Ytukyg

Suppose $Y sim{N(a,b)}$, $X sim{N(c,d)}$, and $Y$ is independent of $X$. After sampling 25 observations from both $Y$ and $X$, I run the following regression model: $Y=beta_{0}+beta_{1}X + epsilon$. I wish to test the hypothesis $H_{0}: beta_{0}=0$ against the alternative $H_{1}: beta_{0}neq 0$.

My question is, since the distributions of $Y$ and $X$ are known, is there an exact 'null distribution' for the parameter $beta_{0}$? If so, what is the distribution? By null distribution, I mean the sampling distribution of $beta_{0}$ under the null hypothesis.

If anyone knows the answer assuming the true correlation coefficient between $Y$ and $X$ is 0.1, rather than assuming independence, that would be a big help also. This is all for a simulation study I'm working on.

Since you have specified that $X$ and $Y$ are independent, the conditional mean of $Y$ given $X$ is:

$$mathbb{E}(Y|X) = mathbb{E}(Y) = c,$$

which implies that:

$$beta_0 = c quad quad quad beta_1 = 0 quad quad quad varepsilon sim text{N}(0, d).$$

In this case there is nothing to test --- your regression parameters are fully determined by the distributional assumptions you have made at the start of the question.

Remember that a regression model is a model designed to describe the conditional distribution of $Y$ given $X$. If you assume independence of these variables then this pre-empts the entire modelling exercise.

In simple linear regression the computation of the estimate of $beta_0$ is:

$$hatbeta_0 = frac {1}{n} S_y + frac {1}{n} S_x frac {n S_{xy} - S_x S_y}{ n S_{xx} - S_x S_x}$$

with $S_x = sum x_i $, $S_y = sum y_i $, $S_{xx} = sum x_i x_i $, $S_{xy} = sum x_i y_i $

You could say it will be a linear sum of the $y_i$

$$hatbeta_0 = frac {1} {n} sum c_i y_i $$

with

$$c_i =left( 1 + frac {n x_i - S_x}{n S_{xx} - S_x S_x} right) $$

This does not seem to follow an easy distribution (or at least not a typical well known distribution) for both random $x_i $ and $y_i$ you have:

$$hatbeta_0 sim N(mu, sigma^2)$$

where $mu$ and $sigma$ are random variables themselves depending on the distribution of $X$ as well. (if every $y_i$ has an identical distribution $N(a,b)$ then $mu = a$, independent from the distribution of $X$)

However if you condition on $x_i$ then $hatbeta_0$ follows a regular normal distribution (note that the $y_i$ do not need to be distributed according to identical Normal distributions) .

In testing you often do not know the variance of this normal distribution and you will estimate it based on the residuals. Then you will use the t-distribution.