Econ 240 C

1
Econ 240 C
1

• Lecture 14

2
Project II
3
I.  Work in Groups II.  You will be graded based
on a PowerPoint presentation and a written
report. III.   Your report should have an
executive summary of one to one and a half pages
that summarizes your findings in words for a
non-technical reader. It should explain the
problem being examined from an economic
perspective, i.e. it should motivate interest in
the issue on the part of the reader. Your report
should explain how you are investigating the
issue, in simple language. It should explain why
you are approaching the problem in this
particular fashion. Your executive report should
explain the economic importance of your
findings.
4
The technical details of your findings you can
attach as an appendix
Technical Appendix 1.      Table of
Contents 2.      Spreadsheet of data used and
sources or, if extensive, a subsample of the
data 3.      Describe the analytical time series
techniques you are using 4.      Show descriptive
statistics and histograms for the variables in
the study 5.      Use time series data for your
project show a plot of each variable against time
5
Group A Group B Group C Tara
Copello Pungdalis Suos Calvin Yeung Zhimin
Zhou Micah Witt Andrew Cahill Andrea
Cardani Charles Rabkin Ashley Hedberg Jonathan
Hester Will Hippen Jesse Smith Evan
Nakano Thomas Bruister Darren Doi Eric
Laschinger Arnaud Piechaud Sarab Khalsa Yana
Ten Kyu-Sang Park Jong Duk Woo Group D Group
E Carl-Einar Thorner Jeffrey Ahlvin Robert
Connor Gleason Russell Ludwick Antung Anthony
Liu Aren Megerdichian Hamid Ghofrani Carrie
Koen Joonho Shin Anthony Kasza Ufook
Sahilliohlu Matthew Stevens
6
Outline

• Exponential Smoothing
• Back of the envelope formula geometric
  distributed lag L(t) ay(t-1) (1-a)L(t-1)
  F(t) L(t)
• ARIMA (p,d,q) (0,1,1) ?y(t) e(t)
  (1-a)e(t-1)
• Error correction L(t) L(t-1) ae(t)
• Intervention Analysis

7
Part I Exponential Smoothing

• Exponential smoothing is a technique that is
  useful for forecasting short time series where
  there may not be enough observations to estimate
  a Box-Jenkins model
• Exponential smoothing can be understood from many
  perspectives one perspective is a formula that
  could be calculated by hand

8
(No Transcript)
9
Three Rates of Growth
10
(No Transcript)
11
Simple exponential smoothing

• Simple exponential smoothing, also known as
  single exponential smoothing, is most appropriate
  for a time series that is a random walk with
  first order moving average error structure
• The levels term, L(t), is a weighted average of
  the observation lagged one, y(t-1) plus the
  previous levels, L(t-1)
• L(t) ay(t-1) (1-a)L(t-1)

12
Single exponential smoothing

• The parameter a is chosen to minimize the sum of
  squared errors where the error is the difference
  between the observation and the levels term e(t)
  y(t) L(t)
• The forecast for period t1 is given by the
  formula L(t1) ay(t) (1-a)L(t)
• Example from John Heinke and Arthur Reitsch,
  Business Forecasting, 6th Ed.

13
observations Sales
1 500
2 350
3 250
4 400
5 450
6 350
7 200
8 300
9 350
10 200
11 150
12 400
13 550
14 350
15 250
16 550
17 550
18 400
19 350
20 600
21 750
22 500
23 400
24 650
14
Single exponential smoothing

• For observation 1, set L(1) Sales(1) 500, as
  an initial condition
• As a trial value use a 0.1
• So L(2) 0.1Sales(1) 0.9Level(1) L(2)
  0.1500 0.9500 500
• And L(3) 0.1Sales(2) 0.9Level(2) L(3)
  0.1350 0.9500 485

15
observations Sales Level
1 500 500
2 350  
3 250  
4 400  
5 450  
6 350  
7 200  
8 300  
9 350  
10 200  
11 150  
12 400  
13 550  
14 350  
15 250  
16 550  
17 550  
18 400  
19 350  
20 600  
21 750  
22 500  
23 400  
24 650  
16
observations Sales Level
1 500 500
2 350 500
3 250 485
4 400  
5 450  
6 350  
7 200  
8 300  
9 350  
10 200  
11 150  
12 400  
13 550  
14 350  
15 250  
16 550  
17 550  
18 400  
19 350  
20 600  
21 750  
22 500  
23 400  
24 650  
a 0.1
17
Single exponential smoothing

• So the formula can be used to calculate the rest
  of the levels values, observation 4-24
• This can be set up on a spread-sheet

18
observations Sales Level
1 500 500
2 350 500
3 250 485
4 400 461.5
5 450 455.4
6 350 454.8
7 200 444.3
8 300 419.9
9 350 407.9
10 200 402.1
11 150 381.9
12 400 358.7
13 550 362.8
14 350 381.6
15 250 378.4
16 550 365.6
17 550 384.0
18 400 400.6
19 350 400.5
20 600 395.5
21 750 415.9
22 500 449.3
23 400 454.4
24 650 449.0
a 0.1
19
Single exponential smoothing

• The forecast for observation 25 is L(25)
  0.1sales(24)0.9(24)
• Forecast(25)Levels(25)0.16500.9449
• Forecast(25) 469.1

20
(No Transcript)
21
Single exponential distribution

• The errors can now be calculated e(t)
  sales(t) levels(t)

22
observations Sales Level error
1 500 500 0
2 350 500 -150
3 250 485 -235
4 400 461.5 -61.5
5 450 455.4 -5.35
6 350 454.8 -104.8
7 200 444.3 -244.3
8 300 419.9 -119.9
9 350 407.9 -57.9
10 200 402.1 -202.1
11 150 381.9 -231.9
12 400 358.7 41.3
13 550 362.8 187.2
14 350 381.6 -31.6
15 250 378.4 -128.4
16 550 365.6 184.4
17 550 384.0 166.0
18 400 400.6 -0.6
19 350 400.5 -50.5
20 600 395.5 204.5
21 750 415.9 334.1
22 500 449.3 50.7
23 400 454.4 -54.4
24 650 449.0 201.0
a 0.1
23
observations Sales Level error error squared
1 500 500 0 0
2 350 500 -150 22500
3 250 485 -235 55225
4 400 461.5 -61.5 3782.25
5 450 455.4 -5.35 28.62
6 350 454.8 -104.8 10986.18
7 200 444.3 -244.3 59698.86
8 300 419.9 -119.9 14376.05
9 350 407.9 -57.9 3353.58
10 200 402.1 -202.1 40852.14
11 150 381.9 -231.9 53780.95
12 400 358.7 41.3 1704.33
13 550 362.8 187.2 35027.05
14 350 381.6 -31.6 996.06
15 250 378.4 -128.4 16487.67
16 550 365.6 184.4 34016.68
17 550 384.0 166.0 27553.51
18 400 400.6 -0.6 0.37
19 350 400.5 -50.5 2554.91
20 600 395.5 204.5 41823.74
21 750 415.9 334.1 111594.53
22 500 449.3 50.7 2565.62
23 400 454.4 -54.4 2960.80
24 650 449.0 201.0 40412.28
a 0.1
24
observations Sales Level error error squared    
1 500 500 0 0    
2 350 500 -150 22500    
3 250 485 -235 55225    
4 400 461.5 -61.5 3782.25    
5 450 455.4 -5.35 28.62    
6 350 454.8 -104.8 10986.18    
7 200 444.3 -244.3 59698.86    
8 300 419.9 -119.9 14376.05    
9 350 407.9 -57.9 3353.58    
10 200 402.1 -202.1 40852.14    
11 150 381.9 -231.9 53780.95    
12 400 358.7 41.3 1704.33    
13 550 362.8 187.2 35027.05    
14 350 381.6 -31.6 996.06    
15 250 378.4 -128.4 16487.67    
16 550 365.6 184.4 34016.68    
17 550 384.0 166.0 27553.51    
18 400 400.6 -0.6 0.37    
19 350 400.5 -50.5 2554.91    
20 600 395.5 204.5 41823.74    
21 750 415.9 334.1 111594.53    
22 500 449.3 50.7 2565.62    
23 400 454.4 -54.4 2960.80    
24 650 449.0 201.0 40412.28    
          sum sq res 582281.2
a 0.1
25
Single exponential smoothing

• For a 0.1, the sum of squared errors is S
  (errors)2 582,281.2
• A grid search can be conducted for the parameter
  value a, to find the value between 0 and 1 that
  minimizes the sum of squared errors
• The calculations of levels, L(t), and errors,
  e(t) sales(t) L(t) for a 0.6

26
observations Sales Levels
1 500 500
2 350 500
3 250 410
4 400 314
5 450 365.6
6 350 416.2
7 200 376.5
8 300 270.6
9 350 288.2
10 200 325.3
11 150 250.1
12 400 190.0
13 550 316.0
14 350 456.4
15 250 392.6
16 550 307.0
17 550 452.8
18 400 511.1
19 350 444.4
20 600 387.8
21 750 515.1
22 500 656.0
23 400 562.4
24 650 465.0
a 0.6
27
Single exponential smoothing

• Forecast(25) Levels(25) 0.6sales(24)
  0.4levels(24) 0.6650 0.4465 776

28
observations Sales Levels error error square    
1 500 500 0 0    
2 350 500 -150 22500    
3 250 410 -160 25600    
4 400 314 86 7396    
5 450 365.6 84.4 7123.36    
6 350 416.2 -66.2 4387.74    
7 200 376.5 -176.5 31150.84    
8 300 270.6 29.4 864.45    
9 350 288.2 61.8 3814.38    
10 200 325.3 -125.3 15699.02    
11 150 250.1 -100.1 10023.67    
12 400 190.0 210.0 44080.13    
13 550 316.0 234.0 54747.14    
14 350 456.4 -106.4 11322.57    
15 250 392.6 -142.6 20324.22    
16 550 307.0 243.0 59036.75    
17 550 452.8 97.2 9445.88    
18 400 511.1 -111.1 12348.55    
19 350 444.4 -94.4 8920.73    
20 600 387.8 212.2 45037.39    
21 750 515.1 234.9 55172.40    
22 500 656.0 -156.0 24349.97    
23 400 562.4 -162.4 26379.58    
24 650 465.0 185.0 34237.15    
          Sum of Sq Res 533961.9
a 0.6
29
Single exponential smoothing

• Grid search plot

30
(No Transcript)
31
Single Exponential Smoothing

• EVIEWS Algorithmic search for the smoothing
  parameter a
• In EVIEWS, select time series sales(t), and open
• In the sales window, go to the PROCS menu and
  select exponential smoothing
• Select single
• the best parameter a 0.26 with sum of squared
  errors 472982.1 and root mean square error
  140.4 (472982.1/24)1/2
• The forecast, or end of period levels mean 532.4

32
(No Transcript)
33
(No Transcript)
34
Forecast L(25) 0.26Sales(24) 0.74L(24)
532.4 0.26650 0.74491.07 532.4
35
(No Transcript)
36
Part II. Three Perspectives on Single Exponential
Smoothing

• The formula perspective
• L(t) ay(t-1) (1 - a)L(t-1)
• e(t) y(t) - L(t)
• The Box-Jenkins Perspective
• The Updating Forecasts Perspective

37
Box Jenkins Perspective

• Use the error equation to substitute for L(t) in
  the formula, L(t) ay(t-1) (1 - a)L(t-1)
• L(t) y(t) - e(t)
• y(t) - e(t) ay(t-1) (1 - a)y(t-1) -
  e(t-1) y(t) e(t) y(t-1) - (1-a)e(t-1)
• or Dy(t) y(t) - y(t-1) e(t) - (1-a) e(t-1)
• So y(t) is a random walk plus MAONE noise, i.e
  y(t) is a (0,1,1) process where (p,d,q) are the
  orders of AR, differencing, and MA.

38
Box-Jenkins Perspective

• In Lab Eight, we will apply simple exponential
  smoothing to retail sales, a process you used for
  forecasting trend in Lab 3, and which can be
  modeled as (0,1,1).

39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
Box-Jenkins Perspective

• If the smoothing parameter approaches one, then
  y(t) is a random walk
• Dy(t) y(t) - y(t-1) e(t) - (1-a) e(t-1)
• if a 1, then Dy(t) y(t) - y(t-1) e(t)
• In Lab Eight, we will use the price of gold to
  make this point

45
(No Transcript)
46
(No Transcript)
47
(No Transcript)
48
(No Transcript)
49
Box-Jenkins Perspective

• The levels or forecast, L(t), is a geometric
  distributed lag of past observations of the
  series, y(t), hence the name exponential
  smoothing
• L(t) ay(t-1) (1 - a)L(t-1)
• L(t) ay(t-1) (1 - a)ZL(t)
• L(t) - (1 - a)ZL(t) ay(t-1)
• 1 - (1-a)Z L(t) ay(t-1)
• L(t) 1/ 1 - (1-a)Z ay(t-1)
• L(t) 1 (1-a)Z (1-a)2 Z2 ay(t-1)
• L(t) ay(t-1) (1-a)ay(t-2) (1-a)2ay(t-3)
  .

50
(No Transcript)
51
The Updating Forecasts Perspective

• Use the error equation to substitute for y(t) in
  the formula, L(t) ay(t-1) (1 - a)L(t-1)
• y(t) L(t) e(t)
• L(t) aL(t-1) e(t-1) (1 - a)L(t-1)
• So L(t) L(t-1) ae(t-1),
• i.e. the forecast for period t is equal to the
  forecast for period t-1 plus a fraction a of the
  forecast error from period t-1.

52
Part III. Double Exponential Smoothing

• With double exponential smoothing, one estimates
  a trend term, R(t), as well as a levels term,
  L(t), so it is possible to forecast, f(t), out
  more than one period
• f(tk) L(t) kR(t), kgt1
• L(t) ay(t) (1-a)L(t-1) R(t-1)
• R(t) bL(t) - L(t-1) (1-b)R(t-1)
• so the trend, R(t), is a geometric distributed
  lag of the change in levels, DL(t)

53
Part III. Double Exponential Smoothing

• If the smoothing parameters a b, then we have
  double exponential smoothing
• If the smoothing parameters are different, then
  it is the simplest version of Holt-Winters
  smoothing

54
Part III. Double Exponential Smoothing

• Holt- Winters can also be used to forecast
  seasonal time series, e.g. monthly
• f(tk) L(t) kR(t) S(tk-12) kgt1
• L(t) ay(t)-S(t-12) (1-a)L(t-1) R(t-1)
• R(t) bL(t) - L(t-1) (1-b)R(t-1)
• S(t) cy(t) - L(t) (1-c)S(t-12)

55
Part V. Intervention Analysis
56
Intervention Analysis

• The approach to intervention analysis parallels
  Box-Jenkins in that the actual estimation is
  conducted after pre-whitening, to the extent that
  non-stationarity such as trend and seasonality
  are removed
• Example preview of Lab 8

57
Telephone Directory Assistance

• A telephone company was receiving increased
  demand for free directory assistance, i.e.
  subscribers asking operators to look up numbers.
  This was increasing costs and the company changed
  policy, providing a number of free assisted calls
  to subscribers per month, but charging a price
  per call after that number.

58
Telephone Directory Assistance

• This policy change occurred at a known time,
  March 1974
• The time series is for calls with directory
  assistance per month
• Did the policy change make a difference?

59
(No Transcript)
60
The simple-minded approach

• 549 - 162
• 387

61
(No Transcript)
62
(No Transcript)
63
(No Transcript)
64
Principle

• The event may cause a change, and affect time
  series characteristics
• Consequently, consider the pre-event period,
  January 1962 through February 1974, the event
  March 1974, and the post-event period, April 1974
  through December 1976
• First difference and then seasonally difference
  the entire series

65
Analysis Entire Differenced Series
66
(No Transcript)
67
(No Transcript)
68
(No Transcript)
69
(No Transcript)
70
Analysis Pre-Event Differences
71
(No Transcript)
72
(No Transcript)
73
(No Transcript)
74
So Seasonal Nonstationarity

• It was masked in the entire sample by the
  variance caused by the difference from the event
• The seasonality was revealed in the pre-event
  differenced series

75
(No Transcript)
76
Pre-Event Analysis

• Seasonally differenced, differenced series

77
(No Transcript)
78
(No Transcript)
79
(No Transcript)
80
(No Transcript)
81
Pre-Event Box-Jenkins Model

• 1-Z12 1 ZAssist(t) WN(t) aWN(t-12)

82
(No Transcript)
83
(No Transcript)
84
(No Transcript)
85
Modeling the Event

• Step function

86
Entire Series

• Assist and Step
• Dassist and Dstep
• Sddast sddstep

87
(No Transcript)
88
(No Transcript)
89
(No Transcript)
90
Model of Series and Event

• Pre-Event Model 1-Z12 1 ZAssist(t) WN(t)
  aWN(t-12)
• In Levels Plus Event Assist(t)WN(t)
  aWN(t-12)/1-Z1-Z12 (-b)step
• Estimate 1-Z12 1 ZAssist(t) WN(t)
  aWN(t-12) (-b) 1-Z12 1 Zstep

91
(No Transcript)
92
(No Transcript)
93
Policy Change Effect

• Simple decrease of 387 (thousand) calls per
  month
• Intervention model decrease of 397 with a
  standard error of 22

94
(No Transcript)
95
Stochastic Trends Random Walks with Drift

• We have discussed earlier in the course how to
  model the Total Return to the Standard and Poors
  500 Index
• One possibility is this time series could be a
  random walk around a deterministic trend
• Sp500(t) expa dt WN(t)/1-Z
• And taking logarithms,

96
Stochastic Trends Random Walks with Drift

• Lnsp500(t) a dt WN(t)/1-Z
• Lnsp500(t) a dt WN(t)/1-Z
• Multiplying through by the difference operator, D
  1-Z
• 1-ZLnsp500(t) a dt WN(t-1)
• LnSp500(t) a dt - LnSp500(t-1) a
  d(t-1) WN(t)
• D Lnsp500(t) d WN(t)

97

• So the fractional change in the total return to
  the SP 500 is drift, d, plus white noise
• More generally,
• y(t) a dt 1/1-ZWN(t)
• y(t) a dt 1/1-ZWN(t)
• y(t) a dt- y(t-1) a d(t-1) WN(t)
• y(t) a dt y(t-1) a d(t-1) WN(t)
• Versus the possibility of an ARONE

98

• y(t) a dtby(t-1)ad(t-1)WN(t)
• Y(t) a dt by(t-1)ad(t-1)WN(t)
• Or y(t) a(1-b)bdd(1-b)tby(t-1)
  wn(t)
• Subtracting y(t-1) from both sides
• D y(t) a(1-b)bd d(1-b)t
  (b-1)y(t-1) wn(t)
• So the coefficient on y(t-1) is once again
  interpreted as b-1, and we can test the null that
  this is zero against the alternative it is
  significantly negative. Note that we specify the
  equation with both a constant,
• a(1-b)bd and a trend d(1-b)t

99
Part IV. Dickey Fuller Tests Trend
100
Example

• Lnsp500(t) from Lab 2

101
(No Transcript)
102
(No Transcript)
103
(No Transcript)
104
(No Transcript)