1 Econ 495 - Econometric Review Contentsecon.arts.ubc.ca/nfortin/econ495/lecdummy49511.pdf• This is perhaps the simplest example of the dummy variable trap. • Since we have choosen

Econ 495 - Econometric Review 1

Contents

2 Dummy Variables 2

2.1 The case of one dummy variable . . . . . . . . . . . . . . 2

2.2 Multiple categories . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Interactions among Dummy Variables . . . . . . . 15

2.2.2 Allowing for Different Slopes . . . . . . . . . . . . 19

2.3 Pooling independent cross-sections over time . . . . . . . 22

2.4 Differences-in-Differences estimator and dummy variables . 27


2 Dummy Variables

2.1 The case of one dummy variable

• Often times, variables that take on continuous values are not available,

instead dummy variables have to be used.

• For example, Statistics Canada often classifies continuous variables,

such as age, into categories to preserve confidentiality; education is

often available in the form of the highest degree or diploma attained.

• A dummy variable is a variable that takes on the value 1 or 0, thus

dummy variables are also called binary variables.


• Examples: male (= 1 if the worker is male, 0 otherwise), part-timework status (= 1 if the worker is part-time, 0 otherwise), etc.

• Consider a simple model with one continuous variable X and onedummy D

Yi = β0 + β1X1 + δ0D + ui

This can be interpreted as an intercept shift

If D = 0, then Yi = β0 + β1X1 + ui

If D = 1, then Yi = β0 + β1X1 + δ0D + ui

where the case of D = 0 is the base or reference group

• Returning to our Canadian wage regression, let’s include a male dummy

log(wagesi) = β0 + β1educi + δ0male + ui


. gen male=(sex==1)

. regress lrwage male schooling [weight=fweight](analytic weights assumed)(sum of wgt is 2.2780e+06)

Source | SS df MS Number of obs = 9720-------------+------------------------------ F( 2, 9717) = 1156.78

Model | 452.291246 2 226.145623 Prob > F = 0.0000Residual | 1899.62851 9717 .19549537 R-squared = 0.1923

-------------+------------------------------ Adj R-squared = 0.1921Total | 2351.91976 9719 .241991949 Root MSE = .44215

------------------------------------------------------------------------------lrwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------male | .2138704 .0089866 23.80 0.000 .1962548 .2314861

schooling | .0841955 .0019724 42.69 0.000 .0803292 .0880618_cons | 1.568616 .0268082 58.51 0.000 1.516066 1.621165

------------------------------------------------------------------------------


• Notice that we have included only 1 dummy when there are two groups.

Since female + male = 1, putting both in would have resulted in

perfect collinearity and one variable would have been dropped!

• This is perhaps the simplest example of the dummy variable trap.

• Since we have choosen females as our base group, β0 represents the

intercept for females and δ0 represents the male advantage.

• In terms of expectations, if we assume the zero conditional mean as-

sumption E(u|male, schooling) = 0, then

δ0 = E(lrwage|male = 1, schooling)−E(lrwage|male = 0, schooling)


• δ = 0.214 is the difference in log hourly wage between males and

females, given the same amount of education (and the same error

term u)

• It says that for the same level of education, men earn about 21%

more than women. The correct calculation is a bit lower (wageM −wageF )/wageF = exp(−0.213704) − 1 = 0.193

• Of course, other factors would have to be taken into account to de-

termine whether this is a discrimination effect

• Instead, we could have constructed a female dummy,

log(wagesi) = α0 + α1educi + γ0female + ui


. gen female=(sex==2)

. regress lrwage female schooling [weight=fweight](analytic weights assumed)(sum of wgt is 2.2780e+06)





-------------+----------------------------------------------------------------female | -.2138704 .0089866 -23.80 0.000 -.2314861 -.1962548

schooling | .0841955 .0019724 42.69 0.000 .0803292 .0880618_cons | 1.782486 .026398 67.52 0.000 1.73074 1.834232

------------------------------------------------------------------------------


• Notice that −γ0 = δ0 the coefficient of the dummy variables are ofthe same magnitude but of opposite sign

• Also α0 = β0 + δ0 and β0 = α0 + γ0

• It does not matter which group is choosen to be the base group, butkeeping track of which group is the base group is important for theinterpretation

2.2 Multiple categories

• We can use dummy variables to control for something with multiplecategories


• In our LFS data, education was initially available in terms of 7 cate-gories, as in Table 1

• In this case, we can construct dummy variables for each category, butwe have to omit one category

• We could let STATA choose the category by including edd* in the listof explanatory variables, but often it is best to choose ourselves

• An intermediate category that has a sufficiently large number of ob-servations is a good choice, in the context of wage regression highschool graduates are often the base group

• Because the base group is absorbed in the intercept, if there are ncategories there should be n − 1 dummy variables


. tab educ90, gen(edd)

highest |educational |attainment | Freq. Percent Cum.------------+-----------------------------------

0 | 547 4.61 4.61 /* 0 to 8 years*/1 | 1,878 15.82 20.43 /*Some secondary*/2 | 2,459 20.72 41.15 /*Grade 11 to 13*/3 | 1,088 9.17 50.31 /*Some post secondary*/4 | 4,002 33.72 84.03 /*Post secondary diploma*/5 | 1,321 11.13 95.16 /*Bachelors */6 | 575 4.84 100.00 /*Graduate degree */

------------+-----------------------------------Total | 11,870 100.00

regress lrwage edd1 edd2 edd4 edd5 edd6 edd7 [weight=fweight] /*edd3 omitted */(analytic weights assumed)(sum of wgt is 2.2780e+06)






-------------+----------------------------------------------------------------edd1 | -.1295773 .0241357 -5.37 0.000 -.1768883 -.0822663edd2 | -.173165 .0157088 -11.02 0.000 -.2039575 -.1423724edd4 | -.0482456 .0173476 -2.78 0.005 -.0822504 -.0142407edd5 | .1495457 .0126066 11.86 0.000 .1248341 .1742573edd6 | .3869386 .0163941 23.60 0.000 .3548028 .4190744edd7 | .5895479 .0226934 25.98 0.000 .5450641 .6340316

_cons | 2.692276 .0097833 275.19 0.000 2.673099 2.711454------------------------------------------------------------------------------


• The interpretation of the wage premiums is by comparison with high

school educated workers

• Since the dependent variable is log(wages), the coefficients of edd7

and edd6 mean that workers with a bachelor’s degree make about 39%

more than workers with a high school degree and that percentage is

about 59% for workers with a graduate degree

• If there are a lot of categories, it may make sense to group some

together

. gen lesshs=0

. replace lesshs=1 if educ90<=1(2425 real changes made). gen hs=0


. replace hs=1 if educ90==2(2459 real changes made). gen somecol=0. replace somecol=1 if educ90==3 | educ90==4(5090 real changes made). gen univ=0. replace univ=1 if educ90==5 | educ90==6(1896 real changes made)

. sum lrwage lesshs hs somecol univ [weight=fweight](analytic weights assumed)

Variable | Obs Weight Mean Std. Dev. Min Max-------------+-----------------------------------------------------------------

lrwage | 9720 2278028 2.783154 .4919268 1.261226 4.371922lesshs | 9720 2278028 .1807976 .3848702 0 1

hs | 9720 2278028 .2177673 .4127496 0 1somecol | 9720 2278028 .4312809 .4952807 0 1

univ | 9720 2278028 .1701542 .3757875 0 1

. regress lrwage lesshs somecol univ [weight=fweight]


(analytic weights assumed)(sum of wgt is 2.2780e+06)





-------------+----------------------------------------------------------------lesshs | -.162843 .0146856 -11.09 0.000 -.1916297 -.1340562somecol | .1029682 .0121338 8.49 0.000 .0791835 .1267529

univ | .4461324 .0149344 29.87 0.000 .4168579 .475407_cons | 2.692276 .0098909 272.20 0.000 2.672888 2.711665

------------------------------------------------------------------------------

• In the case of educational attainment, the dummy variables reflect thechoices of individuals. The question of causality is a central issue. Are


individuals with a university degree earning higher wages because they

are more able individuals or because of their higher productivity?

2.2.1 Interactions among Dummy Variables

• Interacting dummy variables is like subdividing the groups

• Suppose we would like to know whether the returns to education are

the same for men and women, using the 4 above education categories,

there will be 8 groups, so we need 7 dummy variables

• If high school*female is the base group


. gen lhsmale=lesshs*male

. gen scolmale=somecol*male

. gen univmale=univ*male

. regress lrwage male lesshs somecol univ lhsmale scolmale univmale[weight=fweight] (analytic weights assumed)(sum of wgt is 2.2780e+06)




------------------------------------------------------------------------------


lrwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

male | .183935 .0192114 9.57 0.000 .1462766 .2215933lesshs | -.2436651 .0215678 -11.30 0.000 -.2859426 -.2013877somecol | .0798044 .0166586 4.79 0.000 .0471502 .1124587

univ | .457988 .0207182 22.11 0.000 .417376 .4986lhsmale | .1002952 .0288863 3.47 0.001 .0436721 .1569184

scolmale | .0425959 .023568 1.81 0.071 -.0036024 .0887941univmale | -.0310315 .02902 -1.07 0.285 -.0879167 .0258538

_cons | 2.600929 .0135387 192.11 0.000 2.57439 2.627467------------------------------------------------------------------------------

• To illustrate the meaning of the interactions, let’s consider the corre-

spondence between the groups and which dummies are turned on


Male Female True iflesshs 1 0 male=1 & lesshs=1 & lhsmale=1lesshs 0 1 lesshs=1high school 1 0 male=1high school 0 1 no dummy is turned on, the constant is

the intercept for this base groupsomecol 1 0 male=1 & somecol=1 & scolmale=1somecol 0 1 somecol=1univ 1 0 male=1 & univ=1 & univmale=1univ 0 1 univ=1

• The interpretation of the coefficient on univ is the premium (46%)

that a women with a bachelor’s degree gets by comparison with high

school educated women


• The interaction univmale is the premium a university educated male

gets by comparison with a university education female, it is not signif-

icant; but of course, he gets the male premium!

• The significant interaction lhsmale tells us that there was a case for

different education coefficient by gender

2.2.2 Allowing for Different Slopes

• We can also interact a dummy variable with a continuous variable


• Let’s use the continuous version of educational attainment, to estim-

tate

log(wagesi) = β0 + β1educi + δ0male + δ1male ∗ educi + ui

. gen schomale=schooling*male

. regress lrwage male schooling schomale [weight=fweight](analytic weights assumed)(sum of wgt is 2.2780e+06)




------------------------------------------------------------------------------



male | .4436921 .0530265 8.37 0.000 .3397492 .5476351schooling | .0942717 .0030221 31.19 0.000 .0883478 .1001957schomale | -.0175286 .003986 -4.40 0.000 -.0253419 -.0097153

_cons | 1.43575 .0403754 35.56 0.000 1.356606 1.514894------------------------------------------------------------------------------

• In this regression β0 = 1.436 is the intercept for females, and β0+δ0 =

1.436 + 0.447 = 1.883 is the intercept for males

• β1 = 0.094 is the slope for females and β1 + δ1 = 0.094 − 0.0175 =

0.0765 is the slope for males.

• It is typical to find that the wage equation for women has a lower

intercept but a steeper slope. It means that women earn less than


men at low levels of education but that the gap narrows as education

increases.

2.3 Pooling independent cross-sections over time

• Many cross-sectional surveys, such as the Labour Force Survey, are

repeated over time

– We may want to pool cross sections just to get bigger sample sizes

– or to investigate the effect of time

– or to investigate whether relationships have changed over time


• To reflect the fact that the population may have different distributions,

we allow the intercept to differ across the time periods

• This is done by introducing year dummy variables for all but one year

• We can also interact the time dummy with a key variable to look for

changes over time

• For example, let’s see if there has been significant progress in the

gender wage gap in Canada from October 1997 to October 2004

. use c:\data\lfs1004.dta

. append using c:\data\lfs1097.dta


. gen time=0

. replace time=1 if survyear==2004(59140 real changes made)

. gen female=(sex==2)

. gen femtime=female*time

. regress lrwage female schooling time femtime [weight=fweight](analytic weights assumed)(sum of wgt is 2.4898e+07)




------------------------------------------------------------------------------



female | -.2091473 .0042243 -49.51 0.000 -.2174269 -.2008678schooling | .0902508 .0006366 141.76 0.000 .089003 .0914986

time | -.0149797 .0039705 -3.77 0.000 -.0227618 -.0071975femtime | .0077309 .0057311 1.35 0.177 -.003502 .0189639_cons | 1.700167 .0088124 192.93 0.000 1.682895 1.717439

------------------------------------------------------------------------------

An alternative command is:

. xi: regress lrwage schooling i.female*time [weight=fweight]i.female _Ifemale_0-1 (naturally coded; _Ifemale_0 omitted)i.female*time _IfemXtime_# (coded as above)(analytic weights assumed)(sum of wgt is 2.4898e+07)






-------------+----------------------------------------------------------------schooling | .0902508 .0006366 141.76 0.000 .089003 .0914986

_Ifemale_1 | -.2091473 .0042243 -49.51 0.000 -.2174269 -.2008678time | -.0149797 .0039705 -3.77 0.000 -.0227618 -.0071975

_IfemXtime_1 | .0077309 .0057311 1.35 0.177 -.003502 .0189639_cons | 1.700167 .0088124 192.93 0.000 1.682895 1.717439

------------------------------------------------------------------------------

• Since the coefficient of female*time=0.0077 is not statistically sig-

nificant, we do not find a significant improvement in the gender wage

gap controlling for education


2.4 Differences-in-Differences estimator and dummy vari-

ables

• Experimental data are often panel data, that is, observations on the

same individuals before and after treatment, as would be the case in

the medical field

• In economics, althought experimental data now relatively more com-

mon they are not always available

• Often, we have to use data from what we call a quasi-experiment or

a natural experiment


• A natural experiment occurs when some exogenous event–either truly

natural or the result of a policy change–change the environment in

which individuals, economic agents or countries operate

• Like a true experiment, it has a treatment group which is affected

by the policy change and a control group which is thought not to be

affected by the policy change

• Unlike a true experiment, the treatment and control groups are not

choosen randomly

• Sometimes controlling for other variables helps make the group assign-

ment closer to random, sometimes an instrumental variable is needed


• Let YT,before

be the sample average of Y for those in the treatment

group and YT,after

be the sample average for the treatment group

after the experiment.

• Similarly, let YC,before

and YC,after

denote the corresponding pre-

treatment and post-treatment sample averages for the control group

• The average change in Y over the course of the experiment for those

in the treatment group is ∆YT

= YT,after − Y

T,before

• The average change in Y over the course of the experiment for those

in the control group is ∆YC

= YC,after − Y

C,before


• The differences-in-differences estimator is

δ̂DID

=(Y

T,after − YT,before

)−

(Y

C,after − YC,before

)

= ∆YT − ∆Y

C

• If the treatment is randomly assigned, then δ̂DID

is an unbiased and

consistent estimator of the causal effect

• To test whether δ̂DID

is statistically different from zero, we can apply

regression analysis to the pooled data set

Y = β0 + δ0 ∗ after + β1 ∗ treated + δDIDtreated ∗ after + u

• We can also add other X’s to the regression to control for factors that

might make the treatment and control group different


• Consider the policy experiment of the 2001 Pay Equity Law in the

Province of Quebec

• Using women from that province as the treated group, and those in

the Rest of Canada [ROC] as controls, we can run the regression of

the wages of women, controlling for education

. gen quebec=0

. replace quebec=1 if prov>=23 & prov<=25(21320 real changes made)

. gen quetime=quebec*time

. regress lrwage schooling time quebec quetime [weight=fweight] if female==1(analytic weights assumed)(sum of wgt is 1.1970e+07)






-------------+----------------------------------------------------------------schooling | .1004 .0009101 110.32 0.000 .0986162 .1021838

time | -.0109533 .0045211 -2.42 0.015 -.0198147 -.0020918quebec | -.0119871 .0069314 -1.73 0.084 -.0255726 .0015985quetime | .0045126 .0093351 0.48 0.629 -.0137843 .0228095_cons | 1.359245 .0125311 108.47 0.000 1.334684 1.383806

------------------------------------------------------------------------------

• Although positive, the coefficient on quetime is not significant.


• Perhaps the Quebec law had some positive spillovers on women’s wage

in ROC or there may be a specific Quebec trend, we may try to use

men as an additional control group to control.

• To do this, we will look at the change in the gender gap by interacting

quetime with the female dummy, of course, we need to add all other

possible interactions minus 1 (2*2*2-1=7)

• This will be called a triple difference (DDD) estimator

• Without controlling for schooling, this would give the following table

• The regression framework allows us to control for schooling and obtain

standard errors easily.


Table 1: Average Log Wages

1997-2004 Change Women MenQuebec ROC Quebec ROC

(1) 1997 2.676 2.691 2.845 2.891(2) 2004 2.705 2.704 2.860 2.894

1st D (3) Row (2)- Row (1) 0.029 0.013 0.014 0.0032nd D (4) Quebec (3) - ROC (3) 0.016 0.0113rd D (5) Women (4) - Men (4) 0.005


. gen feqtime=female*quetime

. gen femque=female*quebec

. gen feqctime=female*quebec*time

. regress lrwage schooling female time quebec femtime quetime femque /*> */ feqctime [weight=fweight](analytic weights assumed)(sum of wgt is 2.4898e+07)





-------------+----------------------------------------------------------------


schooling | .0902054 .0006366 141.71 0.000 .0889577 .091453female | -.2144925 .0048316 -44.39 0.000 -.2239623 -.2050227

time | -.0158113 .00455 -3.48 0.001 -.0247293 -.0068934quebec | -.0344593 .0067991 -5.07 0.000 -.0477854 -.0211332femtime | .0072471 .0065576 1.11 0.269 -.0056058 .0200999quetime | .0036344 .0092983 0.39 0.696 -.01459 .0218589femque | .0221537 .0099396 2.23 0.026 .0026723 .0416352

feqctime | .0020585 .0134833 0.15 0.879 -.0243687 .0284856_cons | 1.708992 .0089766 190.38 0.000 1.691398 1.726586

------------------------------------------------------------------------------

• Now femque captures the advantage of women in Quebec vs. ROC

and is significant at the 5% level

• But the difference over time, before and after the implementation of

the law, is not significant!

Documents

1 Econ 495 - Econometric Review Contentsecon.arts.ubc.ca/nfortin/econ495/lecdummy49511.pdf• This is perhaps the simplest example of the dummy variable trap. • Since we have choosen