73

JohnRos's answer is very good. In plain English, endogeneity means you got the causation wrong. That the model you wrote down and estimated does not properly capture the way causation works in the real world. When you write: \begin{equation} Y_i=\beta_0+\beta_1X_i+\epsilon_i \end{equation} you can think of this equation in a number of ways. You could ...


43

[The following perhaps seems a little technical because of the use of equations but it builds mainly on the arrow charts to provide the intuition which only requires very basic understanding of OLS - so don't be repulsed.] Suppose you want to estimate the causal effect of $x_i$ on $y_i$ given by the estimated coefficient for $\beta$, but for some reason ...


24

Even though this question already has an accepted answer, I think I can still contribute to this. The Koenker (2005) book will really not get you far because developments in IV quantile regression started to pick up around that time. The early IV quantile regression techniques include the causal chain framework by Chesher (2003), which was further developed ...


24

To answer your first question, you are correct that sample selection is a specific form of endogeneity (See Antonakis et al. 2010 for a good basic review of endogeneity and common remedies), however you are not correct in saying that the likelihood of being treated is the endogenous variable, as it is the treatment variable itself ("non-random treatment ...


20

Let me use an example: Say you want to quantify the (causal) effect of education on income. You take education years and income data and regress one against the other. Did you recover what you wanted? Probably not! This is because the income is also caused by things other than education, but which are correlated to education. Let's call them "skill": We can ...


20

What was proposed to you is sometimes referred to as a forbidden regression and in general you will not consistently estimate the relationship of interest. Forbidden regressions produce consistent estimates only under very restrictive assumptions which rarely hold in practice (see for instance Wooldridge (2010) "Econometric Analysis of Cross Section an Panel ...


17

When you want to estimate a simple model like $$Y_i = \alpha + \beta X_i + \epsilon_i$$ and instead of the true $Y_i$ you only observe it with some error $\widetilde{Y}_i = Y_i + \nu_i$ which is such that it is uncorrelated with $X$ and $\epsilon$, if you regress $$\widetilde{Y}_i = \alpha + \beta X_i + \epsilon_i$$ your estimated $\beta$ is $$ \begin{align} ...


14

Actually the issue of selection bias is the initial motivation for using instruments. The question here is whether the randomized draft lottery gets around this issue. You are perfectly right in asking: what are the limitations of this instrument? If indeed rich kids had better chances to avoid the draft, then the negative effect of service on later earnings ...


14

This presentation provides a decent overview with worked examples. Weak instruments means that the instrument has a low correlation with the endogenous explanatory variable. This could result in a larger variance in the coefficient, and severe finite-sample bias. "The cure can be worse than the disease" (Bound, Jaeger, Baker, 1993/1995). See here for more ...


13

Your case is less problematic than the other way round. The expectations and linear projections operators go through a linear first stage (e.g. OLS) but not not through non-linear ones like probit or logit. Therefore it's not a problem if you first regress your continous endogenous variable $X$ on your instrument(s) $Z$, $$X_i = a + Z'_i\pi + \eta_i$$ and ...


13

The relevant formula is $$\mathbb{Var}(\beta_{IV})=\sigma^2 \cdot (X'P_{Z}X)^{-1},$$ where $$\sigma^2 = (y-X\beta_{IV})'(y-X\beta_{IV})/(n-k_{SS}),$$ and $$P_Z = Z \, (Z'Z)^{-1} Z',$$ and $k_{SS}$ is the number of regressors in the second stage. Some people will just use $n$ or $n-k_{FS}$ since the choice does not matter asymptotically. Kit Baum has code in ...


12

This is a question which appears sometimes in the Statalist. Let me write $x_{1}$ and $x_{2}$ instead of $x$ and $z$ (in the literature $z$ is usually reserved for instruments rather than endogenous variables) and let $x_3 = x_1 \cdot x_2$. Your model then becomes: $$y = ax_1 + bx_2 + cx_3 + e$$ which has three endogenous variables. Assuming that you have ...


12

I would take a gander at the 7 Chernozhukov and Hansen IVQR papers. The 2005 paper is often cited. They also provide links to data and code in MATLAB, OX and Stata. Another frequently cited paper in this literature is Abadie, Angrist, and Imbens (2002). Frolich and Melly (2010) and Kwak (2010) are also worth checking out, especially if you use Stata. Both ...


12

The point of instrumental variable regression is to provide an unbiased estimate of the causal effect of exposure $X$ on outcome $O$, when there is some unmeasured—possibly unmeasureable—variable $U$ confounding the relationship between $X$ and $O$. Here's a DAG of the simplest circumstance under which one would use instrumental variables estimation ($X$, $U$...


11

Yes. For example in the DAG below, the instrumental variable $Z$ causes $X$, while the effect of $X$ on $O$ is confounded by unmeasured variable $U$. The instrumental variable model for this DAG would be to estimate the causal effect of $X$ on $O$ using $E(O|\hat{X})$, where $\hat{X} = E(X|Z)$. This estimate is an unbiased causal estimate if: $Z$ must be ...


10

I am not a fan of the Angrist and Pischke book, but they do have a flair for phrasing, and as they say, fuzzy RD is IV (Sec. 6.2). This fact is obscured by the fact that the instrument is essentially a nonlinear transformation (step function) of one of the included exogenous variables, which by virtue of the conditional exogeneity assumption, is a valid ...


10

It is possible to implement this yourself, as long as you are able to keep the notation straight. I will work through this systematically by first laying out the notation, and then translating that notation transparently into R code. Due to the length of this note, bugs might exist. Additionally, the reference to what you have described is Baum, Schaffer ...


10

There has been a similar question regarding a probit first stage and an OLS second stage. In the answer I have provided a link to notes that contain a formal proof of the inconsistency of this regression which is formally known as "forbidden regression", as it was termed by Jerry Hausman. The main reason for the inconsistency of the probit first stage/OLS ...


10

There are two criteria for good instruments: The instrument $z$ is correlated with the endogenous variable $x$ (relevance). The instrument $z$ affects dependent variable $y$ only through $x$. In other words, $z$ itself does not cause $y$. This is the exclusion restriction. You can check 1 (relevance) statistically, but must make good arguments to support 2 ...


10

Yes, they surely can. As a matter of fact, the SCM/DAG literature has been working on generalized notions of instrumental variables, you might want to check Brito and Pearl, or Chen, Kumor and Bareinboim. The basic IV dag is usually represented as: Where $U$ is unobserved and $Z$ is an instrument for the effect of $X$ on $Y$. Although this is the graph ...


10

While drawing DAGs...what are instrumental and adjusted variables? An instrumental variable is an observed variable that is often used to help obtain an unbiased estimate for a causal effect that is confounded by another variable that is usually unobserved. The classical situation can be depicted in the following DAG: Here, X is our main exposure, and the ...


8

This follows up on gung's comment. Overall average treatment effect is not the point. Suppose you have $1000$ new diabetes cases where the subject is between the ages of $5$ and $15$, and $1000$ new diabetes patients over $30$. You want to assign half to treatment. Why not flip a coin, and on heads, treat all of the young patients, and on tails, treat all ...


8

User25901 is looking for a straight-forward simple, real-world explanation what the terms exogenous and endogenous mean. Responding with arcane examples or mathematical definitions does not really answer the question that was asked. How do I get a gut understanding of these two terms? Here's what I came up with: Exo - external, outside Endo - internal, ...


8

I think you tried to prove consistency of the IV estimator but started the proof with the OLS estimator instead. If you write: $$\newcommand{\Cov}{\operatorname{Cov}}\beta_{1}^{\mathrm{IV}} = \frac{\Cov(Z,Y)}{\Cov(Z,X)} = \frac{\Cov(Z,\beta_{0}+\beta_{1}X+U)}{\Cov(Z,X)}$$ you should get your desired result $$\beta_{1}\frac{\Cov(Z,X)}{\Cov(Z,X)} + \frac{\Cov(...


8

I think there are at least four main sources of instruments: Theory combined with clever data collection (Example: distance from job training center varies the cost of participation in training, and we're interested in the effect of training on wages) Exogenous variation in policies or program implementation, over time or across space (Example: Vietnam ...


8

If you have a weak instrument then the bias of the IV estimator can be large and in some cases it can even be bigger than the bias of the OLS estimator. With their tabulated values Stock and Yogo first fix the largest relative bias of the two stage least squares estimator (2SLS) relative to OLS that is acceptable. In this sense the test answers the question: ...


8

So the vast majority of my field (though not the part I work in most) is concerned with just this - the fitting of GLM-type models to observational data. For the most part, instrumental variables are a rarity, either due to a lack of familiarity with the technique or, as importantly, the lack of a good instrument. To address your questions in order: The ...


8

In contrast to the view from the epidemiologist's side shown by Fomite, instrumental variables are an essential toolkit in economics that is taught fairly early on. The reason for this is that there is a huge focus on trying to answer causal questions in economic research nowadays which goes to an extend where mere correlations are even regarded as ...


8

Following Hernán and Robins' Causal Inference, Chapter 16: Instrumental variable estimation, instrumental variables have four assumptions/requirements: $Z$ must be associated with $X$. $Z$ must causally affect $Y$ only through $X$ There must not be any prior causes of both $Y$ and $Z$. The effect of $X$ on $Y$ must be homogeneous. This assumption/...


7

While Michael Chernick gave a good answer, I do not think that the people who are involved in treatment effect estimation think in terms of finite populations and randomization-based inference. Economists (Angrist and Imbens are well-known econometricians) usually don't; if the OP comes from the same tradition, that is the central issue of this question. ...


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