Question

We have a regression equation:

I) $y_{}i=B_{}1 + B_{}2x_{}2_{}i+ B_{}3x_{}3_{}i +u_{}i$ (restricted form , $H0: \gamma _{1}=\gamma _{2}=\gamma _{3}=0$ )

We want to do a Ramsey-Reset test and form the alternative hypothesis that the true functional form of the regression is:

II) $y_{}i=B_{}1 + B_{}2x_{}2_{}i+ B_{}3x_{}3_{}i + \gamma _{}1\hat{y}_{}i+\gamma _{2}\hat{y}_{i}^{^{2}}+ \gamma _{2}\hat{y}_{i}^{^{3}} + v_{}i$

But in our econometrics class it is said that due to the inclusion of $\gamma _{}1\hat{y}_{}i$ in the alternative form (unrestricted form) we say that this regression can not be estimated and its estimators can not be found. It is said that the reason behind it is multicollinearity and by matrix algebra we say that when  $\gamma _{}1\hat{y}_{}i$ is included  $rank\left ( X \right ) < k$ (k:number of regressors and the intercept term) and  $\left ( X^{T} \right X )^{-1}$ is not invertible. I couldn't do this by myself and couldn't come to this conclusion unfortunately...

So my question is how can I show this in very clear steps by using the matrix algebra form of regression ?

Answer 1

In terms of equations it is easier to understand.

For each y_i in the second equation, we can substitute the first equation.

Let's only partially substitute for y_i in the second equation. Then we will get

y_i=B₁₊ B₂x₂₁+ B₃x_3i + $\gamma$₁(B₁₊ B₂x₂₁+ B₃x_3i) + $\gamma$₂y_i^2 +$\gamma$₃y_i^3+v_i

Now, coefficients of x_2i and x_3i can be clubbed together.

y_i=B₁₊ ($\gamma$₁B₂B₂)x_2i+ ($\gamma$₁B₃B₃)x_3i + $\gamma$₁(B₁)+ $\gamma$₂y_i^2 +$\gamma$₃y_i^3+v_i

This can further also be done for square and cube values of y_i.

Now, clearly we can see that on substitution all RHS variables will get reduced to X_2i and X_3i. Therefore, there is no way to separate out the effect of y_i in this equation. This is due to the problem of multicollinearity which is present due to the fact that the Y_i(S) on the RHS are a linear function of x_i(s) and on doing a regression analysis the effect cannot be separated. Only effect on Y_i due to X_2i and X_3i can be calculated.