3

I know that an OLS model is supposed to be linear in its parameters. I wonder if this model is an OLS model, or can it be turned into one?

Y.L
  • 31
  • 1
  • 1
    Set $\beta = \beta_1 + \beta_2$. Now does this look linear in the parameters? – jcken Mar 26 '21 at 19:51
  • I am not sure because what it means is that there is a constraint between $\beta$ and $\beta_2$ in this case. the parameters affect each other – Y.L Mar 26 '21 at 19:54
  • 1
    The function $(\alpha,\beta_1,\beta_2)\to \alpha + \cdots + \epsilon_i$ is *manifestly* linear from its very form. Nothing needs to be demonstrated. Perhaps reading https://stats.stackexchange.com/questions/148638 will help. – whuber Mar 26 '21 at 20:34

1 Answers1

4

Yes.

$$\begin{align}y_i & =\alpha+(\beta_1+\beta_2)x_1+\beta_2x_2+\epsilon_i\\ \\ & = \alpha+\beta_1x_1+\beta_2x_1+\beta_2x_2+\epsilon_i\\ \\ & =\alpha+\beta_1x_1+\beta_2(x_1+x_2)+\epsilon_i\end{align}$$

Alexis
  • 26,219
  • 5
  • 78
  • 131
Ryan Volpi
  • 1,638
  • 8
  • 17
  • Still not sure since this model will not be BLUE. It's wrong in the sense that if you take $x*=x_1x_2$ then $x1=\frac{x*}{x_2}$. Isn't that a type of collinearity? – Y.L Mar 26 '21 at 20:04
  • 1
    It looks like you want $x_1+x_2$, not $ x_1x_2$. – Dave Mar 26 '21 at 20:09
  • Does that violate an assumption of the gauss-markov theorem? https://en.m.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem – Ryan Volpi Mar 26 '21 at 20:14
  • @RyanVolpi Non-mobile link: https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem – Alexis Mar 26 '21 at 20:21
  • @RyanVolpi it does: https://en.wikipedia.org/wiki/Multicollinearity – Y.L Mar 26 '21 at 21:27
  • note that it also means that if $\beta_1>0$ and $\beta_2>0$, then the first coefficient constrained to be greater than the second – Y.L Mar 26 '21 at 21:34
  • @Y.L I don't think it does. Can you be more specific? Lack of multicollinearity is not an assumption of the gauss markov theorem. – Ryan Volpi Mar 26 '21 at 21:51
  • @RyanVolpi it is. What you find in Wiki is not complete. look at https://stats.stackexchange.com/questions/307298/why-is-the-ols-assumption-no-perfect-multicollinearity-so-vital – Y.L Mar 26 '21 at 21:59
  • @RyanVolpi they typically call it "no perfect colinearity" between regressors – Y.L Mar 26 '21 at 22:02
  • @Y.L Oh yes, I see. *Perfect* multicollinearity is different in that it implies one variable is an exact linear function of the other, which is not true in your case. Additionally, that's a requirement to get a unique solution to OLS, which is different from whether the OLS estimate is BLUE. But in any case, $X_1+X_2$ is not perfectly colinear with $X_1$ except in some trivial case, so I don't think you'll have any issue. – Ryan Volpi Mar 26 '21 at 22:39