Advanced Econometrics Chapter 6: Dummy Variables Chapter DUMMY VARIABLES Hedonic model of housing prices: pi = prices of ith house Si = size (square of feet) 1 if i th house has air conditioning ACi = 0 otherwise I INTERCEPT DUMMY: Regression Model: Ln( pi ) = β1 + β S i + β ACi + ε i β2 = relative change in price Unit change in square footage β = 0.05 : Each extra square foot adds 5% to value of house β = 0.12 : The AC adds 12% to price of house with AC without AC β1 + β β1 o If no AC ( ACi = ) intercept = "reference group" o If no AC ( ACi = ) intercept = + Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables 1 if i th house has no air conditioning Let ACi = 0 otherwise Ln( pi ) = β1 + β S i + β ACi + β NACi + ε i 1 S1 1 S 1 S100 1 0 → Dummy variable trap: ACi + NACi = alway 1 Let: 1 if i th house is brick Bi = 0 if not 1 if i th house is cement block Ci = 0 if not 1 if i th house is wood Wi = 0 if not Dummy variable trap → no reference group Ln( pi ) = β1 + β S i + β ACi + β Bi + β 5Ci + ε i o Reference group (all dummies = 0): houses of wood without AC → intercept = o Houses of wood with AC: intercept = 1+ o Houses of cement without AC: intercept = o Houses of cement with AC: intercept = 1+ 1+ 3+ 5 II INTERCEPT DUMMIES WITH INTERACTIONS: 1 if i th house is cement block & AC Let: CACi = Ci × ACi = 0 if not Ln( pi ) = β1 + β S i + β ACi + β Bi + β 5Ci + β 6Ci ACi + ε i Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables Ln( pi ) = β1 + β S i + β ACi + β Bi + ( β + β ACi )Ci + ε i o Houses of wood with AC: intercept = o Houses of cement with AC: intercept = 1+ 1+ o Houses of cement without AC: intercept = 3+ 1+ 5+ Let Di = distance to a waste site 1 if no AC ACi Di = Di × ACi = 0 if AC Ln( pi ) = β1 + β S i + β ACi + β Di + β ACi Di + ε i o Reference group: Ln( pi ) = β1 + β S i + β ACi + ε i o Non-reference group with AC: Ln( pi ) = β1 + β S i + β ACi + β Di + β Di + ε i Ln( pi ) = β1 + β S i + β ACi + ( β + β ) Di + ε i Not only change in the intercept, but also change in the slope Ln(WAGEi ) = β1 + β EDU i + β AGEi + ε i Ln(WAGEi ) = β1 + β EDU i + β AGEi + β MARi + β ( AGEi × MARi ) + ε i o Reference group: Ln(WAGEi ) = β1 + β EDU i + β AGEi + ε i o Non-reference group: Ln(WAGEi ) = ( β1 + β ) + β EDU i + ( β + β ) AGEi + ε i III SEASONAL EFFECTS: Let: St = retail sales yt = personal income ut = unemployment rate → S t = β1 + β y t + β 3ut + ε t Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables 1 if t = January D1 = 0 if not 1 if t = February D2 = 0 if not 1 if t = Nov D11 = 0 if not → S t = β1 + β y t + β 3ut + γ D1t + γ D2 t + + γ 11 D11t + ε t 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, , 40 Allow for country - specific intercepts (for pooled data) (country fixed effect): 1 if obs from country D1it = 0 if not 1 if obs from country D it = 0 if not Year - specific dummies: 1 if obs from year (1981) Y1 = 0 if not 1 if obs from year (1982) Y2 = 0 if not Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables = elasticity (double log regression) β= % ∆Q ∆Q / Q ∆Q L = = × % ∆L ∆L / L ∆L Q V TEST FOR STRUCTURE BREAK: LQVt = β1 + β LQTt + β LQK t + β LQEt + ε t H : β11 = β12 , β 21 = β 22 , β 31 = β 32 , β 41 = β 42 , r = H A : β11 ≠ β12 , β 21 ≠ β 22 , β 31 ≠ β 32 , β 41 ≠ β 42 From the 1st subsample: obs → ESS1, df = - = From the 2st subsample: 17 obs → ESS2, df = 17 o R-model (ESSR) all 25 obs to estimate single model with parameters o U-model (ESSU) from separate regressions, using and then 17 obs to estimate single models with and parameters ESSU = ESS1 + ESS2, df = 25-8 = 17 F174 = ( ESS R − ESSU ) / ESSU / 17 If : H : β 21 = β 22 , β 31 = β 32 , β 41 = β 42 , r = H A : β 21 ≠ β 22 , β 31 ≠ β 32 , β 41 ≠ β 42 o U-model does not change o R-model use all 25 obs to estimate single model: LQVt = β1 + β1* Dt + β LQTt + β LQK t + β LQEt + ε t 1 if t is obs → 25 Dt = 0 if not Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables VI DIFFERENCES IN DIFFERENCES: Y = β1 + β D1 + β D2 + β D1 D2 + ε ∆2 ∆1 (Y1) 1 (Y2) 0 (Y3) ∆3 (Y4) ∆4 What is the meaning of 4? Y1: intercept = β1 + β + β + β Y2: intercept = β1 + β3 Y2: intercept = β1 + β Y2: intercept = β1 Differences: ∆ = Y1 − Y3 = β + β ∆ = Y2 − Y4 = β ∆ = Y1 − Y2 = β + β ∆ = Y3 − Y = β → Differences in Differences: β = ∆ − ∆ = ∆ − ∆ The co-impact of D1 & D2 on dependent variable makes β (usually β 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 Nam T Hoang University of New England - Australia University of Economics - HCMC - Vietnam ...Advanced Econometrics Chapter 6: Dummy Variables 1 if i th house has no air conditioning Let ACi = 0 otherwise Ln( pi ) =... New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables Ln( pi ) = β1 + β S i + β ACi + β Bi + ( β + β ACi )Ci + ε i o Houses of wood... New England - Australia University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 6: Dummy Variables 1 if t = January D1 = 0 if not 1 if t = February D2 = 0 if not 1 if