Linked Questions

4
votes
2answers
121 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
164 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
2k 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 ...
15
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
301 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
471 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
0answers
838 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 ...
1
vote
2answers
301 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 ...
4
votes
1answer
216 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
127 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
337 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
142 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
101 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 ...

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