Is the regression model Identified? (is it possible to obtain a least squares estimator of the parameters?)

Question

$y_t$ is the dependent variable, $x_t$ and $z_t$ are explanatory variables, and $α$, $β$ and $γ$ are unknown parameters.

$y_t$ = $α$ + $β$$x^3_t$ + $γ$/$log$($x_t$) + $u_t$

$y_t$ = $α$ + $β$$x_t$ + $x_t$/$γ$ + $u_t$

$y_t$ =$√β$$√x_t$ + $u_t$

For the 3 regressions, how would I determine if they are "Identified" or not (basically is it possible to obtain a least squares estimator of the parameters). Furthermore if it can be estimated via linear OLS, how would I write the regressand and regressors for the linear regression to be used?

Answer 1

The first model is multilinear, define $z=x^3$ and $w=\frac{1}{\log(x)}$; so $y=\alpha+\beta z+\gamma w$

The second model is linear but it write $y=\alpha+(\beta+\frac{1}{\gamma})x=\alpha+\delta x$ and you only can reach the value of $\delta$ (no way to get separately $\beta$ and $\gamma$).

The third model is linear : define $z=\sqrt x$ and $\alpha=\sqrt{\beta}$ and the model write $y=\alpha z$.