Under which assumptions a regression can be interpreted causally?

Question

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.

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Consider a data generation process $\text{D}_X(x_1, ... , x_n|\theta)$, where $\text{D}_X(\cdot)$ is a joint density function, with $n$ variables and parameter set $\theta$.

It is well known that a regression of the form $x_n = f(x_1, ... , x_{n-1}|\theta)$ is estimating a conditional mean of the joint distribution, namely, $\text{E}(x_n|x_1,...,x_{n-1})$. In the specific case of a linear regression, we have something like

$$ x_n = \theta_0 + \theta_1 x_1 + ... + \theta_{n-1}x_{n-1} + \epsilon $$

The question is: under which assumptions of the DGP $\text{D}_X(\cdot)$ can we infer the regression (linear or not) represents a causal relationship?

It is well known that experimental data does allow for such interpretation. For what I can read elsewhere, it seems the condition required on the DGP is exogeneity:

$$ \text{E}(x_1, ... x_{n-1}|\epsilon) = 0$$

The nature of randomisation involved in experimental data ensures the above is the case. The story goes then to argue why observational data normally fails in achieving such condition, reasons including omitted variable bias, reverse causality, self-selection, measurement errors, and so on.

I am however uncertain about this condition. It seems too weak to encompass all potential arguments against regression implying causality. Hence my question above.

UPDATE: I am not assuming any causal structure within my DGP. I am assuming that the DGP is complete, in the sense that there must be some causality (an ontological position it could well be debated), and all relevant variables are included. The key is to identify the set of assumptions which ensure me causality goes from certain variables to the other, without assuming from the outset such direction of causality.

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Many similar posts on the site spend time mentioning why correlation does not imply causation, without providing hard arguments about when it does. That is the case, for instance, of this very popular post. Additionally, in the most popular post on the site about the topic, the accepted answer gives the very vague answer:

Expose all hidden variables and you have causation.

I do not know how to translate that to my question above. Neither is the second most upvoted answer. And so on. That is why I believe this post does not have an answer elsewhere.

Answer 1

I made efforts in this direction and I feel myself in charge to give an answer. I written several answers and questions about this topic. Probably some of them can help you. Among others:

Regression and causality in econometrics

conditional and interventional expectation

linear causal model

Structural equation and causal model in economics

regression and causation

What is the relationship between minimizing prediction error versus parameter estimation error?

Difference Between Simultaneous Equation Model and Structural Equation Model

endogenous regressor and correlation

Random Sampling: Weak and Strong Exogenity

Conditional probability and causality

OLS Assumption-No correlation should be there between error term and independent variable and error term and dependent variable

Does homoscedasticity imply that the regressor variables and the errors are uncorrelated?

So, here:

Regression and Causation: A Critical Examination of Six Econometrics Textbooks - Chen and Pearl (2013)

the reply to your question

Under which assumptions a regression can be interpreted causally?

is given. However, at least in Pearl opinion, the question is not well posed. Matter of fact is that some points must be fixed before to “reply directly”. Moreover the language used by Pearl and its colleagues are not familiar in econometrics (not yet).

If you looking for an econometrics book that give you a best reply … I have already made this work for you. I suggest you: Mostly Harmless Econometrics: An Empiricist's Companion - Angrist and Pischke (2009). However Pearl and his colleagues do not consider exhaustive this presentation neither.

So let me try to answer in most concise, but also complete, way as possible.

Consider a data generation process $\text{D}_X(x_1, ... , x_n|\theta)$, where $\text{D}_X(\cdot)$ is a joint density function, with $n$ variables and parameter set $\theta$. It is well known that a regression of the form $x_n = f(x_1, ... , x_{n-1}|\theta)$ is estimating a conditional mean of the joint distribution, namely, $\text{E}(x_n|x_1,...,x_{n-1})$. In the specific case of a linear regression, we have something like $$ x_n = \theta_0 + \theta_1 x_1 + ... + \theta_{n-1}x_{n-1} + \epsilon $$
The question is: under which assumptions of the DGP $\text{D}_X(\cdot)$ can we infer the regression (linear or not) represents a causal relationship? ... UPDATE: I am not assuming any causal structure within my DGP.

The core of the problem is precisely here. All assumptions you invoke involve purely statistical informations only; in this case there are no ways to achieve causal conclusions. At least not in coherently and/or not ambiguous manner. In your reasoning the DGP is presented as a tools that carried out the same information that can be encoded in the joint probability distribution; no more (they are used as synonym). The key point is that, as underscored many times by Pearl, causal assumptions cannot be encoded in a joint probability distribution or any statistical concept completely attributable to it. The root of the problems is that joint probability distribution, and in particular conditioning rules, work well with observational problems but cannot facing properly the interventional one. Now, intervention is the core of causality. Causal assumptions have to stay outside distributional aspects. Most econometrics books fall in confusion/ambiguity/errors about causality because the tools presented there do not permit to distinguish clearly between causal and statistical concepts.

We need something else for pose causal assumptions. The Structural Causal Model (SCM) is the alternative proposed in causal inference literature by Pearl. So, DGP must be precisely the causal mechanism we are interested in, and our SCM encode all we know/assume about the DGP. Read here for more detail about DGP and SCM in causal inference: What's the DGP in causal inference?

Now. You, as most econometrics books, rightly invoke exogeneity, that is a causal concept:

I am however uncertain about this condition [exogeneity]. It seems too weak to encompass all potential arguments against regression implying causality. Hence my question above.

I understand well your perplexity about that. Actually many problems move around "exogeneity condition". It is crucial and it can be enough in quite general sense, but it must be used properly. Follow me.

Exogeneity condition must be write on a structural-causal equation (error), no others. Surely not on something like population regression (genuine concept but wrong here). But even not any kind of “true model/DGP” that not have clear causal meaning. For example, no absurd concept like "true regression" used in some presentations. Also vague/ambiguous concepts like "linear model" are used a lot, but are not adequate here.

No more or less sophisticated kind of statistical condition is enough if the above requirement is violated. Something like: weak/strict/strong exogeneity … predetermiteness … past, present, future … orthogonality/scorrelation/independence/mean independence/conditional independence .. stochastic or non stochastic regressors .. ecc. No one of them and related concepts is enough if them are referred on some error/equation/model that do not have causal meaning since origin. You need structural-causal equation.

Now, you and some econometrics books, invoke something like: experiments, randomization and related concepts. This is one right way. However it can be used not properly as in Stock and Watson manual case (if you want I can give details). Even Angrist and Pischke refers on experiments but them introduce also structural-causal concept at the core of their reasoning (linear causal model - chapter 3 pag 44). Moreover, in my checks, them are the only that introduce the concepts of bad controls. This story sound like omitted variables problem but here not only correlation condition but also causal nexus (pag 51) are invoked.

Now, exist in literature a debate between "structuralists vs experimentalists". In Pearl opinion this debate is rhetorical. Briefly, for him structural approach is more general and powerful … experimental one boil down to structural. Indeed structural equations can be viewed as language for coding a set of hypothetical experiment.

Said that, direct answer. If the equation:

$$ x_n = \theta_0 + \theta_1 x_1 + ... + \theta_{n-1}x_{n-1} + \epsilon $$

is a linear causal model like here: linear causal model

and the exogeneity condition like $$ \text{E}[\epsilon |x_1, ... x_{n-1}] = 0$$ hold.

Then a linear regression like:

$$ x_n = \beta_0 + \beta_1 x_1 + ... + \beta_{n-1}x_{n-1} + v $$

has causal meaning. Or better all $\beta$s identifies $\theta$s and them have clear causal meaning (see note 3).

In Angrist and Pischke opinion, model like above are considered old. Them prefer to distinguish between causal variable(s) (usually only one) and control variables (read: Undergraduate Econometrics Instruction: Through Our Classes, Darkly - Angrist and Pischke 2017). If you select the right set of controls, you achieve a causal meaning for the causal parameter. In order to select the right controls, for Angrist and Pischke you have to avoid bad controls. The same idea is used even in structural approach, but in it is well formalized in the back-door criterion [reply in: Chen and Pearl (2013)]. For some details on this criterion read here: Causal effect by back-door and front-door adjustments

As conclusion. All above says that linear regression estimated with OLS, if properly used, can be enough for identification of causal effects. Then, in econometrics and elsewhere are presented other estimators also, like IV (Instrumental Variables estimators) and others, that have strong links with regression. Also them can help for identification of causal effects, indeed they were designed for this. However the story above hold yet. If the problems above are not solved, the same, or related, are shared in IV and/or other techniques.

Note 1: I noted from comments that you ask something like: "I have to define the directionality of causation?" Yes, you must. This is a key causal assumption and a key property of structural-causal equations. In experimental side, you have to be well aware about what is the treatment variable and what the outcome one.

Note 2:

So essentially, the point is whether a coefficient represents a deep parameter or not, something which can never ever be deduced from (that is, it is not assured alone by) exogeneity assumptions but only from theory. Is that a fair interpretation? The answer to the question would then be "trivial" (which is ok): it can when theory tells you so. Whether such parameter can be estimated consistently or not, that is an entirely different matter. Consistency does not imply causality. In that sense, exogeneity alone is never enough.

I fear that your question and answer come from misunderstandings. These come from conflation between causal and puerely statistical concepts. I’m not surprise about that because, unfortunately, this conflation is made in many econometrics books and it represent a tremendous mistake in econometrics literature.

As I said above and in comments, the most part of mistake come from ambiguous and/or erroneous definition of DGP (=true model). The ambiguous and/or erroneous definition of exogeneity, is a consequence. Ambiguous and/or erroneous conclusion about the question come from that. As I said in comments, the weak points of doubled and Dimitriy V. Masterov answers come from these problems.

I starting to face these problems years ago, and I started with the question: “Exogeneity imply causality? Or not? If yes, what form of exogeneity is needed?” I consulted at least a dozen of books (the more widespread were included) and many others presentations/articles about the points. There was many similarities among them (obvious) but to find two presentations that share precisely the same definitions/assumptions/conclusions was almost impossible.
From them, sometimes seemed that exogenety was enough for causality, sometimes not, sometimes depend from the form of exogeneity, sometimes nothing was said. As resume, even if something like exogeneity was used everywhere, the positions moved from “regression never imply causality” to “regression imply causality”. I feared that some counter circuits was there but … only when I encountered the article cited above, Chen and Pearl (2013), and Pearl literature more in general, I realized that my fear were well founded. I’m econometrics lover and felt disappointment when realized this fact. Read here for more about that: How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?

Now, exogeneity condition is something like $E[\epsilon|X]=0$ but is meaning depend crucially on $\epsilon$. What it is?

The worst position is that it represent something like “population regression error/residual” (DGP=population regression). If linearity is imposed also, this condition is useless. If not, this condition impose a linearity restriction on the regression, no more. No causal conclusions are permitted. Read here: Regression and the CEF

Another position, the most widespread yet, is that $\epsilon$ is something like “true error” but the ambiguity of DGP/true model is shared there too. Here there are the fog, in many case almost nothing is said … but the usual common ground is that it is a “statistical model” or simply a “model”. From that, exogeneity imply unbiasedness/consistency. No more. No causal conclusion, as you said, can be deduced. Then, causal conclusions come from “theory” (economic theory) as you and some books suggest. In this situation causal conclusions can arrive only at the end of the story, and them are founded on something like an, foggy, "expert judgement". No more. This seems me unsustainable position for econometric theory. This situation is inevitable if, as you (implicitly) said, exogeneity stay in statistical side … and economic theory (or other fields) in another.

We must to change perspective. Exogeneity is, also historically, a causal concept and, as I said above, must be a causal assumption and not just statistical one. Economic theory is expressed also in term of exogeneity; them go together. In different words, the assumptions that you looking for and that permit us causal conclusion for regression, cannot stay in regression itself. These assumption must stay outside, in a structural causal model. You need two objects, no just one. The structural causal model stand for theoretical-causal assumptions, exogeneity is among them and it is needed for identification. Regression stand for estimation (under other pure statistical assumption). Sometimes Econometric literature don't distinguish clearly between regression and true model neither, sometimes the the distinction is made but the role of true model (or DGP) is not clear. From here the conflation between causal and statistical assumptions come from; first of all an ambiguous role for exogeneity.

Exogeneity condition must be write on structural causal error. Formally, in Pearl language (formally we need it) the exogeneity condition can be written as:

$E[\epsilon |do(X)]=0$ that imply

$E[Y|do(X)]=E[Y|X]$ identifiability condition

in this sense exogeneity imply causality.

Read also here: Random Sampling: Weak and Strong Exogenity

Moreover in this article: TRYGVE HAAVELMO AND THE EMERGENCEOF CAUSAL CALCULUS – Pearl (2015). Some of the above points above are treated.

For some take away of causality in linear model read here: Linear Models: A Useful “Microscope” for Causal Analysis - Pearl (2013)

For an accessible presentation of Pearl literature read this book: JUDEA PEARL, MADELYN GLYMOUR, NICHOLAS P. JEWELL - CAUSAL INFERENCE IN STATISTICS: A PRIMER http://bayes.cs.ucla.edu/PRIMER/

Note 3: More precisely, is needed to say that $\theta$s surely represent the so called direct causal effects, but without additional assumptions is not possible to say if they represent the total causal effects too. Obviously if there are confusion about causality at all is not possible to address this second round distinction.

Answer 2

Here's a partial answer for when the underlying model is actually linear. Suppose that the true underlying model is $$Y = \alpha + \beta X + v.$$

I'm making no assumptions about $v$, though we have that $\beta$ is THE effect of $X$ on $Y$. A linear regression for $\beta$, which we will denote as $\tilde{\beta}$ is simply just a statistical relationship between $Y,X$ and we have $$\tilde{\beta} = \frac{cov(Y,X)}{var(X)}.$$

So one already 'cheap' answer (which you've mentioned already) is that a linear regression identifies a causal effect when the covariance corresponds to a causal effect and not just a statistical relationship. But let's try to do a bit better.

Focusing on the covariance, we have \begin{align*} cov(Y,X) & = cov(\alpha + \beta X + v, X)\\ & = \beta cov(X,X) + cov(v,X) \\ & = \beta var(X) + cov(v,X), \end{align*}

and so dividing by the variance of $X$, we get that $$ \tilde{\beta} = \beta + \frac{cov(v,X)}{var(X)}.$$

We need $cov(v,X) = 0$ for $\tilde{\beta} = \beta$. We know that $$cov(v,X) = E[vX] - E[v]E[X],$$ and we need that to be zero, which is true if and only if $E[vX] = E[v]E[X]$, which is true if and only if $v$ and $X$ are uncorrelated. A sufficient condition for this is mean independence similar to what you wrote: i.e. that $E[X|v] = E[X]$, so that $E[vX] = E[E[X|v]v] = E[X]E[v]$ (alternatively, you could let $v' = v - E[V]$ and require $E[v'|X]= 0$ so that $E[v'X] - E[v']E[X] = 0$ which is typically done in regression analysis). All the 'intuitive' language you quote from other posts are various ways to think concretely of such assumptions holding in application. Depending on the field, the terms and concepts and approaches will all differ, but they are all trying to get these kind of assumptions to hold.

Your comment also made me realize that it's important to really stress my assumption of "the true underlying model." I am defining $Y$ as I did. In many situations, we may not know what $Y$ is, and depending on the field, this is precisely why things get 'less rigorous' in some sense. Because you're no longer taking the model specification itself for granted. In some fields such as causal inference in statistics, you could think of these issues using DAGs or the idea of d-separation. In others, such as economics, you could start with a model of how individuals or firms behave and back out a true model through that approach, and so on.

As a final side note, note that in this case, the conditional mean independence assumption is stronger than what you need (you 'just' need the covariance to be zero). This stems from the fact that I specified a linear relationship, but it should be intuitive that imposing less structure on the model and departing from a linear regression will need stronger assumptions even closer to the notion of the error term being mean independent (or fully independent) of $X$ for you to get a causal effect (which also becomes trickier to define.. one approach could be to think of the partial of $Y$ wrt $X$).

Answer 3

The question is: under which assumptions of the DGP $\text{D}_X(\cdot)$ can we infer the regression (linear or not) represents a causal relationship?

It is well known that experimental data does allow for such interpretation. For what I can read elsewhere, it seems the condition required on the DGP is exogeneity:

$$ \text{E}(x_1, ... x_{n-1}|\epsilon) = 0$$

Regression by itselve can not be interpreted causaly. Indeed 'correlation ≠ causation'. You can see this with the correlated data in the image below. The image is symmetric (the pairs x,y follow a bivariate normal distribution) and regression does not tell whether Y is caused by X or vice versa.

The regression model can be interpreted as representing a causal relationship when the causality is explicitly part of the related data generating process. This is for instance the case when the experimenter performs an experiment where a variable is controlled/changed by the experimenter (and the rest is kept the same, or assumed to be the same), for instance, a 'treatment study', or in an observational study when we assume there is an 'instrumental variable'.

So it is explicit assumptions about causality in the DGP that make a regression relate to a causal relationship. And not situations where the data follows a certain relationship like $\text{E}(x_1, ... x_{n-1}|\epsilon) = 0$

About the condition $\text{E}(x_1, ... x_{n-1}|\epsilon) = 0$

I believe this should be $\text{E}(\epsilon | x_1, ... x_{n-1}) = 0$. The $\text{E}(x_1, ... x_{n-1}|\epsilon) = 0$ is already easily violated when all $x_i>0$, or if you use standardized data then it is violated when there's heteroscedasticity. Or maybe you switched the meaning of X|Y as conditional on X instead of conditional on Y?

The condition on it's own does not assure that your regression model is to be interpreted causally. In the above example (the image) you can use a regression $x_1 = x_2 +\epsilon$ or $x_2 = x_1 +\epsilon$ and for both cases the condition is true (can be assumed to be true), but that does not make it a causal relationship, at least one (possibly both) of the two regressions can not be interpreted causally.

It is the assumption of the linear model as causal that is the key factor in assuring you that the regression model can be interpreted causally. The condition is necessary when you wish to ensure that the estimate of a parameter in a linear model relates completely to the causal model and not partially to the noise and confounding variables as well. So yes, this condition is related to an interpretation of regression as a causal model, but this interpretation starts with an explicit assumption of a causal mechanism in the data generating process.

The condition is more related to ensuring that the causal effect (whose effect size is unknown) is properly estimated by an ordinary least squares regression (ensure there's no bias), but the condition is not related to a sufficient condition that turns a regression into a causal model.

Maybe the $\epsilon$ referring to some true error in a theoretical/mechanistic/ab-initio model (e.g. some specific random process that creates the noise term like dice rolls, particle counts in radiation, vibration of molecules, etc.)? Then the question might be a bit semantic. If you are defining an $\epsilon$ that is the true error in a linear model, then you are implicitly defining the statistical model as equal to the model that is the data generating process. Then it is not really the exogeneity condition that makes that the linear regression can be interpreted causally, but instead the implicit definition/interpretation of $\epsilon$.

Answer 4

Short answer:

There is no explicit way of proving causality. All claims of causality must be logically derived, i.e. through common sense (theory). Imagine having an operator (like correlation) which would return causality or non-causality between variables: you would be able to perfectly identify the sources and relations of anything in the universe (e.g. what/who would an interest rise have an impact on; which chemical would cure cancer etc.). Clearly, this is idealistic. All conclusions of causality are made through (smart) inferences from observations.

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Long answer:

The question of which variables cause another is a philosophical one, in the sense that it must be logically determined. For me, the clearest way to see this is through the 2 classical examples of a controlled vs non-controlled experiment. I will go through these while emphasizing how much is statistics and how much is common sense (logic).

1. Controlled experiment: fertilizer

Assume you have an agricultural field divided into parcels (squares). There are parcels on which crops $(y)$ grow with and without sunlight $(X_1)$, with and without good nutrients $(X_2)$. We wish to see if a certain fertilizer ($X_3$) has an impact or not on the crop yield $y$. Let the DGP be: $y_i = \beta_0+\beta_1 X_{1i}+\beta_2 X_{2i}+\beta_3 X_{3i} +\varepsilon_i$. Here $\varepsilon_i$ represents the inherent randomness of the process, i.e. the randomness that we would have in predicting crop yield, even if this true DGP were known.

Exogeneity: [skip if clear]

The strong exogeneity assumption $E[\varepsilon_i|\textbf{X}]=0$ that you mention is needed in order for the coefficients estimated by OLS $\hat\beta$ to be unbiased (not causal). If $E[\varepsilon_i|\textbf{X}]=c$ where $c$ is any constant, all $\hat{\beta_j}$ except for the intercept $\hat{\beta_0}$ are still unbiased. Since we are interested in $\beta_3$ this is sufficient. (Side note: other weaker assumptions such as weak exogeneity and orthogonality between $X$ and $\varepsilon$ are sufficient for unbiasedness.) Saying that $E[X|Z]=c$ for any 2 random variables $X$ and $Z$ means that $X$ is not systematically dependent in the mean on $Z$, i.e. if I take the mean ($\to\infty$) of $X$, for any pair of $(X,Z)$ I will get (approx.) the same value each time, so knowing $Z$ does not help at all in predicting the mean of $X$ (e.g. $E[X|Z=10]=E[X|Z=10000]=E[X|Z=-5]=E[X]=c$)

Why is this interesting? Remember, we want to know if the fertilizer $X_3$ has an impact or not ($\beta_3=0?$) on the crop yield $y$. By spraying fertilizer on random parcels, we implicitly "force" exogeneity of $X_3$ compared to all other regressors. How? Well, if we randomly spray fertilizer on a parcel, no matter if it has sunlight or not, if it has good nutrients or not and if we then take the mean value of fertilizer for sunny parcels, it will be the same as the mean value for non-sunny parcels. Same with nutrient-rich parcels. E.g: the results of the table below hold approx. for large numbers. It makes sense after all that, if $X_3$ is independent of $X_1$, its mean should not change (significantly) as $X_1$ changes.

So, in other words $X_3$ is exogenous wrt $X_1,X_2$, i.e. $E[X_3|X_1,X_2]=c$. This means that effectively, if we want to estimate $\beta_3$ unbiasedly, we don't need $X_1,X_2$. Hence these two variables (sun, nutrients) can be treated as randomness and incorporated into the noise term, giving the regression: $y_i = \beta_0 + \beta_3 X_{3i} + \epsilon_i$, where $\epsilon_i = \beta_1 X_{1i} + \beta_2 X_{2i} + \varepsilon_i$. Hence, the noise term can also be interpreted as a collection of all other variables that influence the response $y$, but not in a systematic fashion in the mean. (Note that $\hat\beta_0$ is biased; further note that exogeneity is weaker than independence, since the variables could be related in a higher moment instead of the mean, such as the variance, but exogeneity would still hold, see heteroskedasticity).

Causality:

Now where does causality come into play? So far we have only shown that randomly distributing fertilizer on better or worse parcels lets us look at crop yield and fertilizer alone, without taking into account the other variables (sun, nutrients), i.e. "forcing" exogeneity of fertilizer and thus all other variables into the noise term. Causality itself was and will not be proven. However, if $\hat\beta_3$ turns out to be significant, we can logically conclude that, since the randomization of fertilizer effectively "de-relates" it from all other variables (in the mean), it must have an impact on crop yield, since all other variables have no systematic impact in this setting.

In other words: 1) we used exogeneity to statistically prove that this is the condition we need for unbiased estimators (for OLS); 2) we used randomization to get this exogeneity and get rid of other uninteresting variables; 3) we logically concluded that, since there is a positive relation, it must be a causal one.

Notice that 3) is just a common sense conclusion, no statistics involved as in 1) or 2). It could theoretically be wrong, since e.g. it could have been that the fertilizer was actually a 'placebo' ($\beta_3=0$) but was distributed only on the sunny and nutrient-rich parcels by pure chance. Then the regression would wrongly show a significant coefficient because the fertilizer would get all the credit from the good parcels, when in fact it does nothing. However, with a large number of parcels this is so unlikely that it is very reasonable to conclude causality.

2. Uncontrolled experiment: wage and education

[I will eventually (?) return with an edit to continue here later; topics to be addressed OVB,Granger-causality and instantaneous causality in VAR processes]

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This question is precisely the reason why I started learning statistics/data science - shrinking the real world into a model. Truth/ common sense/ logic are the essence. Great question.

Answer 5

Let the true DGP (to be defined below) be

$$y=\mathbf{X}\beta + \mathbf{z}\alpha + \mathbf{v},$$

where $\mathbf{X}$ and $\mathbf{z}$ are regressors, and $\mathbf{z}$ is a $n \times 1$ for simplicity (you can think of it as an index of many variables if that feels restrictive). $\mathbf{v}$ is uncorrelated with $\mathbf{X}$ and $\mathbf{z}$.

If $z$ is left out of the OLS model,

$$\hat \beta_{OLS} = \beta + (N^{-1}\mathbf{X}'\mathbf{X})^{-1}(N^{-1}\mathbf{X}'\mathbf{z})\alpha+(N^{-1}\mathbf{X}'\mathbf{X})^{-1}(N^{-1}\mathbf{X}'\mathbf{v}).$$

Under the no-correlation assumption, the third term has a $\mathbf{plim}$ of zero, but $$\mathbf{plim}\hat \beta_{OLS}=\beta + \mathbf{plim} \left[ (N^{-1}\mathbf{X}'\mathbf{X})^{-1}(N^{-1}\mathbf{X}'\mathbf{z}) \right] \alpha.$$

If $\alpha$ is zero or $\mathbf{plim} \left[ (N^{-1}\mathbf{X}'\mathbf{X})^{-1}(N^{-1}\mathbf{X}'\mathbf{z}) \right] = 0$, then $\beta$ can be interpreted causally. In general, the inconsistency can be positive or negative.

So you need to get the functional form right, and include all variables that matter and are correlated with the regressors of interest.

There is another nice example here.

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I think this might be a good example to give some intuition about when parameters can have a causal interpretation. This lays bare what it means to have a true DGP or the have the functional form right.

Let's say we have a SEM/DGP like this:

$$y_1 = \gamma_1 + \beta_1 y_2 + u_1,\quad 0<\beta_1 <1, \quad y_2=y_1+z_1$$

Here we have two endogenous variables (the $y$s), a single exogenous variable $z_1$, a random unobserved disturbance $u_1$, a stochastic relationship linking the two $y$s, and a definitional identity linking the three variables. We also have an inequality constraint to avoid dividing by zero below. The variation in $z_1$ is exogenous, so it is like a casual intervention that "wiggles" stuff around. This wriggling has a direct effect on $y_2$, but there is also an indirect one through the first equation.

Suppose a smart student, who has been paying attention to the lessons on simultaneity, writes down a reduced form model for $y_1$ and $y_2$ in terms of $z_1$: $$\begin{align} y_1 =& \frac{\gamma_1}{1-\beta_1} + \frac{\beta_1}{1-\beta_1} z_1 + \frac{u_1}{1-\beta_1} \\ =& E[y_1 \vert z_1] + v_1 \\ y_2 =& \frac{\gamma_1}{1-\beta_1} + \frac{1}{1-\beta_1} z_1 + \frac{u_1}{1-\beta_1} \\ =& E[y_2 \vert z_1] + v_1, \end{align}$$

where $v_1 = \frac{u_1}{1- \beta_1}$. The two coefficients on $z_1$ have a causal interpretation. Any external change in $z_1$ will cause the $y$s to change by those amounts. But in the SEM/DGP, the values of $y$s also respond to $u_1$. In order to separate the two channels, we require $z_1$ and $u_1$ to be independent in order not to confound the two sources. That is the condition under which the causal effects of $z$ are identified. But this is probably not what we care about here.

In the SEM/DGP,

$$\frac{\partial y_1}{\partial y_2} = \beta_1 =\frac{\partial y_1}{\partial z_1} \div \frac{\partial y_2}{\partial z_1} =\frac{ \frac{\beta_1}{1-\beta_1}}{ \frac{1}{1-\beta_1}}.$$

We know that we can recover $\beta_1$ from the two reduced form coefficients (assuming independence of $z_1$ and $u_1$).

But what does it mean for $\beta_1$ to be the causal effect of $y_2$ on $y_1$ when they are jointly determined? All the changes come from $z_1$ and $u_1$ (as the reduced form equation makes clear), and $y_2$ is only an intermediate cause of $y_1.$ So the first structural equation gives us "snapshot" impact, but the reduced form equations give us an equilibrium impact after allowing the endogenous variables to "settle."

Given a system of linear equations, there are formal conditions for when parameters like $\beta_1$ are recoverable. They can be a DAG or a system of equations. But this is all to say that whether something is "causal" cannot be recovered from a single linear equation and some assumptions about exogeneity. There is always some model lurking in the background, even if it is not acknowledged as such. That is what it means to get the DGP "right", and that is a crucial ingredient.

Answer 6

Regression is just a series of statistical technique to strengthen causal inferences between two variables of interest by controlling for alternate causal explanations. Even a perfectly linear relationship (r2=1) is meaningless without first establishing the theoretical basis for causality. Classic example being the correlation between icecream consumption and pool drownings--neither causes the other by both are caused by summer weather.

The point of experiments is to determine causality, which typically requires establishing that: 1) one thing happened before the other, 2) that the putative cause had some explanation mechanism for affecting the outcome, and 3) that there are no competing explanations or alternate causes. Also helps if the relationship is reliable--that the lights go on every time you hit the switch. Experiments are designed to establish these relationships, by controlling conditions to establish chronological sequence and control for possible alternate causes.

Pearl (Pearl, J. (2009). Causality. Cambridge university press) is a good read, but beyond that lies a (fascinating) philosophical rat-hole regarding causation and explanation.