Linked Questions
27 questions linked to/from What are the differences between stochastic and fixed regressors in linear regression model?
39
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6
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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 $\...
- 2,677
21
votes
2
answers
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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 ...
- 997
20
votes
5
answers
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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 ...
- 63.7k
7
votes
2
answers
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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?
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7
votes
3
answers
568
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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 ...
- 1,360
5
votes
1
answer
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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 ...
- 4,176
5
votes
3
answers
174
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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 ...
- 569
4
votes
2
answers
504
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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 ...
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4
votes
1
answer
397
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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^{-}}]=\...
- 698
3
votes
2
answers
2k
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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 ...
3
votes
1
answer
404
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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 ...
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2
votes
1
answer
163
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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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2
votes
1
answer
248
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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'...
- 356
2
votes
1
answer
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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 ...
2
votes
1
answer
184
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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, ...
