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Let us consider Situation 1. Let us assume that $\rho$ is observed. If it does not work when $\rho$ is observed, there is no reason why it (using a proxy of $\rho$ as instrument) should work when $\rho$ is not observed. $\rho$ is endogenous so we can't just include it as a regressor in an equation. Thus, let us consider IV estimation. Obvious instruments for ...

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Endogenous variables are correlated with the error terms and z is correlated with endogenous variable. Doesn't this imply that z is correlated with error terms? No it doesn't. For mean-centered variables for simplicity, we have for linear models, ENDOGENEITY: $E(xu) \neq 0$ RELEVANCE : $E(xz) \neq 0$ The above conditions do not necessarily imply ...

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Endogeneity can arise from several reasons and in each case the explanation will be slightly different. I won't show full review of all possible reasons but just two important examples: Simultaneity For example, following Verbeek a guide to modern econometrics pp 146, suppose that true model is given by system of equations: $$y = \beta_1 + \beta_2 x_{2t} + \... 3 Broadly speaking endogeneity means that something some variable is determined within model as opposed to outside it. More narrowly in econometrics it means correlated with the error term (Wooldridge, (2009). Introductory Econometrics: 4ed. p. 88). These meanings are related since one will often lead to the another. In this paper they actually use it in both ... 3 Assume the simplest linear regression model$$y = bx + u$$The OLS etimator for b is$$\hat b_{OLS} = \frac {\sum x_iy_i}{\sum x_i^2} = b + \frac {\sum x_ie_i}{\sum x_i^2}$$Whatever the true b is (it is zero in your case, as another answer pointed out) , the fact is that$$\text{plim} \frac {n^{-1}\sum x_ie_i}{n^{-1}\sum x_i^2} \neq 0 $$and so ... 3 What you are referring to is two-stage least squares. This is an instrumental variable commonly applied to correct for endogeniety and selection bias. It is a pretty hot topic in economics at the moment and, when applied correctly, can be very useful and will remove the selection bias. There are a few conditions and assumptions - Suppose you want to ... 2 The question is not totally clear, but I will attempt to give you some guidance. To answer your first questions, confounding variables are not a type of endogenous variable. We do not observe nor are we interested in the confounding variables, which means they are not endogenous variables in our model. You later give the correct definition of an ... 2 Windmeijer (2000, Economics Letters) presents a treatment of the estimation of count-data models with fixed effects and endogeneity. http://www.sciencedirect.com/science/article/pii/S0165176500002287 See this slideshow by Wooldridge for a pedagogic and progressive presentation of panel data models with endogeneity. Count-data models are introduced slide ... 2 Your first point is valid for not using fixed effects as you are interested in the entire border not just these 16 roads. Random effects can be advantageous when you have such a small sample compared to the population for your treatment group. Also note, if the unobserved effect has a large variance or T is very large then RE will be close to FE anyway. ... 1 You are completely correct that mathematically endogeneity is defined \text{cov(X,e)}\neq0, which is true not only when it’s correlated with omitted variable but also with measurement error or reverse causation/simultaneity. However, every single variable in economics can be subject to correlation with omitted variables or measurement error, so if you ... 1 No, I don't see how the dummy variable you are proposing would give you the same (or even similar) analysis as in the original model. I would recommend dealing with the endogeneity in another way. IV estimation, or one of the more recent methods, difference-in-difference, matching etc. One of the will be suitable. 1 Is this legit and what is the rational behind this? Yes you will find this even as a recommendation in many textbooks (e.g. see Romer Advanced Macroeconomics pp 376) so it is legit although with a caveat. A good instrument should be correlated with the endogenous variable and be able to through it exert an effect on dependent variable. Well lags are more ... 1 I'm going to assume that by "conditions for instrumental variables to work" you mean "instrumental variables is consistent." However, there are other properties to consider, like performance in small samples, etc.  \newcommand{\Cov}{\text{Cov}}  In this simple case, the IV estimator in sample of size n is$$ \hat \gamma_2 = \frac{\...

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Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and $cov(Z,\epsilon_1)\neq 0$ would violate them (see A Guide to Modern Econometrics by Verbeek). Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have $F$-...

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Adding to the excellent answer by @Alecos Papadopoulos, here are two simple numerical examples with $z=x^2+v$, $z^*$ is mean-centred $z$ and $v$ is independent from $u$, in which $E(xu)\neq 0$, $E(xz^*)\neq 0$, but $E(z^*u)=0$. Example 1 (with $x$ having a symmetric distribution) $E(xu)=1.5$, $E(xz^*)=-0.5$, $E(z^*u)=0$ Example 2 (showing that the ...

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In principle, there is nothing to exclude the case where all regressors are endogenous. My issue is that the Hurlin-Dumitrescu test has to do with Granger-causality, and Granger-causality examines the relation between the regressors and the dependent variable, not the relation between the regressors and the error term. On the other hand, endogeneity refers ...

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In a typical OLS model, $Y=\alpha+\beta X+\epsilon$, endogeneity exists when $E[\epsilon\,|\,X]\ne 0$, which results from $X$ and $\epsilon$ being correlated with one another. In your case, $Y$ uncorrelated with $\epsilon$ implies only that $E[\epsilon]=0$, which is not the same as the exogeneity condition $E[\epsilon\,|\,X]= 0$. Moreover, $Y$ ...

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I looked a bit more into this and I think I found the confirmation of my suspicion into this article, which describes the command ivreg2 in STATA. I am not a super-techy econometrician but from my understanding it can apparently be done using the orthog() option, and under certain conditions it is equivalent to a Hausman test. http://www.stata-journal.com/...

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(i) I think your idea makes sense. Under the null, $[X,Z]$ is orthogonal to $\varepsilon$. Under the alternative, $X$ is correlated with $\varepsilon$. (ii) Your statement that it's "basically a test of whether the OLS residual are orthogonal to $Z$" is exactly what I think. (iii) Your thought about the power depending on the relevance of $Z$ also makes ...

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