# Regression on trivariate data with one coefficient 0

Suppose {$$(x_i,y_i,z_i):i=1,2,...,n$$} is a set of trivariate observations on three variables:$$X,Y,Z$$, where $$z_i=0$$ for $$i=1,2,...,n-1$$ and $$z_n=1$$.Suppose the least squares linear regression equation of $$Y$$ on $$X$$ based on the first $$n-1$$ observations is $$y=\hat{\alpha_0}+\hat{\alpha_1}x$$ and the least squares linear regression equation of $$Y$$ on $$X$$ and $$Z$$ based on all the $$n$$ observations is $$y=\hat{\beta_0}+\hat{\beta_1}x+\hat{\beta_2}z$$.

We need to show that $$\hat{\alpha_1}=\hat{\beta_1}$$.

My approach:

Based on the first $$n-1$$ observations, as $$z_i=0$$, so, we consider a typical linear regression model of $$Y$$ on $$X$$.

Thus,the least square estimate $$\hat{\alpha_1}=\frac{\sum_{i=1}^{n-1} (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n-1} (x_i-\bar{x})^2}$$

And in the second case, we have:

$$y_1=\beta_0+\beta_1 x_1+e_1$$

$$y_2=\beta_0+\beta_1 x_2+e_2$$

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$$y_n=\beta_{0}+\beta_1 x_n+\beta_2+e_n$$

Thus, the error sum of squares:

$$=\sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2+(y_n-\beta_1 x_n -\beta_0 -\beta_2)^2$$

Differentiating this w.r.t. $$\beta_0,\beta_1,\beta_2$$ and equating them to $$0$$, we get the same value of the estimate $$\hat{\beta_1}$$, as the normal equations for $$\beta_0,\beta_1$$ come out to be the same by plugging in $$\hat{\beta_2}=y_n-\hat{\beta_1}x_n-\hat{\beta_0}$$.

So, is my approach correct? Or can you guys see a major flaw? Let me know

By differentiating with respect to $$\beta_2$$, we can see that at the optimal value, we must have
$$\hat{\beta}_2 = y_n -\hat{\beta_1}x_n-\hat{\beta_0}$$
Hence the problem to solve for $$\hat{\beta_0}$$ and $$\hat{\beta_1}$$ is the same as minimizing
$$\sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2$$
Hence, we know that $$\hat{\beta_1}=\hat{\alpha_1}$$ and furthermore, $$\hat{\beta_0}=\hat{\alpha_0}$$.