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This is a follow up question to the question I've posted here.

Suppose $Y \sim N(X\beta, \sigma^2I)$, where $y \in \mathbb{R}^n$. Let $X \in \mathbb{R}^{n \times p}$ denote a full rank design matrix. By ordinary least squares, the residuals are $$\hat{e} = (I - X(X^TX)^{-1}X^T)Y$$ Let $u_1 \in \mathbb{R}^p$ denote a unit vector in $\mathbb{R}^p$. Let $\hat{\beta}_1 = u_1^T (X^TX)^{-1}X^TY$ denote the OLS estimate for the first covariate in $X$. Suppose I'm interested in the conditional distribution of $$\hat{\beta}_1 | f(\hat{e}) < \hat{\beta}_1 < f^*(\hat{e}), \hat{e} = e_0$$

where $f(e_0)$ and $f^*(e_0)$ are some functions of $e_0.$ From the answer to my previous post, $\hat{\beta}_1$ and residuals $\hat{e}$ are independent, so I can drop $\hat{e} = e_0$ from the conditioning to obtain

$$\hat{\beta}_1 | f(e_0) < \hat{\beta}_1 < f^*(e_0)$$

Since $\hat{\beta}_1$ is normal, then truncating $\hat{\beta}_1$ between some fixed quantities $f(e_0)$ and $f^*(e_0)$ leads to a truncated normal. Therefore, $\hat{\beta}_1|f(e_0) < \hat{\beta}_1 < f^*(e_0)$ is a truncated normal with lower truncation limit $f(e_0)$ and upper truncation limit $f^*(e_0)$

My question is, once I condition on $\hat{e} = e_0,$ is there still randomness left in $\hat{\beta}_1$? Based on my previous post, I learned that $$[\hat{\beta}_1|\hat{e} = e_0] \overset{d}{=} \hat{\beta}_1$$

Is the same true in this case? i.e.,

$$[\hat{\beta}_1 | f(e_0) < \hat{\beta}_1 < f^*(e_0)]\overset{d?}{=} \hat{\beta}_1$$

The reason I feel that this is not true is because changing the value of $e_0$ would lead to a different set of truncation limits on $\hat{\beta}_1$. On the other hand, if $\hat{\beta}_1$ is independent of $\hat{e}$, then it's independent from any functions of $\hat{e}$. This might suggest that $[\hat{\beta}_1 | f(e_0) < \hat{\beta}_1 < f^*(e_0)]\overset{d}{=} \hat{\beta}_1$. I am quite confused and would greatly appreciate any insight on this.

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  • $\begingroup$ I'm a little confused by your notation. How does your notation $y\sim N(\mu,\sigma^2 I)$ relate to the rest of the problem. Based on the rest of the problem, I would have guessed you need something like $Y|X\sim N(X\beta,\sigma^2 I)$? Also, what is $f$? Is it assumed known but arbitrary? Does it need to be invertible? And it sounds like it maps from $\mathbb{R}^n$ to $\mathbb{R}$? $\endgroup$
    – psboonstra
    Apr 1, 2021 at 20:33
  • $\begingroup$ Yes, technically $y|X \sim N(X\beta, \sigma^2I)$. $f$ is arbitrary and maps from $\mathbb{R}^n$ to $\mathbb{R}$. $\endgroup$
    – Adrian
    Apr 2, 2021 at 3:47
  • $\begingroup$ Thanks. another question. I see now that my first comment may have misled you, notationally speaking, because you are using $\beta$ in a different sense than I assumed. I had interpreted $\beta_1$ as the regression coefficient corresponding to the first predictor, but you are writing $\beta_1$ as the fitted value of the outcome given $X=u_1$ for some covariate pattern $u_1$. In other words, what you are calling $\hat\beta_1$, I would write as $\hat Y|X=u_1$, or $\hat Y(u_1)$. Is my understanding correct? $\endgroup$
    – psboonstra
    Apr 2, 2021 at 18:42
  • $\begingroup$ $\beta_1$ is the coefficient corresponding to the first covariate. $u_1$ is a unit vector with 1 in the first position and 0 everywhere else. So $\beta_1 = u_1^T\beta$, where $\beta_1$ is a scalar, and $\beta$ is the entire coefficient vector of length $p$. $\endgroup$
    – Adrian
    Apr 2, 2021 at 20:21
  • $\begingroup$ Ok. I understand a unit vector to be any vector with length 1, and I was confused by your stating that $u_1 \in \mathbb{R}^p$ instead of just saying $u_1=\{1, 0, \ldots, 0\}$. $\endgroup$
    – psboonstra
    Apr 2, 2021 at 20:43

1 Answer 1

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Knowing that $\hat\beta_1$ is statistically independent from the residual vector $\hat e$ is not relevant to the question. Your question comes down to: is it true that $[\hat{\beta}_1 | c_l < \hat{\beta}_1 < c_u]\overset{d}{=}\hat{\beta}_1$ for some arbitrary constants $c_l$ and $c_u$ where $c_l<c_u$? The answer is 'no: they are not equal in distribution'. As a counterexample, assuming that $\hat\beta_1$ is normally distributed, $\Pr(\hat\beta_1 < c_l)>0$ but $\Pr(\hat\beta_1 < c_l | c_l < \hat{\beta}_1 < c_u)=0$.

Addendum To your comment, On the other hand, if $\hat\beta_1$ is independent of $\hat e$ , then it's independent from any functions of $\hat e$, that only applies to functions of $\hat e$ alone. You are proposing to condition on a random variable that is a function of both $\hat e$ and $\hat \beta_1$, which is an entirely different kind of dependence. Altogether.

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