Chapter 16: Time Series Forecasting and Index Numbers
1
Chapter 16
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.
Gain a general understanding time series forecasting techniques.
2.
Understand the four possible components of time-series data.
3.
Understand stationary forecasting techniques.
4.
Understand how to use regression models for trend analysis.
5.
Learn how to decompose time-series data into their various elements.
6.
Understand the nature of autocorrelation and how to test for it.
7.
Understand autoregression in forecasting.
CHAPTER TEACHING STRATEGY
Time series analysis attempts to determine if there is something inherent in the
history of the variable that can be captured in a way that will help us forecast the future
for this 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.
Chapter 16: Time Series Forecasting and Index Numbers
2
A full gamet of time series forecasting techniques have 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 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 the
student’s CD-Rom.
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 which includes
trend. A more detailed explanation of Holt’s method is available on the student’s CDRom. 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 16.4 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 hopefully removed.
In regression analysis involving data over time, autocorrelation can be a problem.
Because of this, section 16.5 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.
Chapter 16: Time Series Forecasting and Index Numbers
3
CHAPTER OUTLINE
16.1 Introduction to Forecasting
Time Series Components
The Measurement of Forecasting Error
Error
Mean Absolute Deviation (MAD)
Mean Square Error (MSE)
16.2 Smoothing Techniques
Naïve Forecasting Models
Averaging Models
Simple Averages
Moving Averages
Weighted Moving Averages
Exponential Smoothing
16.3 Trend Analysis
Linear Regression Trend Analysis
Regression Trend Analysis Using Quadratic Models
Holt’s Two-Parameter Exponential Smoothing Method
16.4 Seasonal Effects
Decomposition
Finding Seasonal Effects with the Computer
Winters’ Three-Parameter Exponential Smoothing Method
16.5 Autocorrelation and Autoregression
Autocorrelation
Ways to Overcome the Autocorrelation Problem
Addition of Independent Variables
Transforming Variables
Autoregression
16.6 Index Numbers
Simple Index Numbers
Unweighted Aggregate Price Index Numbers
Weighted Price Index Numbers
Laspeyres Price Index
Paasche Price Index
Chapter 16: Time Series Forecasting and Index Numbers
4
KEY TERMS
Autocorrelation
Autoregression
Averaging Models
Cyclical Effects
Decomposition
Deseasonalized Data
Durbin-Watson Test
Error of an Individual
Forecast
Exponential Smoothing
First-Difference Approach
Forecasting
Forecasting Error
Irregular Fluctuations
Mean Absolute Deviation (MAD)
Mean Squared Error (MSE)
Moving Average
Naïve Forecasting Methods
Quadratic Regression Model
Seasonal Effects
Serial Correlation
Simple Average
Simple Average Model
Time Series Data
Trend
Weighted Moving Average
SOLUTIONS TO PROBLEMS IN CHAPTER 16
16.1
Period
1
2
3
4
5
6
7
8
9
Total
MAD =
MSE =
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

12.30
= 1.367
9

20.43
= 2.27
9
2
no. forecasts
e2
5.29
2.56
1.96
1.21
0.09
0.81
3.61
4.41
0.49
20.43
Chapter 16: Time Series Forecasting and Index Numbers
16.2
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 =
16.3
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

125.00
= 12.5
10
1,919
= 191.9
10
Period Value F
1
2
3
4
5
6
19.4
23.6
24.0
26.8
29.2
35.5
16.6
19.1
22.0
24.8
25.9
28.6
Total
MAD =
MSE =
2.8
4.5
2.0
2.0
3.3
6.9
21.5
e
no. forecasts
e
e
e
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

21.5
= 5.375
4

94.59
= 23.65
4
2
no. forecasts
e2
5
Chapter 16: Time Series Forecasting and Index Numbers
16.4
Year
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Total
Acres
140,000
141,730
134,590
131,710
131,910
134,250
135,220
131,020
120,640
115,190
114,510
MAD =
MSE =
16.5
a.)
b.)
Forecast
140,000
141,038
137,169
133,894
132,704
133,632
134,585
132,446
125,362
119,259
e
no. forecasts
e

2
no. forecasts

e
1730
-6448
-5459
-1984
1546
1588
-3565
-11806
-10172
-4749
-39,319
6
e
1730
6448
5459
1984
1546
1588
3565
11806
10172
4749
49047
49,047
= 4,904.7
10
361,331,847
= 36,133,184.7
10
4-mo. mov. avg.
error
44.75
52.75
61.50
64.75
70.50
81.00
14.25
13.25
9.50
21.25
30.50
16.00
4-mo. wt. mov. avg.
error
53.25
56.375
62.875
67.25
76.375
89.125
5.75
9.625
8.125
18.75
24.625
7.875
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
Chapter 16: Time Series Forecasting and Index Numbers
c.)
7
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.
16.6
Period
Value
F( =.1)
Error
F( =.8)
Error
Difference
1
2
3
4
5
6
7
8
211
228
236
241
242
227
217
203
211
213
215
218
220
221
220
23
26
24
7
-4
-17
225
234
240
242
230
220
11
7
2
-15
-13
-17
12
19
22
22
9
0
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 the same amount of error in forecasting the value for
period 8. There is no strong argument in favor of either model.
16.7
Period
Value
 =.3
Error
 =.7
Error 3-mo.avg. Error
1
2
3
4
5
6
7
8
9
9.4
8.2
7.9
9.0
9.8
11.0
10.3
9.5
9.1
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
9.6
0.5
1.4
1.1
0.4
-0.9
-0.5
An examination of the forecast errors reveals that for periods 4 through 9,
the 3-month moving average has the smallest error for two periods,  = .3 has the
smallest error for three periods, and  = .7 has the smallest error for one period.
The results are mixed.
Chapter 16: Time Series Forecasting and Index Numbers
16.8
16.9
8
(a)
F(a)
(c)
e(a)
(b)
F(b)
193.04
213.78
407.68
562.10
569.10
595.08
397.38
414.06
2852.36
2915.49
3000.63
3161.94
3364.41
3550.76
3740.97
3854.64
Year
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
Orders
2512.7
2739.9
2874.9
2934.1
2865.7
2978.5
3092.4
3356.8
3607.6
3749.3
3952.0
3949.0
4137.0
2785.46
2878.62
2949.12
3045.50
3180.20
3356.92
3551.62
3722.94
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
(c)
e(b)
126.14
176.91
356.17
445.66
384.89
401.24
208.03
282.36
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
 = .9 produces a smaller mean average error.
 e =2023.0
Chapter 16: Time Series Forecasting and Index Numbers
9
16.10 Simple Regression Trend Model:
ŷ = 37,969 + 9899.1 Period
F = 1603.11 (p = .000), R2 = .988, adjusted R2 = .988,
se = 6,861, t = 40.04 (p = .000)
Quadratic Regression Trend Model:
ŷ = 35,769 + 10,473 Period - 26.08 Period2
F = 772.71 (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.
16.11 Trend line:
R2 = 80.9%
Members = 17,206 – 62.7 Year
se = 158.8
F = 63.54, reject the null hypothesis.
Regression Plot
Members = 17206.2 - 62.6814 Year
S = 158.837
R-Sq = 80.9 %
R-Sq(adj) = 79.6 %
17400
17200
Members
17000
16800
16600
16400
16200
16000
0
5
10
Year
15
Chapter 16: Time Series Forecasting and Index Numbers
10
16.12
Trend Model:
Shipments = -12,138,725 + 6115.6 Year
R2 = 88.2
adjusted R2 = 87.3
se = 9725
t = 9.49 (p = .000)
F = 89.97 (p = .000)
Quadratic Model:
Shipments = 2,434,939,619 – 2,451,417 Year + 617.01 Year2
R2 = 99.7
adjusted R2 = 99.7
tyear = -21.51 (p = .000)
tyearsq = 21.56 (p = .000)
F = 2016.66 (p = .000)
se = 1544
The graph indicates a quadratic fit rather than a linear fit. The quadratic model
produced an R2 = 99.7 compared to R2 = 88.2 for linear trend indicating a better
fit for the quadratic model.
Chapter 16: Time Series Forecasting and Index Numbers
11
16.13
Month
Jan.(yr. 1)
Feb.
Mar.
Apr.
May
June
Broccoli
12-Mo. Mov.Tot.
2-Yr.Tot.
TC
SI
3282.8
136.78
93.30
3189.7
132.90
90.47
3085.0
128.54
92.67
3034.4
126.43
98.77
2996.7
124.86
111.09
2927.9
122.00
100.83
2857.8
119.08
113.52
2802.3
116.76
117.58
2750.6
114.61
112.36
2704.8
112.70
92.08
2682.1
111.75
99.69
2672.7
111.36
102.73
132.5
164.8
141.2
133.8
138.4
150.9
1655.2
July
146.6
1627.6
Aug.
146.9
1562.1
Sept.
138.7
Oct.
128.0
Nov.
112.4
Dec.
121.0
Jan.(yr. 2)
104.9
1522.9
1511.5
1485.2
1442.7
1415.1
Feb.
99.3
1387.2
Mar.
102.0
1363.4
Apr.
122.4
1341.4
May
112.1
1340.7
June
108.4
1332.0
July
Aug.
Sept.
Oct.
Nov.
Dec.
119.0
119.0
114.9
106.0
111.7
112.3
Chapter 16: Time Series Forecasting and Index Numbers
12
16.14
Month
Ship
12m tot
2yr tot
TC
SI
TCI
T
Jan(Yr1) 1891
1968.64
2047.09
Feb
1986
1971.49
2054.11
Mar
1987
1945.22
2061.12
Apr
1987
1977.97
2068.14
May
2000
1977.85
2075.16
June
2082
1963.24
2082.18
C
23822
July
1878
Aug
2074
Sept
2086
Oct
2045
Nov
1945
47689
1987.04
94.51 1969.94
2089.19
95.11
47852
1993.83 104.02 2020.52
2096.21
95.11
48109
2004.54 104.06 2006.76
2103.23
95.31
48392
2016.33 101.42 1978.71
2110.25
95.55
48699
2029.13
95.85 2042.25
2117.27
95.84
49126
2046.92
90.92 2002.94
2124.28
96.36
49621
2067.54
93.64 2015.49
2131.30
97.01
49989
2082.88 101.01 2088.63
2138.32
97.41
50308
2096.17 101.42 2081.3
2145.34
97.71
50730
2113.75 100.82 2121.32
2152.35
98.21
51132
2130.50 101.53 2139.04
2159.37
98.66
51510
2146.25 109.31 2212.18
2166.39
99.07
51973
2165.54
2173.41
99.64
52346
2181.08 101.37 2153.99
2180.43 100.03
52568
2190.33 103.55 2181.85
2187.44 100.13
23867
23985
24124
24268
24431
Dec
1861
24695
Jan(Yr2) 1936
24926
Feb
2104
25063
Mar
2126
25245
Apr
2131
25485
May
2163
June
2346
July
2109
Aug
2211
Sept
2268
25647
25863
97.39 2212.25
26110
26236
Chapter 16: Time Series Forecasting and Index Numbers
13
26332
Oct
2285
52852
2202.17 103.76 2210.93
2194.46
100.35
53246
2218.58
94.97 2212.35
2201.48
100.78
53635
2234.79
92.94 2235.42
2208.50
101.19
53976
2249.00
97.07 2272.63
2215.51
101.51
54380
2265.83
98.42 2213.71
2222.53
101.95
54882
2286.75
97.17 2175.28
2229.55
102.56
55355
2306.46 100.54 2308.46
2236.57
103.12
55779
2324.13 101.93 2342.76
2243.59
103.59
56186
2341.08 108.03 2384.75
2250.60
104.02
56539
2355.79
96.23 2377.98
2257.62
104.35
56936
2372.33 103.57 2393.65
2264.64
104.76
57504
2396.00 105.34 2428.12
2271.66
105.47
58075
2419.79 103.40 2420.90
2278.68
106.19
58426
2434.42
95.05 2429.70
2285.69
106.51
58573
2440.54
93.30 2450.67
2292.71
106.45
58685
2445.21
95.53 2431.91
2299.73
106.33
58815
2450.63 100.95 2455.93
2306.75
106.24
58806
2450.25 103.91 2492.47
2313.76
105.90
58793
2449.71 104.75 2554.34
2320.78
105.56
58920
2455.00 100.73 2445.61
2327.80
105.46
59018
2459.08 104.59 2425.29
2334.82
105.32
59099
2462.46
2341.84
105.15
26520
Nov
2107
26726
Dec
2077
26909
Jan(Yr3) 2183
27067
Feb
2230
Mar
2222
Apr
2319
May
2369
27313
27569
27786
27993
June
2529
28193
July
2267
28346
Aug
2457
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
July
2336
29523
29576
94.86 2450.36
Chapter 16: Time Series Forecasting and Index Numbers
Aug
2518
Sept
2454
Oct
2559
14
59141
2464.21 102.18 2453.08
2348.85
104.91
59106
2462.75
99.64 2360.78
2355.87
104.54
58933
2455.54 104.21 2476.05
2362.89
103.92
58779
2449.13
97.34 2503.20
2369.91
103.34
58694
2445.58
94.25 2480.81
2376.92
102.89
58582
2440.92
97.87 2487.08
2383.94
102.39
58543
2439.29 100.97 2445.01
2390.96
102.02
58576
2440.67 103.33 2468.97
2397.98
101.78
58587
2441.13
99.01 2406.02
2405.00
101.50
58555
2439.79 101.16 2440.66
2412.01
101.15
58458
2435.75 102.31 2349.86
2419.03
100.69
58352
2431.33
94.76 2417.63
2468.16
98.51
58258
2427.42 103.44 2435.74
2475.17
98.07
57922
2413.42 103.34 2401.31
2482.19
97.23
57658
2402.42 105.31 2436.91
2489.21
96.51
57547
2397.79
99.30 2478.40
2496.23
96.06
57400
2391.67
92.45 2379.47
2503.24
95.54
57391
2391.29
99.40 2454.31
2510.26
95.26
57408
2392.00
99.54 2368.68
2517.28
95.02
57346
2389.42
94.92 2252.91
2524.30
94.66
57335
2388.96 100.76 2389.32
2531.32
94.38
57362
2390.08
99.03 2339.63
2538.33
94.16
57424
2392.67 102.23 2329.30
2545.35
94.00
29565
29541
29392
Nov
2384
29387
Dec
2305
29307
Jan(Yr5) 2389
29275
Feb
2463
29268
Mar
2522
29308
Apr
2417
May
2468
June
2492
July
2304
29279
29276
29182
29170
Aug
2511
29088
Sept
2494
28834
Oct
2530
28824
Nov
2381
28723
Dec
2211
28677
Jan(Yr6) 2377
28714
Feb
2381
Mar
2268
Apr
2407
May
2367
June
2446
28694
28652
28683
28679
28745
Chapter 16: Time Series Forecasting and Index Numbers
July
Aug
Sept
Oct
Nov
Dec
15
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
Total
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
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
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
Regression Output for Trend Line:
Yˆ = 2035.58 + 7.1481 X
R2 = .682, Se = 102.9
Chapter 16: Time Series Forecasting and Index Numbers
16
16.15 Regression Analysis
The regression equation is:
Food = 0.628 + 0.690 Shelter
Predictor
Coef
Stdev
t-ratio
p
Constant
0.6283
0.7583
0.83
0.416
Shelter
0.6905
0.1055
6.54
0.000
s = 2.018
Food
14.3
8.5
3.0
6.3
9.9
11.0
8.6
7.8
4.1
2.1
3.8
2.3
3.2
4.1
4.1
5.8
5.8
2.9
1.2
2.2
2.4
2.8
3.3
2.6
2.2
2.1
R-sq = 64.1%
Shelter
9.6
9.9
5.5
6.6
10.2
13.9
17.6
11.7
7.1
2.3
4.9
5.6
5.5
4.7
4.8
4.5
5.4
4.5
3.3
3.0
3.1
3.2
3.2
3.1
3.3
2.9
Yˆ
7.2570
7.4642
4.4260
5.1855
7.6713
10.2262
12.7810
8.7071
5.5308
2.2164
4.0117
4.4950
4.4260
3.8736
3.9426
3.7355
4.3569
3.7355
2.9069
2.6997
2.7688
2.8378
2.8378
2.7688
2.9069
2.6307
R-sq(adj) = 62.6%
e
7.04296
1.03581
-1.42599
1.11446
2.22866
0.77382
-4.18103
-0.90709
-1.43079
-0.11640
-0.21169
-2.19504
-1.22599
0.22641
0.15736
2.06451
1.44306
-0.83549
-1.70690
-0.49975
-0.36880
-0.03785
0.46215
-0.16880
-0.70690
-0.53070
e2
49.6033
1.0729
2.0335
1.2420
4.9669
0.5988
17.4810
0.8228
2.0472
0.0135
0.0448
4.8182
1.5031
0.0513
0.0248
4.2622
2.0824
0.6981
2.9135
0.2497
0.1360
0.0014
0.2136
0.0285
0.4997
0.2816
(e  e
t
D =
)2 = 36.09 + 6.06 + 6.45 + 1.24 + 2.12 + 24.55 + 10.72 +
0.27 + 1.73 + 0.01 + 3.93 + 0.94 + 2.11 + 0.00 + 3.64 + 0.39 + 5.19 +
0.76 + 1.46 + 0. 17 + 0.11 + 0.25 + 0.40 + 0.29 + 0.31 = 109.19
t 1
 (e  e
e
t
2
t 1
)2

109.19
= 1.12
97.69
Since D = 1.12 is less than dL, the decision is to reject the null hypothesis. There
is significant autocorrelation.
Chapter 16: Time Series Forecasting and Index Numbers
16.16
17
First Differences
Year
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
Food
5.8
5.5
-3.3
-3.6
-1.1
2.4
0.8
3.7
2.0
-1.7
1.5
-0.9
-0.9
0.0
-1.7
0.0
2.9
1.7
-1.0
-0.2
-0.4
-0.5
0.7
0.4
0.1
Shelter
-0.3
4.4
-1.1
-3.6
-3.7
-3.7
5.9
4.6
4.8
-2.6
-0.7
0.1
0.8
-0.1
0.3
-0.9
0.9
1.2
0.3
-0.1
-0.1
0.0
0.1
-0.2
0.4
The regression equation is:
Predictor
Constant
Shelterdiff
S = 2.069
Coef
0.3647
0.4599
Fooddiff = 0.365 + 0.460 Shelterdiff
StDev
0.4164
0.1692
R-Sq = 24.3%
Analysis of Variance
Source
DF
Regression
1
Residual Error
23
Total
24
SS
31.642
98.504
130.146
T
0.88
2.72
P
0.390
0.012
R-Sq(adj) = 21.0%
MS
31.642
4.283
F
7.39
P
0.012
The resulting model is much weaker than that obtained with the raw data.
Chapter 16: Time Series Forecasting and Index Numbers
18
16.17 The regression equation is:
Failed Bank Assets = 1,379 + 136.68 Number of Failures
for x= 150:
R2 = 37.9%
ŷ = 21,881 (million $)
adjusted R2 = 34.1%
se = 13,833
F = 9.78, p = .006
The Durbin Watson statistic for this model is:
D = 2.49
The critical table values for k = 1 and n = 18 are dL = 1.16 and dU = 1.39. Since
the observed value of D = 2.49 is above dU, the decision is to fail to reject the null
hypothesis. There is no significant autocorrelation.
Failed Bank Assets
8,189
104
1,862
4,137
36,394
3,034
7,609
7,538
56,620
28,507
10,739
43,552
16,915
2,588
825
753
186
27
Number of Failures
11
7
34
45
79
118
144
201
221
206
159
108
100
42
11
6
5
1
ŷ
2,882.8
2,336.1
6,026.5
7,530.1
12,177.3
17,507.9
21,061.7
28,852.6
31,586.3
29,536.0
23,111.9
16,141.1
15,047.6
7,120.0
2,882.8
2,199.4
2,062.7
1,516.0
e
5,306.2
-2,232.1
-4,164.5
-3,393.1
24,216.7
-14,473.9
-13,452.7
-21,314.6
25,033.7
- 1,029.0
-12,372.9
27,410.9
1,867.4
- 4,532.0
- 2,057.8
- 1,446.4
- 1,876.7
- 1,489.0
e2
28,155,356
4,982,296
17,343,453
11,512,859
586,449,390
209,494,371
180,974,565
454,312,622
626,687,597
1,058,894
153,089,247
751,357,974
3,487,085
20,539,127
4,234,697
2,092,139
3,522,152
2,217,144
Chapter 16: Time Series Forecasting and Index Numbers
16.18
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Failure Diff.
4
-27
- 9
-34
-39
-26
-57
-20
15
47
51
8
58
31
5
1
4
19
Asset Diff.
8,085
-1,758
-2,275
-32,257
33,360
- 4,575
71
-49,082
28,113
17,768
-32,813
26,637
14,327
1, 763
72
567
159
Regression Analysis:
The regression equation is:
AssetDiff = 412 + 97 FailureDiff
Predictor
Constant
FailureDiff
Coef
412
96.6
StDev
5458
171.1
s = 22,498
R-Sq = 2.1%
t
0.08
0.56
p
0.941
0.581
R-Sq(adj) = 0.0%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
15
16
SS
161,413,890
7,592,671,226
7,754,085,116
MS
161,413,890
506,178,082
F
p
0.32 0.581
The Durbin-Watson Statistic, D = 2.93. The table critical d values for this test
are: dL = 1.13 and dU = 1.38. Since the observed D = 2.93 is greater than the
upper critical value, the decision is to fail to reject the null. We do not have
enough evidence to declare that there is significant autocorrelation.
While there is no significant autocorrelation in these data, the regression model
is extremely weak (the p-value for F is .581 and the adjusted R2 is zero).
Chapter 16: Time Series Forecasting and Index Numbers
16.19 Starts
311
486
527
429
285
275
400
538
545
470
306
240
205
382
436
468
483
420
404
396
329
254
288
302
351
331
361
364
lag1
*
311
486
527
429
285
275
400
538
545
470
306
240
205
382
436
468
483
420
404
396
329
254
288
302
351
331
361
20
lag2
*
*
311
486
527
429
285
275
400
538
545
470
306
240
205
382
436
468
483
420
404
396
329
254
288
302
351
331
The model with 1 lag:
Housing Starts = 158 + 0.589 lag 1
F = 13.66
p = .001 R2 = 35.3% adjusted R2 = 32.7%
se = 77.55
The model with 2 lags:
Housing Starts = 401 - 0.065 lag 2
F = 0.11
p = .744 R2 = 0.5% adjusted R2 = 0.0% Se = 95.73
The model with 1 lag is the best model with a very modest R2 32.7%. The model
with 2 lags has no predictive ability.
Chapter 16: Time Series Forecasting and Index Numbers
16.20 The autoregression model is:
21
Juice = 552 + 0.645 Juicelagged2
The F value for this model is 27.0 which is significant at alpha = .001.
The value of R2 is 56.2% which denotes modest predictability. The
adjusted R2 is 54.2%. The standard error of the estimate is 216.6. The DurbinWatson statistic is 1.70 which indicates that there is no significant autocorrelation
in this model.
16.21 Year
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
Price
22.45
31.40
32.33
36.50
44.90
61.24
69.75
73.44
80.05
84.61
87.28
16.22 Year Patents
1980
66.2
1981
71.0
1982
63.3
1983
62.0
1984
72.7
1985
77.2
1986
76.9
1987
89.4
1988
84.3
1989 102.5
1990
99.2
1991 106.8
1992 107.4
1993 109.7
1994 124.1
1995 114.4
1996 122.6
1997 125.5
1998 163.1
a.) Index1950
100.0
139.9
144.0
162.6
200.0
272.8
310.7
327.1
356.6
376.9
388.8
Index
66.7
71.6
63.8
62.5
73.3
77.8
77.5
90.1
85.0
103.3
100.0
107.7
108.3
110.6
125.1
115.3
123.6
126.5
164.4
b.) Index1980
32.2
45.0
46.4
52.3
64.4
87.8
100.0
105.3
114.8
121.3
125.1
Chapter 16: Time Series Forecasting and Index Numbers
16.23
22
Year
Totals
1985
1.31
1.99
2.14
2.89
1992
1.53
2.21
1.92
3.38
1997
1.40
2.15
2.68
3.10
8.33
9.04
9.33
Index1987 =
8.33
(100) = 100.0
8.33
Index1992 =
9.04
(100) = 108.5
8.33
Index1995 =
9.33
(100) = 112.0
8.33
16.24
Year
1994
1995
1996
1997
1998
1999
2000
2001
2002
1.06
1.47
1.70
6.65
1.21
1.65
1.70
6.90
1.09
1.60
1.80
7.50
1.13
1.62
1.85
8.10
1.10
1.58
1.80
7.95
1.16
1.61
1.82
7.96
1.23
1.78
1.98
8.24
1.23
1.77
1.96
8.21
1.08
1.61
1.94
8.19
Totals 10.88
11.46
11.99
12.70
12.43
12.55
13.23
13.17
12.82
Index1994 =
10.88
(100) = 87.5
12.43
Index1995 =
11.46
(100) = 92.2
12.43
Index1996 =
11.99
(100) = 96.5
12.43
Index1997 =
12.70
(100) = 102.2
12.43
Chapter 16: Time Series Forecasting and Index Numbers
Index1998 =
12.43
(100) = 100.0
12.43
Index1999 =
12.55
(100) = 101.0
12.43
Index2000 =
13.23
(100) = 106.4
12.43
Index2001 =
13.17
(100) = 106.0
12.43
Index2002 =
12.82
(100) = 103.1
12.43
16.25
Item
Quantity
1995
Price
1995
Price
2000
Price
2001
Price
2002
1
2
3
4
21
6
17
43
0.50
1.23
0.84
0.15
0.67
1.85
0.75
0.21
0.68
1.90
0.75
0.25
0.71
1.91
0.80
0.25
P1995Q1995 P2000Q1995 P2001Q1995
Totals
P2002Q1995
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
Index1997 =
P
P
Q1995
2000
Q1995
(100) =
46.95
(100) = 121.6
38.61
(100) =
49.18
(100) = 127.4
38.61
(100) =
50.72
(100) = 131.4
38.61
1995
Index1998 =
P
P
Q1995
2001
Q1995
1995
Index1999 =
P
P
Q1995
2002
Q1995
1995
23
Chapter 16: Time Series Forecasting and Index Numbers
16.26
Item
Price
1997
Price Quantity Price Quantity
2001
2001
2002
2002
1
2
3
22.50
10.90
1.85
27.80
13.10
2.25
P1997Q2001 P1997Q2002
Totals
24
13
5
41
28.11
13.25
2.35
P2001Q2001 P2002Q2002
292.50
54.50
75.85
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
Index1998 =
P
P
Q2001
2001
Q2001
(100) =
519.15
(100) = 122.8
422.85
(100) =
546.72
(100) = 124.7
438.60
1997
Index1999 =
P
P
Q2002
2002
Q2002
1997
16.27 a) The linear model:
12
8
44
Yield = 9.96 - 0.14 Month
F = 219.24 p = .000
The quadratic model:
R2 = 90.9s = .3212
Yield = 10.4 - 0.252 Month + .00445 Month2
F = 176.21 p = .000 R2 = 94.4% se = .2582
Both t ratios are significant, for x,
t = - 7.93, p = .000 and for x, 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.
Chapter 16: 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
MAD =
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
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
c)
 = .3
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
7.99 8.86 .87
8.12 8.60 .48
 = .7
F
10.08
10.06
9.49
9.31
9.58
9.56
9.43
8.81
8.50
8.56
8.16
e
.03
.82
.26
.38
.03
.19
.88
.45
.09
.57
.04
25
Chapter 16: Time Series Forecasting and Index Numbers
7.91
7.73
7.39
7.48
7.52
7.48
7.35
7.04
6.88
6.88
7.17
7.22
8.46 .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 =
8.13
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
26
.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
64.36
8.05
100.87
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
32.61
8.12
Chapter 16: Time Series Forecasting and Index Numbers
27
31.75
7.91
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
31.15
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
2nd
3rd
4th
total
99.82
101.05
98.64
98.67
398.18
Since the total is not 400, adjust each seasonal index by multiplying by
1.004571 resulting in the final seasonal indexes of:
1st 100.28
2nd 101.51
3rd 99.09
4th 99.12
400
=
398.18
Chapter 16: Time Series Forecasting and Index Numbers
16.28
Year
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
16.29
Item
1
2
3
4
5
6
Totals
Quantity
2073
2290
2349
2313
2456
2508
2463
2499
2520
2529
2483
2467
2397
2351
2308
Index Number
100.0
110.5
113.3
111.6
118.5
121.1
118.8
120.5
121.6
122.0
119.8
119.0
115.6
113.4
111.3
1998
3.21
0.51
0.83
1.30
1.67
0.62
1999
3.37
0.55
0.90
1.32
1.72
0.67
2000
3.80
0.68
0.91
1.33
1.90
0.70
2001
3.73
0.62
1.02
1.32
1.99
0.72
2002
3.65
0.59
1.06
1.30
1.98
0.71
8.14
8.53
9.32
9.40
9.29
Index1998 =
P
P
(100) 
8.14
(100) = 100.0
8.14
P
P
(100) 
8.53
(100) = 104.8
8.14
P
P
(100) 
9.32
(100) = 114.5
8.14
P
P
(100) 
9.40
(100) = 115.5
8.14
P
P
(100) 
9.29
(100) = 114.1
8.14
1998
1998
Index1999 =
1999
1998
Index2000 =
2000
1998
Index2001 =
2001
1998
Index2002 =
28
2002
1998
Chapter 16: Time Series Forecasting and Index Numbers
16.30
Item
1
2
3
1999
P
Q
2.75
0.85
1.33
Laspeyres1998:
12
47
20
2000
P
Q
2001
P
Q
2002
P
Q
2.98 9
0.89 52
1.32 28
3.10 9
0.95 61
1.36 25
3.21 11
0.98 66
1.40 32
P1999Q1999
P2002Q1999
33.00
39.95
26.60
38.52
46.06
28.00
99.55
112.58
Totals
Laspeyres Index2002 =
P
P
Q1999
2002
Q1999
(100) =
1999
Paasche2001:
29
112.58
(100) = 113.1
99.55
P1999Q2001 P2001Q2001
Totals
Paasche Index2001 =
24.75
51.85
33.25
27.90
57.95
34.00
109.85
119.85
P
P
Q2001
12001
Q2001
1999
(100) =
119.85
(100) = 109.1
109.85
Chapter 16: Time Series Forecasting and Index Numbers
16.31
Year
Quantity
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
3654
3547
3285
3238
3320
3294
3393
3946
4588
6204
7041
7031
7618
8214
7936
7667
7474
7244
7173
6832
6912
a) moving average
F
e
3495.33
3356.67
3281.00
3284.00
3335.67
3544.33
3975.67
4912.67
5944.33
6758.67
7230.00
7621.00
7922.67
7939.00
7692.33
7461.67
7297.00
7083.00
257.33
36.67
13.00
109.00
610.33
1043.67
2228.33
2128.33
1086.67
859.33
984.00
315.00
255.67
465.00
448.33
288.67
465.00
171.00
 e =11,765.33
MADmoving average =
MAD=.2 =
c)
e
numberforecasts
e
numberforecasts
=
=
30
b) = .2
F
3654.00
3632.60
3563.08
3498.06
3462.45
3428.76
3421.61
3526.49
3738.79
4231.83
4793.67
5241.14
5716.51
6216.01
6560.01
6781.41
6919.93
6984.74
7022.39
6984.31
e
325.08
178.06
168.45
35.76
524.39
1061.51
2465.21
2809.17
2237.33
2376.86
2497.49
1719.99
1106.99
692.59
324.07
188.26
190.39
72.31
 e =18,973.91
11,765.33
= 653.63
18
18,973.91
= 1054.11
18
The three-year moving average produced a smaller MAD (653.63) than did
exponential smoothing with  = .2 (MAD = 1054.11). Using MAD as the
criterion, the three-year moving average was a better forecasting tool than the
exponential smoothing with  = .2.
Chapter 16: Time Series Forecasting and Index Numbers
31
16.32-16.34
Month
Chem
Jan(91)
Feb
Mar
Apr
May
June
23.701
24.189
24.200
24.971
24.560
24.992
July
22.566
Aug
24.037
Sept
25.047
12m tot 2yr tot
TC
SI
TCI
T
288.00
575.65
23.985
94.08
23.872
23.917
575.23
23.968
100.29
24.134
23.919
576.24
24.010
104.32
24.047
23.921
577.78
24.074
100.17
24.851
23.924
578.86
24.119
95.50
24.056
23.926
580.98
24.208
93.32
23.731
23.928
584.00
24.333
95.95
24.486
23.931
586.15
24.423
98.77
24.197
23.933
587.81
24.492
103.23
23.683
23.936
589.05
24.544
103.59
24.450
23.938
590.05
24.585
102.44
24.938
23.940
592.63
24.693
107.26
24.763
23.943
595.28
24.803
97.12
25.482
23.945
597.79
24.908
99.05
24.771
23.947
601.75
25.073
103.98
25.031
23.950
605.59
25.233
96.41
25.070
23.952
607.85
25.327
94.07
24.884
23.955
287.65
287.58
288.66
Oct
24.115
289.12
Nov
23.034
289.74
Dec
22.590
291.24
Jan(92) 23.347
292.76
Feb
24.122
Mar
25.282
Apr
25.426
May
25.185
June
26.486
July
24.088
293.39
294.42
294.63
295.42
297.21
298.07
Aug
24.672
299.72
Sept
26.072
302.03
Oct
24.328
303.56
Nov
23.826
304.29
Chapter 16: Time Series Forecasting and Index Numbers
Dec
24.373
32
610.56
25.440
95.81
25.605
23.957
613.27
25.553
94.73
25.388
23.959
614.89
25.620
100.59
25.852
23.962
616.92
25.705
107.34
25.846
23.964
619.39
25.808
104.46
25.924
23.966
622.48
25.937
99.93
25.666
23.969
625.24
26.052
109.24
26.608
23.971
627.35
26.140
94.95
26.257
23.974
629.12
26.213
97.51
25.663
23.976
631.53
26.314
103.44
26.131
23.978
635.31
26.471
96.90
26.432
23.981
639.84
26.660
95.98
26.725
23.983
644.03
26.835
94.54
26.652
23.985
647.65
26.985
93.82
26.551
23.988
652.98
27.208
97.16
26.517
23.990
659.95
27.498
106.72
27.490
23.992
666.46
27.769
104.37
27.871
23.995
672.57
28.024
101.43
28.145
23.997
679.39
28.308
106.50
28.187
24.000
686.66
28.611
93.48
28.294
24.002
694.30
28.929
100.13
29.082
24.004
701.34
29.223
105.34
29.554
24.007
706.29
29.429
97.16
29.466
24.009
306.27
Jan(93) 24.207
307.00
Feb
25.772
307.89
Mar
27.591
309.03
Apr
26.958
310.36
May
June
25.920
312.12
28.460
313.12
July
24.821
314.23
Aug
25.560
Sept
27.218
Oct
25.650
Nov
25.589
Dec
25.370
314.89
316.64
318.67
321.17
322.86
Jan(94) 25.316
324.79
Feb
26.435
328.19
Mar
29.346
331.76
Apr
28.983
334.70
May
28.424
337.87
June
30.149
July
26.746
341.52
345.14
Aug
28.966
349.16
Sept
30.783
352.18
Oct
28.594
Chapter 16: Time Series Forecasting and Index Numbers
33
354.11
Nov
28.762
710.54
29.606
97.14
30.039
24.011
715.50
29.813
97.33
30.484
24.014
720.74
30.031
96.34
30.342
24.016
725.14
30.214
100.80
30.551
24.019
727.79
30.325
106.75
30.325
24.021
730.25
30.427
101.57
29.719
24.023
733.94
30.581
100.53
30.442
24.026
738.09
30.754
106.63
30.660
24.028
1992
95.95
98.77
103.23
103.59
102.44
107.26
97.12
99.05
103.98
96.41
94.07
95.81
1993
94.73
100.59
107.34
104.46
99.93
109.24
94.95
97.51
103.44
96.90
95.98
94.54
356.43
Dec
29.018
359.07
Jan(95) 28.931
361.67
Feb
30.456
Mar
32.372
Apr
30.905
May
30.743
June
32.794
363.47
364.32
365.93
368.01
370.08
July
Aug
Sept
Oct
Nov
Dec
29.342
30.765
31.637
30.206
30.842
31.090
Seasonal Indexing:
Month
1991
Jan
Feb
Mar
Apr
May
June
July
94.08
Aug
100.29
Sept
104.32
Oct
100.17
Nov
95.50
Dec
93.32
Total
1994
93.82
97.16
106.72
104.37
101.43
106.50
93.48
100.13
105.34
97.16
97.14
97.33
Adjust each seasonal index by 1200/1199.88 = 1.0001
1995
96.34
100.80
106.75
101.57
100.53
106.63
Index
95.34
99.68
106.74
103.98
100.98
106.96
94.52
99.59
104.15
97.03
95.74
95.18
1199.88
Chapter 16: Time Series Forecasting and Index Numbers
34
Final Seasonal Indexes:
Month
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
Index
95.35
99.69
106.75
103.99
100.99
106.96
94.53
99.60
104.16
97.04
95.75
95.19
Regression Output for Trend Line:
ŷ = 22.4233 + 0.144974 x
R2 = .913
Regression Output for Quadratic Trend:
ŷ = 23.8158 + 0.01554 x + .000247 x2
R2 = .964
In this model, the linear term yields a t = 0.66 with p = .513 but the squared term
predictor yields a t = 8.94 with p = .000.
Regression Output when using only the squared predictor term:
ŷ = 23.9339 + 0.00236647 x2
R2 = .964
Note: The trend model derived using only the squared predictor was used in
computing T (trend) in the decomposition process.
Chapter 16: Time Series Forecasting and Index Numbers
16.35
1999
P
Q
Item
Marg. 0.83
Short. 0.89
Milk 1.43
Coffee 1.05
Chips 3.01
Total 7.21
Index1999 =
21
5
70
12
27
2000
P
Q
2001
P
Q
0.81 23
0.87
3
1.56 68
1.02 13
3.06 29
7.32
0.83 22
0.87
4
1.59 65
1.01 11
3.13 28
7.43
P
P
(100) 
7.21
(100) = 100.0
7.21
P
P
(100) 
7.32
(100) = 101.5
7.21
P
P
(100) 
7.43
(100) = 103.05
7.21
1999
1999
Index2000 =
2000
1999
Index2001 =
2001
1999
P1999Q1999
P2000Q1999
P2001Q1999
17.43
4.45
100.10
12.60
81.27
215.85
17.01
4.35
109.20
12.24
82.62
225.42
17.43
4.35
111.30
12.24
82.62
229.71
Totals
IndexLaspeyres2000 =
P
P
Q1999
2000
Q1999
(100) =
225.42
(100) = 104.4
215.85
(100) =
229.71
(100) = 106.4
215.85
1999
IndexLaspeyres2001 =
P
P
Q1999
2001
Q1999
1999
Total
35
P1999Q2000
P1999Q2001
P2000Q2000
P2001Q2001
19.09
2.67
97.24
13.65
87.29
219.94
18.26
3.56
92.95
11.55
84.28
210.60
18.63
2.61
106.08
13.26
88.74
229.32
18.26
3.48
103.35
11.11
87.64
223.84
Chapter 16: Time Series Forecasting and Index Numbers
IndexPaasche2000 =
P
P
Q2000
2000
Q2000
(100) =
229.32
(100) = 104.3
219.94
(100) =
223.84
(100) = 106.3
210.60
1999
IndexPaasche2001 =
P
P
Q2001
2001
Q2001
1999
16.36
36
ŷ = -7,248,156 + 1,072,187x
ŷ (55) = 51,722,129
R2 = 99.1%
F = 2640.1, p = .000
se = 1,945,100
Durbin-Watson:
n = 26
k=1
 = .05
D = 0.10
dL = 1.30 and dU = 1.46
Since D = 0.10 < dL = 1.30, the decision is to reject the null hypothesis.
There is significant autocorrelation.
Chapter 16: Time Series Forecasting and Index Numbers
16.37 Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
X
Fma
100.2
102.1
105.0
105.9
110.6
115.4
118.6
124.1
128.7
131.9
133.7
133.4
132.0
131.7
132.9
133.0
131.3
129.6
Fwma
103.3
105.9
109.2
112.6
117.2
121.7
125.8
129.6
131.9
132.8
132.7
132.5
132.4
132.2
SEMA
104.3
53.29
107.2
90.25
111.0
88.36
114.8 132.25
119.3 132.25
124.0 104.04
128.1
62.41
131.2
14.44
132.7
0.01
132.8
1.21
132.3
0.04
132.4
0.25
132.6
1.21
132.2
6.76
37
SEWMA
39.69
67.24
57.76
86.49
88.36
62.41
31.36
4.84
0.49
1.21
0.36
0.36
1.69
6.76
SE = 678.80 440.57
MSEma =
MSEwma =
 SE
numberforecasts

 SE
numberforecasts
686.77
= 49.06
14

449.02
= 32.07
14
The weighted moving average does a better job of forecasting the data using
MSE as the criterion.
Chapter 16: Time Series Forecasting and Index Numbers
38
16.38 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 with the four-month lag does not have overall significance and has an
adjusted R2 of 1%. This model has virtually no predictability. The model with
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.
16.39-16.41:
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
Chapter 16: Time Series Forecasting and Index Numbers
3 60.373
492.176 61.522
39
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
3 58.088
4 61.443
Quarter
1
2
3
4
Year1
Year2
Year3
Year4
Year5
Year6
Index
97.68
104.06
98.13
100.58
97.91
105.51
96.51
100.93
99.48
103.30
97.04
104.64
95.22
103.59
94.43
100.38
98.09
102.28
97.02
101.07
97.89
103.65
96.86
100.86
Total
Adjust the seasonal indexes by:
399.26
400
= 1.00185343
399.26
Chapter 16: Time Series Forecasting and Index Numbers
40
Adjusted Seasonal Indexes:
16.42
Quarter
Index
1
2
3
4
98.07
103.84
97.04
101.05
Total
400.00
ŷ = 81 + 0.849 x
R2 = 55.8% F = 8.83 with p = .021
se = 50.18
This model with a lag of one year has modest predictability. The overall F is
significant at  = .05 but not at  = .01.
16.43 The regression equation is:
Equity Funds = -591 + 3.01 Taxable Money Markets
R2 = 97.1%
se = 225.9
Yˆ
Equity TaxMkts
44.4
74.5
-366.69
41.2
181.9
- 43.64
53.7
206.6
30.66
77.0
162.5
-101.99
83.1
209.7
39.98
116.9
207.5
33.37
161.5
228.3
95.93
180.7
254.7
175.34
194.8
272.3
228.28
249.0
358.7
488.17
245.8
414.7
656.62
411.6
452.6
770.62
522.8
451.4
767.01
749.0
461.9
798.59
866.4
500.4
914.40
1,269.0
629.7
1,303.33
et
411.091
84.837
23.040
178.991
43.116
83.533
65.568
5.358
-33.482
-239.170
-410.815
-359.017
-244.207
- 49.591
- 47.997
-34.325
et2
168,996
7,197
531
32,038
1859
6,978
4,299
29
1,121
57,202
168,769
128,893
59,637
2,459
2,304
1,178
et – et-1
(et – et-1)2
-326.254
- 61.797
155.951
-135.875
40.417
-17.965
-60.210
-38.840
-205.688
-171.645
51.798
114.810
194.616
1.594
13.672
106,441.673
3,818.869
24,320.714
18,462.016
1,633.534
322.741
3,625.244
1,508.546
42,307.553
29,462.006
2,683.033
13,181.336
37,875.387
2.541
186.924
Chapter 16: Time Series Forecasting and Index Numbers
1,750.9
2,399.3
2,978.2
4,041.9
3,962.3
761.8
898.1
1,163.2
1,408.7
1,607.2
1,700.68
2,110.66
2,908.07
3,646.52
4,243.60
50.224
288.639
70.131
395.378
-281.301
e
t
D =
 (e  e
e
t 1
t
)2
2
t

2
2,522
83,313
4,918
156,323
79,131
41
84.549
7,148.533
238.415 56,841.712
-218.508 47,745.746
325.247 105,785.611
-676.679 457,894.469
 (e  e
= 969,697
t
t 1
)2 = 961,248.188
961,248.188
= 0.99
969,697
For n = 21 and  = .01, dL = 0.97 and dU = 1.16.
Since dL = 0.97 < D = 0.99 < dU = 1.16, the Durbin-Watson test is
inconclusive.
 = .1
16.44
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
PurPwr
6.04
5.92
5.57
5.40
5.17
5.00
4.91
4.73
4.55
4.34
4.67
5.01
4.86
4.72
4.60
4.48
4.86
5.15
F
6.04
6.03
5.98
5.92
5.85
5.77
5.68
5.59
5.49
5.38
5.31
5.28
5.24
5.19
5.13
5.07
5.05
e
 = .5
 = .8
F
e
F
e
.12
.46
.58
.75
.85
.86
.95
1.04
1.15
.71
.30
.42
.52
.59
.65
.21
.10
6.04
5.98
5.78
5.59
5.38
5.19
5.05
4.89
4.72
4.53
4.60
4.81
4.84
4.78
4.69
4.59
4.73
.12
.41
.38
.42
.38
.28
.32
.34
.38
.14
.41
.05
.12
.18
.21
.27
.42
6.04
5.94
5.64
5.45
5.23
5.05
4.94
4.77
4.59
4.39
4.61
4.93
4.87
4.75
4.63
4.51
4.79
.12
.37
.24
.28
.23
.14
.21
.22
.25
.28
.40
.07
.15
.15
.15
.35
.36
= 10.26 .
e
= 4.83
e
= 3.97
e
Chapter 16: Time Series Forecasting and Index Numbers
MAD1 =
e
MAD2 =
e
MAD3 =
e
N
N
N
=
10.26
= .60
17
=
4.83
= .28
17
=
3.97
= .23
17
42
The smallest mean absolute deviation error is produced using  = .8.
The forecast for 1998 is:
F(1998) = (.8)(5.15) + (.2)(4.79) = 5.08
Bankrupcies = 75,532.436 – 0.016 Year
16.45 The model is:
Since R2 = .28 and the adjusted R2 = .23, this is a weak model.
et
- 1,338.58
- 8,588.28
- 7,050.61
1,115.01
12,772.28
14,712.75
- 3,029.45
- 2,599.05
622.39
9,747.30
9,288.84
- 434.76
-10,875.36
- 9,808.01
- 4,277.69
- 256.80
et – et-1
(et – et-1)2
- 7,249.7
1,537.7
8,165.6
11,657.3
1,940.5
-17,742.2
430.4
3,221.4
9,124.9
- 458.5
- 9,723.6
-10,440.6
1,067.4
5,530.3
4,020.9
52,558,150
2,364,521
66,677,023
135,892,643
3,765,540
314,785,661
185,244
10,377,418
83,263,800
210,222
94,548,397
109,006,128
1,139,343
30,584,218
16,167,637
 (e  e
t
D =
 (e  e
e
t 1
2
t
t
)2

t 1
)2 =921,525,945
et2
1,791,796
73,758,553
49,711,101
1,243,247
163,131,136
216,465,013
9,177,567
6,755,061
387,369
95,009,857
86,282,549
189,016
118,273,455
96,197.060
18,298,632
65,946
e
t
2
=936,737,358
921,525,945
= 0.98
936,737,358
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.