Linked Questions

39 votes
6 answers

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 $\...
luchonacho's user avatar
  • 2,677
21 votes
2 answers

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 ...
Hirek's user avatar
  • 997
20 votes
5 answers

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 ...
Richard Hardy's user avatar
7 votes
2 answers

What does the assumption: "The independent variable is not random." in OLS mean?

What does the assumption: "The independent variable is not random." in OLS mean? How can you verify that hypothesis?
Victor's user avatar
  • 1,035
7 votes
3 answers

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 ...
floyd's user avatar
  • 1,360
5 votes
1 answer

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 ...
ColorStatistics's user avatar
5 votes
3 answers

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 ...
user523384's user avatar
4 votes
2 answers

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 ...
user2879934's user avatar
4 votes
1 answer

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^{-}}]=\...
dandar's user avatar
  • 698
3 votes
2 answers

Regression and the CEF

I recently read in this page ( that: "Regression offers a way of approximating ...
Rafael Hernández Salazar's user avatar
3 votes
1 answer

Why is each observation in a sample considered a random variable in linear regression?

I have the following excerpt in my statistics textbook: I am confused by the sentence: "Another way statisticians treat this model is that, assume $X_1...X_n$ are random variables, we make ...
Nova's user avatar
  • 651
2 votes
1 answer

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, ...
Aditya Agarwal's user avatar
2 votes
1 answer

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'...
Tortar's user avatar
  • 356
2 votes
1 answer

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 ...
orthonormal-stice's user avatar
2 votes
1 answer

Meaning of & intuition behind predictors being fixed in linear regression [duplicate]

My question is a bit naive. I'm trying to get the exact & clear meaning of the phrase "predictor variables are fixed and not random in linear regression". According to my understanding, ...
SRISHTI GUREJA's user avatar

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