Chap19 time series-analysis_and_forecasting

Chap 19-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chapter 19

Time-Series Analysis and Forecasting

Statistics for Business and Economics

6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-2

Chapter Goals

After completing this chapter, you should be able to:

Compute and interpret index numbers Weighted and unweighted price index Weighted quantity index

Test for randomness in a time series Identify the trend, seasonality, cyclical, and irregular

components in a time series Use smoothing-based forecasting models, including

moving average and exponential smoothing Apply autoregressive models and autoregressive

integrated moving average models


Index Numbers

Index numbers allow relative comparisons over time

Index numbers are reported relative to a Base Period Index

Base period index = 100 by definition

Used for an individual item or measurement


Consider observations over time on the price of a single item

To form a price index, one time period is chosen as a base, and the price for every period is expressed as a percentage of the base period price

Let p0 denote the price in the base period

Let p1 be the price in a second period The price index for this second period is

0

1

p

p100

Single Item Price Index


Index Numbers: Example

Airplane ticket prices from 1995 to 2003:

90320

288(100)

P

P100I

2000

19961996

Year PriceIndex

(base year = 2000)

1995 272 85.0

1996 288 90.0

1997 295 92.2

1998 311 97.2

1999 322 100.6

2000 320 100.0

2001 348 108.8

2002 366 114.4

2003 384 120.0

100320

320(100)

P

P100I

2000

20002000

120320

384(100)

P

P100I

2000

20032003

Base Year:


Prices in 1996 were 90% of base year prices

Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year)

Prices in 2003 were 120% of base year prices

Index Numbers: Interpretation

90)100(320

288100

P

PI

2000

19961996

100)100(320

320100

P

PI

2000

20002000

120)100(320

384100

P

PI

2000

20032003


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

Laspeyres Index


Unweighted Aggregate Price Index

Unweighted aggregate price index for period t for a group of K items:

K

1i0i

K

1iti

p

p100

= 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

i = item

t = time period

K = total number of items

K

1i0i

K

1iti

p

p


Unweighted total expenses were 18.8% higher in 2004 than in 2001

Automobile Expenses:Monthly Amounts ($):

Year Lease payment Fuel Repair TotalIndex

(2001=100)

2001 260 45 40 345 100.0

2002 280 60 40 380 110.1

2003 305 55 45 405 117.4

2004 310 50 50 410 118.8

Unweighted Aggregate Price Index: Example

118.8345

410(100)

P

P100I

2001

20042004


Weighted Aggregate Price Indexes

A weighted index weights the individual prices by some measure of the quantity sold

If the weights are based on base period quantities the index is called a Laspeyres price index

The Laspeyres price index for period t is the total cost of purchasing the quantities traded in the base period at prices in period t , expressed as a percentage of the total cost of purchasing these same quantities in the base period

The Laspeyres quantity index for period t is the total cost of the quantities traded in period t , based on the base period prices, expressed as a percentage of the total cost of the base period quantities


Laspeyres Price Index

= quantity of item i purchased in period 0

= price of item i in time period 0

= price of item i in period t

Laspeyres price index for time period t:

0iq

K

1i0i0i

K

1iti0i

pq

pq100

ti

0i

p

p


Laspeyres Quantity Index

K

1i0i0i

K

1i0iti

pq

pq100

= price of item i in period 0

= quantity of item i in time period 0

= quantity of item i in period t

Laspeyres quantity index for time period t:

ti

0i

q

q

0ip


The Runs Test for Randomness

The runs test is used to determine whether a pattern in time series data is random

A run is a sequence of one or more occurrences above or below the median

Denote observations above the median with “+” signs and observations below the median with “-” signs



Consider n time series observations Let R denote the number of runs in the

sequence The null hypothesis is that the series is random Appendix Table 14 gives the smallest

significance level for which the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) as a function of R and n

(continued)



If the alternative is a two-sided hypothesis on nonrandomness,

the significance level must be doubled if it is less than 0.5

if the significance level, , read from the table is greater than 0.5, the appropriate significance level for the test against the two-sided alternative is 2(1 - )

(continued)


Counting Runs

Sales

Time

- - + - - + + + + - - - - - + + + +

Runs: 1 2 3 4 5 6

n = 18 and there are R = 6 runs

Median


Runs Test Example

Use Appendix Table 14

n = 18 and R = 6

the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) at the 0.044 level of significance

Therefore we reject that this time series is random using = 0.05

n = 18 and there are R = 6 runs


Runs Test: Large Samples

Given n > 20 observations Let R be the number of sequences above or below

the median

Consider the null hypothesis H0: The series is random

If the alternative hypothesis is positive association between adjacent observations, the decision rule is:

α20 z

1)4(n2nn

12n

Rz if H Reject


Runs Test: Large Samples

Consider the null hypothesis H0: The series is random

If the alternative is a two-sided hypothesis of nonrandomness, the decision rule is:

α/22α/220 z

1)4(n2nn

12n

Rzorz

1)4(n2nn

12n

Rz if H Reject

(continued)


Example: Large Sample Runs Test

A filling process over- or under-fills packages, compared to the median

OOO U OO U O UU OO UU OOOO UU O UU OOO UUU OOOO UU OO UUU O U OO UUUUU OOO U O UU OOO U OOOO UUU O UU OOO U OO UU O U OO UUU O UU OOOO UUU OOO

n = 100 (53 overfilled, 47 underfilled)

R = 45 runs



A filling process over- or under-fills packages, compared to the median

n = 100 , R = 45

1.2064.975

6

1)4(1002(100)100

12

10045

1)4(n2nn

12n

RZ

22

(continued)


1.960

Rejection Region /2 = 0.025

Since z = -1.206 is not less than -z.025 = -1.96, we do not reject H0

1.96

Rejection Region /2 = 0.025

H0: Fill amounts are random

H1: Fill amounts are not random

Test using = 0.05


(continued)


Time-Series Data

Numerical data ordered over time The time intervals can be annually, quarterly,

daily, hourly, etc. The sequence of the observations is important Example:

Year: 2001 2002 2003 2004 2005

Sales: 75.3 74.2 78.5 79.7 80.2


Time-Series Plot

the vertical axis measures the variable of interest

the horizontal axis corresponds to the time periods

U.S. Inflation Rate

0.002.004.006.008.00

10.0012.0014.0016.00

197

5

197

7

197

9

198

1

198

3

198

5

198

7

198

9

199

1

199

3

199

5

199

7

199

9

200

1

Year

Infl

ati

on

Rat

e (

%)

A time-series plot is a two-dimensional plot of time series data


Time-Series Components

Time Series

Cyclical Component

Irregular Component

Trend Component

Seasonality Component


Upward trend

Trend Component

Long-run increase or decrease over time (overall upward or downward movement)

Data taken over a long period of time

Sales

Time


Downward linear trend

Trend Component

Trend can be upward or downward Trend can be linear or non-linear

Sales

Time Upward nonlinear trend

Sales

Time

(continued)


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

Fall


Cyclical Component

Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to

trough

Sales1 Cycle

Year


Irregular Component

Unpredictable, random, “residual” fluctuations Due to random variations of

Nature Accidents or unusual events

“Noise” in the time series


Time-Series Component Analysis

Used primarily for forecasting Observed value in time series is the sum or product of

components Additive Model

Multiplicative model (linear in log form)

where Tt = Trend value at period t

St = Seasonality value for period t

Ct = Cyclical value at time t

It = Irregular (random) value for period t

ttttt ICSTX

ttttt ICSTX


Smoothing the Time Series

Calculate moving averages to get an overall impression of the pattern of movement over time

This smooths out the irregular component

Moving Average: averages of a designatednumber of consecutivetime series values


(2m+1)-Point Moving Average

A series of arithmetic means over time Result depends upon choice of m (the

number of data values in each average) Examples:

For a 5 year moving average, m = 2 For a 7 year moving average, m = 3 Etc.

Replace each xt with

m

mjjt

*t m)n,2,m1,m(tX

12m

1X


Moving Averages

Example: Five-year moving average

First average:

Second average:

etc.

5

xxxxxx 54321*

5

5

xxxxxx 65432*

6


Example: Annual Data

Year Sales

1

2

3

4

5

6

7

8

9

10

11

etc…

23

40

25

27

32

48

33

37

37

50

40

etc…

Annual Sales

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11

Year

Sa

les

…

…


Calculating Moving Averages

Each moving average is for a consecutive block of (2m+1) years

Year Sales

1 23

2 40

3 25

4 27

5 32

6 48

7 33

8 37

9 37

10 50

11 40

Average Year

5-Year Moving Average

3 29.4

4 34.4

5 33.0

6 35.4

7 37.4

8 41.0

9 39.4

… …

5

322725402329.4

etc…

Let m = 2


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

Sal

es

Annual 5-Year Moving Average

Annual vs. Moving Average

The 5-year moving average smoothes the data and shows the underlying trend


Centered Moving Averages

Let the time series have period s, where s is even number i.e., s = 4 for quarterly data and s = 12 for monthly data

To obtain a centered s-point moving average series Xt

*:

Form the s-point moving averages

Form the centered s-point moving averages

(continued)

s/2

1(s/2)jjt

*.5t )

2

sn,2,

2

s1,

2

s,

2

s(txx

)2

sn,2,

2

s1,

2

s(t

2

xxx

*.5t

*.5t*

t


Centered Moving Averages Used when an even number of values is used in the moving

average Average periods of 2.5 or 3.5 don’t match the original

periods, so we average two consecutive moving averages to get centered moving averages

Average Period

4-Quarter Moving

Average

2.5 28.75

3.5 31.00

4.5 33.00

5.5 35.00

6.5 37.50

7.5 38.75

8.5 39.25

9.5 41.00

Centered Period

Centered Moving

Average

3 29.88

4 32.00

5 34.00

6 36.25

7 38.13

8 39.00

9 40.13

etc…


Calculating the Ratio-to-Moving Average

Now estimate the seasonal impact Divide the actual sales value by the centered

moving average for that period

*t

t

x

x100


Calculating a Seasonal Index

Quarter Sales

Centered Moving Average

Ratio-to-Moving Average

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…

…

…

83.7

84.4

94.1

132.4

86.5

94.9

92.2

etc…

…

…

83.729.88

25(100)

x

x100

*3

3


Calculating Seasonal Indexes

Quarter Sales

Centered Moving Average

Ratio-to-Moving Average

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…

…

…

83.7

84.4

94.1

132.4

86.5

94.9

92.2

etc…

…

…

1. Find the median of all of the same-season values

2. Adjust so that the average over all seasons is 100

Fall

Fall

Fall

(continued)


Interpreting Seasonal Indexes

Suppose we get these seasonal indexes:

SeasonSeasonal

Index

Spring 0.825

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:


Exponential Smoothing

A weighted moving average Weights decline exponentially

Most recent observation weighted most

Used for smoothing and short term forecasting (often one or two periods into the future)


Exponential Smoothing

The weight (smoothing coefficient) is Subjectively chosen Range from 0 to 1 Smaller gives more smoothing, larger

gives less smoothing The weight is:

Close to 0 for smoothing out unwanted cyclical and irregular components

Close to 1 for forecasting

(continued)


Exponential Smoothing Model

Exponential smoothing model

where: = exponentially smoothed value for period t = exponentially smoothed value already

computed for period i - 1 xt = observed value in period t = weight (smoothing coefficient), 0 < < 1

11 xx ˆ

n),1,2,t1;(0)x(1xx t1tt ααα ˆˆ

tx̂

1-tx̂


Exponential Smoothing Example Suppose we use weight = .2

Time Period

(i)

Sales(Yi)

Forecast from prior

period (Ei-1)

Exponentially Smoothed Value for this period (Ei)

1

2

3

4

5

6

7

8

9

10

etc.

23

40

25

27

32

48

33

37

37

50

etc.

--

23

26.4

26.12

26.296

27.437

31.549

31.840

32.872

33.697

etc.

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

etc.

= x1 since no prior information exists

1x̂

t1tt 0.2)x(1x0.2x ˆˆ


Sales vs. Smoothed Sales

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 10Time Period

Sa

les

Sales Smoothed


Forecasting Time Period (t + 1)

The smoothed value in the current period (t) is used as the forecast value for next period (t + 1)

At time n, we obtain the forecasts of future values, Xn+h of the series

)1,2,3(hxx nhn ˆˆ


Exponential Smoothing in Excel

Use tools / data analysis /

exponential smoothing

The “damping factor” is (1 - )


To perform the Holt-Winters method of forecasting: Obtain estimates of level and trend Tt as

Where and are smoothing constants whose values are fixed between 0 and 1

Standing at time n , we obtain the forecasts of future values, Xn+h of the series by

Forecasting with the Holt-Winters Method: Nonseasonal Series

tx̂

12221 xxTxx ˆ

n),3,4,t1;α(0α)x(1)Txα(x t1t1tt ˆˆ

nnhn hTxx ˆˆ

n),3,4,t1;β(0)xxβ)((1βTT 1tt1tt ˆˆ


Assume a seasonal time series of period s

The Holt-Winters method of forecasting uses a set of recursive estimates from historical series

These estimates utilize a level factor, , a trend factor, , and a multiplicative seasonal factor,

Forecasting with the Holt-Winters Method: Seasonal Series


The recursive estimates are based on the following equations


1)α(0F

xα)(1)Txα(x

st

t1t1tt

ˆˆ

1)(0x

x)(1FF

t

tstt γγγ

ˆ

1)β(0)xxβ)((1βTT 1tt1tt ˆˆ

Where is the smoothed level of the series, Tt is the smoothed trend of the series, and Ft is the smoothed seasonal adjustment for the series

tx̂

(continued)


After the initial procedures generate the level, trend, and seasonal factors from a historical series we can use the results to forecast future values h time periods ahead from the last observation Xn in the historical series

The forecast equation is

shttthn )FhTx(x ˆˆ

where the seasonal factor, Ft, is the one generated for the most recent seasonal time period


(continued)


Autoregressive Models

Used for forecasting Takes advantage of autocorrelation

1st order - correlation between consecutive values 2nd order - correlation between values 2 periods

apart pth order autoregressive model:

Random Error

tptp2t21t1t xxxx εφφφγ



Let Xt (t = 1, 2, . . ., n) be a time series A model to represent that series is the autoregressive

model of order p:

where , 1 2, . . .,p are fixed parameters

t are random variables that have mean 0 constant variance and are uncorrelated with one another

tptp2t21t1t xxxx εφφφγ

(continued)



The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of , 1 2, . . .,p for which the sum of squares

is a minimum

2ptp2t21t1

n

1ptt )xxx(xSS

φφφγ

(continued)


Forecasting from Estimated Autoregressive Models

Consider time series observations x1, x2, . . . , xt Suppose that an autoregressive model of order p has been fitted to

these data:

Standing at time n, we obtain forecasts of future values of the series from

Where for j > 0, is the forecast of Xt+j standing at time n and

for j 0 , is simply the observed value of Xt+j

tptp2t21t1t xxxx εφφφγ ˆˆˆˆ

)1,2,3,(hxxxx phtp2ht21ht1ht ˆˆˆˆˆˆˆˆ φφφγ

jnx ˆ

jnx ˆ


Autoregressive Model: Example

Year Units

1999 4 2000 3 2001 2 2002 3 2003 2 2004 2 2005 4 2006 6

The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order autoregressive model.


Autoregressive Model: Example Solution

Year xt xt-1 xt-2

99 4 -- -- 00 3 4 -- 01 2 3 4 02 3 2 3 03 2 3 2 04 2 2 3 05 4 2 2 06 6 4 2

CoefficientsIntercept 3.5X Variable 1 0.8125X Variable 2 -0.9375

Excel Output

Develop the 2nd order table

Use Excel to estimate a regression model

2t1tt 0.9375x0.8125x3.5x ˆ


Autoregressive Model Example: Forecasting

Use the second-order equation to forecast number of units for 2007:

4.625

0.9375(4)0.8125(6)3.5

)0.9375(x)0.8125(x3.5x

0.9375x0.8125x3.5x

200520062007

2t1tt

ˆ

ˆ


Autoregressive Modeling Steps

Choose p

Form a series of “lagged predictor” variables xt-1 , xt-2 , … ,xt-p

Run a regression model using all p variables

Test model for significance

Use model for forecasting


Chapter Summary

Discussed weighted and unweighted index numbers Used the runs test to test for randomness in time series

data Addressed components of the time-series model Addressed time series forecasting of seasonal data

using a seasonal index Performed smoothing of data series

Moving averages Exponential smoothing

Addressed autoregressive models for forecasting

Technology

Chap19 time series-analysis_and_forecasting