lecture 14 econometrics

7/29/2019 Lecture 14 econometrics

1/35

Lecture 15. Dummy variables, continued

Seasonal effects in time series

Consider relation between electricity consumption Y and electricityprice X .

The data are quarterly time series.

First model

tttuXY ++= lnln 21

What is the interpretation of 2 ?


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Because electricity consumption depends on the weather and specialcircumstances (Christmas, summer holidays) we expect is to be

different in the quarters, even if the price is constant.

Solution: Define

11 =tD ift is the first quarter of a year

01=

tD if not

Define 432 ,, ttt DDD analogously for the other quarters.


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To allow for differences in the average consumption betweenquarters we write

44332211 DDD +++=

Substitution in the regression model gives

ttttttuXDDDY +++++= lnln 24433221

Why not 1tD in model?


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Average electricity consumption in the four quarters (given price tX )

Quarter 1ttt

XXYE ln)|(ln 21 +=

Quarter 2ttt

XXYE ln)|(ln 221 ++=

Quarter 3ttt


Quarter 4ttt


Interpretation 432 ,, : Relative change (relative to quarter 1) of

electricity consumption in quarters 2,3,4.


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Change of reference quarter to quarter 2:

44331211 DDD +++=

Intercept in the four quarters

Reference quarter is quarter 1

4131211 ,,, +++

Reference quarter is quarter 2

4131121 ,,, +++


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Hence

24423322211 ,,, ===+=

The same relations hold for the OLS estimates. Change of referencequarter does not require re-estimation.

Same result holds for change in reference category for any

qualitative variable with more than two values (e.g. earlier example

with type of work)


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If we want to investigate whether price elasticity depends on seasonwe write

48372652 DDD +++=

Substitution gives

ttttttt

ttttt

uXDXDXD

XDDDY

+++

+++++=

lnlnln

lnln

483726

54433221


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Price elasticities in the four quarters

Quarter 1 5

Quarter 2 65 +

Quarter 3 75 + Quarter 4 85 +


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Tests:

Average demand does not change with the season

Price elasticity constant over seasons

Derive price elasticities if we choose period 2 as reference period.


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Structural change

Events may change the relation between economic variables.

Consider time series data on dependent variableY

andindependent variable X for e.g. years nt ,,1K= .

In year 0nt= some event happens.

This event induces a structural change if the regression

coefficients change due to the event.


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Original model (no structural change)

ntuXYttt

,,1, K=++=

Model with structural change in0

n :

011 ,,1, ntuXY ttt K=++=

nntuXYttt

,,1,)( 02121 K+=++++=


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This is equivalent to introducing the dummy variable

0=t

D for 0,,1 nt K=

1=

t

D

fornnt

,,10K+=

with the model

ntuXDXDYtttttt

,,1,2121

K=++++=

Test for structural change can be done in two ways

Estimate separate models and compare ESS

Estimate model with dummy and test 0,0 22 ==

This gives the same value for the test statistic.


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Outliers

There may be individual observations that do not fit the relation

See output/graphs

Reason:

Omitted variables

Error in the data

Some unknown event/circumstance

How to check this?


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Introduce dummy variable

123, =iD for observation 23 (and 0 otherwise)

Include this in the regression model and test whether coefficient is 0.

See output.


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Dependent Variable: LNWAGEMethod: Least SquaresDate: 11/01/01 Time: 08:42

Sample: 1 49Included observations: 49

Variable Coefficient Std. Error t-Statistic Prob.

C 6.864366 0.186127 36.88002 0.0000EDUC 0.052987 0.017107 3.097432 0.0034

EXPER 0.020776 0.006321 3.286999 0.0020AGE -0.002250 0.003804 -0.591382 0.5574

RACE 0.071479 0.081543 0.876575 0.3856GENDER 0.242610 0.071645 3.386300 0.0015

R-squared 0.470916 Mean dependent var 7.454952Adjusted R-squared 0.409395 S.D. dependent var 0.312741S.E. of regression 0.240344 Akaike info criterion 0.100786Sum squared resid 2.483904 Schwarz criterion 0.332438Log likelihood 3.530733 F-statistic 7.654508Durbin-Watson stat 1.708658 Prob(F-statistic) 0.000032


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-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

5 10 15 20 25 30 35 40 45

LNWAGE Residuals


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obs Actual Fitted Residual Residual Plot

1 7.20415 7.20983 -0.00568

2 7.79770 7.64738 0.150323 7.44717 7.47824 -0.03107

4 7.28688 7.44899 -0.16212

5 7.40184 7.57869 -0.17685

6 7.20415 7.27025 -0.06610

7 7.37901 7.36391 0.01509

8 7.04229 7.06440 -0.02211

9 7.35628 7.78801 -0.43173

10 7.31055 7.61880 -0.30825

11 7.11802 7.16206 -0.0440412 7.20415 7.19236 0.01179

13 7.20415 7.36341 -0.15926

14 8.12829 7.90676 0.22153

15 7.51698 7.67094 -0.15396

16 6.88857 7.34406 -0.45549

17 7.20415 7.54399 -0.33984

18 7.35628 7.14941 0.20687

19 7.07918 7.21830 -0.13911

20 7.20415 7.41777 -0.21362

21 7.20415 7.35979 -0.15564

22 7.68110 7.56638 0.11472

23 7.24566 7.69937 -0.45372

24 7.65681 7.51358 0.14323

25 7.70436 7.68690 0.01746

26 8.18172 7.69434 0.48738

27 7.58680 7.32344 0.26337

28 7.11802 7.09935 0.01866

29 7.56320 7.43966 0.12354

30 7.68018 7.43267 0.2475131 7.76853 7.35286 0.41568

32 7.20415 7.41537 -0.21122

33 7.51698 7.28394 0.23304

34 7.86825 7.65101 0.21724

35 7.83716 7.72690 0.11026

36 7.37901 7.39163 -0.01262

37 7.51698 7.69670 -0.17973

38 7.70436 7.45990 0.24446

39 7.33237 7.18733 0.1450440 7.28688 7.29798 -0.01110

41 8.10380 8.01117 0.09263

42 8.25140 7.75389 0.49751

43 7.51698 7.48605 0.03093

44 7.28688 7.29015 -0.00328

45 7.26753 7.58319 -0.31567


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C 6.789626 0.182398 37.22432 0.0000GENDER 0.261107 0.069438 3.760286 0.0005

AGE -0.001271 0.003687 -0.344724 0.7320EXPER 0.018787 0.006149 3.055107 0.0039

EDUC 0.061945 0.016981 3.647842 0.0007RACE 0.065118 0.078464 0.829904 0.4113D23 -0.530696 0.249938 -2.123314 0.0397

R-squared 0.522205 Mean dependent var 7.454952Adjusted R-squared 0.453948 S.D. dependent var 0.312741S.E. of regression 0.231101 Akaike info criterion 0.039638Sum squared resid 2.243118 Schwarz criterion 0.309898Log likelihood 6.028867 F-statistic 7.650623Durbin-Watson stat 1.653329 Prob(F-statistic) 0.000014


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C 7.401588 0.160656 46.07096 0.0000GENDER 0.276639 0.074299 3.723311 0.0006EXPER 0.017241 0.004698 3.669938 0.0007EDUC 0.022429 0.013386 1.675632 0.1018

AGE -0.002105 0.002676 -0.786674 0.4362RACE 0.095861 0.060972 1.572208 0.1240D23 -0.293381 0.187884 -1.561503 0.1265

CLERICAL -0.419411 0.083845 -5.002245 0.0000CRAFTS -0.342397 0.081728 -4.189469 0.0002MAINT -0.525459 0.092669 -5.670253 0.0000

R-squared 0.778514 Mean dependent var 7.454952Adjusted R-squared 0.727402 S.D. dependent var 0.312741S.E. of regression 0.163285 Akaike info criterion -0.606736Sum squared resid 1.039816 Schwarz criterion -0.220650Log likelihood 24.86503 F-statistic 15.23148Durbin-Watson stat 1.985725 Prob(F-statistic) 0.000000


20/35

Application: Election 2000 in Florida

Effect of butterfly ballot in Palm Beach County on Buchanan vote

Data for all Florida counties

Votes candidates

Size and demographic composition of counties (census). What isrelevant?


21/35

Model

Dependent variable?

Independent variables?

How do we check whether Palm Beach is different?


22/35

Election 2000 in Florida: Butterfly ballot in Palm Beach county

Outcome of 2000 presidential election disputed.

Claims of voting irregularities in Florida.

One issue was a confusing ballot design in Palm Beach county, thebutterfly ballot.

Order of punch holes different from order of the two main candidates,Bush and Gore.

Claim: Many voters mistakenly voted for Buchanan, the candidate of

the Reform Party.


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Research question: Did Buchanan get an unusually large fraction of thevotes in Palm Beach county?


24/35

Regression model

Dependent variable: log of fraction votes for Buchanan.

Independent variables

Percentage of population Hispanic

Percentage of population Black

Percentage of population over 65

Percentage of population with college degree

Income (1000$ per year)

Population (10000)


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Descriptive statistics

Date:04/06/05Sample: 1 67

Time: 22:14

FRACBUCHA FRACGORE FRACBUSH PERCBLACK PERCHISPAN PERCOVER6 PERCCOLLE INCOME1000 POPULATION

Mean 0.004697 0.428125 0.551544 15.89701 6.288060 16.80293 13.89701 26.18864 21.87156Median 0.003976 0.430705 0.549881 14.40000 3.500000 14.60939 11.90000 25.71800 8.191900Maximum 0.017452 0.676075 0.741084 61.80000 54.40000 33.43856 37.10000 38.13000 204.4600Minimum 0.000897 0.241105 0.310129 2.300000 0.900000 6.974674 5.200000 17.09800 0.628900Std. Dev. 0.003218 0.091383 0.092058 11.07191 8.186436 7.011421 6.588534 4.794646 36.05383Skewness 1.912928 0.348309 -0.282254 1.926733 3.699223 0.846034 1.203425 0.446578 3.023152

Kurtosis 7.448237 3.331569 3.187830 7.627821 19.94750 2.718083 4.690722 2.364601 13.30769

J arque-Bera 96.10031 1.661643 0.988110 101.2424 954.6232 8.214687 24.15201 3.354070 398.6674Probability 0.000000 0.435691 0.610147 0.000000 0.000000 0.016451 0.000006 0.186927 0.000000

Observations 67 67 67 67 67 67 67 67 67


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OLS results: Basis equation


27/35

Signs of coefficients plausible?

Interpretation of coefficients: dependent variable is log!


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OLS residuals: Graph

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

5 10 15 20 25 30 35 40 45 50 55 60 65

LNFRACBUCHANAN Residuals


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OLS residuals: Table


30/35


31/35

OLS results: Palm Beach dummy

To check whether Palm Beach is special include dummy that is 1 for PalmBeach (observation 50) and 0 otherwise


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Interpretation of Palm Beach dummy

dy 794.1521.2ln ++= "

Hence

794.1lnln = normalobserved yy

so that

103.51

794.1

==

ey

yy

normal

normalobserved

i.e. fraction 5 times higher than expected. Fraction is .00789.


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Sensitivity check: Include log fraction Bush vote


34/35

Effect on Bush and Gore vote


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lecture 14 econometrics

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