Business statistics, 6e, 2005, groebner CH15

Chapter GoalsAfter completing this chapter, you should be able to:  Develop and implement basic forecasting models  Identify the components present in a time series  Compute and inte

Chapter Goals

After completing this chapter, you should be

able to:

 Develop and implement basic forecasting models

 Identify the components present in a time series

 Compute and interpret basic index numbers

 Use smoothing-based forecasting models, including single and double exponential smoothing

 Apply trend-based forecasting models, including linear

trend, nonlinear trend, and seasonally adjusted trend

The Importance of Forecasting

rates, and expected revenues from income taxes for policy purposes

consumer preferences for strategic planning

 College administrators forecast enrollments to plan for facilities and for faculty recruitment

 Retail stores forecast demand to control inventory levels, hire employees and provide training

Time-Series Data

intervals

daily, hourly, etc.

Time Series Plot

 the vertical axis

measures the variable

A time-series plot is a two-dimensional

plot of time series data

Time-Series Components

Time-Series

Cyclical Component

Random Component

Trend

Component

Seasonal Component

Upwar d trend

Trend Component

(overall upward or downward movement)

Sales

Downward linear trend

Trend Component

Seasonal Component

Cyclical Component

trough

Sales

1 Cycle

Year

Random Component

 Nature

 Accidents or unusual events

Index Numbers

comparisons over time

Base Period Index

measurement

Index Numbers

(continued)

100 y

y I

0

t

t 

where

Index Numbers: Example

90 )

100

( 320

288 100

y

y I

100

( 320

320 100

y

y I

2000

2000

2000   

120 )

100

( 320

384 100

y

y I

2000

2003

2003   

Base Year:

( 320

288 100

y

y I

2000

1996

1996   

100 )

100

( 320

320 100

y

y I

2000

2000

2000   

120 )

100

( 320

384 100

y

y I

2000 2003

2003   

Aggregate Price Indexes

 An aggregate index is used to measure the rate

of change from a base period for a group of items

Aggregate Price Indexes

Unweighted

aggregate price index

Weighted

aggregate price indexes

Paasche Index Laspeyres Index

Unweighted Aggregate Price

Index

) 100

( p

p I

0

t t

 



where

 Combined expenses in 2004 were 18.8%

410 (100)

Weighted Aggregate Price

Indexes

) 100

( p

q

p

q I

0 t

t

t t

 



qt = weighting percentage at q0 = weighting percentage at

pt = price in time period t

p0 = price in the base period

) 100

( p

q

p

q I

0 0

t

0 t

 



Commonly Used Index

Numbers

 Dow Jones Industrial Average

 S&P 500 Index

 NASDAQ Index

Deflating a Time Series

equivalent

) 100

( I

y y

t

t

where = adjusted time series value at time t

yt = value of the time series at time t

It = index (such as CPI) at time t

t

adj y

Deflating a Time Series:

Example

(in real terms)?

Year

Movie Title

Total Gross $

Deflating a Time Series:

Example

as much as Star Wars, and about 4 times as much as Titanic when measured in

7 1431 )

100

( 9 13

Total Gross

Trend-Based Forecasting

 Estimate a trend line using regression analysis

Year

Time Period (t) Sales (y) 1999

2000 2001 2002 2003 2004

1 2 3 4 5 6

20 40 30 50 70 65

t b b

yˆ  0  1

 Use time (t) as the independent variable:

Trend-Based Forecasting

 The linear trend model is:

Sales trend

0 10 20 30 40 50 60 70 80

20 40 30 50 70 65

t 5714

9 333

12

yˆ  

(continued)

Trend-Based Forecasting

 Forecast for time period 7:

Sales

0 10 20 30 40 50 60 70 80

20 40 30 50 70 65

??

(continued)

33 79

(7) 5714

9 333

12

yˆ







Comparing Forecast Values

to Actual Data

 The forecast error or residual is the difference

between the actual value in time t and the forecast value in time t:

 Error in time t:

t t

Two common Measures of

Fit

the forecasts match the actual values

MSE (mean squared error)

 Average squared difference between y t and F t

MAD (mean absolute deviation)

 Average absolute value of difference between y t and F t

 Less sensitive to extreme values

MSE vs MAD

Mean Square Error

n

) F y

( MSE

2 t t



n

| F y

| MAD  t  t



where:

y t = Actual value at time t

F t = Predicted value at time t

n = Number of time periods

Mean Absolute Deviation

 Autocorrelation is correlation of the error terms

(residuals) over time

(continued)

residuals are random and independent

Time (t) Residual Plot

-15 -10 -5 0 5 10 15

a cyclic pattern, not

random

Testing for Autocorrelation

 The Durbin-Watson Statistic is used to test for autocorrelation

H 0 : ρ = 0 (residuals are not correlated)

2 t

n

1 t

2 1 t t

e

) e e

( d

Testing for Positive

Autocorrelation

 Calculate the Durbin-Watson test statistic = d

(The Durbin-Watson Statistic can be found using PHStat or Minitab)

Decision rule: reject H 0 if d < d L

H 0 : ρ = 0 (positive autocorrelation does not exist)

H A : ρ > 0 (positive autocorrelation is present)

Reject H0 Do not reject H0

 Find the values d L and d U from the Durbin-Watson table

(for sample size n and number of independent variables p)

Inconclusive

1 98 3279

18 3296 e

) e e

(

2 t

n 1 t

2 1 t t

 Here, n = 25 and there is one independent variable

 Using the Durbin-Watson table, d L = 1.29 and d U = 1.45

 d = 1.00494 < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists

 Therefore the linear model is not the appropriate model

Nonlinear Trend Forecasting

the time series exhibits a nonlinear trend

if this is an improvement

t

2 1 0

Multiplicative Time-Series Model

components

t t

t t

Finding Seasonal Indexes

Ratio-to-moving average method:

cyclical components (T t and C t )

Moving Average: averages of consecutive

time series values

Moving Averages

period for computing means)

be equal to the number of seasons

 For quarterly data, L = 4

 For monthly data, L = 12

Q1 average

4

Q5 Q4

Q3

Q2 average

Quarterly Sales

0 10 20 30 40 50 60

Calculating Moving Averages

 Each moving average is for a consecutive block of 4 quarters

4-Quarter Moving Average

4

27 25

40 23

etc…

Centered Moving Averages

quarters, so we average two consecutive moving

Average Period

4-Quarter Moving Average

Centered Moving Average

Calculating the Ratio-to-Moving Average

 Now estimate the S t x I t value

moving average for that quarter

t t

t t

t

C T

y I

S







Calculating Seasonal

Indexes

Quarter Sales

Centered Moving Average

Moving Average

…

29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc…

… …

0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc…

…

…

88 29

25 837

.

Calculating Seasonal

Indexes

Quarter Sales

Centered Moving Average

Moving Average

Ratio-to-1 2 3 4 5 6 7 8 9 10 11

23 40 25 27 32 48 33 37 37 50 40

29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc…

…

0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc…

…

Average all of the Fall values to get Fall’s seasonal index

Summer 1.310

Fall 0.920

Winter 0.945

 = 4.000 four seasons, so must sum to 4

Spring sales average 82.5% of the annual average sales

Summer sales are 31.0% higher than the annual average sales etc…

 Interpretation:

observed value by its seasonal index

t

t t

t t

S

y I

C

T   

variation

Quarter Sales

Seasonal Index

0.825 1.310 0.920 0.945 0.825 1.310 0.920 0.945 0.825 1.310 0.920 …

27.88 30.53 27.17 28.57 38.79 36.64 35.87 39.15 44.85 38.17 43.48

Unseasonalized vs

Seasonalized

Sales: Unseasonalized vs Seasonalized

0 10 20 30 40 50 60

Forecasting Using Smoothing Methods

Exponential Smoothing Methods

Single Exponential Smoothing

Double Exponential Smoothing

Single Exponential

Smoothing

 Weights decline exponentially

 Most recent observation weighted most

forecasting

 Close to 0 for smoothing out unwanted cyclical and irregular components

 Close to 1 for forecasting

(continued)

Exponential Smoothing

Model

) F y

( F

t t

1

t y ( 1 ) F

F      

where:

 = alpha (smoothing constant)

or:

Forecast from prior period

Forecast for next period

(Ft+1)

1 2 3 4 5 6 7 8 9 10

23 40 25 27 32 48 33 37 37 50

NA 23 26.4 26.12 26.296 27.437 31.549 31.840 32.872 33.697

23 (.2)(40)+(.8)(23)=26.4 (.2)(25)+(.8)(26.4)=26.12 (.2)(27)+(.8)(26.12)=26.296 (.2)(32)+(.8)(26.296)=27.437 (.2)(48)+(.8)(27.437)=31.549 (.2)(48)+(.8)(31.549)=31.840 (.2)(33)+(.8)(31.840)=32.872 (.2)(37)+(.8)(32.872)=33.697 (.2)(50)+(.8)(33.697)=36.958

t t

1 t

F ) 1 ( y

Sales vs Smoothed Sales

Double Exponential

Smoothing

may need adjustment

for trend

Double Exponential Smoothing

Model

where:

yt = actual value in time t

 = constant-process smoothing constant

 = trend-smoothing constant

Ct = smoothed constant-process value for period t

Tt = smoothed trend value for period t

Ft+1= forecast value for period t + 1

) T

C )(

1 ( y

1 t 1

t t

t t

1

Double Exponential

Smoothing

by computer

Chapter Summary

time-series model

forecasting

 linear and nonlinear models

 moving averages

 single and double exponential smoothing