What does it exactly mean by linearity in parameters in the assumptions for classical linear regression model and Gauss-Markov theorem?

My understanding is:

  1. y = b0 + b1 x + e - Linear in parameters
  2. y = b3 + b4^2 x + e - Non-linear in parameters => Can it be re-written like y = b3 + b5 x + e where b5 = b4^2

In case 2, since final values of all b's are nothing but constants ultimately, we can rewrite b4^2 as some other constant b5. So if we rewrite this, case 2 also becomes linear in parameters. Isn't it?


  1. y = 10 + 16 x + e (Linear in parameter)
  2. y = 10 + 5^2 x + e (Non-linear in parameter) but can be re-written as y = 10 + 25 x + e which is linear in parameter.

Please help me to clarify what I'm lacking to undestand.


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First thing is that we're essentially dealing with a vector of parameters, so that we can consider that we have $\boldsymbol{\beta} = (\beta_0,\beta_1,...,\beta_p)^\top$.

See https://en.wikipedia.org/wiki/Linear_algebra and/or https://en.wikipedia.org/wiki/Linear_map and notice (working in the notation of the second link, and writing $\sum_{j=0}^p \beta_j x_j$ as $X\boldsymbol{\beta}$), that $f(\boldsymbol{\beta})=E(Y|x)=X\boldsymbol{\beta}$ satisfies the two conditions in the definition of a linear map: $f(\boldsymbol{\beta}_1+\boldsymbol{\beta}_2)=f(\boldsymbol{\beta}_1)+f(\boldsymbol{\beta}_2)$ and for some scalar constant $\alpha$, that $f(\alpha\boldsymbol{\beta})=\alpha f(\boldsymbol{\beta})$.

If it is not immediately clear (or if you're unused to working with vectors and matrices), you can check those conditions for each of your example models.

For example your first model has $E(Y|x) = \beta_0\cdot 1+\beta_1 \cdot x = f(\beta_0,\beta_1)$. Now consider

\begin{eqnarray} f((\beta_0+\gamma_0),(\beta_1+\gamma_1))&=& (\beta_0+\gamma_0)\cdot 1+(\beta_1+\gamma_1) \cdot x \\ &=& \beta_0\cdot 1+\beta_1 \cdot x + \gamma_0\cdot 1+\gamma_1 \cdot x\\ &=&f(\beta_0,\beta_1)+f(\gamma_0,\gamma_1) \end{eqnarray}

That's the first condition; you can check the second condition in a similar way.

No doubt that seems trivial to the point of banality - but once you have linearity, a lot follows from that. Linear algebra underlies a large number of widely used statistical models and it's worth learning some of the basics before trying to understand a theorem like Gauss-Markov.

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