Chapter16 - Time-Series Forecasting and Index

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 16

Learning Objectives

In this chapter, you learn:



_{About seven different time-series forecasting}

models: moving averages, exponential smoothing,

the linear trend, the quadratic trend, the exponential

trend, the autoregressive, and the least-squares

models for seasonal data.



_{To choose the most appropriate time-series}

forecasting model



_{About price indexes and the difference between}

The Importance of Forecasting



_{Governments forecast unemployment, interest rates,}

and expected revenues from income taxes for policy

purposes



_{Marketing executives forecast demand, sales, and}

consumer preferences for strategic planning



_{College administrators forecast enrollments to plan}

for facilities and for faculty recruitment



_{Retail stores forecast demand to control inventory}

Time Series Plot



_{the vertical axis}

measures the variable

of interest



_{the horizontal axis}

corresponds to the

time periods

U.S. Inflation Rate

0.00

A time-series plot is a

Time-Series Components

Time Series

Cyclical

Component

Irregular

Component

Trend

Component

Trend Component



_{Long-run increase or decrease over time (overall}

upward or downward movement)



_{Data taken over a long period of time}

Upwar

d trend

Sales

Trend Component



_{Trend can be upward or downward}



_{Trend can be linear or non-linear}

Downward linear trend

Sales

Time

Upward nonlinear trend

Sales

Seasonal Component



_{Short-term regular wave-like patterns}



_{Observed within 1 year}



_{Often monthly or quarterly}

Sales

Time (Quarterly)

Winter

Spring

Summer

Fall

Winter

Spring

Summer

Cyclical Component



_{Long-term wave-like patterns}



_{Regularly occur but may vary in length}



_{Often measured peak to peak or trough to trough}

Sales

1 Cycle

Irregular Component



_{Unpredictable, random, “residual”}

fluctuations



_{Due to random variations of}



_Nature



_{Accidents or unusual events}

Multiplicative Time-Series

Model for Annual Data



_{Used primarily for forecasting}



_{Observed value in time series is the product of}

components

i

i

i

i

T

C

I

Y

_

_

_

where

T

_i

= Trend value at year i

C

_i

= Cyclical value at year i

Multiplicative Time-Series Model

with a Seasonal Component



_{Used primarily for forecasting}



_{Allows consideration of seasonal variation}

where

T

_i

= Trend value at time i

S

_i

= Seasonal value at time i

C

_i

= Cyclical value at time i

i

i

i

i

i

T

S

C

I

Smoothing the

Annual Time Series

Calculate moving averages to get an overall

impression of the pattern of movement over

time

Moving Averages



_{Used for smoothing}



_{A series of arithmetic means over time}



_{Result dependent upon choice of L (length of}

period for computing means)



_Examples:

Moving Averages



_{Example: Five-year moving average}



_{First average:}



_{Second average:}

5

Y

Y

Y

Y

Y

MA(5)

1



2



3



4



5



5

Y

Y

Y

Y

Y

MA(5)

2



3



4



5



6

Example: Annual Data

Year

Sales

1

Annual Sales

Example: Annual Data



_{Each moving average is for a consecutive}

block of 5 years

Year

Sales

1

23

Average

Year

Annual vs. Moving Average

Annual vs. 5-Year Moving Average

0

10

20

30

40

50

60

1

2

3

4

5

6

7

8

9

10 11

Year

S

a

le

s

Annual

5-Year Moving Average



_{The 5-year}

Exponential Smoothing



_{Used for smoothing and short term}

forecasting (often one period into the future)



_{A weighted moving average}



_{Weights decline exponentially}

Exponential Smoothing



_{The weight (smoothing coefficient) is W}



_{Subjectively chosen}



_{Range from 0 to 1}



_{Smaller W gives more smoothing, larger W}

gives less smoothing



_{The weight is:}



_{Close to 0 for smoothing out unwanted cyclical}

and irregular components

Exponential Smoothing Model



_{Exponential smoothing model}

1

1

Y

E

_

1

i

i

i

WY

(

1

W

)

E

E

_

_

_

_

where:

E

_i

= exponentially smoothed value for period i

E

_i-1

= exponentially smoothed value already

computed for period i - 1

Y

_i

= observed value in period i

W = weight (smoothing coefficient), 0 < W < 1

Exponential Smoothing

Example



_{Suppose we use weight W = .2}

Time

Period (i)

Sales

(Y

_i

)

Forecast from

prior period

(E

_i-1

)

Exponentially Smoothed Value for

this period (E

_i

)

Exponential Smoothing

Example



_{Fluctuations have}

been smoothed



_{NOTE: the}

smoothed value in

this case is

generally a little

low, since the

trend is upward

sloping and the

weighting factor is

only .2

0

10

20

30

40

50

60

1

2

3

4

5

6

7

8

9

10

Time Period

S

a

le

s

Forecasting Time Period i + 1



_{The smoothed value in the current period}

(i) is used as the forecast value for next

period (i + 1) :

i

1

i

E

Least Squares Linear

Trend-Based Forecasting



_{Estimate a trend line using regression analysis}

Year

Time

Period

(X)

Sales

_(Y)

Least Squares Linear

Trend-Based Forecasting



_{The linear trend forecasting equation is:}

Sales trend

0

i

21.905

9.5714

X

Yˆ

_

_

Year

Time

Period (X) Sales (Y)

Least Squares Linear

Trend-Based Forecasting



_{Forecast for time period 6}

Year

Time

Period

(X)

Sales

_(y)

Sales trend

Least Squares Quadratic

Trend-Based Forecasting



_{A nonlinear regression model can be used}

when the time series exhibits a nonlinear

trend



_{Quadratic Trend Forecasting Equation:}



_{Test quadratic term for significance}



_{Can try other functional forms to get best fit}

2

2

1

0

ˆ

i

i

b

X

X

b

b

Least Squares Exponential

Trend-Based Forecasting



_{Exponential trend forecasting equation:}

i

1

0

i

)

b

b

X

Yˆ

log(

_

_

where b

₀

= estimate of log(β

₀

)

b

₁

= estimate of log(β

₁

)

Interpretation:

%

100

)

1

βˆ

Model Selection Using

Differences



_{Use a linear trend model if the first differences are}

approximately constant



_{Use a quadratic trend model if the second}

differences are approximately constant

Model Selection Using

Differences



_{Use an exponential trend model if the percentage}

differences are approximately constant

Autoregressive Modeling

i

p

-i

p

2

-i

2

1

-i

1

0

i

A

A

Y

A

Y

A

Y

Y

_

_

_

_



_

_

δ



_{Used for forecasting}



_{Takes advantage of autocorrelation}



_{1st order - correlation between consecutive values}



_{2nd order - correlation between values 2 periods apart}



_p

th

order Autoregressive models:

Autoregressive Modeling

Example

Year Units

97 4

98

3

99 2

00 3

01

2

02

2

03

4

04 6

Autoregressive Modeling

Example

Year Y

_i

Y

_i-1

Y

_i-2

97

4

--

98

3

4

99 2

3 4

00 3

2 3

01 2 3 2

02

2 2 3

03

4 2 2

04

6 4 2

Coefficients

Intercept

3.5

X Variable 1

0.8125

X Variable 2

-0.9375

Excel Output



_{Develop the 2nd order table}



_{Use Excel to estimate a}

regression model

2

i

1

i

i

3.5

0.8125Y

0.9375Y

Autoregressive Modeling

Example

Use the second-order equation to

forecast number of units for 2005:

Autoregressive Modeling

Steps

1. Choose p (note that df = n – 2p – 1)

2. Form a series of “lagged predictor” variables

Y

_i-1

, Y

_i-2

, … ,Y

_i-p

3. Use Excel to run regression model using all p

variables

4. Test significance of A

_p

If null hypothesis rejected, this model is selected

Selecting A Forecasting Model



_{Perform a residual analysis}



_{Look for pattern or direction}



_{Measure magnitude of residual error using}

squared differences



_{Measure residual error using MAD}



_{Use simplest model}

Residual Analysis

Random errors

Trend not accounted for

Cyclical effects not accounted for

Seasonal effects not accounted for

T

T

T

T

e

e

e

_e

0

0

Measuring Errors

Choose the model that gives the smallest SSE and/or MAD



_{Mean Absolute Deviation}

(MAD)



_{Not sensitive to outliers}



_{Sum of squared errors}

(SSE)



_{Sensitive to outliers}

Principal of Parsimony



_{Suppose two or more models provide a good}

fit for the data



_{Select the simplest model}



_{Simplest model types:}



_{Least-squares linear}



_{Least-squares quadratic}



_{1st order autoregressive}



_{More complex types:}



_{2nd and 3rd order autoregressive}

Forecasting With Seasonal

Data

Recall the classical time series model with seasonal

variation:

Suppose the seasonality is quarterly

Define three new dummy variables for quarters:

Q

1

= 1 if first quarter, 0 otherwise

Q

₂

= 1 if second quarter, 0 otherwise

Q

3

= 1 if third quarter, 0 otherwise

(Quarter 4 is the default if Q

₁

= Q

₂

= Q

₃

= 0)

i

i

i

i

i

T

S

C

I

Exponential Model with

Quarterly Data

Transform to linear form:

i

(β

₁

-1)*100% = estimated quarterly compound growth rate (in %)

β

₂

= estimated multiplier for first quarter relative to fourth quarter

β

₃

= estimated multiplier for second quarter relative to fourth quarter

Estimating the Quarterly Model

Exponential forecasting equation:

3

(

₁

_

_

_{= estimated quarterly compound growth rate (in %)}

= estimated multiplier for first quarter relative to fourth quarter

= estimated multiplier for second quarter relative to fourth quarter

= estimated multiplier for third quarter relative to fourth quarter

Quarterly Model Example

Suppose the forecasting equation is:

Quarterly Model Example

Unadjusted trend value for first quarter of first year

4.0% = estimated quarterly compound growth rate

Ave. sales in Q

₂

are 82.7% of average 4

th

quarter sales,

after adjusting for the 4% quarterly growth rate

Ave. sales in Q

₃

are 84.5% of average 4

th

quarter sales,

after adjusting for the 4% quarterly growth rate

Ave. sales in Q

₄

are 105.2% of average 4

th

_quarter

sales, after adjusting for the 4% quarterly growth rate

Index Numbers



_{Index numbers allow relative comparisons over}

time



_{Index numbers are reported relative to a base}

period index

Simple Price Index

Simple Price Index:

100

P

P

I

base

i

i





where

I

i

= index number for year i

P

i

= price for year i

Index Numbers: Example

Airplane ticket prices from 1998 to 2006:

2

1998

272

92.2

1999

288

97.6

2000

295

100

2001

311

105.4

2002

322

109.2

2003

320

108.5

2004

348

118.0

2005

366

124.1

2006

384

130.2

Index Numbers: Interpretation



_{Prices in 1998 were 92.2% of}

base year prices



_{Prices in 2000 were 100% of}

base year prices (by

definition, since 2000 is the

base year)



_{Prices in 2006 were 130.2%}

of base year prices

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

Unweighted

Aggregate Price Index

Unweighted aggregate price index formula:

100

= unweighted price index at time t

= sum of the prices for the group of items at time t

= sum of the prices for the group of items in time period 0



Unweighted Aggregate Price

Index: Example

Unweighted total expenses were 18.8%

higher in 2006 than in 2003

Automobile Expenses:

Monthly Amounts ($):

Year

Lease payment

Fuel

Repair

Total

Index (2003=100)

2003

260

45

40

345

100.0

2004

280

60

40

380

110.1

2005

305

55

45

405

117.4

2006

310

50

50

410

118.8

118.8

(100)

345

410

100

P

P

I

2003

2006

2006









Weighted

Aggregate Price Indexes

Paasche index

100

Laspeyres index

Common Price Indexes



_{Consumer Price Index (CPI)}



_{Producer Price Index (PPI)}



_{Stock Market Indexes}



_{Dow Jones Industrial Average}



_{S&P 500 Index}

Pitfalls in

Time-Series Analysis



_{Assuming the mechanism that governs the}

time series behavior in the past will still hold

in the future



_{Using mechanical extrapolation of the trend}

Chapter Summary



_{Discussed the importance of forecasting}



_{Addressed component factors of the time-series}

model



_{Performed smoothing of data series}



_{Moving averages}



_{Exponential smoothing}



_{Described least squares trend fitting and forecasting}



_{Linear, quadratic and exponential models}

Chapter Summary



_{Addressed autoregressive models}



_{Described procedure for choosing appropriate}

models



_{Addressed time series forecasting of monthly or}

quarterly data (use of dummy variables)



_{Discussed pitfalls concerning time-series analysis}



_{Discussed index numbers}