Is this model a linear regression model?

Question

We have,

$$Y_i = log_{10}(\beta_1X_{i1}) + \beta_2X_{i2} + \epsilon_i$$

So,

$$Y_i = log_{10}(\beta_1) + log_{10}(X_{i1}) + \beta_2X_{i2} + \epsilon_i$$

I don't think this is a general linear regression model because of the term $log_{10}(\beta_1)$ as I believe this makes the model non-linear in the parameters. However, I also think that since $\beta_1$ is a constant, $log_{10}(\beta_1)$ is also a constant, and thus the model should still be linear in the parameters. In addition, I am not sure how to think about the term $log_{10}(X_{i1})$ since it does not have a coefficient. What is its role in the model?

Answer 1

$$Y_i = \log_{10}(\beta_1) + \log_{10}(X_{i1}) + \beta_2X_{i2} + \epsilon_i$$ is not linear since $$\frac{\partial Y_i}{\partial \beta_1} \neq 0$$ But $$Y_i = \beta_3+ \log_{10}(X_{i1}) + \beta_2X_{i2} + \epsilon_i$$ is linear.

When done, make $\beta_1=10^{\beta_3}$.