Statistics for Business and Economics

1
Statistics for Business and Economics

• Chapter 13
• Time SeriesDescriptive Analyses, Models,
  Forecasting
• Lyn Noble
• Revisions by Peter Jurkat

2
Learning Objectives

• Describe Time Series
• Explain Descriptive Analyses
• Define Time Series Components
• Explain Forecasting
• Describe Measures of Accuracy
• Define Autocorrelation
• Explain DurbinWatson Test

3
Time Series

• Data generated by processes over time
• Describe and predict output of processes
• Descriptive analysis
• Understanding patterns
• Inferential analysis
• Forecast future values

4
Index Number

• Measures change over time relative to a base
  period
• Price Index measures changes in price
• e.g. Consumer Price Index (CPI)
• Quantity Index measures changes in quantity
• e.g. Number of cell phones produced annually

5
Simple Index Number

• Based on price/quantity of a single commodity

where Yt value at time t Y0 value at time 0
(base period)
6
Simple Index Number Example
Year 1990 1.2991991 1.0981992 1.08719
93 1.0671994 1.0751995 1.1111996 1.2241997 1.1
991998 1.031999 1.1362000 1.4842001 1.422002
1.3452003 1.5612004 1.8522005 2.272006 2.572

• The table shows the price per gallon of regular
  gasoline in the U.S for the years 1990 2006.
  Use 1990 as the base year (prior to the Gulf
  War). Calculate the simple index number for 1990,
  1998, and 2006.

7
Simple Index Number Solution

• 1990 Index Number (base period)

1998 Index Number
Indicates price had dropped by 20.7 (100 79.3)
between 1990 and 1998.
8
Simple Index Number Solution

• 2006 Index Number

Indicates price had risen by 98 (100 198)
between 1990 and 2006.
9
Simple Index Numbers 19902006
10
Simple Index Numbers 19902006
11
Class Exercise
Example US copper and steel prices production
Calculate the simple (un-weighted) copper price
index for the current period to closest
10 Enter A for 90, B for 100, C for 110
12
Composite Index Number

• Made up of two or more commodities
• A simple index using the total price or total
  quantity of all the series (commodities)
• Disadvantage Quantity of each commodity
  purchased is not considered

13
Composite Index Number Example

• The table on the next slide shows the closing
  stock prices on the last day of the month for
  DaimlerChrysler, Ford, and GM between 2005 and
  2006. Construct the simple composite index using
  January 2005 as the base period. (Source
  Nasdaq.com)

14
Simple Composite Index Solution
First compute the total for the three stocks for
each date.
15
Simple Composite Index Solution
Now compute the simple composite index by
dividing each total by the January 2005 total.
For example, December 2006
16
Simple Composite Index Solution
17
Simple Composite Index Solution
18
Class Exercise
Example US copper and steel prices production
Calculate the simple (un-weighted) composite
index for copper and steel for the current period
to nearest 10. Enter A for 90, B for 100, C
for 110
19
Weighted Composite Price Index

• Weights prices by quantities purchased before
  computing totals
• Weighted totals used to compute composite index
• Laspeyres Index
• Uses base period quantities as weights
• Paasche Index
• Uses quantities from each period as weights

20
Laspeyres Index

• Uses base period quantities as weights
• Appropriate when quantities remain approximately
  constant over time period
• Example Consumer Price Index (CPI)

21
Calculating a Laspeyres Index
Note t0 subscript stands for base period
where
Pit price for each commodity at time t Qit
quantity of each commodity at time t t0 base
period
22
Laspeyres Index Number Example

• The table shows the closing stock prices on
  1/31/2005 and 12/29/2006 for DaimlerChrysler,
  Ford, and GM. On 1/31/2005 an investor purchased
  the indicated number of shares of each stock.
  Construct the Laspeyres Index using 1/31/2005 as
  the base period.

23
Base Value
Weighted total for base period (1/31/2005)
Weighted total for current period 12/29/2006
24
Laspeyres Index Solution
Indicates portfolio value had decreased by 13.3
(10086.7) between 1/31/2005 and 12/29/2006.
25
Class Exercise
Example US copper and steel prices production
Calculate the Laspeyres price index for the
current period to nearest 1. Enter A for
93.6, B for 95.5, C for 102.3
26
Paasche Index

• Uses quantities for each period as weights
• Appropriate when quantities change over time
• Compare current prices to base period prices at
  current purchase levels
• Disadvantages
• Must know purchase quantities for each time
  period
• Difficult to interpret a change in index when
  base period is not used

27
Calculating a Paasche Index
Weights are quantities for time period t
where
Pit price for each commodity at time t Qit
quantity of each commodity at time t t0 base
period
28
Laspeyres Index Number Example

• The table shows the 1/31/2005 and 12/29/2006
  prices and volumes in millions of shares for
  DaimlerChrysler, Ford, and GM. Calculate the
  Paasche Index using 1/31/2005 as the base period.
  (Source Nasdaq.com)

29
Paasche Index Solution
30
Paasche Index Solution
12/29/2006 prices represent a 24.8 (100 75.2)
decrease from 1/31/2005 (assuming quantities were
at 12/29/2006 levels for both periods)
31
Class Exercise
Example US copper and steel prices production
Calculate the Paasche price index for the current
period (enter rounded whole number) Enter 1 for
93.5, 2 for 95.5, 3 for 102.3
32
Exponential Smoothing
33
Exponential Smoothing

• Type of weighted average
• Removes rapid fluctuations in time series (less
  sensitive to shortterm changes in prices)
• Allows overall trend to be identified
• Used for forecasting one period future values
• Exponential smoothing constant (w) affects
  smoothness of series

34
Exponential Smoothing Constant

• Exponential smoothing constant, 0 lt w lt 1
• w close to 0
• More weight given to previous values of time
  series
• Smoother series
• w close to 1
• More weight given to current value of time series
• Series looks similar to original (more variable)

35
Calculating an Exponential Smoothed Series

• E1 Y1 (same as original series or a given
  value)
• E2 wY2 (1 w)E1 E w(Y2 E1)
• E3 wY3 (1 w)E2
• Et wYt (1 w)Et1

See Intrate30.xls
36
Exponential Smoothing Example

• The closing stock prices on the last day of the
  month for DaimlerChrysler in 2005 and 2006 are
  given in the table. Create an exponentially
  smoothed series using w .2.

37
Exponential Smoothing Solution

• E1 45.51
• E2 .2(46.10) .8(45.51) 45.63
• E3 .2(44.72) .8(45.63) 45.45
• E24 .2(61.41) .8(53.92) 55.42

38
Exponential Smoothing Solution

• E1 45.51
• E2 .2(46.10) .8(45.51) 45.63
• E3 .2(44.72) .8(45.63) 45.45
• E24 .2(61.41) .8(53.92) 55.42

39
Exponential Smoothing Solution
Actual Series
Smoothed Series (w .2)
40
Class Exercise
Example US copper production
Construct the exponentially smoothed price series
using January as the base period The March value
is A. 1000 B. 1006 C. 1010
41
Class Exercise
Example US copper production
Construct the exponentially smoothed price series
using January as the base period The March value
is A.1000 B.1016 C.1040
42
Measuring Forecast Accuracy
43
Mean Absolute Deviation (MAD)

• Mean absolute difference between the forecast and
  actual values of the time series
• where m number of forecasts used

44
Mean Absolute Percentage Error (MAPE)

• Mean of the absolute percentage of the difference
  between the forecast and actual values of the
  time series
• where m number of forecasts used

45
Root Mean Squared Error (RMSE)

• Square root of the mean squared difference
  between the forecast and actual values of the
  time series
• where m number of forecasts used

46
Forecasting Accuracy Example

• Using the DaimlerChrysler data from 1/31/2005
  through 8/31/2006, three time series models were
  constructed and forecasts made for the next four
  months.
• Model I Exponential smoothing (w .2)
• Model II Exponential smoothing (w .8)
• Model III HoltWinters (w .8, v .7)

47
Forecasting Accuracy Example

• Model I

48
Forecasting Accuracy Example

• Model II

49
Forecasting Accuracy Example

• Model III

50
Forecasting Trends Simple Linear Regression
51
Time Series Components

• Additive Time Series Model Yt Tt Ct St
  Rt
• Tt secular trend (describes longterm movements
  of Yt)
• Ct cyclical effect (describes fluctuations
  about the secular trend attributable to business
  and economic conditions)
• St seasonal effect (describes fluctuations that
  recur during specific time periods)
• Rt residual effect (what remains after other
  components have been removed)

52
Simple Linear Regression

• Model E(Yt) ß0 ß1t
• Relates time series, Yt, to time, t
• Cautions
• Risky to extrapolate (forecast beyond observed
  data)
• Does not account for cyclical effects

53
Simple Linear Regression Example

• The data shows the average undergraduate tuition
  at all 4year institutions for the years
  19962004 (Source U.S. Dept. of Education). Use
  leastsquares regression to fit a linear model.
  Forecast the tuition for 2005 (t 11) and
  compute a 95 prediction interval for the
  forecast.

54
Simple Linear Regression Solution

• From Excel

55
Simple Linear Regression Solution
56
Simple Linear Regression Solution

• Forecast tuition for 2005 (t 11)

95 prediction interval
57
Seasonal Regression Models
58
Seasonal Regression Models

• Takes into account secular trend and seasonal
  effects (seasonal component)
• Uses multiple regression models
• Dummy variables to model seasonal component
• E(Yt) ß0 ß1t ß2Q1 ß3Q2 ß4Q3 where

See QtrGDPAnalyzed.xls
59
Autocorrelation and The DurbinWatson Test
60
Autocorrelation

• Time series data may have errors that are not
  independent
• Time series residuals
• Correlation between residuals at different points
  in time (autocorrelation)
• 1st order correlation Correlation between
  neighboring residuals (times t and t 1)
• If present can investigate use of models
• AR autoregressive predict DV from precious DVs
  only
• ARMA autoregressive moving average predict DV
  based on prior values of DVs and IVs
• ARIMA autoregressive integrated moving average
  (more so)

61
Autocorrelation

• Plot of residuals v. time for tuition data shows
  residuals tend to group alternately into positive
  and negative clusters

62
DurbinWatson Test

• Ho No firstorder autocorrelation of residuals
• Ha Positive firstorder autocorrelation of
  residuals
• Test Statistic

63
Interpretation of d Statistic

• 0 d 4
• If residuals uncorrelated, then d 2
• If residuals positively autocorrelated, then d lt
  2
• If residuals negatively autocorrelated, then d gt2

See Sales35Analyzed.xls to determine serial
correlation
64
Rejection Region for the DurbinWatson d Test
Rejection region evidence of positive
autocorrelation
d
3
2
4
0
1
dL
dU
Nonrejection region insufficient evidence of
positive autocorrelation
Possibly significant autocorrelation
65
DurbinWatson Test Example

• Use the DurbinWatson test to test for the
  presence of autocorrelation in the tuition data.
  Use a .05.

66
DurbinWatson Test Solution

• H0
• Ha
• ? ? n k
• Critical Value(s)

Test Statistic Decision Conclusion
.05 10 1
67
DurbinWatson Solution

• Test Statistic

68
DurbinWatson Test Solution
No 1storderautocorrelation

• H0
• Ha
• ? ? n k
• Critical Value(s)

Test Statistic Decision Conclusion
d .51
Positive 1storderautocorrelation
.05 10 1
Reject at ? .05
There is evidence of positive autocorrelation
d
2
4
0
.88
1.32
69
Conclusion

• Described Time Series
• Explained Descriptive Analyses
• Defined Time Series Components
• Explained Forecasting
• Described Measures of Accuracy
• Defined Autocorrelation
• Explained DurbinWatson Test