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xtoaxaca - Extending the Kitagawa-Oaxaca-Blinder Decomposition Approach to longitudinal data analyses Hannes Kr oger, J org Hartmann German Institute for Economic Research (DIW), Berlin University of G ottingen 24.05.2019 1 / 27 Who

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xtoaxaca - Extending the Kitagawa-Oaxaca-Blinder Decomposition Approach to longitudinal data analyses

Hannes Kr¨

  • ger, J¨
  • rg Hartmann

German Institute for Economic Research (DIW), Berlin University of G¨

  • ttingen

24.05.2019

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Who would benefit from using xtoaxaca

You have at least two time points You want a flexible way to decompose the level over time You want a (counterfactual) decomposition of change over time

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Prior approaches

Based on different research questions very many different decompositions can be chosen At the moment we cannot make a systematic comparison We present our preferred solution

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB)

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB) Yl = Xlβl + ǫl, E(ǫl) = 0, l ∈ {A, B}

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB) Yl = Xlβl + ǫl, E(ǫl) = 0, l ∈ {A, B} R = E + C + I

R = [E(XB) − E(XA)]βA + E(XA)(βB − βA) + [E(XB) − E(XA)](βB − βA)

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB) Yl = Xlβl + ǫl, E(ǫl) = 0, l ∈ {A, B} R = E + C + I

R = [E(XB) − E(XA)]βA + E(XA)(βB − βA) + [E(XB) − E(XA)](βB − βA)

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB) Yl = Xlβl + ǫl, E(ǫl) = 0, l ∈ {A, B} R = E + C + I

R = [E(XB) − E(XA)]βA + E(XA)(βB − βA) + [E(XB) − E(XA)](βB − βA)

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Traditional approach Kitagawa-Oaxaca-Blinder decomposition

R = E(YA) − E(YB) Yl = Xlβl + ǫl, E(ǫl) = 0, l ∈ {A, B} R = E + C + I

R = [E(XB) − E(XA)]βA + E(XA)(βB − βA) + [E(XB) − E(XA)](βB − βA)

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Research questions for the parts of the decomposition

E: How much smaller/bigger would the gap be, if the endowments of group A were the same as for group B?

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Research questions for the parts of the decomposition

E: How much smaller/bigger would the gap be, if the endowments of group A were the same as for group B? C: How much smaller/bigger would the gap be, if the effect of the explanatory variables of group A were the same as the effects for group B?

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GC prediction approach

R(t) = E(YB(t) − YA(t)) Latent growth curve model for change in happiness parametric, semi-parametric, non-parametric

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Predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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Counterfactual predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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Counterfactual predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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Predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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Counterfactual predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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Counterfactual predictions of happiness

4 4.5 5 5.5 6 6.5 7 7.5 8 1 2 3 4 5 6 7 8 9 10 Time

High education Low education Low education - High E Low education - High E+C 95%-CI

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GC decomposition approach

R(t) = E(t) + C(t) + I(t) Decomposition is time dependent

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GC decomposition approach

R(t) = E(t) + C(t) + I(t) Decomposition is time dependent

R = [E(Xht) − E(Xlt)]βlt + E(Xlt)(βht − βlt) + [E(Xht) − E(Xlt)](βht − βlt)

Conditional on functional form of time

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GC decomposition approach

R(t) = E(t) + C(t) + I(t) Decomposition is time dependent

R = [E(Xht) − E(Xlt)]βlt + E(Xlt)(βht − βlt) + [E(Xht) − E(Xlt)](βht − βlt)

Conditional on functional form of time

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GC decomposition approach

R(t) = E(t) + C(t) + I(t) Decomposition is time dependent

R = [E(Xht) − E(Xlt)]βlt + E(Xlt)(βht − βlt) + [E(Xht) − E(Xlt)](βht − βlt)

Conditional on functional form of time

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GC decomposition approach

R(t) = E(t) + C(t) + I(t) Decomposition is time dependent

R = [E(Xht) − E(Xlt)]βlt + E(Xlt)(βht − βlt) + [E(Xht) − E(Xlt)](βht − βlt)

Conditional on functional form of time

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Decomposition of change

Decomposing the change in happiness it always needs two time points to compare

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Decomposition of change

∆Y l = Y l

t − Y l s

= ¯ X l

t βl t − ¯

X l

sβl s

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Decomposition of change

∆Y l = Y l

t − Y l s

= ¯ X l

t βl t − ¯

X l

sβl s

and the change of the group difference over time then is ∆Y = ∆Y A − ∆Y B = ( ¯ X A

t βA t − ¯

X A

s βA s ) − ( ¯

X B

t βB t − ¯

X B

s βB s )

= ¯ X A

t βA t − ¯

X A

s βA s − ¯

X B

t βB t + ¯

X B

s βB s

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Change due to endowment

∆YE = ( ¯ X A

t − ¯

X A

s )βA s − ( ¯

X B

t − ¯

X B

s )βB s

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Change due to endowment

∆YE = ( ¯ X A

t − ¯

X A

s )βA s − ( ¯

X B

t − ¯

X B

s )βB s

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Change due to coefficients

∆YC = (βA

t − βA s ) ¯

X A

s − (βB t − βB s ) ¯

X B

s

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Change due to coefficients

∆YC = (βA

t − βA s ) ¯

X A

s − (βB t − βB s ) ¯

X B

s

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Research questions for the parts of the decomposition

dE: How much smaller/bigger would the change in the gap be, if the endowments of group A had changed in the same way as for group B (and the difference in coefficients had stayed the same)?

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Research questions for the parts of the decomposition

dE: How much smaller/bigger would the change in the gap be, if the endowments of group A had changed in the same way as for group B (and the difference in coefficients had stayed the same)? dC: How much smaller/bigger would the change in the gap be, if the coefficients

  • f group A had changed in the same way as for group B (and the difference in

endowments had stayed the same)?

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Choose data, model and estimator

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Choose data, model and estimator Data Stored estimates Group, time, decomposition variables and time points

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Choose data, model and estimator Data

xtoaxaca

Stored estimates Group, time, decomposition variables and time points margins mean β(g,t) E(X(g,t))

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Choose data, model and estimator Data

xtoaxaca

Stored estimates Group, time, decomposition variables and time points Level Change margins mean β(g,t) E(X(g,t))

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Quick note on the use of margins

all decompositions are done at means or other specified values calculation of average over the population not necessary margins conducted only on one observation speed of the calculation of margins (not mean) independent of sample size

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

xtoaxaca exp, groupvar(group2) groupcat(1 2) timevar(time) times(1 3 5 ) /// model1(base) model2(control) /// timebandwidth(1) basecontrols(edu) change timeref(1)

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xtoaxaca in action

Stata example

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General problems with decomposition

Reference group Reference group of decomposition variables

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Extensions?

Different forms of change decomposition (e.g. Kim, Makepiece) Different counterfactual scenarios Reduce to one model estimation Extension for SEM

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