P
P
2008
Q2007
2007
Q2007
P
P
2009
Q2007
2007
Q2007
(100) =
23123
.
(100) = 105.2
219.73
(100) =
243.92
(100) = 111.0
219.73
P2007Q2008
P2007Q2009
P2008Q2008
P2009Q2009
28.98
2.82
97.24
13.65
81.49
224.18
27.726
3.76
92.95
11.55
78.68
214.66
30.36
2.91
106.08
13.26
82.94
235.55
30.58
4.48
105.30
13.75
83.72
237.83
© 2010 John Wiley & Sons Canada, Ltd.
2009
Price Quantity
1.39
22
1.12
4
1.62
65
1.25
11
2.99
28
8.37
499
Chapter 15: Time Series Forecasting and Index Numbers
IndexPaasche2008 =
IndexPaasche2009 =
15.36
P
P
2008
Q2008
2007
Q2008
P
P
2009
Q2009
2007
Q2009
(100) =
23555
.
(100) = 105.1
224.18
(100) =
237.83
(100) = 110.8
214.66
ŷ = 9.5382 – 0.2716 x
ŷ (7) = 7.637
R2 = 40.2%
F = 12.78, p = .002
se = 0.264862
Durbin-Watson:
n = 21
k=1
 = .05
D = 0.44
dL = 1.22 and dU = 1.42
Since D = 0.44 < dL = 1.22, the decision is to reject the null hypothesis.
There is significant autocorrelation.
15.37
Month
January
(2005)
February
March
April
May
June
July
August
September
October
November
CPI
Fma
Fwma
SEma
SEwma
104.5
104.8
105.2
105.2
105.4
105.4
105.4
105.6
105.9
105.9
106.3
104.93
105.15
105.30
105.35
105.45
105.58
105.70
105.05
105.24
105.34
105.38
105.48
105.66
105.79
0.226
0.063
0.010
0.063
0.202
0.106
0.360
0.123
0.026
0.004
0.048
0.176
0.058
0.260
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Chapter 15: Time Series Forecasting and Index Numbers
December
January
(2006)
February
March
April
May
June
July
August
September
October
November
December
January
(2007)
February
March
April
May
June
July
August
September
October
November
December
MSEma =
MSEwma =
106.2
105.93
106.03
0.076
0.029
106.2
106.6
107
106.9
107.5
107.2
107.5
107.7
108.3
108.4
108.6
108.4
106.08
106.15
106.33
106.50
106.68
107.00
107.15
107.28
107.48
107.68
107.98
108.25
106.14
106.19
106.37
106.64
106.8
107.13
107.21
107.35
107.52
107.85
108.14
108.39
0.016
0.203
0.456
0.160
0.681
0.040
0.123
0.181
0.681
0.526
0.391
0.023
0.004
0.168
0.397
0.068
0.490
0.005
0.084
0.123
0.608
0.303
0.212
0.000
108.6
109.1
109.5
109.6
109.9
109.9
110
110.1
110.5
110.3
110.3
110
108.43
108.50
108.68
108.90
109.20
109.53
109.73
109.85
109.98
110.13
110.23
110.30
108.45
108.52
108.76
109.09
109.37
109.65
109.8
109.91
110.01
110.22
110.29
110.32
Total
0.031
0.360
0.681
0.490
0.490
0.141
0.076
0.063
0.276
0.031
0.006
0.090
7.313
0.022
0.336
0.548
0.260
0.281
0.063
0.040
0.036
0.240
0.006
0.000
0.102
5.118
SE
7.313

= 0.2285
No. Forecasts
32
SE
5118
.

= 0.1599
No. Forecasts
32
The weighted moving average does a better job of forecasting the data using
MSE as the criterion.
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
15.38 The regression model with one-month lag is:
Cotton Prices = - 61.24 + 1.1035 LAG1
F = 130.46 (p = .000), R2 = .839, adjusted R2 = .833,
se = 17.57, t = 11.42 (p = .000).
The regression model with four-month lag is:
Cotton Prices = 303.9 + 0.4316 LAG4
F = 1.24 (p = .278), R2 = .053, adjusted R2 = .010,
se = 44.22, t = 1.11 (p = .278).
The model for the four-month lag does not have overall significance and has an
adjusted R2 of 1%. This model has virtually no predictability. The model for
the one-month lag has relatively strong predictability with adjusted R2 of 83.3%.
In addition, the F value is significant at  = .001 and the standard error of the
estimate is less than 40% as large as the standard error for the four-month lag
model.
15.39
Qtr TSCI 4qrtot
8qrtot
TC
SI
TCI
T
Year1 1 54.019
2 56.495
213.574
3
50.169
425.044 53.131
94.43
51.699 53.722
421.546 52.693 100.38
52.341 55.945
423.402 52.925
98.09
52.937 58.274
430.997 53.875 102.28
53.063 60.709
440.490 55.061
97.02
55.048 63.249
453.025 56.628 101.07
56.641 65.895
467.366 58.421
97.68
58.186 68.646
480.418 60.052 104.06
60.177 71.503
211.470
4
52.891
210.076
Year2 1
51.915
213.326
2
55.101
217.671
3
53.419
222.819
4
57.236
230.206
Year3 1 57.063
237.160
2 62.488
243.258
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Chapter 15: Time Series Forecasting and Index Numbers
3 60.373
492.176 61.522
98.13
62.215 74.466
503.728 62.966 100.58
62.676 77.534
512.503 64.063
97.91
63.957 80.708
518.498 64.812 105.51
65.851 83.988
524.332 65.542
96.51
65.185 87.373
526.685 65.836 100.93
65.756 90.864
526.305 65.788
99.48
66.733 94.461
526.720 65.840 103.30
65.496 98.163
521.415 65.177
97.04
65.174 101.971
511.263 63.908 104.64
66.177 105.885
501.685 62.711
95.22
60.889 109.904
491.099 61.387 103.59
61.238 114.029
248.918
4 63.334
254.810
Year4 1 62.723
257.693
2 68.380
260.805
3 63.256
263.527
4 66.446
263.158
Year5 1 65.445
263.147
2 68.011
263.573
3 63.245
257.842
4 66.872
253.421
Year6 1 59.714
248.264
2 63.590
242.835
3 58.088
4 61.443
Quarter
1
2
3
4
Year1
Year2
Year3
Year4
Year5
Year6
Index
94.43
100.38
98.09
102.28
97.02
101.07
97.68
104.06
98.13
100.58
97.91
105.51
96.51
100.93
99.48
95.22
103.30 103.59
97.04
104.64
97.89
103.65
96.86
100.86
Total
399.26
Adjust the seasonal indexes by:
400
= 1.00185343
399.26
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
Adjusted Seasonal Indexes:
15.40
Quarter
Index
1
2
3
4
98.07
103.84
97.04
101.05
Total
400.00
Time Period
Q1(yr1)
Q2
Q3
Q4
Q1(yr2)
Q2
Q3
Q4
Q1(yr3)
Q2
Q3
Q4
Q1(yr4)
Q2
Q3
Q4
Q1(yr5)
Q2
Q3
Q4
Q1(yr6)
Q2
Q3
Q4
Deseasonalized Data
55.082
54.406
51.699
52.341
52.937
53.063
55.048
56.641
58.186
60.177
62.215
62.676
63.957
65.851
65.185
65.756
66.733
65.496
65.174
66.177
60.889
61.238
59.860
60.805
15.41 Linear Model: ŷ = 53.41032 + 0.532488 x
R2 = 55.7% F = 27.65 with p = .000
se = 3.43
Quadratic Model: ŷ = 47.68663 + 1.853339 x –0.052834 x2
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
R2 = 76.6% F = 34.37 with p = .000
se = 2.55
In the quadratic regression model, both the linear and squared terms have
significant t statistics at alpha .001 indicating that both are contributing. In
addition, the R2 for the quadratic model is considerably higher than the R2 for the
linear model. Also, se is smaller for the quadratic model. All of these indicate
that the quadratic model is a stronger model.
15.42
The autoregression model is: Yt = 14561.6 + 672.1 Yt-1
The very low value of R2 (1.5%) and the very high value of se indicate that this
regression model has poor predictability. There is no evidence to the presence of
first-order autocorrelation.
15.43
Foreign Inflows = -8385.35672401229+1.23873557314474*Foreign Outflows
Durbin Watson test = 1.443 Because we used a simple linear regression, the value
of k is 1. The sample size, n, is 12, and 
Table A.9 start at n=15, the actual values for an n of 12 can be easily obtained on
the internet by using Google. For example
http://www.nd.edu/~wevans1/econ30331/Durbin_Watson_tables.pdf lists dU =
dL
Because the computed D statistic 1.443, is greater than the value of dL
the null hypothesis is accepted. No autocorrelation is present in this example.
 = .1
15.44
Year
1
2
3
4
5
6
7
8
9
10
11
PurPwr
6.04
5.92
5.57
5.40
5.17
5.00
4.91
4.73
4.55
4.34
4.67
F
6.04
6.03
5.98
5.92
5.85
5.77
5.68
5.59
5.49
5.38
 = .5
e
.12
.46
.58
.75
.85
.86
.95
1.04
1.15
.71
 = .8
F
e
F
e
6.04
5.98
5.78
5.59
5.38
5.19
5.05
4.89
4.72
4.53
.12
.41
.38
.42
.38
.28
.32
.34
.38
.14
6.04
5.94
5.64
5.45
5.23
5.05
4.94
4.77
4.59
4.39
.12
.37
.24
.28
.23
.14
.21
.22
.25
.28
© 2010 John Wiley & Sons Canada, Ltd.
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Chapter 15: Time Series Forecasting and Index Numbers
12
13
14
15
16
17
18
5.01
4.86
4.72
4.60
4.48
4.86
5.15
5.31
5.28
5.24
5.19
5.13
5.07
5.05
e
MAD1 =
e
MAD2 =
e
MAD3 =
e
N
N
N
.30
.42
.52
.59
.65
.21
.10
4.60
4.81
4.84
4.78
4.69
4.59
4.73
.41
.05
.12
.18
.21
.27
.42
4.61
4.93
4.87
4.75
4.63
4.51
4.79
.40
.07
.15
.15
.15
.35
.36
= 10.26 .
e
= 4.83
e
= 3.97
=
10.26
= .60
17
=
4.83
= .28
17
=
3.97
= .23
17
The smallest mean absolute deviation error is produced using  = .8.
The forecast for year 19 is:
F(19) = (.8)(5.15) + (.2)(4.79) = 5.08
15.45 The model is:
Bankruptcies = 75,532.436 – 0.016 Year
Since R2 = .28 and the adjusted R2 = .23, this is a weak model.
et
et2
et – et-1
(et – et-1)2
-1338.6
1791849.96
-8588.3
73758896.89
-7249.7
52558150.09
-7050.6
49710960.36
1537.7
2364521.29
1115
1243225.00
8165.6
66677023.36
12772.3
163131647.29
11657.3
135892643.29
14712.8
216466483.84
1940.5
3765540.25
-3029.4
9177264.36
-17742
314785660.84
-2599.1
6755320.81
430.3
185158.09
622.4
387381.76
3221.5
10378062.25
9747.3
95009857.29
9124.9
83263800.01
9288.8
86281805.44
-458.5
210222.25
-434.8
189051.04
-9723.6
94548396.96
-10875
118274325.16
-10441
109006128.36
-9808
96196864.00
1067.4
1139342.76
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Chapter 15: Time Series Forecasting and Index Numbers
-4277.7
-256.8
Total
D =
18298717.29
65946.24
936739596.73
 (e  e
e
t 1
t
2
t
)2

5530.3
4020.9
30584218.09
16167636.81
921526504.70
921526504.70
= 0.98
936739596.73
For n = 16,  = .05, dL = 1.10 and dU = 1.37
Since D = 0.98 < dL = 1.10, the decision is to reject the null hypothesis and
conclude that there is significant autocorrelation.
© 2010 John Wiley & Sons Canada, Ltd.
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