Lecture 15. Dummy variables, continued
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1 Lecure 15. Dummy variables, coninued Seasonal effecs in ime series Consider relaion beween elecriciy consumpion Y and elecriciy price X. The daa are quarerly ime series. Firs model ln α 1 + α2 Y = ln X + u Wha is he inerpreaion of α 2?
2 Because elecriciy consumpion depends on he weaher and special circumsances (Chrismas, summer holidays) we expec is o be differen in he quarers, even if he price is consan. Soluion: Define D 1 = 1 if is he firs quarer of a year D 1 = 0 if no Define D 2 D 3, D 4 analogously for he oher quarers.,
3 To allow for differences in he average consumpion beween quarers we wrie α + 1 = β1 + β2d2 + β3d3 β4d4 Subsiuion in he regression model gives lny = β 2 ln X + u 1 + β2d 2 + β3d3 + β4d 4 + α Why no D 1 in model?
4 Average elecriciy consumpion in he four quarers (given price X ) Quarer 1 E(lnY X ) = β 1 + α2 ln X Quarer 2 E(lnY X ) = β 1 + β2 + α2 ln X Quarer 3 E(lnY X ) = β 1 + β3 + α2 ln X Quarer 4 E(lnY X ) = β 1 + β4 + α2 ln X Inerpreaion β 2, β3, β4: Relaive change (relaive o quarer 1) of elecriciy consumpion in quarers 2,3,4.
5 Change of reference quarer o quarer 2: α + 1 = γ 1 + γ 2D1 + γ 3D3 γ 4D4 Inercep in he four quarers Reference quarer is quarer 1 β, β + β 1 β1 + β2, β1 + β3, Reference quarer is quarer 2 γ, γ + γ 1 + γ 2 γ 1, γ 1 + γ 3,
6 Hence γ 1 = β1 + β2 γ 2 = β2, γ 3 = β3 β2,, γ = β β The same relaions hold for he OLS esimaes. Change of reference quarer does no require re-esimaion. Same resul holds for change in reference caegory for any qualiaive variable wih more han wo values (e.g. earlier example wih ype of work) 4 4 2
7 If we wan o invesigae wheher price elasiciy depends on season we wrie D D D β β β β α = Subsiuion gives u X D X D X D X D D D Y = ln ln ln ln ln β β β β β β β β
8 Price elasiciies in he four quarers Quarer 1 β 5 Quarer 2 β 5 + β6 Quarer 3 β 5 + β7 Quarer 4 β 5 + β8
9 Tess: Average demand does no change wih he season Price elasiciy consan over seasons Derive price elasiciies if we choose period 2 as reference period.
10 Srucural change Evens may change he relaion beween economic variables. Consider ime series daa on dependen variable Y and independen variable X for e.g. years = 1, K, n. In year = n0 some even happens. This even induces a srucural change if he regression coefficiens change due o he even.
11 Original model (no srucural change) Y = α + β X + u, = 1, K, n Model wih srucural change in n 0: Y = α β 1 + 1X + u = 1, K,, n 0 Y = α 1 + α2 + ( β1 + β2 ) X + u, = n0 + 1, K, n
12 This is equivalen o inroducing he dummy variable D = 0 for = 1, K, n0 D = 1 for = n + 1,, n wih he model 0 K Y = α 1 + α2d + β1x + β2d X + u, = 1, K, n Tes for srucural change can be done in wo ways Esimae separae models and compare ESS Esimae model wih dummy and es α =, β 0 This gives he same value for he es saisic =
13 Ouliers There may be individual observaions ha do no fi he relaion See oupu/graphs Reason: Omied variables Error in he daa Some unknown even/circumsance How o check his?
14 Inroduce dummy variable D 1 for observaion 23 (and 0 oherwise) i, 23 = Include his in he regression model and es wheher coefficien is 0. See oupu.
15 Dependen Variable: LNWAGE Mehod: Leas Squares Dae: 11/01/01 Time: 08:42 Sample: 1 49 Included observaions: 49 Variable Coefficien Sd. Error -Saisic Prob. C EDUC EXPER AGE RACE GENDER R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)
16 LNWAGE Residuals
17 obs Acual Fied Residual Residual Plo E
18 Dependen Variable: LNWAGE Mehod: Leas Squares Dae: 10/29/01 Time: 22:21 Sample: 1 49 Included observaions: 49 Variable Coefficien Sd. Error -Saisic Prob. C GENDER AGE EXPER EDUC RACE D R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)
19 Dependen Variable: LNWAGE Mehod: Leas Squares Dae: 11/01/01 Time: 08:47 Sample: 1 49 Included observaions: 49 Variable Coefficien Sd. Error -Saisic Prob. C GENDER EXPER EDUC AGE RACE D CLERICAL CRAFTS MAINT R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)
20 Applicaion: Elecion 2000 in Florida Effec of buerfly ballo in Palm Beach Couny on Buchanan voe Daa for all Florida counies Voes candidaes Size and demographic composiion of counies (census). Wha is relevan?
21 Model Dependen variable? Independen variables? How do we check wheher Palm Beach is differen?
22 Elecion 2000 in Florida: Buerfly ballo in Palm Beach couny Oucome of 2000 presidenial elecion dispued. Claims of voing irregulariies in Florida. One issue was a confusing ballo design in Palm Beach couny, he buerfly ballo. Order of punch holes differen from order of he wo main candidaes, Bush and Gore. Claim: Many voers misakenly voed for Buchanan, he candidae of he Reform Pary.
23 Research quesion: Did Buchanan ge an unusually large fracion of he voes in Palm Beach couny?
24 Regression model Dependen variable: log of fracion voes for Buchanan. Independen variables Percenage of populaion Hispanic Percenage of populaion Black Percenage of populaion over 65 Percenage of populaion wih college degree Income (1000$ per year) Populaion (10000)
25 Descripive saisics Dae: 04/06/05 Sample: 1 67 Time: 22:14 FRACBUCHA FRACGORE FRACBUSH PERCBLACK PERCHISPAN PERCOVER6 PERCCOLLE INCOME1000 POPULATION Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera Probabiliy Observaions
26 OLS resuls: Basis equaion
27 Signs of coefficiens plausible? Inerpreaion of coefficiens: dependen variable is log!
28 OLS residuals: Graph LNFRACBUCHANAN Residuals
29 OLS residuals: Table
30
31 OLS resuls: Palm Beach dummy To check wheher Palm Beach is special include dummy ha is 1 for Palm Beach (observaion 50) and 0 oherwise
32 Inerpreaion of Palm Beach dummy ln y = d Hence so ha ln y ln y = observed normal y observed y y normal normal = e = i.e. fracion 5 imes higher han expeced. Fracion is
33 Sensiiviy check: Include log fracion Bush voe
34 Effec on Bush and Gore voe
35
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