# 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 interpreted, not even as a simple correlation (i.e., the magnitude of the effect could be wrong as could be the sign).

I was puzzled by the part that I have put in bold. How can the coefficient not be interpreted as a correlation? A correlation simply gives the relationship between two variables, without accounting for confounding factors. This is exactly what could be obtained with the coefficient of x that they mention (even though the coefficient would not be technically a proper correlation coefficient if x and y are not standardized first, of course)

The article that you cited exist even in published version: On making causal claims: A review and recommendations - J. Antonakis et al. - The Leadership Quarterly 21 (2010) 1086–1120. However the phrase that you cited remain unmodified (maybe all article). Let me rewrite it:

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 interpreted, not even as a simple correlation (i.e., the magnitude of the effect could be wrong as could be the sign). (pag 1088)

I disagree with this affirmation that seems me wrong.

It seems me that correlation is a “free concept”. We can calculate the linear correlation coefficient between almost any pair of random variables and the number that we obtain remain a respectable misure of linear dependence. “Almost” because we need of one random variable with proper joint probability distribution that admit finite second order moments. Not else. For regression coefficients counterpart, the same is valid. In fact regression coefficients maintain the same information of correlation coefficients (total or partial).

Then, in my opinion your surprise in reading the above sentence is comprehensible.

Now we have to note that the article is focused on the meaning of causality and related problems. In fact the misinterpretation of coefficients and their possible causal meaning is common in regression analysis. At the other side the correlational meaning always remain; endogeneity argument is irrelevant about that.

Now we have to note that later the authors affirm:

If x is endogenous the coefficient of x simply has no meaning. The true coefficient could be higher, lower, or even of a different sign. (pag 1088)

In my opinion this sentence reveal the origin of the confusion.

At first, the authors themselves affirm, rightly, that exogeneity is a causal concept (pag 1088). Now, endogeneity is the opposite of exogeneity and, then, it has causal meaning also (read here What do "endogeneity" and "exogeneity" mean substantively?). Therefore endogeneity problem preclude causal meaning of regression coefficients but not correlational one.

Then, it seems me that the only possibility to find some sense at affirmation of authors stay in the word “true”. The concept of “true coefficient” come from that of “true model”; concept that authors refers on. More precisely, in the phrase cited, the authors talk about the omitted variables problem that come from the fact of to estimate a, sometimes called, short regression. Formally, and very briefly, we have a true model like that:

$$y = \beta_0 + \beta_1 x + \beta_2 z + u$$

But, omitting $$z$$, we estimate a (short) regression like

$$y = \theta_0 + \theta_1 x + \epsilon$$

So, in general, we achieve parameters ($$\theta_0$$ and $$\theta_1$$) that are biased (respect to $$\beta_0$$ and $$\beta_1$$). Authors consider them with neither causal nor correlational meaning because them are simply “not true” (= no meaning).

But so, what is the difference between causal and correlational meaning? Authors conflate them? Considering the fact that authors use the paradigm of true model in causal sense, seem me that the answer is yes. In fact parameters of short regression ($$\theta_0$$ and $$\theta_1$$) lose causal meaning (biased) but maintain genuine correlational one. More precisely, taking aside any standardization problem, $$\theta_1$$ have the same meaning of $$\rho(x,y)$$.

At the other side It seem me that only another possible explanation exist. It stay into a sort of vague and empty concept of true model. Unfortunately great problems come from this concept: What meaning “true” and/or “untrue” parameters and/or correlations have?

I stay focused on the meaning of econometrics models since several years and I disagree with this delusional paradigm of true model used yet in several textbooks. In fact It seem me that, at least in econometrics, coefficients can be “true” only in causal sense. Let me pose some point about that:

If the so called true model have clear causal meaning no problems like above occur. In my opinion a notable example is: To explain or to predict - Shmueli (2010).

However the usual concept of true model can remain problematic also in causal inference. For some idea read here: Regression and causality in econometrics (see also the comments on accepted answer)

Maybe the concept of true model remain useful if it stay for structural model with clear causal meaning. Read also here: Difference Between Simultaneous Equation Model and Structural Equation Model

Unfortunately sometimes the concept of true model is used in ambiguous manner and sometimes else without any causal argument. In this setting is possible to achieve concepts like: true parameter, biased parameter, ecc; without clear causal meaning. Sometimes clearly without. At the same way is possible to have concept like true correlation. This story can hold in pure abstract statistical theory but the problem is that is not clear what meaning so kind of true parameters and/or correlations have in empirical work; surely this meaning is not substantive. A clear example is ARMA models. See here: Structural equation and causal model in economics

Moreover note that, as pointed out in Shmueli 2010, if our goal is only forecasting, as in ARMA models case, the empirical counterpart of true model is not necessarily the best (bias-variance tradeoff argument). In practice we can build effective forecasting model without refers on any sort of true model. In causal analysis this is not the case.

EDIT: From endogeneity the estimated parameters become biased/incorrect, but biased in comparison to what? To $$\beta$$s, but what they are? They are exactly the structural-causal parameters of interest. In the perspective that I explain above the word “true” stand exactly, and only, for structural-causal. There are no others meaning for the word “true”. Concept like “true but not causal” are obscure for me, it can have abstract/theoretical justification but in practical work this meaning is, at best, useless. Worse, in the context above (causal claims) concept like those can produce only misunderstanding. Worse again, It seem me that concept like “true correlation” and/or “true regression” are completely delusional concepts. What they should be “untrue/incorrect/biased regression/correlations”. We can find concept like these in some Econometrics textbooks, I know. By now I convinced myself that concept like these, and related that move around them, can produce only ambiguity and confusion. Correlations and any others probabilistic/distributional quantities are completely “free concept” that we can study about the data we want.

• I agree in general. Perhaps the authors meant that one cannot take the correlation as a guide to the true strength or nature of the relation between two variables. Also, they authors might have been warning against making forecasts or predictions on the strength of a correlation, or directing your model-building on the basis of such a correlation. Or they might have said, "Such correlations are not useful for the purposes to which correlations are typically put." Commented Apr 29, 2020 at 21:22
• All of you seem like you definitely know what you're talking about but here's my take. If $x$ is endogenous, then it's not just the causaility that is an issue. The actual estimate of the $x$ coefficient will be incorrect. The coefficient being incorrect implies that the implied correlation between $x$ and the response is also incorrect. So, I'm pretty sure that it's a matter of incorrect versus correct rather than an issue of causailty. The reason for the incorrectness is that, if $x$ is endogenous, then it is correlated with the error term and all the standard OLS theory goes out the window. Commented Apr 29, 2020 at 22:28
• First of all let me precise that my explanation above address an specific issue. The article referred on can remain interesting and useful in several respect. Now, your comments are interesting and in some extent related also. Thanks for these. Commented Apr 30, 2020 at 8:20
• @Ed Rigdon you write: “Or they might have said, "Such correlations are not useful for the purposes to which correlations are typically put."”. I completely agree. Infact this phrase can appear similar while it is completely different for the other questioned above. My explanation make efforts to disentangle causal meaning from correlational one and, eventually, others. Then, you write: “… correlation as a guide to the true strength or nature of the relation between two variables” but what is this “true relation”? Stands for causal relation? I suppose so, otherwise what it is? Commented Apr 30, 2020 at 8:21
• @Ed Rigdon; About: “By "true" relation I meant the relation in the DGP, which I suppose is what the causal analysis people mean by causal relation”. This is exactly the meaning that I adopted in the explanation. About forecasting: you understand my perspective and I understand your. This story is too large for comments and this thread is not the right place. I add something here (stats.stackexchange.com/questions/202278/…) for render complete my perspective. Commented May 4, 2020 at 13:38