Suppose that $\mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x$ where $X, Y$ are random varibles. Given a set of observations consisting pairs of $X,Y$, is it possible to attach it as probabiltiy density function?
The motivation of this question is a more fundamental thing that I am seeking help with. I want to know if it is possible to perform a hypothesis test testing whether the true specification of the equation is in fact linear without any transformation of the random vaibles, that is, $\mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x$, given a set of pairs $X,Y$. To do this, I have to assume the specification is true and compute the p-value for theset of pairs $X,Y$. However, to do this, I have to first compute the null distribution. And I am pretty much lost at this point. My professor told me that such test does not exist, would any one help me continue my thought process please? Thanks.
some futher thoughts The problem is it is hard to come up with a null distribution. In a one-sample mean test, we suppose the true population mean and deduce the distribution of the sample mean. Similarly, in this problem, we should suppose a true specification, or the realtionship between $X$ and $Y$ for populationand, then deduce the distribution of possible investigated relationship between them for a random sample. But how can we actually do that?
My idea is what characterize the need for a regression is the random variable $\epsilon$. If we can characterize the variance this $\epsilon$, assuming it is normal, we can in some sense show how legit does a particular pair $(x_k,y_k)$ fits the equation $y_i=\beta_0 + \beta_1 x_i+\epsilon_i$. In other words, if we have a pair of realized $X$ and $Y$ and suppose we know $\beta_0$ and $\beta_1$, we can treat the $\epsilon_i$ as a realized value of the random varible $\epsilon$. And it is relatively easy to come up with a distribution of $\epsilon$ and test the p-value of $\epsilon_i$. However, the problem is we don't really know what $\beta_0$ and $\beta_1$ in the null distribution, since we testing for the specifcition for all possible $\beta_0$ and $\beta_1$, not just a single pair of $\beta$ (Let us assume for the time being $\epsilon$ is known for simplification.) Then is it possible to come up witha way to synthesis all possible pair of $\beta$ and test for the null?