Deriving the mean and variance of the least squares slope estimator in simple linear regression

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Recommends Hakan Baba · 2020-03 · Cross Validated Stack Exchange · on “Proof for the standard error of parameters in linear regression”

Once we have @ThorGirl's derivation for $\mathrm{SE}\left(\hat{\beta}_1\right)^2$ we can use that to derive the Standard Error for $\hat{\beta}_0$ , i.e. $\mathrm{SE}\left(\hat{\beta}_0\right)^2 $ . Note: If you are looking for a step-by-step explanation of @ThorGirl's answer take a look at this video [this video] . We are going to use the following assumptions / observations: Each output $y_i$ is assumed to have the form: $\beta_0 + \beta_1 x_i + \epsilon_i$ $\overline{y} = \frac{1}{n}\sum_{i=1}^{n} y_i$ The error terms $\epsilon_i$ are uncorrelated with constant variance $\sigma^2$ $y_i$ , $\hat{\beta}_0$ and $\hat{\beta}_1$ are random variables. $x_i$ are known constants. $\beta_0$ and $\beta_

Recommends Hakan Baba · 2020-03 · Cross Validated Stack Exchange · on “Proof for the standard error of parameters in linear regression”

If anyone are looking for a step-by-step explanation of @ThorGirl's answer take a look at this video [this video] .

Cites trder · 2019-10 · Cross Validated Stack Exchange · on “Expected value of the parameters in linear regression (trying to understand which part is constant and which isn't and why)”

I'm studying Linear Regression and trying to proof/demonstrate some properties of the parameters. When I started working with the expected value of the slope, I got confused with something. I actually couldn't reach the expected result, which is beta 1. So, I found a demonstration very close to what I was doing and I was hoping someone could explain to me the part where I got stuck. First, we have beta 1 parameter: [LINK] The author then uses Ci to make his demonstration less messy: [LINK] The following step is where I got confused. When I see the next part of the demonstration (see picture below), I have no idea why is Ci constant, but Yi isn't! Why is the expected value of Ci equal to Ci,

Cites LJNG · 2022-06 · Cross Validated Stack Exchange · on “is the $Var(\hat{\beta_1}|X)$ same as the variance of $Var(\hat{\beta_1})$ in simple linear regression?”

Here are two captures from different videos on youtube, one obtains $Var(\hat{\beta_1})$ , the other obtains $Var(\hat{\beta_1}|X)$ but they yields the same equations when the second capture further comments that $Var(\hat{\beta_1}|X)\neq Var(\hat{\beta_1})$ [LINK] [LINK] capture1 vedio [this video] capture2 vedio [LINK]