Advanced Econometrics - Part I - Chapter 6: Dummy Variables

Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 6 DUMMY VARIABLES Hedonic model of housing prices: pi = prices of ith house. Si = size (square of feet).    = otherwise 0 ngconditioniair has house i if1 th iAC I. INTERCEPT DUMMY: Regression Model: iiii ACSpLn εβββ +++= 321)( footage square in changeUnit price in change relative 2 =β 05.02 =β : Each extra square foot adds 5% to value of house. 12.03 =β : The AC adds 12% to price of house. o If no AC ( 0=iAC ) intercept = β1 "reference group". o If no AC ( 1=iAC ) intercept = β1 + β3 1β 31 ββ + ACwithout ACwith Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam Let    = otherwise 0 ngconditioniair no has house i if1 th iAC iiiii NACACSpLn εββββ ++++= 4321)(             101 011 101 100 2 1 S S S  → Dummy variable trap: ACi + NACi = 1 alway. Let:    = not if0 brick is house i if1 th iB    = not if0 block cement is house i if1 th iC    = not if0 woodis house i if1 th iW Dummy variable trap → no reference group. iiiiii CBACSpLn εβββββ +++++= 54321)( o Reference group (all dummies = 0): houses of wood without AC → intercept = β1. o Houses of wood with AC: intercept = β1 + β3. o Houses of cement without AC: intercept = β1+ β5 o Houses of cement with AC: intercept = β1+ β3+ β5 II. INTERCEPT DUMMIES WITH INTERACTIONS: Let:    =×= not if0 AC&blockcement is house i if1 th iii ACCCAC iiiiiiii ACCCBACSpLn εββββββ ++++++= 654321)( Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam iiiiiii CACBACSpLn εββββββ ++++++= )()( 654321 o Houses of wood with AC: intercept = β1 + β3. o Houses of cement with AC: intercept = β1+ β3+ β5 + β6 o Houses of cement without AC: intercept = β1+ β5 Let Di = distance to a waste site.    =×= AC if0 AC no if1 iiii ACDDAC iiiiiii DACDACSpLn εβββββ +++++= 54321)( o Reference group: iiii ACSpLn εβββ +++= 321)( o Non-reference group with AC: iiiiii DDACSpLn εβββββ +++++= 54321)( iiiii DACSpLn εβββββ +++++= )()( 54321 Not only change in the intercept, but also change in the slope. iiii AGEEDUWAGELn εβββ +++= 321)( iiiiiii MARAGEMARAGEEDUWAGELn εβββββ +×++++= )()( 54321 o Reference group: iiii AGEEDUWAGELn εβββ +++= 321)( o Non-reference group: iiii AGEEDUWAGELn εβββββ +++++= )()()( 53241 III. SEASONAL EFFECTS: Let: St = retail sales. yt = personal income. ut = unemployment rate. → tttt uyS εβββ +++= 321 Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam    = = not if0 January tif1 1D    = = not if0 February tif1 2D ...    = = not if0 Nov tif1 11D → ttttttt DDDuyS εγγγβββ +++++++= 11112211321  IV. POOLED DATA: (Time series and cross sectional data). fit = fertility rate of country i at year t. yit = per capital income of country i at year t. Eit = Female education of country i at year t. i = 1, 2, 3... , 40 Allow for country - specific intercepts (for pooled data) (country fixed effect):    = not if0 1country from obs if1 1itD    = not if0 2country from obs if1 2itD ... Year - specific dummies:    = not if0 (1981) 1 year from obs if1 1Y    = not if0 (1982) 2 year from obs if1 2Y ... Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam β = elasticity (double log regression). Q L L Q LL QQ L Q × ∆ ∆ = ∆ ∆ = ∆ ∆ = / / % % β V. TEST FOR STRUCTURE BREAK: ttttt LQELQKLQTLQV εββββ ++++= 4321 2 4 1 4 2 3 1 3 2 2 1 2 2 1 1 10 ,,,: ββββββββ ====H , r = 4 2 4 1 4 2 3 1 3 2 2 1 2 2 1 1 1 ,,,: ββββββββ ≠≠≠≠AH From the 1st subsample: 8 obs → ESS1, df = 8 - 4 = 4 From the 2st subsample: 17 obs → ESS2, df = 17. o R-model (ESSR) all 25 obs to estimate single model with 4 parameters. o U-model (ESSU) from 2 separate regressions, using 8 and then 17 obs to estimate single models with 4 and 4 parameters. ESSU = ESS1 + ESS2, df = 25-8 = 17 17/ 4/)(4 17 U UR ESS ESSESSF −= If : 2 4 1 4 2 3 1 3 2 2 1 20 ,,: ββββββ ===H , r = 3 2 4 1 4 2 3 1 3 2 2 1 2 ,,: ββββββ ≠≠≠AH o U-model does not change. o R-model use all 25 obs to estimate single model: tttttt LQELQKLQTDLQV εβββββ +++++= 432 * 11    → = not if0 259 obs is t if1 tD Advanced Econometrics Chapter 6: Dummy Variables Nam T. Hoang University of New England - Australia 6 University of Economics - HCMC - Vietnam VI. DIFFERENCES IN DIFFERENCES: εββββ ++++= 21423121 DDDDY ∆2 1 1 0 1 ∆1 (Y1) (Y2) ∆3 1 0 0 0 (Y3) (Y4) ∆4 What is the meaning of β4? Y1: intercept = 4321 ββββ +++ Y2: intercept = 31 ββ + Y2: intercept = 21 ββ + Y2: intercept = 1β Differences:    =−=∆ +=−=∆ 3423 43311 β ββ YY YY    =−=∆ +=−=∆ 2434 42212 β ββ YY YY → Differences in Differences: 42314 ∆−∆=∆−∆=β The co-impact of D1 & D2 on dependent variable makes 4β (usually 4β is negative). → The impact of marriage on wages of the group is different with the impact of marriage on wages of the non-union group. → To capture the differences in differences → including the interaction of two dummy variables.