CHAPTER 2 STATISTICAL FUNDAMENTALS FOR FORECASTING

1
CHAPTER 2STATISTICAL FUNDAMENTALS FOR FORECASTING

• THE IMPORTANCE OF PATTERN
• Descriptive and Graphical Tools
• Probability Distributions
• UNIVARIATE SUMMARY STATISTICS
• Using Mean, Median, or Mode
• Properties of Central Values
• Mean Forecast Error

2

• MEASURING ERRORS - STANDARD DEVIATION/MAD
• NORMAL DISTRIBUTION
• Characteristics of the Normal Distribution (ND)
• Describing All Normal Distributions
• Prediction Intervals
• MAD - Measure of Scatter
• An Example Using Sales of Product A
• Frequency Distribution Solution

3

• FITTING VERSUS FORECASTING
• Absolute Error Measures
• Relative Measures of Error
• Cautions in Using Percentages
• Other Error Measures
• STATISTICAL SIGNIFICANCE TEST FOR BIAS
• CORRELATION AND COVARIANCES
• Correlation - Big City Bookstore
• Statistical Significance
• Cause and Effect

4

• AUTOCORRELATIONS AND ACF(k)
• ACFs of Random Series, Random Walk Series,
  Trending Series, Seasonal Series
• Which Measure of Correlation?
• ACFs of Births and Marriages
• SUPPLEMENT 2-A E(VALUES), WN, AND CORR.
• SUPPLEMENT 2-B Q-STATISTIC

5
Chapter 2

• THE IMPORTANCE OF PATTERN
• Actual Value Pattern Error
• DESCRIPTIVE STATISTICS
• Statistical analysis models the past to predict
  the future
• FIGURE 2-1.
• Demand for Product A (Boxes of Computer Paper) A
  series with Random Sales
• The past may be a representative sample of past
  and future values.

6

• Descriptive and Graphical Tools
• Raw data ? Information (i.e. Pattern) ?
  Forecasts ? Decisions
• Series A Distribution
• Demands from 675 to 1,024 100.00
• Demands from 825 and 874 29.20
• Demands above 974 2.08
• Demands above 1,024 0.00
• Demand from 775 to 924 79.20

7

• TABLE 2-1. Frequency/Probability Distribution of
  Demands for Product A Probabilities
• Past percentage frequencies
• P(Rain) 600/1,000 .6 60
• Subjective judgment
• Theoretical structure (e.g., Construction of a
  Die)

8
Probability Distributions in Forecasting

• Assumption 1 The past repeats.
• Assumption 2 The sample is accurate.
• FIGURE 2-2.
• Probability Distribution of Demands for Product A

9
UNIVARIATE SUMMARY STATISTICS

• Predicting Using Mean, Median, or Mode
• Time t 1 2 3 4 5 6 7
• Values Xt 11 4 5 12 9 2 6
• Mean
• SXt 114512926 49
• t 7 (2-1)
• n 7 7
• where SXt is the sum from t 1 to t 7

10

• Deviations are
• Xt - xt xt
• (11 - 7) 4
• ( 4 - 7) -3
• ( 5 - 7) -2
• (12 - 7) 5
• ( 9 - 7) 2
• ( 2 - 7) -5
• ( 6 - 7) -1
•  
• Sxt 4 - 3 - 2 5 2 - 5 - 1 0

11

• Median is the middle value
• 2 4 5 6 9 11 12
• Mode is most frequent
• 1 2 5 6 6 7 10 11
• Comparisons of Measures
• Symmetrical Distributions
• Mean, Median, and Mode are all equal
• Properties of Central Values
• Mean - greatly affected by extremes.
• Outliers greatly affect it.
• Median less affected by extremes.
• Mode not affected by extremes.

12

• Forecast period 9.
•  
• t 1 2 3 4 5 6
  7 8
• Xt 100 70 90 110 1,200 110 130 80
•  
• Ranking, the sales are
• 70 80 90 100 110 110 130 1,200
•  

13
Which is best to forecast Period 9?

• 10070901101,20011013080
• Mean
• 8
• 236.25
• Median (100 110)/2 105
• Mode 110
•  
• 1,200 Greatly Influences the Mean. If 1,200 is a
  typical value, then use the mean of 236.25.

14

• If 1,200 is abnormal, then Median. Better Yet,
  Correct the 1,200 Value.  
• 100709011013011013080
• Mean
• 8
• 102.5
• Median (100 110)/2 105
• Mode 110
• Outliers should normally be replaced. But Retain
  the Value of the Outlier for Planning Purposes.  

15
Outlier Adjustments Are Essential

• Normality is achieved through elimination of the
  abnormal.

16
Mean Forecast Error

• Good Models ME 0, MSE 0
• Over and under forecasting the same.
• Nonzero mean errors yield large cumulative
  errors.

17
Dispersion - Standard Dev. and MAD

• FIGURE 2-3
• Four Distributions with the Same Mean but
  Different Scatter
• Standard deviation Mean Squared Error
• S standard deviation from sample
• s standard deviation from census

18

• s is estimated using the Xbart and S of large
  samples (i.e., sample sizes greater than 30)
  provide accurate estimates of s.
• S(Xt m)2
• s (2-3)
• N
• where m population mean
• N population size.
• S sum over all obs (i.e., a census)

19

• The best estimate of s is S
• S(Xt x)2
• S (2-4)
• n - 1
• where x sample mean
• n-1 better estimates s.

20

• An Example
• t 1 2 3 4 5 6 7
• Xt 11 4 5 12 9 2 6
• x 7
• (11-7)2(4-7)2(5-7)2(12-7)2(9-7)2(2-7)2(6
  -7)2
• S2 7 - 1
• S 3.742, used to estimate s.
• The meaning of S is clearer with ND

21
NORMAL DISTRIBUTION

• FIGURE 2-4. Normal Distribution Area
• Characteristics of the Normal Distribution (ND)
• Is symmetrical, bell-shaped.
• Describes Events with relatively large number of
  minor, independent,chance (random) influences.
• The errors or deviations of large samples are ND
  large gt 30.
• Is defined totally by m and s.

22

• TABLE 2-2.
• Some Standard Confidence Intervals for the
  Normal Distribution
• Describing All Normal Distributions
• The m /-s contains 68 of the ND
• The m /- 1.96s contains 95 of the ND.
• The m /- 2.58s contains 99 of the ND.

23

• Forecast error Actual - Forecast
• Actual Forecast Error
• e.g., forecast 1000 and S of 40
• Actuals Forecast Mean Error /- ZS
• The m /-s contains 68 of the ND
• Actuals 1,000 /- 40 960 to 1,040.
• The m /- 1.96s contains 95 of the ND.
• Actual 1,000 /- 80 920 to 1,080.
• The m /- 2.58s contains 99 of the ND.
• Other common intervals are in Table 2-2.

24
Prediction Intervals

• Based on assumption the past will repeat.
• P(ActualgtForecast3S a good process) .0027/2
  .00135
• P(ActualltForecast-3S a good process) .0027/2
  .00135
• where P is the probability
• means given

25

• If Actual - Forecast gt 3S
• Then infer the process is out of control.
• Only 27/10,000 times will this occur when in
  control.

26
MAD Another Measure of Scatter

• S Xt x
• MAD (2-5)
• n
• Using data from the S calculation,
• 11 4 5 12 9 2 6 with x 7
• 11-74-75-712-79-72-7
  6-7
• MAD
• 7
• 3.143

27

• For the ND
• MAD .80S or S 1.25MAD (2-6)
• In terms of S the following results
• Mean /- 3.00S Mean and - 3.00(40)
• Mean and - 120
• In terms of MAD
• Mean /- 3.75MAD Mean and -3.75(32)
• Mean and - 120

28
An Example Using Sales of Product A

• The series varies randomly about a constant mean
  of 850 without any systematic pattern, an
  effective forecast is the mean of 850.
• Forecast 850
• Error Actual - Forecast Actual - 850
• Table 2-3 Frequency Distribution of Forecast
  Errors for Product A

29
FIT VERSUS FORECAST ? FIT THEN FORECAST

• Fit - use past to fit model
• Forecast - Forecast unknown future values
• WWFATAL.DAT from 1970 to 1989
• Fit 1970 to 1979 then
• Forecast 1980 to 1989. Assume Deaths are Random
  About Mean
• YtYt et 830.2 et
• Where Yt mean of 1970 to 1979 830.2

30

• Table 2-4 Worldwide Airline Deaths - Actual and
  Fitted Values
• DATE DEATHS FITTED ERROR ERROR
  ERROR2
• 1970 700 830.2 -130.2
  130.2 16952.04
• 1971 884 830.2 53.8
  53.8 2894.44
• 1972 1209 830.2 378.8
  378.8 143489.44
• 1973 862 830.2 31.8
  31.8 1011.24
• 1974 1299 830.2 468.8
  468.8 219773.44
• 1975 467 830.2 -363.2
  363.2 131914.24
• 1976 734 830.2 -96.2
  96.2 9254.44
• 1977 516 830.2 -314.2
  314.2 98721.64
• 1978 754 830.2 -76.2
  76.2 5806.44
• 1979 877 830.2 46.8
  46.8 2190.24
• TOTAL 8302 8302 0.0
  1960 632007
• MEAN 830.2 830.2 0.0
  196.0 63200.7

31

• Absolute Error Measures, et Yt - Yt
• n
• ME Set/n 0/10 0 (2-7)
• t1
• n
• MAD Set/n 1960/10 196 (2-8)
• t1
• n
• SSE Se2t 632,007 (2-9)
• t1
• n
• MSE Se2t/n 63,200.7 (2-10)
• t1

32

• RESIDUAL STANDARD ERROR
• RSE Se2t/(n-1)
• 632,007/(10-1) 265 (2-11)

33

• Table 2-5 Worldwide Airline Deaths - Actual and
  Forecasted Values
• DATE DEATHS FORECAST ERROR ERROR
  ERROR2
• 1980 817 830.2
  -13.2 13.2 174.24
• 1981 362 830.2
  -468.2 468.2 219211.24
• 1982 764 830.2
  -66.2 66.2 4382.44
• 1983 809 830.2
  -21.2 21.2 449.44
• 1984 223 830.2
  -607.2 607.2 368691.84
• 1985 1066 830.2
  235.8 235.8 55601.64
• 1986 546 830.2
  -284.2 284.2 80769.64
• 1987 901 830.2
  70.8 70.8 5012.64
• 1988 729 830.2
  -101.2 101.2 10241.44
• 1989 825 830.2
  -5.2 5.2 27.04

34

• The forecast error measures are
• n
• ME Set/n -1260/10 -126 (2-7a)
• t1
• n
• MAD Set/n1873.2/10187.32 (2-8a)
• t1
• n
• SSE Se2t 744,561.6 (2-9a)
• t1
• n
• MSE Se2t/n 74,456.16 (2-10b)
• t1

35

• RSE Se2t/(n-1)
• 744561.6/9 287.6 (2-11b)

36
Relative Measures of Error

• S (Yt - Yt)
• PEt (100) (2-12)
• Yt
• n
• MPE SPEt/n (2-13)
• t1
• n
• MAPE SPet/n (2-14)
• t1

37

• TABLE 2-6 RELATIVE MEASURES OF FIT AND FORECAST
  ERRORS
• DATE DEATHS FIT ERROR PE
  APE
• 1970 700 830.2 -130.2
  -18.60 18.60
• 1971 884 830.2
  53.8 6.09 6.09
• 1972 1209 830.2 378.8
  31.33 31.33
• 1973 862 830.2
  31.8 3.69 3.69
• 1974 1299 830.2 468.8
  36.09 36.09
• 1975 467 830.2 -363.2
  -77.77 77.77
• 1976 734 830.2
  -96.2 -13.11 13.11
• 1977 516 830.2 -314.2
  -60.89 60.89
• 1978 754 830.2
  -76.2 -10.11 10.11
• 1979 877 830.2
  46.8 5.34 5.34
• MEAN 830.2 830.2 0.0
  -9.80 26.30

38

• DATE DEATHS FORECAST ERROR PE
  APE
• 1980 817 830.2
  -13.2 - 1.62 1.62
• 1981 362 830.2
  -468.2 -129.34 29.34
• 1982 764 830.2
  -66.2 -8.66 8.66
• 1983 809 830.2
  -21.2 -2.63 2.62
• 1984 223 830.2
  -607.2 -272.29 272.29
• 1985 1066 830.2
  235.8 22.12 22.12
• 1986 546 830.2
  -284.2 -52.05 52.05
• 1987 901 830.2
  70.8 7.86 7.86
• 1988 729 830.2
  -101.2 -13.88 13.88
• 1989 825 830.2
  -5.2 -.63 .63
• MEAN 704.2 830.2
  -126.0 -45.11 51.11

39

• Cautions in Using Percentages
• PE ? Infinity with small denominators in eq.
  2-12, when the actual is very low.
• OCCAM'S RAZOR and PARSIMONY
• All other things equal, the simplest theory or
  model is the best.

40
Statistical Significance Test for Bias or
Non-zero Mean Error

• et - 0
• t-calc.
• Se/ n
• ( et - et)2
• Se this is the Std Dev.
• n-1
• If t-calc lt t-table, not statistically
  significant Bias
• If t-calc gt t-table, statistically significant
  Bias

41

• CORRELATION MEASURES
• Table 2-7. Big City Bookstore Demand,
  Advertising, and Competition. (BIGCITY.DAT)
• Year Demand (Y) Advertising (X1)
  Competition (X2)
• 1984 27 20
  10
• 1985 23 20
  15
• 1986 31 25
  15
• 1987 45 28
  15
• 1988 47 29
  20
• 1989 42 28
  25
• 1990 39 31
  35
• 1991 45 34
  35
• 1992 57 35
  20
• 1993 59 36
  30
• 1994 73 41
  20
• 1995 84 45
  20
• Demand for Books Sales in 1000.
• Advertising Expenditures in 1000.

42
CORRELATIONS AND COVARIANCES

• Association can be measured by the degree that
  variables covary (e.g. high values of Y with high
  values of X and low values of Y with low values
  of X). Covariance
• S (Xt- X)(Yt-Y)
• COV(X,Y) (2-16)
• n - 1

43

• Y Y X
• 3 1 2
• 2 4
• 2 3 6
• 1
• 1 2 3 4 5 6 X
• Y 2 X 4

44

• Figure 2-7. Example 1 Data
• Covariance Example 1
• X Y X-X Y-Y (X-X)(Y-Y)
• 6 3 2 1 2
  4
• 4 2 0 0 0 COV
  2
• 2 1 -2 -1 2
  3-1
• 4

45

• 4 Y X
• 4 2
• 3 1 4
• 4 6
• 2
• 1
• 1 2 3 4 5 6 X
• Y 3 X 4

46

• Figure 2-8. Example 2 Data
• Covariance Example 2
• X Y X-X Y-Y (X-X)(Y-Y)
• 6 4 2 1 1
  0
• 4 1 0 -2 0 COV
  0
• 2 4 -2 1 -1
  3-1
• 0

47
Correlation A Relative Measure of Association

• Pearson Correlation Coefficient
• If r -1, then there is a perfect negative
  relationship.
• If r 0, then there is no relationship.
• If r 1, then there is a perfect positive
  relationship.

48

• Pearson Correlation
• COV(X,Y)
• rxy (2-17)
• SxSy
• where S(X - X)2 S(Y - Y)2
• Sx Sy
• n-1 n-1

49

• The correlation coefficient for Example 1 with
  perfectly related variables
• 2202(-22) 404
• Sx 4 2
• 3-1 2
• 120212
• Sy 1
• 3-1
• COV(X,Y) 2
• r(xy) 1
• SxSy 21
• X and Y have a perfect correlation of 1.

50

• rxy of 1 means that 1Std Dev D in X is associated
  with a 1Std Dev D in Y and vice versa.
• Example 2, independent variables
• COV(X,Y) 0
• rxy 0
• SxSy SxSy

51

• Correlation Coefficient
• Table 2-8. Big City Bookstore Sums of Squares.
• DEMAND ADVERTISING
  
• YEAR Y X (Y-Y) (X-X)
  (Y-Y)2 (X-X)2 (Y-Y)(X-X)
• 1984 27 20 -20.67 -11.00
  427.11 121.00 227.33
• 1985 23 20 -24.67 -11.00
  608.44 121.00 271.33
• 1986 31 25 -16.67 -6.00
  277.78 36.00 100.00
• 1987 45 28 -2.67 -3.00
  7.11 9.00 8.00
• 1988 47 29 -.67 -2.00
  .44 4.00 1.33
• 1989 42 28 -5.67 -3.00
  32.11 9.00 17.00
• 1990 39 31 -8.67 .00
  75.11 .00 .00
• 1991 45 34 2.67 3.00
  7.11 9.00 -8.00
• 1992 57 35 9.33 4.00
  87.11 16.00 37.33
• 1993 59 36 11.33 5.00
  128.44 25.00 56.67
• 1994 73 41 25.33 10.00
  641.78 100.00 253.33
• 1995 84 45 36.33 14.00
  1320.11 196.00 508.67
• SUM
  3612.67 646.00 1473.00
• MEAN 47.67 31

52

• From Table 2-8 we have
• S (Y - Y)2 3612.67
• Sy 18.1225
• n - 1 12 - 1
• S (X - X)2 646.00
• Sx 7.663
• n - 1 12 - 1
  
• S (Y-Y)(X-X) 1473
• COV(X,Y) 133.91
• n-1 12-1
  

53

• COV(X,Y) 133.91
• rxy .9644
• SxSy 18.127.663
  
• FIGURE 2-9. SEVERAL CORRELATIONS COEFFICIENTS.

54
Statistical Sign. of the Cor. Coef.

• Bivariate relationship is ND with a standard
  deviation called a standard error equal to
• Ser (1 - r2)/(n - 2)
• where r equals rxy and n-2 makes Ser a better
  estimate of the population Fer.
• tr (r - 0)/ (1-r2)/(n-2) (2-18)

55

• The statistical hypotheses are
• H0 r 0 The variables are not related.
• H1 r ? 0 The variables are related.
• If tr lt t from the table with n-2 degrees of
  freedom, then infer no relationship between Y and
  X and accept H0 and H1.
• If tr gt t from the table with n-2 degrees of
  freedom, then infer statistical significance and
  reject H0 and accept H1.

56

• TABLE 2-9
• t and Z for .05 and .01 Probabilities
• df t-value Z-value
• n-k .05 .01 .05
  .01
• 1 12.706 63.657 n.a. n.a.
• 2 4.303 9.925 n.a.
  n.a.
• 3 3.182 5.841 n.a.
  n.a.
• 4 2.776 4.604 n.a.
  n.a.
• 5 2.571 4.032 n.a.
  n.a.
• 6 2.447 3.707 n.a.
  n.a.
• 7 2.363 3.499 n.a.
  n.a.
• 8 2.306 3.355 n.a.
  n.a.
• 10 2.228 3.169 n.a.
  n.a.
• 15 2.131 2.947 n.a.
  n.a.
• 20 2.086 2.845 n.a. n.a.

57

• TABLE 2-9
• t and Z for .05 and .01 Probabilities
• df t-value
  Z-value
• n-k .05 .01 .05
  .01
• 30 2.042 2.750 n.a.
  n.a.
• 40 2.02 2.70 1.96
  2.58
• 50 2.01 2.68 1.96
  2.58
• 100 1.98 2.63 1.96
  2.58
• 500 1.96 2.58 1.96
  2.58
• df effective no. of observations)
• k 2 for correlation coefficient significance
  tests.,
• n.a. not applicable, use the t value.

58

• When a sample size of 10 is used, 95 of the
  calculated r's will be within 2.23 times Ser.
  Assuming that r .9644, the Ser is
• 1-r2 1-.96442
• Ser .08367
• n-2 12-2
• r - 0 .9644 - 0
• tr 11.536
• Ser .08367

59

• t -2.228 t 2.228
  t 11.536
• -2.228.084 2.228.084
• -.187 .187
• when true r 0,
• then, 95 of
• sample rlt.0156
• rxy
• -.187 0
  .187 .9644
• Figure 2-10. Normally Distributed rxy when
  population rxy 0, n-2 10. Since 11.536 is
  greater than the critical value of 2.228 and 3.17
  from Table 2-7 for n-k of 10, then conclude that
  advertising and demand are significantly
  correlated.

60

• Cause and Effect X causes Y
• X and Y are correlated.
• Changes in X precede changes in Y
• There are no other possible explanations for
  changes in Y.
• All other influences have been eliminated as
  causes of Y.
• Correlation Coefficients Measure Linear
  Association
• X' 13 - 6X .75X2

61
AUTOCORRELATIONS AND ACF(k)

• Detecting Univariate Patterns Using Correlations.
• A way to detect association is to graph Yt and
  Yt-7.
• However, this may not be an objective way of
  detection.

62

• Table 2-10. Lagged Values of Yt
• t Yt Yt-1 Yt-2 Yt-3
• 1 3
• 2 6 3
• 3 8 6 3
• 4 4 8 6 3
• 5 4 4 8 6
• 6 8 4 4 8
• Yt6 Yt-15 both for observations 1 to 6

63

• Pearson Autocorrelation
• COV(Yt,Yt-k)
• rYtYt-k (2-19)
• SYtSYt-k

64

• Table 2-11. Calculation of Pearson
  Autocorrelation - rYtYt-1
• PRODUCT
• t ytYt-Yt yt-1Yt-1-Yt-1 ytyt-1 (Yt-Yt)2
  (Yt-1-Yt-1)2
• 1 n.a.
• 2 0 -2 0 0
  4
• 3 2 1 2 4
  1
• 4 -2 3 -6 4
  9
• 5 -2 -1 2 4
  1
• 6 2 -1 -2 4
  1
• SUM 0 0 -4 16
  16

65

• Using the results of Table 2-9
• 16 16
  
• SYt 2 Syt-1 2
• 5-1 5-1
  
• -4
• COV(Yt,Yt-1) -1
• 5-1
• -1 -1
• rYtYt-1 -.2500
• 22 4

66

• AutoCorrelation Function (ACF)
• n-k
• S (Yt-y)(Yt-k-y)
• t1k
• ACF(k) (2-20)
• n
• S (Yt-y)2
• t1
• ACF uses an overall mean without adjusting the
  denominator, as shown Table 2-12 it's less
  accurate as k increases.
• Note that Pearson and ACFs are symmetrical about
  lag of 0 (i.e. YtYt).

67

• Table 2-12. Pearson Correlations VS. ACFs
• YtYt-2 YtYt-1 YtYt YtYt1 YtYt2
• PEARSON -.9113 -.250 1.00 -.250 -.9113
• ACF -.6170 -.223 1.00 -.223
  -.6170

68

• Approximate Standard Error of ACFs
• SeACF 1/ n (2-21)
• where SeACF standard error of ACF
• n no. of obs. in series
• ACF t-test
• ACF(k)
• t (2-22)
• SeACF

69

• Consider a simple example of SeACF for n100,
  ACF(1) .5
• SeACF 1/100.5 1/.1 .10
• t .5/.1 5 t-calculated
• This t value gtgtgt 2, Infer ACF gt 0
• Infer there is a statistically significant
  autocorrelation between Yt and Yt-1.

70

• Consider ACFs for SERIESB.DAT, stock prices in
  Table 2-13.
• The ACFs are very high starting at .9274 at lag 1
  to .0873 through lag 12.
• An approximate SeACF is
• 1 1
• SeACF .1443
• 48.5 6.9282

71

• and the appropriate t-test is
• ACF(1) .9274
• t 6.425
• SeACF .1443
• The first 6 ACFs are statistically significant.
• Remember that the SeACF is only an approximation,
  valid for n/4 where n is at least 50.

72

• Table 2-13. ACFs of Series B
• BtBt-1 BtBt-2 BtBt-3 BtBt-4 BtBt-5
  BtBt-6
• .9274 .8250 .6932 .5636 .4485 .3415
• BtBt-7 BtBt-8 BtBt-9 BtBt-10 BtBt-11
  BtBt-12
• .2503 .1740 .1154 .0870 .0855 .0873

73
Pattern Recognition with ACFs

• ACFs of Random Series and White Noise
• Table 2-15 ACFs of Series A
• AtAt-1 AtAt-2 AtAt-3 AtAt-4 AtAt-5 AtAt-6
• .1784 .2296 .0738 .1237 .0193 .1718
• AtAt-7 AtAt-8 AtAt-9 AtAt-10 AtAt-11 AtAt-12
• .0809 -.0055 .0241 .1467 .1711 .1892
• Figure 2-11. ACFs for Series A. Here

74

• ACFs of Random Walk Series
• Figure 2-12. ACFs of Series B. Here

75

• ACFs of Trending Series
• Table 2-16. ACFs of Series C
• CtCt-1 CtCt-2 CtCt-3 CtCt-4 CtCt-5
  CtCt-6
• .8558 .8199 .7611 .6948 .6417 .5705
• CtCt-7 CtCt-8 CtCt-9 CtCt-10 CtCt-11
  CtCt-12
• .5481 .4691 .4197 .3614 .3156
  .2704
• Figure 2-13. ACFs Series C. Here

76

• Trends Versus Random Walks
• The ACFs of random walks and trends behave
  similarly. Thus, determining which involves
  tests on the series.

77

• ACFs of Seasonal Series
• SERIESD.DAT is demand for diet soft drinks in
  Figure 1-7.
• Table 2-17Autocorrelations of Series D.
• DtDt-1 DtDt-2 DtDt-3 DtDt-4 DtDt-5
  DtDt-6
• .7916 .4837 .0857 -.2920 -.5732
  -.6690
• DtDt-7 DtDt-8 DtDt-9 DtDt-10 DtDt-11
  DtDt-12
• -.5895 -.3541 -.0733 .2111 .4449
  .5107
• DtDt-13 DtDt-14 DtDt-15 DtDt-16 DtDt-17
  DtDt-18
• .4707 .2855 .0261 -.2200 -.4122
  -.5082
• DtDt-19 DtDt-20 DtDt-21 DtDt-22 DtDt-23
  DtDt-24
• -.4427 -.2942 -.0819 .1456 .3043
  .3794
• Figure 2-14 ACFs of Series D. HERE

78
Which Measure of Correlation?

• May have to use many lags and therefore violate
  n/4 rule.
• ACFs are approximations, Pearson is not sensitive
  to the lag.
• Thus, use Pearson when insufficient n.

79
AUTOCORRELATION APPLICATIONS

• Autocorrelations of Births and Marriages
• Figure 2-15. Quarterly Births in U.S. Here
• Table 2-12. ACFs of Marriages. Quarterly Data
  From 198501 To 199204 n32.
• Lag k 1 2 3 4
• ACF -.125 -.711 -.109 .875
• 5 6 7 8
• -.103 -.613 -.107 .743

80

• 2SeACFS .36, (i.e. 21/(32).5).
• Consider the Following Model
• Xt Xt-4 for t 5 to 32 (2-23)
• et Xt - Xt-4 for t 5 to 32 (2-24)
• Figure 2-16. ACFs of Marriage in the U.S. Here
• Figure 2-17. Mars(t) and Mars(t-4) Here
• Figure 2-18. Errors Mars(t) - Mars(t-4). Here
• Figure 2-19. ACFs Mars(t)-Mars(t-4). Here

81

• Table 2-13. ACFs of Errors Marriages(t) -
  Marriages(t-4).
• Qtrly Data From 198601 To 199204
• Lag 1 2 3 4
• ACF .0027 .2458 .1350 -.2034
• 5 6 7 8
• -.0759 -.1055 -.2506 -.1890