Power 14 - UCSB Economics

1

Econ 240 C Lecture 14

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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 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.

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

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

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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 Reitsch, Business Forecasting, 6th Ed. 

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

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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

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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

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Observation

15

16

17

18

19

20

21

22

23

24

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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

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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

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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

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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

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33

Forecast = L(25) = 0.26*Sales(24) + 0.74L(24) = 532.4 =0.26*650 + 0.74*491.07 =532.4

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35

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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 

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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.

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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).

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40

41

Retail Sales: Simple Exponential Smoothing

170000

160000

150000

140000

130000 90

91

92 RETAIL

93

94

95

RETAILSM

96

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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 

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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) + ….

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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.

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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>=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

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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>=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) 

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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 

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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.

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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? 

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60

The simple-minded approach

D=549 - 162 D=387

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62

63

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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

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67

68

69

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Analysis: Pre-Event Differences

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72

73

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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 

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Pre-Event Analysis 

Seasonally differenced, differenced series

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78

79

80

81

Pre-Event Box-Jenkins Model 

[1-Z12 ][1 –Z]Assist(t) = WN(t) – a*WN(t-12)

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84

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Modeling the Event 

Step function

86

Entire Series Assist and Step  Dassist and Dstep  Sddast sddstep 

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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 

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92

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Policy Change Effect Simple: decrease of 387 (thousand) calls per month  Intervention model: decrease of 397 with a standard error of 22 

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Stochastic Trends: Random Walks with Drift

We have discussed earlier in the course how to model the Total Return to the 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

So the fractional change in the total return to the S&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: 

98      



[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

99

Part IV. Dickey Fuller Tests: Trend

100

Example 

Lnsp500(t) from Lab 2

101

102

103

104