# Dummies that take only one non-zero value

According to this blog post in the section «2. Dummies That Take Only One Non-Zero Value», the author states that when the dummy has only 1 observation where it's different from zero, then we can drop that observation without altering the OLS estimates for the remaining coefficients... Why is this true?

I've tried doing some calculations, and it will always depend on the dummy(I've interchanged some columns and rows so that the first column represents the dummy, and the 1st row the observation where the dummy is non-null.) The OLS estimates seems to depend on 1st row...

My design matrix is $\begin{bmatrix} D & X \end{bmatrix}$. For OLS estimates I get $\hat\beta=\begin{bmatrix} 1 & -x_{1}(X'X)^{-1} \\ -(X'X)^{-1} x_{1}' & (X'X)^{-1} \end{bmatrix}\begin{bmatrix} y_1 \\ X'Y \end{bmatrix}$,where $x_1$ is the 1st row of $X$ matrix.

Any help would be appreciated

• Maybe if you could show us your calculations we could point out what's wrong. Possible: This statement is conditional on having a constant in your model. Intuitively: if you observe that older people walk slower, how could that observation be altered if I told you that they all have had 3 hip replacements. If there was variation in the number of hip replacements, you might check for (and find) a link between that number and the walking pace. If not, then not – sheß May 20 '16 at 8:24
• @sheß I've added my calculations. – An old man in the sea. May 20 '16 at 8:44

## 1 Answer

To quote the Blog you cited:

You probably know already that if you have a dummy variable that is zero for all but one of the sample values, then your OLS estimates of the regression model's coefficients will be identical to those that you'd get if you simply dropped the "special"observation (for which the dummy is non-zero) from the regression altogether. I often set the proof of this as an exercise for my students. In addition, the residual for that one special observation will be exactly zero.

I think this is different to what you say:

the author states that when the dummy has only 1 observation where it's different than one, then we can drop that observation without altering the OLS estimates for the remaining coefficients... Why is this true?

• You're right. I corrected the 'english'. However, I think the calculations shown in the question are still valid, and I don't see how they are compatible with the blog... – An old man in the sea. Oct 20 '16 at 16:11