Linked Questions

6
votes
2answers
24k views

How is the formula for the Standard error of the slope in linear regression derived? [duplicate]

As stated in many textbooks, the Standard error of the slope in linear regression with one variable is $\sqrt{\frac{s^2}{SSX}}$ or some rewrite, ${s^2}$ being the error variance and ${SSX}$ being ...
8
votes
1answer
8k views

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 + \...
12
votes
2answers
498 views

Using lm for 2-sample proportion test

I have been using linear models to perform 2-sample proportion tests for a while, but have realized that might not be completely correct. It appears that using a generalized linear model with a ...
0
votes
2answers
1k views

Variance of a regression coefficient

$$\mathrm{Var}(\beta) = E [ (\beta- E\hat{\beta})^2 ] $$ My question is: Why is there "$E$" instead of summation?
1
vote
1answer
943 views

Standard deviation of least-squares standard error

I do a measurement where I collect a set of data and fit it to a linear model using ordinary least squares. From that I get a slope, b and the standard error of it, s. Now I repeat the measurement N ...
5
votes
2answers
147 views

confidence intervals in linear regression

I am trying to understand the confidence interval for linear regression parameters. At this link Derive Variance of regression coefficient in simple linear regression an answer is provided. However, I ...
2
votes
2answers
54 views

Understanding simplification of constants in derivation of variance of regression coefficient

In looking over TooTone's answer in Derive Variance of regression coefficient in simple linear regression, there's a step in line 3 below where $(\beta_0 + \beta_1x_i + u_i )$ is simplified to $u_i$ ...
0
votes
1answer
144 views

OLS variance estimators - multiple linear regression

I got a three-dimensional point set ($X1$, $X2$, $Y$) which can be described by a regression equation of the form $y_c = a + b_1x_1 + b_2x_2$. Now I want to calculate $Var(b_1)$ and $Var(b_2)$ but I ...
0
votes
1answer
133 views

Perform a t-test in multiple regression matrix

I have the following matrices: $$ X'X $$ $$X'y$$$$ y'y $$ I know that the B matrix can be computed as follows : $$ B = (X'X)^{-1}X'y $$ If I want to perform a t test for a specific B, say $$B_{1}$$, ...
0
votes
2answers
67 views

Found an expression I haven't encountered before

I was reading the book an introduction to statistical learning with R (http://www-bcf.usc.edu/~gareth/ISL/data.html), and came across this expression that I haven't seen before. Can anyone tell me ...
1
vote
0answers
48 views

Easiest way to derive standard error of linear regression model slope directly?

Assume that we want to derive the standard error of $\beta_2$ in: $$y=\beta_1 + \beta_2x_2 +u$$ $u$ iid with mean $0$ variance $\sigma ^2$. The way I know how to do it, is convert it in matrix form, ...