# Difference between Ramsey Reset test and link test

My question is simple, what is the difference between reset test and link test? My understanding of reset test is that it tests the correctness of the functional form by incorporating square, cube etc. of explanatory variables $X$. Link test on the other hand regress the dependent variable $Y$ on predicted variable $\hat{Y}$ and $\hat{Y}^2$.

In various places I have seen that reset test and link test is testing omitted variable, but I am not sure if this is true. To me they are testing whether functional form is correct or not. However, I do not see the difference between these two. For instance, would it be enough to run only reset or would one still need to check link? If yes, why? I hope somehow these questions make some sense to somebody.

I also expected to find some discussions on the comparisons of the two tests online. I think that, if we didn't find them, it is due to the fact the Link test is not very popular. However, I try to address the issue below. My understanding is that the main version of the Ramsey test includes powers of fitted values of $\hat{Y}$. Then there is the modified version including powers of explanatory variables $X$, as you say. My reasoning is that the Link test coincides with the Ramsey Reset test using just the squared value (the Ramsey test may also be performed using higher order polynomials), in case there is just 1 regressor; otherwise, the Ramsey test is more flexible since it includes all original regressors $(Y=\alpha+\beta_1*X_1+...+\beta_k*X_k+\hat{Y^2}+... )$, thus allowing for different parameter values as compared to the first regression, while the Link test only includes the predicted value from the previous model. Also, the Link test is more parsimonious than the Ramsey model specified using powers of regressors (if they are more than one) while at the same time taking non-additivity into account (as the "regular" Ramsey test): $\hat{Y}^2$ also includes $2*\beta_1\beta_2*XZ$, if $\hat{Y}=\beta_0+\beta_1*X+\beta_2*Z$. Both tests evaluates the hypotesis: $E(y | x)=F(x\theta)$, thus are generical tests for functional form. In case of rejectance, it could be that we just need to allow for non-linearity and/or non-additivity using the same model, covariates and observations, but there may be many alternative explanations, for example: the link function (or, if you prefer, the dependent variable) is misspecifed, relevant variables are omitted, measurement (or classification) errors. Sometimes Ramsey and Link are presented as tests for omitted variables. I think the reason (apart from the fact the Ramsey test was originally introduced as such, as well as the Link test suggests to be a test on the link function by its very name) is that, unless such variables are available, you can't test for their inclusion directly. On the contrary, one can test for heteroskedasticity, or compare models with different link functions/outcome specification, or nested (for example: linear models vs fractional or order polynomials). Moreover, it is not a test against heteroskedasticity (where squared residuals should be used) nor, in case they are not correlated with the available predictors, omitted variables.