357 views

### Regression's population parameters

Suppose I've specified a linear regression model: $$Y = \beta_0 + \beta_1 X + \epsilon$$ where $\beta_0$, $\beta_1$ are the population parameters. My question is: why are these parameters ...
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### Why is the likelihood function for linear regression based on $𝑝(𝑦|𝑥)$ instead of $𝑝(𝑥,𝑦)$

Likelihood functions typically based on the joint probabilities of the variables involved. In linear regression we have variables 𝑥 and 𝑦, but derivations for MLE under a zero meaned gaussian ...
224 views

### Why does regression model theory not use measure-theoretic sigma-field type notation but counting process models do?

I have been studying counting process theory for time to recurrent event processes and am interested in the explicit use of the conditioning set in the model notation; E[dN(t)|\mathcal{F}_{t^{-}}]=\...
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### Proof that Regression Sum of Squares and Residual Sum of Squares are independent random variables

Having consulted a number of sources, I still can't find a complete proof that Regression Sum of Squares ($SS_{regression}$) and ($SS_{residual}$) are independent random variables. I'll be doubly ...
79 views

### Confused with the fundamental assumptions of Frequentist and Bayesian Linear Regression

In Frequentist Linear Regression, I have seen 2 approaches which lead to basically similar models. We have $W,y,X,\epsilon$ related as $y=W^TX+\epsilon$, where $y$ is the dependent random variable, ...
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### Deriving the posterior distribution over the model parameters: are the model parameters and data independent?

We are told (in Section 9.2.3, Deisenroth et al.: Mathematics for Machine Learning) that we can compute the posterior over a model's parameters $\boldsymbol\theta$ (here in the context of linear ...
342 views

### non stochastic regressors and causation

Randomized controlled experiment is base case for causality (also) in regression. However currently I’m analyzing the role of causality in linear regression as shown in many econometrics textbook. ...
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### Notation of the Likelihood Term in Bayesian Neural Networks

I see that in Bayesian neural networks likelihood function is defined in two ways: $p(W|D) = Z^{-1} p(D|W)p(W)$ or $p(W|y,x)=Z^{-1}p(y|x,W)p(W)$ Are there a slight difference in interpreting $p(D|W)$ ...
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### Clarification on the assumptions $E[u|x]=0$ and the $x_i$ being fixed in repeated samples in Wooldridge Introductory Econometrics

The author is writing on the assumption $E[u|x]=0$. The part of the text which is not clear to me is this (the red lines emphasize where the critical portions are located) : In the first piece I don'...
177 views

### Random Explanatory / Independent Variables [duplicate]

Are explanatory variables in regression always considered non-stochastic? If the explanatory variables are random or stochastic will the regression be still valid? What are the implications on the ...
41 views

### Do we assume that the regressors are uncorrelated with the unobserved error $\epsilon$ for least squares?

I recall seeing sources in the past state that the Gauss-Markov assumptions assume that the regressors are uncorrelated with $\epsilon$ in order to make $E[\hat{\beta}] = \beta$. But is this ...
104 views

### In least square linear regression model, why does the test t-statistic of $\hat{\beta}$ follow a t distribution?

In the least square linear regression model, if the explanatory variables and the error term are independent, and the error term is normal, why does the t-statistic of $\hat{\beta}$ follow a t ...
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### Non-stochastic vs Stochastic regressors and sampling distributions and causation?

I was wondering if I understand these correctly. Would an example of a stochastic regressor be weather? so when thinking about the sampling distribtuion and causality, I would think of repeated ...

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