Linked Questions
52 questions linked to/from Under which assumptions a regression can be interpreted causally?
105
votes
17
answers
73k
views
Under what conditions does correlation imply causation?
We all know the mantra "correlation does not imply causation" which is drummed into all first year statistics students. There are some nice examples here to illustrate the idea.
But sometimes ...
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76
votes
6
answers
9k
views
Criticism of Pearl's theory of causality
In the year 2000, Judea Pearl published Causality. What controversies surround this work? What are its major criticisms?
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22
votes
6
answers
68k
views
Does simple linear regression imply causation?
I know correlation does not imply causation but instead the strength and direction of the relationship. Does simple linear regression imply causation? Or is an inferential (t-test, etc.) statistical ...
- 221
24
votes
3
answers
17k
views
Unconfoundedness in Rubin's Causal Model- Layman's explanation
When implementing Rubin's causal model, one of the (untestable) assumptions that we need is unconfoundedness, which means
$$(Y(0),Y(1))\perp T|X$$
Where the LHS are the counterfactuals, the T is the ...
- 1,229
21
votes
5
answers
4k
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 ...
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30
votes
2
answers
8k
views
do(x) operator meaning?
I have seen the $do(x)$ operator everywhere in some literature review I am doing on Causality (see, for instance this wikipedia entry). However, I cannot find a formal and general definition of this ...
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17
votes
3
answers
1k
views
Which OLS assumptions are colliders violating?
The following webpage says that:
We should not control for a collider variable!
Which OLS assumptions are colliders violating?
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9
votes
2
answers
1k
views
Incorrectly Using the Word "Causal" to Describe a Regression Model?
Suppose we take the classical linear regression model:
$$y_i = \beta_0 + \beta_1 x_i + \epsilon_i$$
Over the years, I have heard so many people say that such an interpretation can be drawn from this ...
15
votes
5
answers
11k
views
What are the differences between stochastic and fixed regressors in linear regression model?
If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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19
votes
4
answers
9k
views
Difference Between Simultaneous Equation Model and Structural Equation Model
Can anybody please help me to understand the differences between simultaneous equations models and structural equation models (SEM)? It will be great if somebody can provide me some literature on it.
...
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15
votes
2
answers
2k
views
Is a regression causal if there are no omitted variables?
A regression of $y$ on $x$ need not be causal if there are omitted variables which influence both $x$ and $y$. But if not for omitted variables and measurement error, is a regression causal? That is, ...
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18
votes
1
answer
4k
views
Causal effect by back-door and front-door adjustments
If we wanted to calculate the causal effect of $X$ on $Y$ in the causal graph below, we can use both the back-door adjustment and front-Door adjustment theorems, i.e.,
$$P(y | \textit{do}(X = x)) = \...
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13
votes
3
answers
3k
views
Regression and causality in econometrics
In regression in general and in linear regression in particular, causal interpretation of parameters is sometimes permitted. At least in econometrics literature, but not only, when causal ...
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6
votes
2
answers
12k
views
Strict exogeneity and lagged variables
I am confused why strict exogeneity must be violated when we have lagged time series variables. My understanding of strict exogeneity is that a variable must be uncorrelated with error terms in all ...
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4
votes
3
answers
2k
views
Conditional probability and causality
I would like to understand the link between conditional probabilities and causality. More precisely:
Assume we have two variables $A=\{0,1\}$ and $B=\{0,1\}$ and we observe:
$P(A=1|B=1)>P(A=1|B=0)...
- 497
9
votes
2
answers
2k
views
What is the relationship between minimizing prediction error versus parameter estimation error?
With the advent of statistical learning techniques, people are talking a lot about prediction error, while in classical statistics, one is focusing on parameter estimation error. What is the ...
- 3,184
5
votes
3
answers
2k
views
Measurement error in one indep variable in OLS with multiple regression
Suppose I regress (with OLS) $y$ on $x_1$ and $x_2$. Suppose I have i.i.d. sample of size n, and that $x_1$ is observed with error but $y$ and $x_2$ are observed without error. What is the probability ...
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18
votes
2
answers
1k
views
How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?
In their article, Chen and Pearl (2013), critically examined 6 econometric textbooks, among these the textbooks written by Wooldridge (2009) {the introductory book}, and Stock & Watson (2011). ...
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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 ...
8
votes
3
answers
887
views
Are all statistical models also causal models?
I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus.
So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a ...
- 241
5
votes
2
answers
738
views
regression and causation
In the Chen and Pearl (2013) article there are several critics about econometrics textbooks. Currently I try to understand more about it.
In particular the Authors written (pag 4, footnote 5):
From ...
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3
votes
2
answers
2k
views
Interpreting interaction term when X1 effect on Y depends on X2 but X2 effect on Y does not depend on X1
Imagine a set of variables, X1, X2, and Y, all continuous variables.
There is a simple case where X1 and X2 affect Y such that:
Y = alpha + β1 X1 + β2 X2 + error
Using R syntax, a model to analyze ...
- 327
2
votes
3
answers
2k
views
Endogeneity testing using correlation test
I am currently testing my linear model using OLS method. The last thing I have to test is endogeneity issue. Is it enough if I test each explanatory variable for correletion with error term? Than ...
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8
votes
1
answer
1k
views
What's the DGP in causal inference?
This question come from this discussion (Under which assumptions a regression can be interpreted causally? ). That discussion touch too arguments and is not possible to speak about all things there. ...
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3
votes
2
answers
1k
views
Is it possible for the zero conditional mean assumption to fail?
I have a questions about the so-called "zero conditional mean" assumption often made in the context of regression analysis. I am struggling to see how it could be violated, or rather where ...
- 303
1
vote
2
answers
1k
views
Regression's population parameters
Suppose I've specified a linear regression model:
$$
Y = \beta_0 + \beta_1 X + \epsilon
$$
where $\beta_0$, $\beta_1$ are the population parameters. My question is: why are these parameters ...
- 137
2
votes
2
answers
358
views
Why are error properties in linear regression assumptions if they are true by construction?
The following two results on the residuals ($\epsilon$) in the case of linear regression get stated as assumptions of the linear regressions
$E(\epsilon) = 0$
$cov(X, \epsilon) = 0$
Here is MIT 18....
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0
votes
2
answers
3k
views
Zero conditional expectation of error in OLS regression
Suppose we have a dependent variable $Y$ and an independent variable $X$ in a population, and we want to estimate the linear model
$$
Y = \beta_{0} + \beta_{1}X + \varepsilon
$$
Using the least-...
- 245
0
votes
1
answer
3k
views
OLS Assumption-No correlation should be there between error term and independent variable and error term and dependent variable
My question is that does endogeneity exists if there is correlation between dependent variable and error term, but not in between error term and independent variable. So for Ex, we know there should ...
- 161
8
votes
1
answer
847
views
conditional and interventional expectation
Conditional expectation $E[Y|X]$ and interventional expectation $E[Y|do(X)]$ are related but conceptually very different things.
I know that if $X$ is a randomly assigned by an experiment, we have ...
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5
votes
1
answer
948
views
linear causal model
Currently I’m focused on linear causal model expressed as a structural equation like this:
$y = \beta_1 x_1 + \beta_2 x_2 + … + \beta_k x_k + u$
where $E[u|x_1,x_2,…,x_k]=0$ (exogenous error)
we ...
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1
vote
1
answer
2k
views
Does homoscedasticity imply that the regressor variables and the errors are uncorrelated?
By OLS regression equation:
$$Y = a + bX + e$$
My thoughts are that homoscedasticity by definition imply that $Var(Y|X) = Var(e|X)=$ constant, then this would imply that $Var(e|X) = Var(e)$ which ...
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3
votes
1
answer
515
views
endogenous regressor and correlation
In a widely cited paper by Antonakis et al. (2010), they mention:
If the relation between x and y is due, in part, to other reasons,
then x is endogenous, and the coefficient of x cannot be ...
- 497
3
votes
2
answers
138
views
Does Pr(Y | X=x) equal Pr(Y | do(X=x)) in a randomized experiment?
On page 435 of Cosma Shalizi's advanced data analysis book (link: https://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf), he states the following about randomized experiment for causal inference:
...
- 31
10
votes
1
answer
606
views
About the meaning of ARMA parameters
I suppose that the main scope of an econometric models should be predictive or causal inference. Following this perspective it was shown that underspecified model can perform better than the correct ...
- 5,779
3
votes
2
answers
759
views
How to test whether OVB by examining two regressors (X_1, X_2) using hypothesis test with null hypothesis H0: corr(X_1,X_2) = 0
Suppose you have an i.i.d. sample ${(𝑌_i , 𝑋_{1,i} , 𝑋_{2,i} ): 𝑖 = 1, ... , 𝑛}$. You want to estimate the causal effect
of $𝑋_1$ on $𝑌$. You first run a regression $𝑌_i = 𝛼_0 + 𝛼_1𝑋_{1,i} +...
- 31
1
vote
1
answer
678
views
OLS Estimation, Bias and Causality
I wish to ask about the bias of an OLS estimator. In what follows I assume that the regression that we are dealing with is an approximation to a linear conditional expectations function. That is we ...
- 303
1
vote
2
answers
436
views
Model causality: graphical models and PCA
If we build a graphical model (DAG) we (may) interpret the arrows as causal dependences.
If we build a graphical model based on the variables returned by principal component analysis (PCA) we should ...
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2
votes
1
answer
373
views
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
1
vote
1
answer
489
views
Random Sampling: Weak and Strong Exogenity
$Y \ = \ X' \beta \ + \ e $
Where $Y = (y_1, ..., y_n)$ and $\beta = (\beta_0,..., \beta_k)$.
Why would Weak Exogenity under random sampling produce Strong Exogenity?
I know that weak exogenity is ...
- 242
1
vote
1
answer
360
views
Econometrics meaning of structural versus regression model
I want to make sure my understanding is correct. Particularly in econometrics, when authors write down a model:
$Y_i = \beta_0 + \beta_1 X_i + \epsilon$
Can I think of this as a 'structural model'- or ...
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0
votes
2
answers
527
views
How do endogenous variables relate with the error term?
Are endogenous variables stochastic or non-stochastic? If they are stochastic,can we say they are uncorrelated or correlated with the error term?
I read this in Basic Econometrics by Gujarati (5th ...
4
votes
3
answers
177
views
Is the physical impact / effect necessarily the independent variable / dependent variable of the regression model
A regression analysis (RA) is often explained as follows:
"...Regressions analyses are statistical methods, by which you can calculate, whether an independent variable (IV) impacts a dependent ...
- 143
0
votes
1
answer
297
views
Endogeneity and Regression Interpretation: how does it work? is it possible?
My question is not particularly straightforward. I will explain my reasoning, and then give the question at the end to avoid any confusion.
$\ Y_{i} = \beta_{0}+\beta_{1}X_{i}+\varepsilon_{i} $
...
- 303
2
votes
2
answers
338
views
Endogenous controls in linear regression - Alternative approach?
I have a cross-section of $x$, $y_1$, and $y_2$. These are individual level data used in labor economics. I have random variation in $x$ and I'm interested in the effect of $x$ on $y_1$. It is well ...
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2
votes
1
answer
261
views
Causal discovery for pairwise independent joint dependent variables
Consider the standard example for variables that are pairwise independent but joint dependent.
$$
(x,y,z)=
\begin{cases}
(0,0,0) & \text{probability 1/4} \\
(1,1,0) & \text{probability 1/4} \\
...
1
vote
2
answers
169
views
Cov(e,X) in the population regression
Say the population regression function is:
$$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i $$
(In the econometrics context) While I can't just assume that $E[\varepsilon_i | X_i] = 0$, can I not say ...
- 53
3
votes
1
answer
123
views
What are the main methods for estimating the Average Treatment Effect in Observational Studies outside of matching?
I am wondering what the main methods for estimating the Average Treatment Effect in Observational Studies are outside of matching. In matching, there are weighting, stratification, propensity score ...
- 4,260
0
votes
1
answer
144
views
simple linear regression causality
lets say we have a perfect linear regression, i.e. we have included all relevant variables (to prevent OVB: omitted variable bias), and also such that there is no other problems like mutli-...
- 185
1
vote
2
answers
192
views
What exactly is a "true" population model in linear regression?
What do we mean by a true population model when talking about linear regression? Say I want to study the effects of years of schooling $S$ on wages. I posit the following two models:
$log(wage)=β_0+...
- 55