# Sampling distribution of regression coefficients for normally distributed random variables

Based on $N$ realizations of two random variables $X \sim N(0,\sigma_X^2)$ and $Y \sim N(0, \sigma_Y^2)$ with correlation $\rho$, I conduct a simple linear regression $Y = \beta_0 + X\beta_1 + \epsilon$.

Using the known distributions of $X$ and $Y$, their correlation $\rho$ and the number of realizations $N$, I am trying to find the sampling distribution of the regression coefficients.

Estimates of the coefficients can be caluclated as:
$\hat \beta_1=r\frac{s_Y}{s_X}$ where $r$ is an estimate of $\rho$ and $s_X, s_Y$ are estimates of $\sigma_X, \sigma_Y$
$\hat \beta_0=\overline{Y} - \hat \beta_1 \overline{X}$ where $\overline{X}, \overline{Y}$ are estimates of the means of $X$ and $Y$

Intuitively, I thought the regression coefficients should follow a normal distribution. However, after some additional thoughts, I am not sure of anymore. Plotting the distribution of $\beta_1$ from 10,000 experimental runs with $N=12, \sigma_X^2=1, \sigma_Y^2=2, \rho =0.6$ (black) vs. fitted normal distribution, a deviation from normal distribution can be seen. This is also the case more iterations (I tested up to 100,000) for which the normal distribution should be quite stable. For more detail on the experiment, see below.

Finding the specific distribution of $\beta_0, \beta_1$ - if there is one - is however only the second step. In the first step, I am trying to find the expected value and variance of the regression coefficients.

It is clear that $E[\hat \beta_0]= 0$ and $E[\hat \beta_1] = \rho \frac{\sigma_Y}{\sigma_X}$.

However, I have trouble deriving the variance of both parameters. I have found some definitions for the coefficient variance in the literature. $Var[\hat \beta_0] = \frac{s_\epsilon^2\sum x_i^2}{N \sum (x_i - \overline{x})^2}$ and $Var[\hat \beta_1] = \frac{s_\epsilon^2}{\sum (x_i - \overline{x})^2} = \frac{s_\epsilon^2}{N \sigma_X^2}$.

However, as stated correctly by whuber, these definitions are conditioned on $X$. A calculation is therefore only possible for specific relizations and the definitions are therefore not applicable to my use case. In the literature, I found that the restriction on fixed $X$ is going back to Fisher (e.g. his work 'Asymptotic distribution of the reduced-rank regression estimator under general conditions' from 1922). However, I did not find a consideration of the case with random $X$ in more recent literature. I have only found refrences to the (non central) Wishart distribution. I'm not sure whether and how the Wishart distribution can be used in my use case, though.

Overall, I am completly stuck on how to derive the variances in the described case, $Var[\hat \beta_0], Var[\hat \beta_0]$ with $X, Y$ being random variables.

For cases with conditioning on $X$, I have found various answers on how to derive the variance of the regression coefficients, for instance https://stats.stackexchange.com/a/89155/48067. However, since not only $\epsilon_i$ is a random variable, the approach described in the answer is not easily transferable to my problem.

Consdering that $\hat \beta = r \frac{cov(X, Y)}{s_X^2}= r \frac{s_Y}{s_X}$, I assume that the distributions of $r, s_Y, s_X$ may for some reason be problematic for the calculation of the variance. I have found that $s_X^2, s_Y^2$ follow a gamma distribution. $s_X, s_Y$ should therefore follow a generalized gamma distribution, which have a well defined variance. However, I am not sure about the quotient of two generalized gamma distributed random variables. $r$ has a skewed distribution on which the Fisher transformation can be used - I am however not sure whether this helps to calculate the variance. Lastly, the sample covariance has a strange distribution, approximately like a shifted gamma distribution.

Overall, I have already derived some information about the distributions of characteristics of the samples. Yet, I have not found a way to use these information to derive the variance of the sample coefficients in my use case.

Can someone point me in the right direction?

--

• Samples of size N (e.g. N=12) are drawn from $X \sim N(0, \sigma_X^2)$ and $Y \sim N(0,\sigma_Y^2)$ (e.g. $\sigma_X^2 = 1$, $\sigma_Y^2 = 2$) so that the samples have a specific correlation $\rho$ (e.g. $\rho = 0.6$). This is achieved by using two uncorrelated random variables $A, B \sim N(0,1)$, and constructing a new (correlated) one as $C = \rho^2 A + \sqrt{1- \rho^2} B$ and correcting the variance of $A$ and $C$ in order to have $X$ and $Y$
• A regression of $Y$ on $X$ is executed leading to coefficients $b_0$ and $b_1$
• Steps one and two are repeated (for instance 10,000 times)
• The distribution resulting of $b_0$ and $b_1$ resulting from the runs are analyzed

The distributions resulting from the last step are the distributions for which I want to derive exact distribution parameters in order to analyze the uncertainty in the regression parameters resulting from the sampling of the random variables.

• (A) You are running into trouble in part due to not distinguishing parameters from their estimates. Two levels of that are needed in this context: (1) you seek the variances of the estimates of $\beta_0$ and $\beta_1$ and (2) because the expressions of those variances involve a parameter ($\sigma_\epsilon$), you need to estimate that parameter. Thus I am suggesting that a careful re-writing of your question along these lines will show you the answer. (B) Your expressions for the variances appear only to be conditional on $X$. – whuber Jun 10 '14 at 13:28
• Thank you very much for your comment. Concerning (A): I have tried to rewrite my question with a differentiation between parameter and estimates. It is now clear to me that a differentiation between $\sigma_\epsilon$ and $s_\epsilon$ is necessary. However, I am still having trouble when using the estimator for $s_\epsilon$ (as described in my edits of the initial post). Concerning (B): Could you further clarify this point? – sebastianb Jun 11 '14 at 7:11
• Re (B): The formulae you quote for the variances of the estimators assume $X$ is fixed: they do not account for the random variation in $X$ itself. A formula for the full variance could be a function of only $\sigma_X, \sigma_Y,$ and $\rho$. Since this distinction (between conditional and unconditional variances) is fundamental, it would help for you to describe your simulation in more detail to make it clear what exactly your are simulating as random and what you are keeping fixed. – whuber Jun 11 '14 at 13:42
• The distinction between conditional and unconditional variance indeed seems to be a problem. The definitions of the coefficient variance I found in the literature always seemed kind of odd for my use case. Thank you for pointing that out! I will try to derive a formula for my specific use case. Meanwhile I have added more details about the experiments in my description in my initial post. In short: samples of size $N$ are drawn randomly from $X$ and $Y$ on which the linear regression is conducted. In contrast, $\sigma_X^2, \sigma_Y^2, \rho$ are fixed. – sebastianb Jun 11 '14 at 14:18
• I have just finished rewriting major parts of my initial posts. Thanks to @whuber, some points got clearer for me, I updated the post appropriately. However, I have not achieved any new steps in my derivation of the coefficient variance, since the general case without fixed $X$ is even harder for me to derive. I would very much appreciate any help. – sebastianb Jun 12 '14 at 9:36

After lots of further paper reading, I have found an answer which is acutally quite simple.
The problem was solved independently by Pearson [1] and Romanovsky [2] in 1926.

Bot found that $var(\hat \beta) = \frac{\sigma_Y^2}{\sigma_X^2} \frac{1-\rho^2}{N-3}$,
which matches my simulation results.

[1] Pearson, Karl. "Researches on the mode of distribution of the constants of samples taken at random from a bivariate normal population." Proceedings of the Royal Society of London. Series A 112.760 (1926): 1-14.
[2] Romanovsky, Vsevolod Ivanovich. "On the distribution of the regression coefficient in samples from normal population." Bulletin de l'Académie des Sciences de l'URSS 20.9 (1926): 643-648.

• you should include the calculation (or at least a sketch of the main ideas) here, so that the question is self contained – WetlabStudent Jan 17 '15 at 23:35
• @sebastianb Thank you very much for you answer, helped me a lot with Bayesian model selection. Can you please tell me if the same methodology for finding mean and variance of beta coefficients works for logistic regression too? – olejnik_ Dec 14 '16 at 20:50
• What was the sampling distribution? The accepted answer seems to only relate to part of the original question (variance of the sampling distribution). – sammosummo Aug 1 '17 at 15:25
• Your answer shows $var(\hat{\beta_1})$; what about for $\beta_0$? – Jonathan May 8 '19 at 22:41