Chapter 7, Dummy Variable 1. A dummy variable takes on 1 and 0 only. The number 1 and 0 have no numerical (quantitative) meaning. The two numbers are used to represent groups. In short dummy variable is categorical (qualitative). (a) For instance, we may have a sample (or population) that includes both female and male. Then a dummy variable can be defined as D = 1 for female and D =0 for male. Such a dummy variable divides the sample into two subsamples (or two sub-populations): one for female and one for male. (b) Dummy variable follows Bernoulli distribution. The distribution is characterized by the parameter p D = { 1, with probability p 0, with probability 1 - p (1) 2. Consider using dummy variable as regressor Y = β 0 + β 1 D + u (2) Regression (2) can be broken into two separate regressions as Y = { β 0 + u, when D =0 (β 0 + β 1 )+ u, when D =1 (3) Taking expectation of (3) leads to E(Y |D = 0) = β 0 (4) E(Y |D = 1) = β 0 + β 1 (5) and β 0 = E(Y |D = 0) (6) β 1 = E(Y |D = 1) - E(Y |D = 0) (7) Therefore β 0 is the mean of Y conditional on D = 0 (or mean of Y in the subpopulation with D = 0), β 1 is the difference in mean Y between the two sub-populations. 1
Transcript
Chapter 7, Dummy Variable
1. A dummy variable takes on 1 and 0 only. The number 1 and 0 have no numerical
(quantitative) meaning. The two numbers are used to represent groups. In short
dummy variable is categorical (qualitative).
(a) For instance, we may have a sample (or population) that includes both female
and male. Then a dummy variable can be defined as D = 1 for female and D = 0
for male. Such a dummy variable divides the sample into two subsamples (or two
sub-populations): one for female and one for male.
(b) Dummy variable follows Bernoulli distribution. The distribution is characterized
by the parameter p
D =
{1, with probability p
0, with probability 1− p(1)
2. Consider using dummy variable as regressor
Y = β0 + β1D + u (2)
Regression (2) can be broken into two separate regressions as
Y =
{β0 + u, when D = 0
(β0 + β1) + u, when D = 1(3)
Taking expectation of (3) leads to
E(Y |D = 0) = β0 (4)
E(Y |D = 1) = β0 + β1 (5)
and
β0 = E(Y |D = 0) (6)
β1 = E(Y |D = 1)− E(Y |D = 0) (7)
Therefore β0 is the mean of Y conditional on D = 0 (or mean of Y in the subpopulation
with D = 0), β1 is the difference in mean Y between the two sub-populations.
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3. Sample mean is the estimate for population mean, so we have the following interpre-
tation for the estimated coefficients in (2)
β0 = yD=0 (8)
β1 = yD=1 − yD=0 (9)
where yD=0 denotes the average Y in the sub-sample for which D = 0, yD=1 denotes
the average Y in the sub-sample for which D = 1. Equation (2) provides a simple way
to carry out a comparison of means test (or two sample t test) between the two groups.
The null hypothesis of two-sample t test says that there is no difference between two
groups:
H0 : β1 = 0
This hypothesis is rejected when the p-value for β1 is less than 0.05.
4. For example, let Y be wage, and D = 1 for female, and D = 0 for male. Then consider
the regression
wage = β0 + β1D + u,
and we know β0 is the average wage for male, and β1 equals average female wage minus
average male wage. The two wages are significantly different if β1 is significant.
5. Now consider a regression with regressor X
Y = β0 + β1D + β2X + u (10)
which can be rewritten as
Y =
{β0 + β2X + u, when D = 0
(β0 + β1) + β2X + u, when D = 1(11)
It follows that
E(Y |X,D = 0) = β0 + β2X (12)
E(Y |X,D = 1) = (β0 + β1) + β2X (13)
β1 = E(Y |X,D = 1)− E(Y |X,D = 0) (14)
so β1 measures the change in mean Y across two groups, holding X constant (or given
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the same level of X). For instance, if X is edu(cation), in the regression
wage = β0 + β1D + β2edu+ u,
β1 equals the average female wage minus average male wage, given the same level of
education.
6. From (11) we can show
dE(Y |X)
dX=
{β2 when D = 0
β2 when D = 1(15)
So regression (10) is restrictive by assuming that the marginal effect of X on Y does
not depend on D. Go back to the wage example. This restriction assumes that when
education changes, wage changes at the same rate for female and male.
7. In chapter 6 we know interaction term can be used to allow the marginal effect of X to
depend on another regressor. The regression with both dummy and interaction term
of dummy and X is
Y = β0 + β1D + β2X + β3(X ∗D) + u (16)
which can be rewritten as
Y =
{β0 + β2X + u, when D = 0
(β0 + β1) + (β2 + β3)X + u, when D = 1(17)
The last equation makes it clear that
Dummy variable allows for different intercepts (or intercept shift)
Interaction term of dummy variable and X allows for different slopes
see Figure 7.2 in textbook.
8. Note regression (16) contains the same amount of information as two separate regressions
of Y on X, one using subsample D = 0, and one using subsample D = 1.
9. Exercise : derive the marginal effect of X on Y implied by (16)
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10. Suppose we have two subsamples, one for female and one for male. We want to estimate
the effect of education on wage. We have two options. Option 1 is to run two separate
regressions, one for female and one for male. Option two is pool (merge) the two
subsamples together and just run one regression. Which option is better?
(a) Essentially this problem is about whether the relationship between education and
wage depends on gender
(b) To answer this question, we just pool the two subsample, and run regression (16).
The point is, we need to use dummy variable and interaction term. The null
hypothesis is gender does not matter, so
β1 = β3 = 0 (18)
We can use F test (called Chow test in this context) for this hypothesis.
i. If p-value is less than 0.05, H0 is rejected, so gender matters. We need to keep
the dummy and interaction term in (16). That means, running two separate
regressions, one for female and one for male, is better idea.
ii. If p-value is greater than 0.05, H0 is not rejected, so gender does not matter.
We need to drop the dummy and interaction term from (16). That means,
running one regression using both subsamples is better idea.
11. What if we have information about gender and marital status? Option one is to define
two dummy variables as
D1 =
{1, female
0, male(19)
D2 =
{1, married
0, unmarried(20)
and use them to run the regression of
Y = β0 + β1D1 + β2D2 + u (21)
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For this regression we can show
E(Y ) =
β0, if D1 = 0, D2 = 0
β0 + β1, if D1 = 1, D2 = 0
β0 + β2, if D1 = 0, D2 = 1
β0 + β1 + β2, if D1 = 1, D2 = 1
Now we can see regression (22) is restrictive because it assumes
