# Parameter identification v. causal identification

As others, it seems (Identification of parameters problem), I get confused about the use of the word "identification" in econometrics. It seems some people talk about "identification" in the sense described in Identification of parameters problem, i.e., a parameter vector $$\theta_1$$ is identified provided :

"there does not exist a different parameter vector $$\theta_1$$ that would induce the same data generating process, given our model specification $$M$$".

If I understand this correctly, it means that in a simple linear model $$y = x'\beta + \epsilon$$, for example, $$\beta$$ is identified under identification assumption $$E[x\epsilon] = 0$$ as well as under the stronger $$E[\epsilon|x] = 0$$

(I guess I should add $$E(\epsilon^2)<\infty$$ and the invertibility of $$E(xx')$$ as auxiliary identification assumptions here? Maybe I should also note that it's not obvious stronger assumptions achieve identification when weaker ones do, as stronger assumptions may lead to "inexistence" of the desired parameters when weaker ones do not.)

At the same time, many people seem to talk about an "identification strategy" as a way to single out causal effects. For example, I feel like I often hear people ask "what is your identification strategy?" as a shortcut for asking "what is your strategy for convincing people that the effect you measured is causal?

In that sense, if I again understand things correctly, $$E[\epsilon|x] = 0$$ would be enough to (causally) identify $$\beta$$, but $$E[x\epsilon] = 0$$ would not be sufficient anymore.

My questions are:

• Does that seem like a fair description of econometricians' use of the term "identification"?
• Is there really such a ubiquitous ambiguity in people's use of the word "identification"?
• Or is there no ambiguity, one commonly accepted use of the term "identification", and I am the one who is missing something?
• You may find this 2019 Lewbel JEL paper useful. Commented Mar 12, 2020 at 21:08

Let's take the simple linear model and discuss three different definitions of the parameter of interest in the linear model, with very different identification settings. All three scenarios are very common in empirical work in economics and it's typically only clear from context of them characterise the analysis.

Here's the linear model.

$$y_i = x_i^T \beta + \epsilon_i$$

It's clear what $$y_i$$ and $$x_i$$ are. But what are $$\beta$$ and $$\epsilon$$ supposed to be? The answer determines how we should think about identification.

1.) Prediction

We could simply be interested in predicting the observable $$y_i$$ when we see the observable $$x_i$$, using the best linear function of $$x_i$$ to do so. Then the target parameter should be defined as $$\beta := \arg \min_{b \in R^k} E[ (y_i - x_i^T b)^2]$$ and the error as $$\epsilon_i := y_i - x_i^T\beta$$. The expectation $$E[\cdot]$$ is with respect to the distribution of observables. This distribution is identified if we observe $$y_i$$ and $$x_i$$. The solution for $$\beta$$ is the well known formula $$E[x_ix_i^T]^{-1} E[x_iy_i]$$. Note that $$E[\epsilon_i x_i^T]=0$$ holds by construction. If we find the best linear predictor, then the prediction error doesn't have a linear relationship with the predictor. If it did, we would have failed at finding the best linear predictor! What about $$E[\epsilon_i |x_i]$$? That will be zero only if $$E[y_i|x_i] = x_i ^T \beta$$, in which case $$x_i ^T \beta$$ isn't just the best linear predictor but the best predictor. The only failure of identification for this definition of $$\beta$$ would be that $$E[x_ix_i^T]^{-1}$$ might not exist, perhaps because some of the $$x_i$$ are linear functions of others. Then $$\beta$$ is not identified, there are different $$\beta$$ that all do the job of forming the best linear predictor. But that wouldn't worry us because we just want to predict and they are all equally good at that.

2.) Prediction with the underlying, not the measured covariate

Often, $$x_i$$ is a proxy for something we are interested in. If $$x_i$$ is the body mass index we might be interested in how it predicts an outcome but we only get to work with a self reported estimate of it, $$w_i$$. If we are content with finding out the predictive relationship between that and an outcome we are back in the first scenario. If not, we face an identification problem. We are interested in $$\beta$$ as defined in the previous scenario, but we do not observe $$x_i$$. We observe only $$w_i = x_i + u_i$$. Now $$\beta$$ is not identified. We have $$y_i = (w_i - u_it)^T \beta+ \epsilon_i$$ or $$y_i = w_i^T \beta+ \epsilon_i - u_i \beta$$. And now $$E[(\epsilon_i - u_i \beta) w_i^T]$$ is no longer $$0$$ by construction and OLS won't estimate $$\beta$$ The most common identification strategy for this problem is to find an instrumental variable: Something that correlates with $$x_i$$, but not with $$u_i$$, and not with $$\epsilon_i$$ either. Maybe a friend's anonymous estimate of that person's body mass index. Note how nothing so far has been about causality, just about prediction, but there is an identification problem anyway, because an important variable is unobserved.

3.) Causal inference

Probably the most common scenario: We actually want to know how $$x_i$$ affects $$y_i$$ not just how it predicts it. How should we define that? We can use structural equations to do it, which express how a variable really, is determined: $$y = f(x, \epsilon)$$. This unknown function tells us what the value $$y$$ would be if we set $$x$$ for any possible value, just like proper scientific models do. It also depends on $$\epsilon$$ which is now the effect of other causes of $$y$$ and this may well be correlated with $$x$$ in the data we see. Let's suppose $$f$$ is linear: $$y_i = x_i^T \beta + \epsilon_i$$ as before. This $$\beta$$ is the one from the first scenario only if $$\epsilon_i$$ is uncorrelated with $$x_i$$. In most empirical analyses, one can think of reasons why $$x_i$$ and $$\epsilon_i$$ might be correlated, and so an identification strategy is needed.

To clearly distinguish this scenario from the first one, different notation is sometimes used: The potential outcomes notation, or causal graphs, or structural models. The best way of thinking about causality in economics is an ongoing area of research and there is no consensus yet. As it stands most researchers will will simply use the standard regression equation, expected values etc and you need to infer from the context whether the goal is prediction, prediction with an imperfectly measured covariate, or causal inference.

A very good textbook covering this material is Mostly Harmless Econometrics.

I think that the way you explained your two different takes on "identification" was clear but I'm not familiar with your second take.

My understanding ( I wouldn't even call myself a novice in this. I'd call myself super novice ) is that the notion you use in your second description, "identification strategy", is the part where you try to figure out which variables should be in the model and which should not be. OTOH, in the identification of causal effects ( again, this is just my experience ), you try to figure out whether the variables you chose in the model are really CAUSING the response or rather just correlated with the response.

In econometrics, Clive Granger came up with "Granger Causality" as a method for addressing whether one variable is truly "causing" the other but his definition is only way of defining it. At the same time, Judea Pearl says that the causal relationship work done in econometrics and statistics is mostly (if not completely ? ) junky.

I recommend trying to read Judea Pearl's work. He's kind of world-renown as far as his work on the notion of causality. Unfortunately, I tried to read one of his papers a long time ago and it seemed that it might take the rest of my life to possibly understand what it was saying so I stopped. I hope this helped some.

• Granger causality is a very particular type of causality, which is why it is usually prefaced with the G-adjective. Many, if not most, would not even consider it as such. A bit more on that here. I think Granger did his work at a time when these ideas were not as well developed as they are now, but the terminology stuck. Commented Mar 13, 2020 at 4:04
• @Dimitry V. Masterov Thanks for your comment. And I agree, That's why I mentioned Judea Pearl. Granger was no slouch in any sense of the word ( nobel in 2003 with Engle ) but when it comes to causaility, I'm pretty certain that Pearl is the person these days. Commented Mar 13, 2020 at 4:37
• I am not sure the latter is the case in economics. I think there is some disagreement about value of Pearl's approach among economists, though also some recent dialogue about the merits of the potential outcomes approach versus the do-calculus for research and for teaching. I think Imbens' working paper is good start on the difference. Commented Mar 13, 2020 at 4:47
• good to know. thanks. Commented Mar 13, 2020 at 14:51

https://autobox.com/pdfs/PREFERRED.pdf presents single equation- multivariable procedures to identify a useful model. http://www.autobox.com/pdfs/WHY-WE-FILTER.ppt also sheds some light on model identification. Parameter identification can lead to revising tentative model identification via necessity and sufficiency checks culminating in a SARMAX MODEL https://autobox.com/pdfs/SARMAX.pdf incorporating the memory of Y and any needed structure from both known(user-suggested X's ) and latent X's ( the I series … Intervention Detected ) .

If your problem is multi-equation the VARIMA methods are available and if no known candidate X's are available then SARIMA is suggested https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf .

• It might be helpful if the down-voter suggested a viable alternative or a pointed critique so that we could all become "smarter" . Commented Mar 13, 2020 at 8:27
• I suspect people object to posts like this one that consist solely of links to proprietary software and do not even answer the question.
– whuber
Commented Apr 25, 2022 at 2:02
• AS no viable alternative was suggested , I think not . Commented Apr 25, 2022 at 18:50