Linked Questions

4
votes
2answers
135 views

Regression with random X [duplicate]

Suppose we have a standard regression model $$Y= X\beta + \epsilon$$ with $$\epsilon \sim \sigma^2$$ $$X \sim N(\mu,\gamma^2)$$ Are the estimated coefficients the same as if $X$ was fixed? Is ...
1
vote
1answer
176 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 ...
2
votes
0answers
27 views

Regression: What is the difference between assuming the covariates are random or not random? [duplicate]

I often see regression expressed in two ways. The covariates are random: In this scenario, we have $(x_i,y_i) \sim G$ for some distribution $G$ and are i.i.d. for $i = 1, \cdots, n$. We then posit $...
32
votes
6answers
3k views

Under which assumptions a regression can be interpreted causally?

First, don't panic. Yes, there are many similar question on this site. But I believe none gives a conclusive answer to the question below. Please bear with me. Consider a data generation process $\...
16
votes
3answers
2k views

Definition and delimitation of regression model

An embarrassingly simple question -- but it seems it has not been asked on Cross Validated before: What is the definition of a regression model? Also a support question, What is not a regression ...
16
votes
2answers
4k views

What is the difference between conditioning on regressors vs. treating them as fixed?

Sometimes we assume that regressors are fixed, i.e. they are non-stochastic. I think that means all our predictors, parameter estimates etc. are unconditional then, right? Might I even go so far that ...
7
votes
3answers
313 views

Which likelihood function is used in linear regression?

When trying to derive the maximum likelihood estimation for a linear regression, We start by a likelihood function. Does it matter if we use either of these 2 forms? $P(y|x,w)$ $P(y,x|w)$ All pages ...
3
votes
2answers
541 views

Regression and the CEF

I recently read in this page (https://www.timlrx.com/2018/02/26/notes-on-regression-approximation-of-the-conditional-expectation-function/#fn1) that: "Regression offers a way of approximating ...
1
vote
2answers
351 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 ...
1
vote
0answers
880 views

When can we use fixed design regression results for the random design setting? [closed]

Suppose I have an independent vector $X$ and a dependent scalar random variable $Y$ and I wish to construct a regression model to predict $Y$ using $X$ given data $\{(x_i,y_i)\}_{i=1}^{n}$. For ...
4
votes
1answer
223 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^{-}}]=\...
5
votes
3answers
129 views

Foundations behind Linear Regression / Statistical Modelling

I've always struggled with the foundations behind the concept of modelling (and specifically regression) - what is random, what is not, what we are modelling. I think I have a grasp of it - but I'd ...
2
votes
1answer
340 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. ...
3
votes
1answer
184 views

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 ...
2
votes
2answers
129 views

Does the OLS estimator in simple linear regression converge a.s.?

Consider the following model. Assume $(x_i, u_i)$ is sequence of independent identically distributed random vectors in $\mathbf{R}^{d+1}:$ $x_i$ are $\mathbf{R}^d$-value random vectors, which will ...
2
votes
1answer
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, ...
1
vote
1answer
76 views

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'...
0
votes
0answers
131 views

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 ...
0
votes
1answer
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 ...
2
votes
1answer
45 views

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 ...
1
vote
2answers
73 views

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 ...
0
votes
1answer
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 ...
1
vote
1answer
25 views

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 $p(D|W)$ ...