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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.

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    $\begingroup$ I wonder whether you mean the distribution of $\hat \beta_0$ rather than of $\beta_0$? You have specified that you are 100% sure that $\beta_0 = 0$, so that is the rather degenerate distribution that it has! But it sounds to me that you might be rather more interested in the distribution of $\hat \beta_0$, which is the estimate of $\beta_0$ that you would make from your random sample - and since different random samples will produce slightly different estimates, your estimator has a non-degenerate probability distribution $\endgroup$ – Silverfish Dec 23 '18 at 9:18
  • $\begingroup$ This question would be more interesting if you drop the independence assumption on $X$ and $Y$, and add an assumption on joint normal distribution. $\endgroup$ – kjetil b halvorsen Dec 23 '18 at 9:59
  • $\begingroup$ Yes, I meant that if I was to test $\beta _{0}=0$ (for a simulation exercise I'm working on... I know the true value is $c$), I would have to generate the sampling distribution of $\hat{\beta _{0}}$ under the null that $\beta _{0}=0$. I know asymptotically this distribution is normal. But since X and Y are both normal and n is relatively small, am I able to use the t-distribution (for example) to form an 'exact' null distribution of $\hat{\beta _{0}}$, rather than using the asymptotic approximation. The true value of the parameter is 0 (obviously), but this is not what I'm after! $\endgroup$ – Anna Efron Dec 24 '18 at 6:16
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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.

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  • $\begingroup$ Thank you. I meant that if I was to test $\beta _{0}=0$ (for a simulation exercise I'm working on... I know the true value is $c$) the usual way, I would have to generate the sampling distribution of $\hat{\beta _{0}}$ under the null that $\beta _{0}=0$. I know asymptotically this sampling distribution is normal. But since $X$ and $Y$ are both normal and $n$ is quite small, am I able to use the t-distribution (for example) to form an 'exact' null distribution of $\hat{\beta _{0}}$, s.t. the the coverage probability is exactly $(1-\alpha)$? And what if $\rho_{XY}=0.1$ (say) instead of 0? $\endgroup$ – Anna Efron Dec 24 '18 at 6:21
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    $\begingroup$ Once you remove the assumption that $X$ and $Y$ are independent, the regression model is your specification of their conditional relationship. Much of the information you have given in your comment unfortunately contradicts your original question. It is also unclear why you would test $H_0: \beta_0 = 0$ if you know from some other source (your simulation) that $\beta_0 = c$. I think at this point you will probably need to ask a new question where all this information is made clear. $\endgroup$ – Ben Dec 24 '18 at 6:44
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In simple linear regression the computation of the estimate of $\beta_0$ is:

$$\hat\beta_0 = \frac {1}{n} S_y + \frac {1}{n} S_x \frac {n S_{xy} - S_x S_y}{ n S_{xx} - S_x S_x}$$

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 $y_i$

$$\hat\beta_0 = \frac {1} {n} \sum c_i y_i $$

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:

$$\hat\beta_0 \sim N(\mu, \sigma^2)$$

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 $\hat\beta_0$ 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.

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