# Power 14 - UCSB Economics

```1
Econ 240 C
Lecture 14
2
Project II
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 nontechnical reader. It should explain the problem
being examined from an economic perspective,
i.e. it should motivate interest in the issue on the
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
The technical details of your findings you can attach as an appendix
Technical Appendix
1.
2.
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
4
5
Group A
Group B
Group C
Tara Copello
Zhimin Zhou
Andrea Cardani
Jonathan Hester
Evan Nakano
Eric Laschinger
Yana Ten
Pungdalis Suos
Micah Witt
Charles Rabkin
Will Hippen
Thomas Bruister
Arnaud Piechaud
Kyu-Sang Park
Calvin Yeung
Andrew Cahill
Ashley Hedberg
Jesse Smith
Darren Doi
Sarab Khalsa
Jong Duk Woo
Group D
Group E
Carl-Einar Thorner
Robert Connor Gleason
Antung Anthony Liu
Hamid Ghofrani
Joonho Shin
Ufook Sahilliohlu
Jeffrey Ahlvin
Russell Ludwick
Aren Megerdichian
Carrie Koen
Anthony Kasza
Matthew Stevens
6
Outline

Exponential Smoothing
 Back
of the envelope formula: geometric distributed
lag: L(t) = a*y(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) + a*e(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
Santa Barbara South Coast Median House Price in Nominal Thousands
1000
HSEPRC
800
600
400
200
0
1970
1980
1990
2000
2010
Three Rates of Growth
9
7
LNHSEPRC
6
5
4
3
1970
1980
1990
Y EAR
2000
2010
10
Santa Barbara South Coast House Price, 000 04 \$
1000
HSEPRC04
800
600
400
200
0
1970
1980
1990
YEAR
2000
2010
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) = a*y(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 t+1 is given by the
formula: L(t+1) = a*y(t) + (1-a)*L(t)
 Example from John Heinke and Arthur

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
13
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) = 0.1*Sales(1) + 0.9*Level(1)
L(2) = 0.1*500 + 0.9*500 = 500
 And L(3) = 0.1*Sales(2) + 0.9*Level(2)
L(3) = 0.1*350 + 0.9*500 = 485

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
Level
500
15
observations
Sales
16
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

observations
Sales
18
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)
= 0.1*sales(24)+0.9*(24)
 Forecast(25)=Levels(25)=0.1*650+0.9*449
 Forecast(25) = 469.1

Single Exponential Sm oothing
800
700
600
Value
500
Sales
400
Levels
300
200
100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Observation
15
16
17
18
19
20
21
22
23
24
21
Single exponential distribution

The errors can now be calculated:
e(t) = sales(t) – levels(t)
observations
Sales
Level
22
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
observations
Sales
Level
23
error
squared
error
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
observations
Sales
Level
24
error
squared
error
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
sum sq res
582281.2
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

observa
tions
26
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) = 0.6*sales(24) +
0.4*levels(24) = 0.6*650 + 0.4*465 = 776
observa
tions
Sales
Levels
28
error
square
error
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
a = 0.6
Sum of Sq Res
533961.9
29
Single exponential smoothing

Grid search plot
Grid Search for Sm oothing Param eter
590000
580000
Sum of Squared Residuals
570000
560000
550000
540000
530000
520000
510000
500000
490000
0
0.2
0.4
0.6
Sm oothing Param eter
0.8
1
1.2
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
33
Forecast = L(25) = 0.26*Sales(24) + 0.74L(24) = 532.4
=0.26*650 + 0.74*491.07 =532.4
34
35
36
Part II. Three Perspectives on
Single Exponential Smoothing

The formula perspective
 L(t)
= a*y(t-1) + (1 - a)*L(t-1)
 e(t) = y(t) - L(t)
The Box-Jenkins Perspective
 The Updating Forecasts Perspective

Box Jenkins Perspective

37
Use the error equation to substitute for L(t) in
the formula, L(t) = a*y(t-1) + (1 - a)*L(t-1)
 L(t)
= y(t) - e(t)
 y(t) - e(t) = a*y(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
40
41
Retail Sales: Simple Exponential Smoothing
170000
160000
150000
140000
130000
90
91
92
RETAIL
93
94
95
RETAILSM
96
43
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
Weekly Closing Price of Gold , Nov. 14, 2003-April 29, 2005
460
440
420
400
380
360
10
20
30
40
GOLD
50
60
70
46
47
48
Box-Jenkins Perspective

49
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)
= a*y(t-1) + (1 - a)*L(t-1)
 L(t) = a*y(t-1) + (1 - a)*ZL(t)
 L(t) - (1 - a)*ZL(t) = a*y(t-1)
 [1 - (1-a)Z] L(t) = a*y(t-1)
 L(t) = {1/ [1 - (1-a)Z]} a*y(t-1)
 L(t) = [1 +(1-a)Z + (1-a)2 Z2 + …] a*y(t-1)
 L(t) = a*y(t-1) + (1-a)*a*y(t-2) + (1-a)2a*y(t-3)
+ ….
50
51
The Updating Forecasts Perspective

Use the error equation to substitute for y(t) in
the formula, L(t) = a*y(t-1) + (1 - a)*L(t-1)
 y(t)
= L(t) + e(t)
 L(t) = a*[L(t-1) + e(t-1)] + (1 - a)*L(t-1)

So L(t) = L(t-1) + a*e(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
k 1
 f(t+k) = L(t) + k*R(t),
k&gt;=1
 L(t) = a*y(t) + (1-a)*[L(t-1) + R(t-1)]
 R(t) = b*[L(t) - L(t-1)] + (1-b)*R(t-1)

k 1
 so
the trend, R(t), is a geometric distributed
lag of the change in levels, DL(t)
Part III. Double Exponential 53
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 HoltWinters smoothing

Part III. Double Exponential
Smoothing
54
Holt- Winters can also be used to forecast
seasonal time series, e.g. monthly
 f(t+k) = L(t) + k*R(t) + S(t+k-12)
k&gt;=1
 L(t) = a*[y(t)-S(t-12)]+ (1-a)*[L(t-1) + R(t-1)]
 R(t) = b*[L(t) - L(t-1)] + (1-b)*R(t-1)
 S(t) = c*[y(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 prewhitening, to the extent that nonstationarity such as trend and seasonality
are removed
 Example: preview of Lab 8

57
Telephone Directory Assistance

A telephone company was receiving
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
60
The simple-minded approach
D=549 - 162
D=387
61
62
63
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
67
68
69
70
Analysis: Pre-Event Differences
71
72
73
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 preevent differenced series

75
76
Pre-Event Analysis

Seasonally differenced, differenced series
77
78
79
80
81
Pre-Event Box-Jenkins Model

[1-Z12 ][1 –Z]Assist(t) = WN(t) – a*WN(t-12)
82
83
84
85
Modeling the Event

Step function
86
Entire Series
Assist and Step
 Dassist and Dstep
 Sddast sddstep

87
88
89
90
Model of Series and Event
Pre-Event Model: [1-Z12 ][1 –Z]Assist(t) =
WN(t) – a*WN(t-12)
 In Levels Plus Event: Assist(t)=[WN(t) –
a*WN(t-12)]/[1-Z]*[1-Z12] + (-b)*step
 Estimate: [1-Z12 ][1 –Z]Assist(t) = WN(t) –
a*WN(t-12) + (-b)* [1-Z12 ][1 –Z]*step

91
92
93
Policy Change Effect
Simple: decrease of 387 (thousand) calls
per month
 Intervention model: decrease of 397 with
a standard error of 22

94
95
Stochastic Trends: Random Walks
with Drift
We have discussed earlier in the course
Standard and Poor’s 500 Index
 One possibility is this time series could be
a random walk around a deterministic
trend”
 Sp500(t) = exp{a + d*t +WN(t)/[1-Z]}
 And taking logarithms,

96
Stochastic Trends: Random Walks
with Drift
Lnsp500(t) = a + d*t + WN(t)/[1-Z]
 Lnsp500(t) –a –d*t = WN(t)/[1-Z]
 Multiplying through by the difference operator, D
= [1-Z]
 [1-Z][Lnsp500(t) –a –d*t] = WN(t-1)

[LnSp500(t) – a –d*t] - [LnSp500(t-1) – a –d*(t1)] = WN(t)
 D Lnsp500(t) = d + WN(t)

97
the S&amp;P 500 is drift, d, plus white noise
 More generally,
 y(t) = a + d*t + {1/[1-Z]}*WN(t)
 [y(t) –a –d*t] = {1/[1-Z]}*WN(t)
 [y(t) –a –d*t]- [y(t-1) –a –d*(t-1)] = WN(t)
 [y(t) –a –d*t]= [y(t-1) –a –d*(t-1)] + WN(t)
 Versus the possibility of an ARONE:

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






[y(t) –a –d*t]=b*[y(t-1)–a–d*(t-1)]+WN(t)
Y(t) = a + d*t + b*[y(t-1)–a–d*(t-1)]+WN(t)
Or y(t) = [a*(1-b)+b*d]+[d*(1-b)]*t+b*y(t-1)
+wn(t)
Subtracting y(t-1) from both sides’
D y(t) = [a*(1-b)+b*d] + [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)+b*d] and a trend [d*(1-b)]*t
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Part IV. Dickey Fuller Tests: Trend
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Example

Lnsp500(t) from Lab 2
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