Statistics for business economics 7th by paul newbold chapter 16

Chapter GoalsAfter completing this chapter, you should be able to:  Compute and interpret index numbers  Weighted and unweighted price index  Weighted quantity index  Test for random

Chapter 16

Time-Series Analysis and

Statistics for Business and Economics

7 th Edition

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

 Base period index = 100 by definition

 Used for an individual item or measurement

16.1

Single Item Price Index

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 p 0 denote the price in the base period

 Let p 1 be the price in a second period

 The price index for this second period is

Index Numbers: Example

 Airplane ticket prices from 2000 to 2008:

90 320

288 (100)

P

P 100

320 (100)

P

P 100

Index Numbers: Interpretation

 Prices in 2001 were 90% of base year prices

 Prices in 2005 were 100%

of base year prices (by definition, since 2005 is the base year)

 Prices in 2008 were 120%

of base year prices

90

(100) 320

288 100

P

P I

320 100

P

P I

384 100

P

P I

2005 2008

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:

1 i

0i

K

1 i

ti

p

p 100

= 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

K1iti

p p

Unweighted Aggregate Price

410 (100)

P

P 100

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

1 i

0i 0i

K

1 i

ti 0i

p q

p

q 100

= quantity of item i purchased in period 0

= price of item i in time period 0

 Laspeyres price index for time period t:

0i

q

0i

p

Laspeyres Quantity Index

1 i

0i 0i

K

1 i

0i ti

p q

p

q 100

0i

p = 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

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

16.2

The Runs Test for Randomness

 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)

The Runs Test for Randomness

 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)

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 H 0 : The series is random

 If the alternative hypothesis is positive association

between adjacent observations, the decision rule is:

α 2

1) 4(n

2n n

1 2

n R

z if

Runs Test: Large Samples

Consider the null hypothesis H 0 : The series is random

 If the alternative is a two-sided hypothesis of

nonrandomness, the decision rule is:

α/2 2

α/2 2

1) 4(n

2n n

1 2

n R

z or

z 1)

4(n

2n n

1 2

n R z

if H

Example: Large Sample

Runs Test

 A filling process over- or under-fills packages,

compared to the median

Example: Large Sample

Runs Test

1.206 4.975

6

1) 4(100

2(100) 100

1 2

100 45

1) 4(n

2n n

1 2

n R

Example: Large Sample

H 0 : Fill amounts are random

H 1 : 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

Sales: 75.3 74.2 78.5 79.7 80.2

16.3

Time-Series Plot

 the vertical axis

measures the variable

of interest

 the horizontal axis

corresponds to the

time periods

plot of time series data

Time-Series Components

Time Series

Cyclical Component

Irregular Component

Trend

Component

Seasonality 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 linear or non-linear

Seasonal Component

 Short-term regular wave-like patterns

 Observed within 1 year

 Often monthly or quarterly

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

 “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

t t

t t

t t t t

t T S C I

X 

Moving Averages:

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 designated

number of consecutive time series values

16.4

(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

 Replace each x t with

x x

x

x * 1 2 3 4 5 5

x x

x

x * 2 3 4 5 6 6

Example: Annual Data

Annual Sales

0 10 20 30 40 50 60

Calculating Moving Averages

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

5-Year Moving Average

25 40

23

etc…

 Let m = 2

Annual vs Moving Average

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 X t*:

 Form the s-point moving averages

 Form the centered s-point moving averages

j t

* 5

2

s n ,

2, 2

s 1, 2

s , 2

s (t

x

) 2

s n ,

2, 2

s 1, 2

s (t

2

x x

x

* 5 t

* 5 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

Centered Moving Average

Calculating the Ratio-to-Moving Average

 Divide the actual sales value by the centered

moving average for that period

* t

t

x x 100

Calculating a Seasonal Index

Quarter Sales

Centered Moving Average

Moving Average

29.88 32.00 34.00 36.25 38.13 39.00 40.13

83.7 84.4 94.1 132.4 86.5 94.9 92.2

83.7 29.88

25 (100)

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…

… …

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

Interpreting Seasonal Indexes

 Suppose we get these

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)

16.5

 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

x t = observed value in period t  = weight (smoothing coefficient), 0 <  < 1

1

1 x

t 1

xˆ

n) , 1,2,

t 1;

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)

23 26.4 26.12 26.296 27.437 31.549 31.840

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

= x1since no prior information exists

1

xˆ

t 1

t

t 0.2 x (1 0.2)x

x ˆ  ˆ   

Sales vs Smoothed Sales

generally a little low,

since the trend is

upward sloping and

the weighting factor

is only 2

0 10 20 30 40 50 60

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, X n+h of the series

) 1,2,3

(h x

Exponential Smoothing in Excel

exponential smoothing

 The “damping factor” is (1 - )

 To perform the Holt-Winters method of forecasting:

 Obtain estimates of level and trend T t 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, X of the series by

Forecasting with the Holt-Winters Method: Nonseasonal Series

t

xˆ

1 2

2 2

x ˆ   

n) , 3,4, t

1;

α (0

αx )

T x

α)(

(1

x ˆ t   ˆ t  1  t  1  t    

n) , 3,4, t

1;

β (0

) x x

β(

β)T (1

T t   t  1  ˆ t  ˆ t  1    

 Assume a seasonal time series of period s

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

Forecasting with the Holt-Winters

Method: Seasonal Series

1) α

(0 F

x α ) T x

α)(

(1

x

s t

t 1

t 1

1)

(0 x

x )F

(1

F

t

t s

t

ˆ

1) β

(0 )

x x

β(

β)T (1

T t   t  1  ˆ t  ˆ t  1  

Where is the smoothed level of the series, T xˆ is the smoothed trend

(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 X n in the historical series

 The forecast equation is

s h t t

t h

n ( x hT )F

where the seasonal factor, F t , is the one generated for

the most recent seasonal time period

Forecasting with the Holt-Winters

Method: Seasonal Series

(continued)

Autoregressive Models

 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 model :

t p

t p 2

t 2 1

t 1

16.6

Autoregressive Models

 Let X t (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

t p

t p 2

t 2 1

t 1

x  γ  φ   φ     φ   ε

(continued)

Autoregressive Models

 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

2 p t p 2

t 2 1

t 1

n

1 p t

Forecasting from Estimated

Autoregressive Models

 Consider time series observations x 1 , x 2 , , x t

 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 X t+j standing at time n and for j  0 , is simply the observed value of X t+j

t p

t p 2

t 2 1

t 1

x  γ ˆ  φ ˆ   φ ˆ     φ ˆ   ε

) 1,2,3, (h

x x

x

x ˆ t  h  γ ˆ  φ ˆ 1 ˆ t  h  1  φ ˆ 2 ˆ t  h  2    φ ˆ p ˆ t  h  p  

j n

x ˆ 

j n

x ˆ 

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.

t 3.5 0.8125x 0.9375x

Autoregressive Model Example: Forecasting

Use the second-order equation to forecast

number of units for 2010:

4.625

0.9375(4) 0.8125(6)

3.5

) 0.9375(x

) 0.8125(x

3.5 x

0.9375x 0.8125x

3.5 x

2008 2009

2010

2 t 1

t t

Autoregressive Modeling Steps

 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