Linear Models - Regression Analysis

Question

As a student learning Applied Regression Analysis, I come from a background with very little information about this topic.

I understand that given $y = \beta_0 + \beta_1x_1 + \epsilon$

$E(y|x) = \beta_0 + \beta_1x_1$ is an exact linear relationship.

However, if we use a function as log($x$), sin($x$), or cos($x$), will the relationship continue to be linear?

For instance, in the parameters $\beta$,

$E(Y|x_1,x_2) = \beta_1 x_1 + \beta_2 \log(x_2)$, is this linear?

Clearly, $\beta_1x_1$ is linear; however, is the part $\beta_2$log($x_2$) also linear? From a calculus point of view, we know that the logarithm function isn't exactly linear in the sense it is a polynomial of degree one.

Any hints are much appreciated.

Answer 1

By linear model we shall mean a mathematical equation that involves random variables , mathematical variables and parameters and that is linear in parameters.

Your $E(Y|x_1,x_2) = \beta_1x_1 + \beta_2 \log x_2$ is linear. Here your parameters are $\beta_1$ and $\beta_2$ and they are linear . Also note that $x_1$ and $x_2$ if given they are fixed and consequently $x_1$ and $\log x_2$ are fixed.

NB: $y=\beta_0 + \beta_1 \exp (-x_1) + \beta_2 \log ~x_2 +e$ ,where $e$~$N(0,\sigma^2)$, is Linear Model.

However $y=\beta_0 \exp(-x_1 \beta_1)+e$ ,where $e$~$N(0,\sigma^2)$, is not linear in parameters and is not a Linear Model.