Linked Questions
27 questions linked to/from What are the differences between stochastic and fixed regressors in linear regression model?
4
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2
answers
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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 ...
- 523
1
vote
1
answer
338
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 ...
- 587
2
votes
1
answer
147
views
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, ...
2
votes
0
answers
29
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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 $...
35
votes
6
answers
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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 $\...
- 2,649
20
votes
5
answers
3k
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 ...
- 62.1k
20
votes
2
answers
6k
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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 ...
- 987
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?
- 1,025
7
votes
3
answers
523
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 ...
- 1,350
3
votes
2
answers
2k
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 ...
4
votes
1
answer
966
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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 ...
- 4,128
1
vote
2
answers
888
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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2
votes
2
answers
726
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 ...
- 402
1
vote
0
answers
1k
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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
1
answer
373
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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