Student’s Solutions Manual and Study Guide: Chapter 15Page1
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:
- Differentiate among various measurements of forecasting error, including mean absolute deviation and mean square error, in order to assess which forecasting method to use.
- Describe smoothing techniques for forecasting models, including naive, simple average, moving average, weighted moving average, and exponential smoothing.
- Determine trend in time-series data by using linear regression trend analysis, quadratic model trend analysis, and Holt’s two-parameter exponential smoothing method.
4. Account for seasonal effects of time-series data by using decomposition and Winters’ three-parameter exponential smoothing method.
5. Test for autocorrelation using the Durbin-Watson test, overcoming it by adding independent variables and transforming variables and taking advantage of it with autoregression.
6. Differentiate among simple index numbers, unweighted aggregate price index numbers, weighted aggregate price index numbers, Laspeyres price index numbers, and Paasche price index numbers by defining and calculating each.
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
Unweighted Aggregate Price Index Numbers
Weighted Aggregate Price Index Numbers
Laspeyres Price Index
Paasche Price Index
KEY TERMS
AutocorrelationMoving Average
AutoregressionNaïve Forecasting Methods
Averaging ModelsPaasche Price Index
CyclesSeasonal Effects
Cyclical EffectsSerial Correlation
DecompositionSimple Average
Deseasonalized DataSimple Average Model
Durbin-Watson TestSimple Index Number
Error of an IndividualForecast Smoothing Techniques
Exponential SmoothingStationary
First-Difference ApproachTime-Series Data
ForecastingTrend
Forecasting ErrorUnweighted Aggregate Price
Index Number Index Number
Irregular FluctuationsWeighted Aggregate Price
Laspeyres Price Index Index Number
Mean Absolute Deviation (MAD)Weighted Moving Average
Mean Squared Error (MSE)
STUDY QUESTIONS
1.Shown below are the forecast values and actual values for six months of data:
Month Actual Values Forecast Values
June29 40
July51 37
Aug.60 49
Sept.57 55
Oct. 48 56
Nov.53 52
The mean absolute deviation of forecasts for these data is ______. The mean square
error is ______.
2.Data gathered on a given characteristic over a period of time at regular intervals are referred to as ______.
3.Time series data are thought to contain four elements: ______, ______, ______, and ______.
4.Patterns of data behaviour that occur in periods of time of less than 1 year are called ______effects.
5.Long-term time series effects are usually referred to as ______.
6.Patterns of data behaviour that occur in periods of time of more than 1 years are called ______effects.
7. Consider the time series data below. The equation of the trend line to fit these data is
______.
Year Sales
199728
199831
199939
200050
200155
200258
200366
200472
200578
200690
200797
2008 104
2009 112
8.Time series data are deseasonalized by dividing the each data value by its associated value of ______.
9.Perhaps the simplest of the time series forecasting techniques are ______models in which it is assumed that more recent time periods of data represent the best predictions.
10.Consider the time-series data shown below:
Month Volume
Jan.1230
Feb. 1211
Mar. 1204
Apr. 1189
May 1195
The forecast volumes for April, May, and June are ______, ______, and ______using a three-month moving average on the data shown above and starting in January. Suppose a three-month weighted moving average is used to predict volume figures for April, May, and June. The weights on the moving average are 3 for the most current month, 2 for the month before, and 1 for the other month. The forecasts for April, May, and June are ______, ______, and ______._ using a three-month moving average starting in January.
11.Consider the data below:
Month Volume
Jan. 1230
Feb. 1211
Mar. 1204
Apr. 1189
May 1195
If exponential smoothing is used to forecast the Volume for May using = .2 and using the January actual figure as the forecast for February, the forecast is ______. If = .5 is used, the forecast is ______. If = .7 is used, the forecast is ______. The alpha value of ______produced the smallest error of forecast.
12.______occurs when the error terms of a regression forecasting
model are correlated. Another name for this is ______.
13.The Durbin-Watson statistic is used to test for ______.
14.Examine the data given below.
Year y x
1994 126 34
1995 203 51
1996 211 60
1997 223 57
1998 238 64
1999 255 66
2000 269 80
2001 271 93
2002 276 92
2003 286 97
2004 289 101
2005 294 108
2006 305 110
2007 311 107
2008 324 109
2009 338 116
The simple regression forecasting model developed from this data is
______. The value of R2 for this model is ______. The
Durbin-Watson D statistic for this model is ______. The critical value of
dL for this model using = .05 is ______and the critical value of dU for this
model is ______. This model (does, does not, inconclusive) ______
contain significant autocorrelation.
15.One way to overcome the autocorrelation problem is to add ______
to the analysis. Another way to overcome the autocorrelation problem is to transform
variables. One such method is the ______approach.
16.A forecasting technique that takes advantage of the relationship of values to previous period
values is ______. This technique is a multiple regression
technique where the independent variables are time-lagged versions of the dependent
variable.
17. Examine the price figures shown below for various years.
Year Price
2005 23.8
2006 47.3
2007 49.1
2008 55.6
2009 53.0
The simple index number for 2008 using 2005 as a base year is ______.
The simple index number for 2009 using 2006 as a base year is ______.
18.Examine the price figures given below for four commodities.
Year
Item 2000 2007 2008 2009
11.891.901.871.84
2 .41 .48 .55 .69
3 .76 .73 .79 .82
The unweighted aggregate price index for 2007 using 2000 as a base year is
______. The unweighted aggregate price index for 2008 using 2000 as
a base year is ______. The unweighted aggregate price index for 2009 using
2000 as a base year is ______.
19.Weighted aggregate price indexes that are computed by using the quantities for the year of interest rather than the base year are called ______price indexes.
20.Weighted aggregate price indexes that are computed by using the quantities for the base year are called ______price indexes.
21.Examine the data below.
Quantity Quantity Price Price
Item 2007 2009 2007 2009
1 23 27 1.33 1.45
2 8 6 5.10 4.89
3 61 72 .27 .29
4 17 24 1.88 2.11
Using 2007 as the base year
The Laspeyres price index for 2009 is ______.
The Paasche price index for 2009 is ______.
ANSWERS TO STUDY QUESTIONS
1. 7.83, 84.5,13. Autocorrelation
2. Time Series Data14. , .916,
1.004, 1.10, 1.37, Does
3. Seasonal, Cyclical, Trend, Irregular
15. Independent Variables,
4. Seasonal First-Differences
5. Trend16. Autoregression
6. Cyclical17. 233.6, 112.05
7. = -14,030.35 + 7.038462 x18. 101.6, 104.9, 109.5
8. S19. Paasche
9. Naive Forecasting20. Laspeyres
10. 1215, 1201.3, 1196, 1210.7, 21. 105.18, 106.82
1197.7, 1194.5
11. 1215.21, 1200.63, 1194.64, .7
12. Autocorrelation, Serial Correlation
SOLUTIONS TO PROBLEMS IN CHAPTER 15
15.1 Periodi
1 2.302.30 5.29
2 1.601.60 2.56
3-1.401.40 1.96
4 1.101.10 1.21
5 0.300.30 0.09
6-0.900.90 0.81
7-1.90 1.90 3.61
8-2.10 2.10 4.41
90.70 0.70 0.49
Total -0.30 12.30 20.43
MAD = = 1.367
MSE = = 2.270
15.3 Period Value F
1 19.4 16.6 2.8 2.8 7.84
2 23.6 19.1 4.5 4.5 20.25
3 24.0 22.0 2.0 2.0 4.00
4 26.8 24.8 2.0 2.0 4.00
5 29.2 25.9 3.3 3.3 10.89
6 35.5 28.6 6.9 6.9 47.61
Total 21.5 21.5 94.59
MAD = = 3.583
MSE = = 15.765
15.5a.
Time Period / Value / 4-Month Moving Average /1 / 27 / - / -
2 / 31 / - / -
3 / 58 / - / -
4 / 63 / - / -
5 / 59 / 44.75 / 14.25
6 / 66 / 52.75 / 13.25
7 / 71 / 61.50 / 9.50
8 / 86 / 64.75 / 21.25
9 / 101 / 70.50 / 30.50
10 / 97 / 81.00 / 16.00
Total / 104.75
MAD = = 17.46
b.
Time Period / Value / 4-Month Weighted Moving Average /1 / 27 / - / -
2 / 31 / - / -
3 / 58 / - / -
4 / 63 / - / -
5 / 59 / 53.250 / 5.750
6 / 66 / 56.375 / 9.625
7 / 71 / 62.875 / 8.125
8 / 86 / 67.250 / 18.750
9 / 101 / 76.375 / 24.625
10 / 97 / 89.125 / 7.875
Total / 74.750
MAD = = 12.46
c. Comparing MAD in a. with MAD in b., we conclude that the four-month
moving average produces greater errorof forecast than the four-month
weighted moving average.
15.7
Time Period / Value / Forecast/
1 / 9.4 / - / -
2 / 8.2 / 9.4 / -
3 / 7.9 / 9.0 / 1.1
4 / 9.0 / 8.7 / 0.3
5 / 9.8 / 8.8 / 1.0
6 / 11.0 / 9.1 / 1.9
7 / 10.3 / 9.7 / 0.6
8 / 9.5 / 9.9 / 0.4
9 / 9.1 / 9.8 / 0.7
Total / 6.0
MAD = = 0.86
Time Period / Value / Forecast/
1 / 9.4 / - / -
2 / 8.2 / 9.4 / -
3 / 7.9 / 8.6 / 0.7
4 / 9.0 / 8.1 / 0.9
5 / 9.8 / 8.7 / 1.1
6 / 11.0 / 9.5 / 1.5
7 / 10.3 / 10.6 / 0.3
8 / 9.5 / 10.4 / 0.9
9 / 9.1 / 9.8 / 0.7
Total / 6.1
MAD = = 0.87
Time Period / Value / 3-Month Moving Average /1 / 9.4 / - / -
2 / 8.2 / - / -
3 / 7.9 / - / -
4 / 9.0 / 8.5 / 0.5
5 / 9.8 / 8.4 / 1.4
6 / 11.0 / 8.9 / 2.1
7 / 10.3 / 9.9 / 0.4
8 / 9.5 / 10.4 / 0.9
9 / 9.1 / 10.3 / 1.2
Total / 6.5
MAD = = 1.08
15.9
Year / Number of Issues / Forecast/
1 / 332 / - / -
2 / 694 / 332 / -
3 / 518 / 404 / 114
4 / 222 / 427 / 205
5 / 209 / 386 / 177
6 / 172 / 351 / 179
7 / 366 / 315 / 51
8 / 512 / 325 / 187
9 / 667 / 362 / 305
10 / 571 / 423 / 148
11 / 575 / 453 / 122
12 / 865 / 477 / 388
13 / 609 / 555 / 54
Total / 1,930
MAD = = 175.5
Year / Number of Issues / Forecast/
1 / 332 / - / -
2 / 694 / 332 / -
3 / 518 / 658 / 140
4 / 222 / 532 / 310
5 / 209 / 253 / 44
6 / 172 / 213 / 41
7 / 366 / 176 / 190
8 / 512 / 347 / 165
9 / 667 / 496 / 171
10 / 571 / 650 / 79
11 / 575 / 579 / 4
12 / 865 / 575 / 290
13 / 609 / 836 / 227
Total / 1,661
MAD = = 151.0
Comparing MAD for α = 0.2 with MAD for α = 0.9, we conclude that the
exponential smoothing with α = 0.2 produces greater errorof forecast than the
exponential smoothing withα = 0.9.
15.11 Let x = Year and y = Consumer Price Index for Food.
Linear Trend Analysis:= – 4,663.3 + 2.3803x
F = 483.77 (p-value = 0.000) R2 = 0.96 adjusted R2 = 0.96 se = 3.00
tx = 21.99 (p = .000)
Quadratic Trend Analysis:
= 334,863–336.810x+0.0847128x2
F = 3,931.4 (p = .000) R2 = .998 adjusted R2 = .997 se = 0.76
tx = -16.61 (p = .000), = 16.72 (p = .000)
A comparison of thescatterplots indicates a quadratic fit rather than a linear fit. The quadratic model produced R2 = 0.998 which is greater thanR2 = 0.96 for linear trend. It also confirms a better fit for the quadratic model. In addition, the standard error of the estimate drops from 3.00 to 0.76 with the quadratic model. The t values for both predictors in the quadratic model are significant.Thus,the quadratic regression model is superior; the x2 variable is a significant addition to the model.
15.13
15.15 Regression Analysis: Let X = U.S. Rate and Y = Canadian Rate.
The regression equation is Canadian Rate = 2.425 + 0.4797 U.S. Rate
Predictor Coef t-ratio p
Constant2.425 2.273 0.041
U.S. Rate 0.4797 1.736 0.106
se= 1.180 R-sq = 0.188 R-sq(adj) = 0.126
Year / U.S.Rate / Canadian
Rate / Ŷ / et / et2 / et - et-1 / (et - et-1)2
1996 / 4.2 / 7.2 / 4.440 / 2.760 / 7.618
1997 / 3.7 / 5 / 4.200 / 0.800 / 0.640 / -1.960 / 3.842
1998 / 4.5 / 4.9 / 4.584 / 0.316 / 0.100 / -0.484 / 0.234
1999 / 3.2 / 4.1 / 3.960 / 0.140 / 0.020 / -0.176 / 0.031
2000 / 3.0 / 4.8 / 3.864 / 0.936 / 0.876 / 0.796 / 0.634
2001 / 2.8 / 5.3 / 3.768 / 1.532 / 2.347 / 0.596 / 0.355
2002 / 3.7 / 3.5 / 4.200 / -0.700 / 0.490 / -2.232 / 4.982
2003 / 3.8 / 2.7 / 4.248 / -1.548 / 2.396 / -0.848 / 0.719
2004 / 3.5 / 3.2 / 4.104 / -0.904 / 0.817 / 0.644 / 0.415
2005 / 1.7 / 2.2 / 3.240 / -1.040 / 1.082 / -0.136 / 0.018
2006 / 2.7 / 3.6 / 3.720 / -0.120 / 0.014 / 0.920 / 0.846
2007 / 2.4 / 2.9 / 3.576 / -0.676 / 0.457 / -0.556 / 0.309
2008 / 5.5 / 4.0 / 5.063 / -1.063 / 1.130 / -0.387 / 0.15
2009 / 5.3 / 4.7 / 4.967 / -0.267 / 0.071 / 0.796 / 0.634
2010 / 5.5 / 4.9 / 5.063 / -0.163 / 0.027 / 0.104 / 0.011
Total / 18.085 / 13.180
Using Table A.9 for n = 15, k = 1(1 independent variable), and = .05,
we find the critical values: dL = 1.08 and dU = 1.36. Since D = 0.729is belowdL =
1.08, we reject the null hypothesis. At = .05, there is enough evidence that
significant autocorrelation is present in the model.
15.17The simple regression forecasting model is:
CPI = -41.677 + 1.4200Industry Price Index
R2 = 0.967 adjusted R2 = 0.965 se = 2.093 F = 467.83 p = .000
Year / Industry Price Index / CPI / Ŷ / et / et2 / et- et-1 / (et- et-1)21995 / 91.9 / 87.6 / 88.821 / -1.221 / 1.491
1996 / 92.3 / 88.9 / 89.389 / -0.489 / 0.239 / 0.732 / 0.536
1997 / 92.9 / 90.4 / 90.241 / 0.159 / 0.025 / 0.648 / 0.420
1998 / 93.3 / 91.3 / 90.809 / 0.491 / 0.241 / 0.332 / 0.110
1999 / 94.9 / 92.8 / 93.081 / -0.281 / 0.079 / -0.772 / 0.596
2000 / 99.0 / 95.4 / 98.903 / -3.503 / 12.271 / -3.222 / 10.381
2001 / 100.0 / 97.8 / 100.323 / -2.523 / 6.366 / 0.980 / 0.960
2002 / 100.0 / 100.0 / 100.323 / -0.323 / 0.104 / 2.200 / 4.840
2003 / 98.8 / 102.8 / 98.619 / 4.181 / 17.481 / 4.504 / 20.286
2004 / 102.0 / 104.7 / 103.163 / 1.537 / 2.362 / -2.644 / 6.991
2005 / 103.6 / 107.0 / 105.435 / 1.565 / 2.449 / 0.028 / 0.001
2006 / 106.0 / 109.1 / 108.843 / 0.257 / 0.066 / -1.308 / 1.711
2007 / 107.6 / 111.4 / 111.115 / 0.285 / 0.081 / 0.028 / 0.001
2008 / 112.3 / 114.1 / 117.789 / -3.689 / 13.609 / -3.974 / 15.793
2009 / 108.4 / 114.4 / 112.251 / 2.149 / 4.618 / 5.838 / 34.082
2010 / 109.5 / 116.5 / 113.813 / 2.687 / 7.220 / 0.538 / 0.289
2011 / 114.6 / 119.9 / 121.055 / -1.155 / 1.334 / -3.842 / 14.761
2012 / 115.2 / 121.7 / 121.907 / -0.207 / 0.043 / 0.948 / 0.899
Total / 70.080 / 112.657
From Table A.9 the critical table values for k = 1,n = 18, and α = 0.05 are
dL = 1.16 and dU = 1.39. Sincethe observed value of D = 1.608 is abovedU, we
fail to reject the nullhypothesis. There is no significant autocorrelation.
15.19
Crude Oil / One-Period / Two-PeriodProduction Yt / Lagged Yt-1 (X1) / Lagged Yt-2 (X2)
84.1 / – / –
85.5 / 84.1 / –
90.3 / 85.5 / 84.1
93.9 / 90.3 / 85.5
91.8 / 93.9 / 90.3
91.6 / 91.8 / 93.9
92.0 / 91.6 / 91.8
96.4 / 92.0 / 91.6
101.3 / 96.4 / 92
105.3 / 101.3 / 96.4
110.3 / 105.3 / 101.3
113.5 / 110.3 / 105.3
119.0 / 113.5 / 110.3
124.7 / 119.0 / 113.5
119.9 / 124.7 / 119
124.8 / 119.9 / 124.7
126.6 / 124.8 / 119.9
132.9 / 126.6 / 124.8
140.4 / 132.9 / 126.6
145.8 / 140.4 / 132.9
143.4 / 145.8 / 140.4
151.3 / 143.4 / 145.8
158.0 / 151.3 / 143.4
153.8 / 158.0 / 151.3
152.6 / 153.8 / 158
161.2 / 152.6 / 153.8
168.9 / 161.2 / 152.6
The model with 1 lagged variable:
F = 1.095.56 p = .000 R2 = 0.979 adjusted R2 = 0.978se = 3.87
For the predictor variable Yt-1: t = 33.10 ( p = .000)
The model with 2 lagged variables:
F = 460.30 p = .000 R2 = 0.977 adjusted R2 = 0.975 se = 4.03
For the predictor variable Yt-1: t = 4.56 ( p = .0002)
For the predictor variable Yt-2: t = 0.098 ( p = .923)
Both models have fairly strong predictability. However, the two-period lagged
variable is not significant in the second model, indicating the presence of first-
order autocorrelation.
15.21
a. Using formulae we calculate, for example, the simple index
number for year 1955:
b. Using formulae we calculate, for example, the simple index
number for year 1955:
The table below displays all the index numbers for given data. Note that in a. 1950 is
the base year, Xo = X1950 = 22.45; in b. 1980 is the base year, Xo = X1980 = 69.75.
Year / Price / a. Index(Xo = X1950) / b. Index
(Xo = X1980)
1950 / 22.45 / 100.0 / 32.2
1955 / 31.40 / 139.9 / 45.0
1960 / 32.33 / 144.0 / 46.4
1965 / 36.50 / 162.6 / 52.3
1970 / 44.90 / 200.0 / 64.4
1975 / 61.24 / 272.8 / 87.8
1980 / 69.75 / 310.7 / 100.0
1985 / 73.44 / 327.1 / 105.3
1990 / 80.05 / 356.6 / 114.8
1995 / 84.61 / 376.9 / 121.3
2000 / 87.28 / 388.8 / 125.1
2005 / 89.56 / 398.9 / 128.4
2010 / 93.22 / 415.2 / 133.6
15.23 Year
1995 / 2002 / 20133.37 / 3.08 / 4.77
4.86 / 4.73 / 5.52
4.22 / 5.9 / 5.72
7.44 / 6.82 / 8.80
Total / 19.89 / 20.53 / 24.81
Index2002 = = 103.2
Index2013 = = 124.7
15.25 Quantity Price
Item 2000 2000 2011 2012 2013
1 21 0.50 0.67 0.68 0.71
2 6 1.23 1.85 1.90 1.91
3 17 0.84 0.75 0.75 0.80
4 43 0.15 0.21 0.25 0.25
P2000Q2000 P2007Q2000 P2008Q2000 P2009Q2000
10.50 14.07 14.28 14.91
7.38 11.10 11.40 11.46
14.28 12.75 12.75 13.60
6.45 9.03 10.75 10.75
Totals 38.6146.95 49.18 50.72
Index2011 = = = 121.6
Index2012 = = = 127.4
Index2013 = = = 131.4
15.27a) The linear model:Yield = 9.956 - 0.1403 Month
F = 219.24 p = .000 R2 = .909 adjusted R2 = .905se = .3212
tMonth = -14.807 (p = .000)
The quadratic model: Yield = 10.44 - 0.2516 Month + .004455 Month2
F = 176.21 p = .000 R2 = .944 adjusted R2 = .938 se = .2582
In the quadratic model, both t ratios are significant,
for Month: t = - 7.925, p = .000 and for Month2: t = 3.613, p = .002
Both models are very strong. The quadratic term adds some
predictability but has a smaller t ratio than does the linear term.
b) xF │e │
10.08 - -
10.05 - -
9.24 - -
9.23 - -
9.69 9.65 .04
9.55 9.55 .00
9.37 9.43 .06
8.55 9.46 .91
8.36 9.29 .93
8.59 8.96 .37
7.99 8.72 .73
8.12 8.37 .25
7.91 8.27 .36
7.73 8.15 .42
7.39 7.94 .55
7.48 7.79 .31
7.52 7.63 .11
7.48 7.53 .05
7.35 7.47 .12
7.04 7.46 .42
6.88 7.35 .47
6.88 7.19 .31
7.17 7.04 .13
7.22 6.99 .23
= 6.77
MAD = = .3385
c)
= .3 = .7
x F F
10.08 - - - -
10.0510.08 .0310.08 .03
9.2410.07 .8310.06 .82
9.23 9.82 .59 9.49 .26
9.69 9.64 .05 9.31 .38
9.55 9.66 .11 9.58 .03
9.37 9.63 .26 9.56 .19
8.55 9.551.00 9.43 .88
8.36 9.25 .89 8.81 .45
8.59 8.98 .39 8.50 .09
7.99 8.86 .87 8.56 .57
8.12 8.60 .48 8.16 .04
7.91 8.46 .55 8.13 .22
7.73 8.30 .57 7.98 .25
7.39 8.13 .74 7.81 .42
7.48 7.91 .43 7.52 .04
7.52 7.78 .26 7.49 .03
7.48 7.70 .22 7.51 .03
7.35 7.63 .28 7.49 .14
7.04 7.55 .51 7.39 .35
6.88 7.40 .52 7.15 .27
6.88 7.24 .36 6.96 .08
7.17 7.13 .04 6.90 .27
7.22 7.14 .08 7.09 .13
= 10.06 = 5.97
MAD=.3 = = .4374MAD=.7 = = .2596
= .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) 4-Month 8-Month
T.C.S. I Moving Total Moving Total T.C S. I__
10.08
10.05
38.60
9.2476.819.60 96.25
38.21
9.2375.929.49 97.26
37.71
9.6975.559.44102.65
37.84
9.5575.009.38101.81
37.16
9.3772.999.12102.74
35.83
8.5570.708.84 96.72
34.87
8.3668.368.55 97.78
33.49
8.5966.558.32103.25
33.06
7.9965.678.21 97.32
32.61
8.1264.368.05100.87
31.75
7.9162.907.86100.64
31.15
7.7361.667.71100.26
30.51
7.3960.637.58 97.49
30.12
7.4859.997.50 99.73
29.87
7.5259.707.46100.80
29.83
7.4859.227.40101.08
29.39
7.3558.147.27101.10
28.75
7.0456.907.11 99.02
28.15
6.8856.127.02 98.01
27.97
6.8856.127.02 98.01
28.15
7.17
7.22
1st Period 102.65 97.78 100.64 100.80 98.01
2nd Period 101.81 103.25 100.26 101.08 98.01
3rd Period 96.25 102.74 97.32 97.49 101.10
4th Period 97.26 96.72 100.87 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
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
15.29 Item 2009 2010 2011 2012 2013
1 3.21 3.37 3.80 3.73 3.65
2 0.51 0.55 0.68 0.62 0.59
3 0.83 0.90 0.91 1.02 1.06
4 1.30 1.32 1.33 1.32 1.30
5 1.67 1.72 1.90 1.99 1.98
6 0.62 0.67 0.70 0.72 0.71
Totals 8.14 8.53 9.32 9.40 9.29
Index2010 = = 104.8
Index2011 = = 114.5
Index2012 = = 115.5
Index2013 = = 114.1
15.31 a) and b)
3-year / α =0.2Year / Emissions / Moving Average / │ei│ / F / │ei│
1977 / 406 / – / – / – / –
1978 / 409 / – / – / 406.0 / –
1979 / 423 / – / – / 406.6 / 16.4
1980 / 428 / 412.7 / 15.3 / 409.9 / 18.1
1981 / 411 / 420.0 / 9.0 / 413.5 / 2.5
1982 / 393 / 420.7 / 27.7 / 413.0 / 20.0
1983 / 385 / 410.7 / 25.7 / 409.0 / 24.0
1984 / 402 / 396.3 / 5.7 / 404.2 / 2.2
1985 / 403 / 393.3 / 9.7 / 403.8 / 0.8
1986 / 394 / 396.7 / 2.7 / 403.6 / 9.6
1987 / 406 / 399.7 / 6.3 / 401.7 / 4.3
1988 / 437 / 401.0 / 36.0 / 402.6 / 34.4
1989 / 453 / 412.3 / 40.7 / 409.5 / 43.5
1990 / 429 / 432.0 / 3.0 / 418.2 / 10.8
1991 / 423 / 439.7 / 16.7 / 420.4 / 2.6
1992 / 435 / 435.0 / 0.0 / 420.9 / 14.1
1993 / 435 / 429.0 / 6.0 / 423.7 / 11.3
1994 / 450 / 431.0 / 19.0 / 426.0 / 24.0
1995 / 461 / 440.0 / 21.0 / 430.8 / 30.2
1996 / 476 / 448.7 / 27.3 / 436.8 / 39.2
1997 / 493 / 462.3 / 30.7 / 444.6 / 48.4
1998 / 498 / 476.7 / 21.3 / 454.3 / 43.7
1999 / 508 / 489.0 / 19.0 / 463.0 / 45.0
2000 / 533 / 499.7 / 33.3 / 472.0 / 61.0
2001 / 526 / 513.0 / 13.0 / 484.2 / 41.8
2002 / 533 / 522.3 / 10.7 / 492.6 / 40.4
2003 / 557 / 530.7 / 26.3 / 500.7 / 56.3
2004 / 554 / 538.7 / 15.3 / 512.0 / 42.0
2005 / 559 / 548.0 / 11.0 / 520.4 / 38.6
2006 / 544 / 556.7 / 12.7 / 528.1 / 15.9
2007 / 569 / 552.3 / 16.7 / 531.3 / 37.7
2008 / 551 / 557.3 / 6.3 / 538.8 / 12.2
2009 / 525 / 554.7 / 29.7 / 541.2 / 16.2
2010 / 537 / 548.3 / 11.3 / 538.0 / 1.0
Total / 529.1 / 808.2
MADmoving average =
MAD=.2 =
c)The three-year moving average produced a smaller MAD (17.1) than did
exponential smoothing with = .2 (MAD = 25.3). Using MAD as the criterion, the three-year moving average was a better forecasting tool than the exponential smoothing with = .2.
15.33
Year / Month / Actual Values / Seasonal Indexes S / Deseasonalized Data2010 / January / 1,591 / 77.73 / 2,047
February / 1,337 / 76.17 / 1,755
March / 2,122 / 85.52 / 2,481
April / 2,781 / 125.79 / 2,211
May / 2,216 / 109.35 / 2,027
June / 1,518 / 102.81 / 1,477
July / 1,167 / 77.87 / 1,499
August / 1,998 / 106.33 / 1,879
September / 2,565 / 103.15 / 2,487
October / 2,702 / 119.69 / 2,257
November / 2,224 / 115.12 / 1,932
December / 2,477 / 100.48 / 2,465
2011 / January / 1,478 / 77.73 / 1,901
February / 2,031 / 76.17 / 2,666
March / 2,220 / 85.52 / 2,596
April / 3,436 / 125.79 / 2,732
May / 3,917 / 109.35 / 3,582
June / 2,913 / 102.81 / 2,833
July / 2,415 / 77.87 / 3,101
August / 3,165 / 106.33 / 2,977
September / 2,504 / 103.15 / 2,428
October / 2,994 / 119.69 / 2,501
November / 3,732 / 115.12 / 3,242
December / 2,887 / 100.48 / 2,873
2012 / January / 1,715 / 77.73 / 2,206
February / 2,862 / 76.17 / 3,757
March / 2,324 / 85.52 / 2,717
April / 4,191 / 125.79 / 3,332
May / 2,500 / 109.35 / 2,286
June / 2,488 / 102.81 / 2,420
July / 4,344 / 77.87 / 5,579
August / 3,004 / 106.33 / 2,825
September / 3,632 / 103.15 / 3,521
October / 4,121 / 119.69 / 3,443
November / 3,626 / 115.12 / 3,150
December / 2,963 / 100.48 / 2,949
2013 / January / 3,044 / 77.73 / 3,916
February / 2,128 / 76.17 / 2,794
March / 2,726 / 85.52 / 3,188
April / 3,760 / 125.79 / 2,989
May / 3,805 / 109.35 / 3,480
June / 3,829 / 102.81 / 3,724
July / 2,209 / 77.87 / 2,837
August / 4,482 / 106.33 / 4,215
September / 3,021 / 103.15 / 2,929
October / 3,698 / 119.69 / 3,090
November / 3,888 / 115.12 / 3,377
December / 3,215 / 100.48 / 3,200
2014 / January / 4,097 / 77.73 / 5,271
February / 2,511 / 76.17 / 3,297
March / 3,064 / 85.52 / 3,583
April / 3,879 / 125.79 / 3,084
May / 3,555 / 109.35 / 3,251
June / 3,505 / 102.81 / 3,409
July / 4,715 / 77.87 / 6,055
August / 4,088 / 106.33 / 3,845
September / 3,179 / 103.15 / 3,082
October / 4,210 / 119.69 / 3,517
November / 4,226 / 115.12 / 3,671
December / 2,776 / 100.48 / 2,763
15.35
2011 2012 2013
ItemPrice QuantityPrice QuantityPrice Quantity
Margarine (500 g)1.26 211.32 231.39 22
Shortening (500 g)0.94 50.97 31.12 4
Milk (2 L)1.43 701.56 681.62 65
Cola (2 litres)1.05 121.02 131.25 11
Potato Chips (220 g)2.81 272.86 292.99 28
Total 7.49 7.73 8.37
Index2011 = = 100.0
Index2012 = = 103.2
Index2013 = = 111.7
P2011Q2011 P2012Q2011 P2013Q2011
26.46 27.72 29.19
4.70 4.85 5.60
100.10 109.20 113.40
12.60 12.24 15.00
75.87 77.22 80.73
Totals 219.73 231.23 243.92
IndexLaspeyres2012 = = = 105.2
IndexLaspeyres2013 = = = 111.0
P2011Q2012 P2011Q2013 P2012Q2012 P2013Q2013
28.98 27.72 30.36 30.58
2.82 3.76 2.91 4.48
97.24 92.95 106.08 105.30
13.65 11.55 13.26 13.75
81.49 78.68 82.94 83.72
Total 224.18 214.66 235.55 237.83
IndexPaasche2012 = = 105.1
IndexPaasche2013 = = 110.8
15.37
Year / Month / CPI / 4-Month Moving Average, F / 4-Month WeightedMoving Average, F / et2
in case a.:
4-Month Moving Average, / et2
in case b.
4-Month Weighted
Moving Average
2010 / July / 115.5 / - / - / - / -
August / 115.6 / - / - / - / -
September / 115.8 / - / - / - / -
October / 116.3 / - / - / - / -
November / 116.3 / 115.80 / 115.93 / 0.250 / 0.137
December / 116.0 / 116.00 / 116.13 / 0.000 / 0.017
2011 / January / 116.0 / 116.10 / 116.13 / 0.010 / 0.017
February / 116.2 / 116.15 / 116.09 / 0.002 / 0.012
March / 117.0 / 116.13 / 116.11 / 0.757 / 0.792
April / 117.2 / 116.30 / 116.46 / 0.810 / 0.548
May / 117.8 / 116.60 / 116.82 / 1.440 / 0.960
June / 117.1 / 117.05 / 117.30 / 0.002 / 0.040
July / 117.3 / 117.28 / 117.32 / 0.000 / 0.000
August / 117.8 / 117.35 / 117.33 / 0.203 / 0.221
September / 118.4 / 117.50 / 117.51 / 0.810 / 0.792
October / 118.7 / 117.65 / 117.87 / 1.102 / 0.689
November / 118.8 / 118.05 / 118.29 / 0.563 / 0.260
December / 118.2 / 118.43 / 118.59 / 0.053 / 0.152
2012 / January / 118.4 / 118.53 / 118.50 / 0.017 / 0.010
February / 118.9 / 118.53 / 118.45 / 0.137 / 0.203
March / 119.2 / 118.58 / 118.60 / 0.384 / 0.360
April / 119.7 / 118.68 / 118.85 / 1.040 / 0.723
May / 119.9 / 119.05 / 119.26 / 0.723 / 0.410
June / 119.4 / 119.43 / 119.6 / 0.001 / 0.040
July / 119.3 / 119.55 / 119.59 / 0.063 / 0.084
August / 119.7 / 119.58 / 119.49 / 0.014 / 0.044
September / 119.9 / 119.58 / 119.54 / 0.102 / 0.130
October / 120.2 / 119.58 / 119.67 / 0.384 / 0.281
November / 120.2 / 119.78 / 119.92 / 0.176 / 0.078
December / 119.5 / 120.00 / 120.09 / 0.250 / 0.348
2013 / January / 119.6 / 119.95 / 119.89 / 0.123 / 0.084
February / 120.6 / 119.88 / 119.75 / 0.518 / 0.722
March / 120.9 / 119.98 / 120.04 / 0.846 / 0.740
April / 121.0 / 120.15 / 120.41 / 0.722 / 0.348
May / 121.2 / 120.53 / 120.75 / 0.449 / 0.203
June / 121.0 / 120.93 / 121.02 / 0.005 / 0.000
Total / 11.956 / 9.445
MSEMoving Average =
MSEWeightedMoving Average =
The weighted moving average does a better job of forecasting the data using MSE as the criterion.
15.39
Time Period / Actual Value / 4-QuarterMoving
Total / 4-Quarter
2-Year
Moving
Total / Ratios of Actual Centered Moving Average / Values to Moving
Averages
1st quarter (year 1) / 54.019
2nd quarter / 56.495
213.574
3rd quarter / 50.169 / 425.044 / 53.131 / 94.43
211.470
4th quarter / 52.891 / 421.546 / 52.693 / 100.38
210.076
1st quarter (year 2) / 51.915 / 423.402 / 52.925 / 98.09
213.326
2nd quarter / 55.101 / 430.997 / 53.875 / 102.28
217.671
3rd quarter / 53.419 / 440.49 / 55.061 / 97.02
222.819
4th quarter / 57.236 / 453.025 / 56.628 / 101.07
230.206
1st quarter (year 3) / 57.063 / 467.366 / 58.421 / 97.68
237.160
2nd quarter / 62.488 / 480.418 / 60.052 / 104.06
243.258
3rd quarter / 60.373 / 492.176 / 61.522 / 98.13
248.918
4th quarter / 63.334 / 503.728 / 62.966 / 100.58
254.810
1st quarter (year 4) / 62.723 / 512.503 / 64.063 / 97.91
257.693
2nd quarter / 68.38 / 518.498 / 64.812 / 105.51
260.805
3rd quarter / 63.256 / 524.332 / 65.542 / 96.51
263.527
4th quarter / 66.446 / 526.685 / 65.836 / 100.93
263.158
1st quarter (year 5) / 65.445 / 526.305 / 65.788 / 99.48
263.147
2nd quarter / 68.011 / 526.72 / 65.840 / 103.30
263.573
3rd quarter / 63.245 / 521.415 / 65.177 / 97.04
257.842
4th quarter / 66.872 / 511.263 / 63.908 / 104.64
253.421
1st quarter (year 6) / 59.714 / 501.685 / 62.711 / 95.22
248.264
2nd quarter / 63.59 / 491.099 / 61.387 / 103.59
242.835
3rd quarter / 58.088
4th quarter / 61.443
Seasonal Indexes
Quarter / Year 1 / Year 2 / Year 3 / Year 4 / Year 5 / Year 6 / Final Index / Adjusted
Index
1 / 98.09 / 97.68 / 97.91 / 99.48 / 95.22 / 97.89 / 98.07
2 / 102.28 / 104.06 / 105.51 / 103.30 / 103.59 / 103.65 / 103.84
3 / 94.43 / 97.02 / 98.13 / 96.51 / 97.04 / 96.86 / 97.04
4 / 100.38 / 101.07 / 100.58 / 100.93 / 104.64 / 100.86 / 101.05
Total / 399.86
Adjust the seasonal indexes by:
The adjusted final seasonal indexes are shown in the table above.
15.41Let x = Time Period (from the 1st quarter (1st of year 1) to the 24th quarter (4th of
year 6) and y = Industrial machinery and equipment shipments.
Linear Model: = 53.410 + 0.53249x
F = 27.65 p = .000 R2 = 0.557 adjusted R2 = 0.537 se = 3.43
For the predictor variable x: t = 5.26 ( p = .000)
Quadratic Model: = 47.687 + 1.8533 x –0.052834 x2
F = 34.37 p = .000 R2 = 0.766 adjusted R2 = 0.744 se = 2.55
For the predictor variable x: t = 5.90 ( p = .000)
For the predictor variable x 2: t = -4.33 ( p = .0003)
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.
15.43The following regression equation was obtained:
Foreign Inflows = -7,985.50 +1.111508 Foreign Outflows.
Note that Y = Foreign Inflows and X = Foreign Outflows.
Year / US $ millions Inflows / US $millionsOutflows / Ŷ / et / et2 / et-et-1 / (et-et-1) 2
1996 / 9,633 / 13,094 / 6,568.6 / 3,064.4 / 9,390,547.36
1997 / 11,522 / 23,059 / 17,644.8 / -6,122.8 / 37,488,679.84 / -9,187.2 / 84,404,643.84
1998 / 22,803 / 34,349 / 30,193.7 / -7,390.7 / 54,622,446.49 / -1,267.9 / 1,607,570.41
1999 / 24,747 / 17,250 / 11,188.0 / 13,559.0 / 183,846,481.00 / 20,949.7 / 438,889,930.09
2000 / 66,796 / 44,678 / 41,674.5 / 25,121.5 / 631,089,762.25 / 11,562.5 / 133,691,406.25
2001 / 27,670 / 36,037 / 32,069.9 / -4,399.9 / 19,359,120.01 / -29,521.4 / 871,513,057.96
2002 / 22,146 / 26,761 / 21,759.6 / 386.4 / 149,304.96 / 4,786.3 / 22,908,667.69
2003 / 7,619 / 21,526 / 15,940.8 / -8,321.8 / 69,252,355.24 / -8,708.2 / 75,832,747.24
2004 / 1,533 / 43,248 / 40,085.0 / -38,552.0 / 1,486,256,704.00 / -30,230.2 / 913,864,992.04
2005 / 33,824 / 34,084 / 29,899.1 / 3,924.9 / 15,404,840.01 / 42,476.9 / 1,804,287,033.61
2006 / 59,765 / 44,404 / 41,369.9 / 18,395.1 / 338,379,704.01 / 14,470.2 / 209,386,688.04
2007 / 114,642 / 57,719 / 56,169.6 / 58,472.4 / 3,419,021,561.76 / 40,077.3 / 1,606,189,975.29
2008 / 57,147 / 79,752 / 80,659.5 / -23,512.5 / 552,837,656.25 / -81,984.9 / 6,721,523,828.01
2009 / 21,438 / 41,728 / 38,395.5 / -16,957.5 / 287,556,806.25 / 6,555.0 / 42,968,025.00
2010 / 23,412 / 38,583 / 34,899.8 / -11,487.8 / 131,969,548.84 / 5,469.7 / 29,917,618.09
2011 / 40,929 / 49,566 / 47,107.5 / -6,178.5 / 38,173,862.25 / 5,309.3 / 28,188,666.49
Total / 7,274,799,380.52 / 12,985,174,850.05
From Table A.9 the critical table values for k = 1,n = 16, and α = 0.01 are
dL = 0.84 and dU = 1.09. Sincethe observed value of D = 1.785 is abovedU, we
fail to reject the nullhypothesis. At α = 0.01, there is not sufficient sample
evidence that significant autocorrelation is present in the model.
15.45The model is: Business Bankruptcy = 75,532.43621 – 0.01574Consumer Bankruptcy
Since R2 = .28 and the adjusted R2 = .23, the model has very modest predictability. Since F = 5.44 with p = 0.0351, the model is significant overall at α = .05, but not significant at α = .01.
et / et2 / et – et-1 / (et – et-1)2-1,338.6 / 1,791,849.96
-8,588.3 / 73,758,896.89 / -7,249.7 / 52,558,150.09
-7,050.6 / 49,710,960.36 / 1,537.7 / 2,364,521.29
1,115.0 / 1,243,225.00 / 8,165.6 / 66,677,023.36
12,772.3 / 163,131,647.29 / 11,657.3 / 135,892,643.29
14,712.8 / 216,466,483.84 / 1,940.5 / 3,765,540.25
-3,029.4 / 9,177,264.36 / -17,742.2 / 314,785,660.84
-2,599.1 / 6,755,320.81 / 430.3 / 185,158.09
622.4 / 387,381.76 / 3,221.5 / 10,378,062.25
9,747.3 / 95,009,857.29 / 9,124.9 / 83,263,800.01
9,288.8 / 86,281,805.44 / -458.5 / 210,222.25
-434.8 / 189,051.04 / -9,723.6 / 94,548,396.96
-10,875.4 / 118,274,325.16 / -10,440.6 / 109,006,128.36
-9,808.0 / 96,196,864.00 / 1,067.4 / 1,139,342.76
-4,277.7 / 18,298,717.29 / 5,530.3 / 30,584,218.09
-256.8 / 65,946.24 / 4,020.9 / 16,167,636.81
Total / 936,739,596.73 / 921,526,504.70
D = = 0.98
From Table A.9 for k = 1, n = 16, = .05: dL = 1.10 and dU = 1.37;
= .01: dL = 0.84 and dU = 1.09.
Since D = 0.98 < dL = 1.10, the decision is to reject the null hypothesis and
conclude that there is significant autocorrelation at = .05.
In case of = .01, we have 0.84 < D = 0.98 < 1.09. Then, the Durbin-Watson test
is inconclusive.
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(MMXIII xii FI)
Black, Chakrapani, Castillo: Business Statistics, Second Canadian Edition