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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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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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### 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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### 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 ...
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### 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. ...
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 ...
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 ...