Dummy Variable Coefficient Proof

[i_not_alone]

Hi to all

I need to seek help with regard to this question.

Show that the OLS estimator of the dummy coefficient ($$\delta$$) in the regression model given by

Y$$_{i}$$=$$\beta_{1}$$ + $$\delta$$D$$_{i}$$ + $$\upsilon$$$$_{i}$$

is equal to the difference between the sample mean of the observations for which D$$_{i}$$ = 1 and the sample mean of the observations for which D $$_{i}$$ =0.

You can click on the GIF file to see the question more properly.

So how do we go about solving this proof, and in mathematical form, how do we express the sample mean for observation which Di = 1 and Di = 0?

Hope I have presented myself clear! Help really needed. Thanks!

 

[EnumaElish]

Did you study matrix algebra? Can you write the definition of the delta coefficient? You can start from the general definition of OLS coefficients in a regression equation.

 

[i_not_alone]

oh hi EnumaElish!

I did not study matrix algebra but I do know about the definition of delta coefficient, using the proof for OLS slope coeffecient proof, which we derive it from the differentiation of RSS/b1 and RSS/b2.

Now, I am stuck at this stage where I have proofed delta = cov (Di,Yi) / Var (Di).. haha.. so how do i carry on to express it into the sample mean for observation which Di = 1 and Di = 0?

 

[EnumaElish]

Did you try to expand the numerator and the denominator? It could help your intuition if you assume four observations (say, three 1's and one 0) then apply the formula. Then you can generalize.