I am reading Bain/Engelhardt's "Introduction to Probability and Math Statistics" about maximum likelihood estimate (MLE) for linear regression (p. 519 - 522).

I first summarize 3 key points from the textbook that I am interested in.

(1) The MLE for $ \hat{\beta} = (X^TX)^{-1}X^TY $ and $ \hat{\sigma}^2 = \frac{(Y-X\hat{\beta})^T(Y-X\hat{\beta})}{n} $. For this part, I totally get it.

(2) Further, the book mentions the following: $\tilde{\sigma} = \frac{(Y-X\hat{\beta})^T(Y-X\hat{\beta})}{n-p-1} $ is the UMVUE of $\sigma^2$. Okay, for this, I understand that it is unbiased estimate. So no problem.

(3) Furhter, it says the following: $T=\frac{\hat{\beta_j}-\beta_j}{\sqrt{\tilde{\sigma}^2 a_{jj}}} \sim t(n-p-1)$ where $ \{a_{jj} \}$ is the number on the diagonal of $(X^TX)^{-1}$ corresponding to $\beta_j$. Basically, this can be used to do hypothesis test. For $H_0: \beta_j = \beta_{j0}$, we reject it if $|t|> t_{1-\alpha/2} (n-p-1)$.

If I understand correctly, typically we assume $\beta_{j0} =0$, and thus we can calcuate t-statistic as $t(n-p-1)=\frac{\hat{\beta_j}}{\sqrt{\tilde{\sigma}^2 a_{jj}}}$.

OK, my two questions as follows.

(1) If we use MLE to estimate regression coefficients for linear regressions, it seeems we can use t-test to evaluate the significance level, right? It seems yes based on this textbook. If so, what is the degree of freedom for this t-test, it seems it should be n-p-1 (p not including intercept). Thus, for simple linear regression, it will be n-1-1=n-2. Correct?

(2) We know that for MLE, it typically also estimates the the variance for the noise term, $ \hat{\sigma}^2$. I know it counts as one parameter. Does it consider use one more degree of freedom? If so, why no need to use n-3 in the t-test for regression coefficcient t-test? Is it because $\beta $ and $ \sigma^2 $ are independent, and thus the estimate of $\sigma^2$ does not impact the df of t-test for regression coefficients.

Thank you so much! Look forward to any feedback and helps.

Added Content:

Even though I added it at the comment section in Rachel's answer, I think it is better to put it here as here allows code colors.

In particular, there is another post, link below, about the degree of freedom in MLE for linear regression.

What does the degree of freedom (df) mean in the results of log-likelihood `logLik`

> m <- lm(mpg ~ hp, data = mtcars) 

logLik(m) 'log Lik.' -87.61931 (df=3)

Regarding the R code output shown above, the following is my additional question, namely question (3):

(3) I understand the t-test's df for regression coefficients is n-p-1. Thus, for simple linear regression, it will be n-2. If so, why does logLik in R return df=3? Is it because 3 here means 3 parameters, and not necessarily 3 df per se? Thank you.

To answer the (3) question by myself, based on the discussion with Rachel and others (see all the comments under this main question and under Rachel's answer):

logLik(m) returns 3 df means that it estimates 3 parameters(intercept, slope and variance). However, since the estimation of $\sigma^2$ is based on a formula with estimated intercept and slope, it does not cost 1 more df. Thus, the actual df in t-test for regression coefficients is still n-2. (2 represents one df for intercept and one df for slope, in the context of simple linear regression.)

  • 1
    $\begingroup$ Hi: The answer to the first question is yes. The answer to the second question is that the estimation of $\sigma^2$ doesn't take away a degree of freedom because, once the $\hat{\beta}$ are known, the $\hat{\sigma}^2$ is known because it's only a function of the estimated residuals. Note that $\bar{X}$ and $s^2$ are independent when normality is assumed but $\beta$ and $\sigma^2$ are not independent. $\endgroup$
    – mlofton
    Commented Jul 17, 2023 at 1:55
  • $\begingroup$ @mlofton thanks for your answer. I got your first answer. But, for the second part, I am not sure. In the textbook (p. 521), it mentions $ \hat{\beta}$ and $\hat{\sigma}^2$ are independent. Are you suggesting the relationship between estimated $ \hat{\beta}$ and $\hat{\sigma}^2$ is different from the relationship between $ \beta $ and $\sigma^2$? Again, thank you! $\endgroup$
    – Will
    Commented Jul 17, 2023 at 3:27
  • $\begingroup$ Re. your point (3): You should correct your SE in the denominator. The term $(X'X)^{-1}$ is a matrix; you need to extract the diagonal entry corresponding to $\beta_j$. $\endgroup$ Commented Jul 17, 2023 at 6:02
  • $\begingroup$ @RachelAltman You are right. I just corrected it. Thank you! $\endgroup$
    – Will
    Commented Jul 17, 2023 at 11:58
  • 1
    $\begingroup$ There are several ways. A common one is a Wald test, where your question is discussed at stats.stackexchange.com/questions/115360. Other tests are the likelihood ratio test and a "score test." All rely on Normal approximations to the distributions of the estimates or of the log likelihood when the number of observations is large. Using a Student t distribution to test MLEs can be done, but it's generally an ad hoc procedure that is foreign to the spirit and theory of maximum likelihood estimation. $\endgroup$
    – whuber
    Commented Jul 17, 2023 at 23:10

1 Answer 1

  1. No. The significance level is always chosen by the practitioner in advance of conducting the hypothesis test. It is the chosen probability of making a Type I error. But yes, you have specified the df correctly.

  2. When $\sigma^2$ is known, $T\sim N(0,1)$. Otherwise, letting $a_j^2$ be the $j^{th}$ diagonal entry of $(X'X)^{-1}$, $$\begin{eqnarray*} T &=& \frac{\hat{\beta}_j-\beta_{j0}}{\tilde{\sigma}a_j} \\ &=& \frac{\frac{\hat{\beta}_j-\beta_{j0}}{\sigma a_j}}{\frac{\tilde{\sigma}a_j}{\sigma a_j}} \\ &=& \frac{\frac{\hat{\beta}_j-\beta_{j0}}{\sigma a_j}}{\sqrt{\frac{\tilde{\sigma}^2}{\sigma^2}}} \end{eqnarray*} $$ The numerator is distributed as $N(0,1)$ and is independent of the denominator. The square of the denominator is distributed as $\chi^2$ on $n-p-1$ df. By definition, this ratio is distributed as $t$ on $n-p-1$ df.

  • $\begingroup$ Thank you so much Rachel! For your first part of the answer, I understand. However, for part 2, what you said makes sesne; but, are you suggesting that the estimtation of $\sigma^2$ does not use one more degree of freedom? $\endgroup$
    – Will
    Commented Jul 17, 2023 at 13:11
  • $\begingroup$ No. When $\sigma$ is known, $T$ has a normal distribution, i.e., the notion of df doesn't apply. When we estimate $\sigma$ with $\tilde \sigma$ (note my correction in my answer...originally I wrote $\hat\sigma$), the distribution of $T$ is $t$ with df determined (by definition) by the df of the $\chi^2$ random variable in the denominator ($n-p-1$, in this case). $\endgroup$ Commented Jul 17, 2023 at 15:44
  • $\begingroup$ Hi Rachel: Thank you for your clarification. Yes, now I understand that the estimation of $\sigma^2$ does not cost 1 df, since it is from the a function of $\hat{\beta}$. However, in R, if you run "m <- lm(mpg ~ hp, data = mtcars) > logLik(m)," it will return 'log Lik.' -87.61931 (df=3)." Is it because 3 means parameters (i.e., intercept, slope, and variance), and not necessarily 3 df, per se? $\endgroup$
    – Will
    Commented Jul 17, 2023 at 17:58
  • $\begingroup$ logLik calls the number of estimated parameters "df" (see documentation). But I don't see why it uses that label. The relevant df (i.e., those associated with the denominator of the expression I wrote above) are correctly listed as $n-2=30$ in the summary output (summary(m)): "Residual standard error: 3.863 on 30 degrees of freedom". $\endgroup$ Commented Jul 17, 2023 at 18:13
  • $\begingroup$ Thank you, Rachel. I got it now. $\endgroup$
    – Will
    Commented Jul 17, 2023 at 19:37

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