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Natural experiments are usually a setting for causal inference rather than a causal inference tool per se. You often need to employ something like difference-in-difference or instrumental variables anyway even when you have a natural experiment. Here a list of statistical causal inference approaches (Approach: Lay description) Instrumental Variables: ...

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"But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?" I don't want to emphasize this too much, but it's worth mentioning that this is not true in general. The following derivation will hopefully provide some understanding of the "contamination" you mention....

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Actual availability of regressors may be an issue here, but if all four mentioned variables are available, the situation is as @Michael mentioned in a comment: Since $X_2$ is correlated with $Y$, it should be included in the regression specification as a "control". $$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + u$$ This is intuitive, but it also takes care ...

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All is too strong, but probably some. This problem is called "smearing". Take a look at the proof in Greene's lecture notes on slide 5. Emily Oster has a nice working paper (and Stata command psacalc) that can help bound the bias.

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This is an example of what statistician Andrew Gelman calls "the fallacy of controlling for an intermediate outcome". Here is his description of this fallacy popping up when researchers ask if having more daughters changes your politics. The decision to have a second child is necessarily conditional on the previous decision to have the first child, and so ...

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In the context of Least-squares estimation, the way we have to (attempt to) deal with possible endogeneity of regressors is through Instrumental Variables estimation. This approach does not depend on having just one endogenous regressor -you may have many. In such a case of course you need to find more instruments which make things harder -but in principle, ...

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The problem is to distinguish between changes along the production function, from $f(K_1, L_1)$ to $f(K_2, L_2)$, and changes of the production function, from $f(K_1, L_1)$ to $g(K_1, L_1)$. A very simple example/model. Imagine that the only input is labor and that this input doubles between t and t+1, but output more than doubles : $L_{t+1} = 2 L_t$, $Y_{t+... 4 Think about a regression where you have two factors. Normally when you have a constant in the regression you would put one level of each factor to 0 as a reference level. Leaving out the constant you have two factors and put one reference level to 0 for one of these. The situation is similar here in the sense that you have firm and individual level fixed ... 4 "Identification" is the most loaded term in econometrics. There are multiple cheap talk equilibria with regard to its meaning. It is used with different intended (but related and overlapping) meanings, in different contexts, by people with different orientations, with different levels of precision. Therefore you will get a range of correct answers. ... 3 You are right. Ordering a variable first reflects the identifying assumption that this variable will not respond contemporeneously to the other shocks in the system. You could obtain the exact same shock series via an OLS regression that includes as explanatory variables only the lags of all the variables in the VAR. Since you were happy to "assume away&... 3 I think the best way how to explain this is to first quickly explain what identification actually is. As mentioned in this thread: For example, in the John Stachurski "A Primer in Econometric Theory" the identification is a process of finding out if the parameters are identifiable and identifiability is defined as “Identifiability means that the ... 2 Firstly, the diff-in-diff already is a fixed effects estimation. If you run a regression with FE and your time variant terms ($S_i*T_t$and$T_t$) as in the plain and simple diff-in-diff setup, as such: $$Y_{it}=B_0+B_1T_t+B_2S_i+B_3S_i*T_t+e_{it} (1)$$ You should obtain similar results. Secondly, your specification -- since it will be on a multiple ... 1 "Identification" is the professional jargon in econometrics for "asserting that the outputs from an econometric model do indeed estimate what we want and declare that they estimate". "Identification" does not include an assertion that a specific estimate coming from combining a specific estimation method with a data sample, will ... 1 This looks challenging, and the following would only work in circumstances where: Reliable estimates can be made of how long each completed project should have taken to complete. Good records are maintained of the progress of individual projects, enabling assessments to be made of the form "Project A was X% complete after n years". Suppose two projects ... 1 I've run RDDs on similar sample sizes and got the paper accepted at a respectable conference. It's important to run McCrary Density Tests on your sample to look for people using knowledge on the treatment assignment rule to manipulate results and - even more important - to have a clear identification strategy, a qualitative justification for what you're ... 1 I am not exactly sure what you're asking. So...I hope we both understand identification strategy to mean the same thing. If so: The identification strategy serves as the bedrock upon which you build your argument. For example, consider some policy implemented by a school district that focuses on improving academic outcomes for low performing students. ... 1 To follow up the comment by @EnergyNumbers, causality flows from your theory. A key distinction is this: the math in any of the methods in @BKay's answer is designed to spit out numbers at the end of the procedure. For example, consider a diff-in-diff where your treatment is something silly like being licked in the face by a dog. You can always set up a ... 1 Regression discontinuity design just as difference in differences is a method for exploiting natural experiments. It builds on arbitrary rules that give different "treatments" to otherwise similar units. An example from Wikipedia: If all students above a given grade—for example 80%—are given the scholarship, it is possible to elicit the local treatment ... 1 Difference in Difference is probably the favourite method in econometrics (although it requires bootstraping, i.e. correcting the data from self-correlation). It basically compares the evolution of two groups, from a point at which none is subject to the given factor to a point at which one of them is subject to the factor. A famous example is Card and ... 1 As other posts have pointed out, endogenous regressors may contaminate all parameter estimates in regression when regressors are correlated. Moreover, it may seem difficult to conceive a situation where, say,$X_1$and$X_2$are correlated and$X_2$is endogenous but$X_1$is not. However, less than that is required to guarantee consistency of$\hat{\beta}...

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After searching for a while, this is the best answer I can so far come up with. 1) A formalized argument for why identification could break down under BBL is from Srisuma ('13). He gives two specific examples in the online appendix where identification is lost because of using additive rather than multiplicative perturbations to construct off-equilibrium ...

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