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Jason Wichert <[email protected]> |

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[email protected] |

Subject |
Re: st: RE: Testing for instrument relevance and overidentification when the endogeneous variable is used in interaction terms |

Date |
Tue, 11 Jun 2013 10:53:40 +0200 |

Apologies in advance for the long post. But since both Mark and I were about to lose track of the issue and following a bunch of additional more or less profound thoughts I put into the matter, here’s a rather extensive synthesis of the matter. Maybe it will attract some additional inputs by other readers. At the very least, it might help other PhD students with similar issues. To summarize: In my model, the dependent variable “y” is influenced by two exogenous variables (ex1, ex2). ex1 and ex2 are distinct constructs, so at the very least interactions between the two can be ruled out. However, the effects of both ex1 and ex2 on y are moderated by my endogenous variable “en” in different fashion. For some further empirical background: There’s an inverted u-shaped association between ex1 and y, and a negative association between ex2 and y. The effect of “en” on y seems negative, if significant at all. Taking interactions into account, the relationship (listing only significant coefficients) looks somewhat like the following according to preliminary OLS and 2SLS results: y = 0.363 ex1 – 0.032 (ex1)^2 – 0.183 ex2 – 0.007 ex1_en + 0.080 ex2_en + 0.001 (ex1)^2_en – 0005 (ex2)^2_en + controls where: y = dependent variable ex1 = exogenous variable 1 ex2 = exogenous variable 2 (ex1)^2 = exogenous variable 1 squared (ex2)^2 = exogenous variable 2 squared en = endogenous variable ex1_en = interaction of ex1 and en ex2_en = interaction of ex2 and en (ex1)^2_en = interaction of exogenous variable 1 squared and en (ex2)^2_en = interaction of exogenous variable 2 squared and en controls = a bunch of control variables including fixed industry- and year or industry-year effects Thus, the endogenous variable seems to have a negative moderating effect on the relationship between ex1 and y, which is attenuated at higher levels of ex1. Furthermore, the endogenous variable seems to have an overall positive moderating effect on the relationship between ex2 and y, which is attenuated at higher levels of ex2. Apart from control variables and instruments (the strength and exogeneity of which stands to debate), the endogenous variable is highly influenced by both ex1 and ex2, predominantly by ex1. Hence, while I can’t rule out omitted variable bias, at the very least my analyses seem subject to be simultaneity bias, calling for instrumental variable approaches. The basic problem is the multiple non-linear endogenous interaction terms, i.e. ex1_en, ex2_en, (ex1)^2_en, and (ex2)^en. Simply predicting “en” as “enhat” (to denote first stage predicted value of the endogenous variable) in a first stage regression and plugging the fitted values into the second stage equation as well as simply generating interaction terms of enhat with the exogenous variables and their squared values is out of question; this would be a classical case of what Wooldridge refers to as “forbidden regression”. Since the interaction terms are nonlinear functions of an endogenous variable, they need to be predicted/instrumented as well. Hence, for up to five endogenous variables (i.e., “en” as well as the two interactions each with ex1 and ex2), we need a total of at least five instruments. In the following, I try to summarize the different approaches discussed here on statalist and found in the literature, as well as some caveats and pitfalls which we already discussed, I have encountered or thought of. a) let’s call it the “ignorance is bliss” approach: I found a comment stating endogeneity bias to be attenuated when the endogenous variable is included in interaction terms with continuous exogenous variables. The cited paper the author of this comment referred to is an unpublished working paper I couldn’t get access to so far, and absent any proof or examinations, my intuition has a hard time taking this statement at face value. In my case, where ex1 has a largely positive effect on both the dependent variable as well as the endogenous variable, the significantly negative interaction between the two seems to indicate a lesser problem of endogeneity. In contrast, endogeneity pretty much does seem to be an issue considering ex2 has a negative effect on the dependent variable, whereas the interaction of ex2 and the endogenous variable has a positive effect as well. In particular, I’m afraid this positive interaction effect might be caused by the positive association between ex1 and “en”. If anyone has any further evidence or intuition concerning the “endogeneity is less of a problem when the endogenous variable is interacted with continuous exogenous variables” statement, please chime in. b) the standard approach: denoting a potential instrument as “z”, the main approach discussed here on statalist for interactions of endogenous variables is instrumenting both the endogenous variable as such, as well as its interaction term. As additional instrument(s), interaction terms of the exogenous variable and the instrument are constructed: ivreg2 y ex (en en_ex = z ex_z) In my case, this would translate to the following stata command, leaving control variables aside and using just 1 instrument: [1] ivreg2 y ex1 ex2 (en ex1_en ex2_en (ex1)^2_en (ex2)^2_en = z ex1_z ex2_z (ex1)^2_z (ex2)^2_z) In the case of say two instruments, this already expands to [2] ivreg2 y ex1 ex2 (en ex1_en ex2_en (ex1)^2_en (ex2)^2_en = z1 ex1_z1 ex2_z1 (ex1)^2_z1 (ex2)^2_z1 z2 ex1_z2 ex2_z2 (ex1)^2_z2 (ex2)^2_z2) To start my analyses, I’m planning to continuously extend the presented model, from incorporating the endogenous variable alone, such as [3] ivreg2 y ex1 ex2 (en = z1 z2) – let’s dub this the „base case“ to the models incorporating the interactions with the linear and then also the squared values of ex1 and ex2. When including solely the endogenous variable without any interaction terms, i.e. running [3], all test statistics are more than fine; instruments seem strong (not just by the F-stat, but also by the significance levels of the respective instruments in the first stage regression), exogenous, and not redundant (although I didn’t look into the latter too much). In particular, the plan is to present F-statistics, the Sargan/Hansen statistic (when more than one instrument is being used), as well as an endogeneity test (Durbin/Wu/Hausman) for “en”. In the “extended cases” [1] and [2], there are a total of five endogenous variables and first stage regressions. So simply reporting individual first stage F-statistics is out of the question. Apart from the endogeneity tests as well as the Sargan/Hansen statistic (in the case of overidentification), Cragg/Donald F-statistics for weak identification and – depending on those results – Kleibergen/Paap F-statistics for underidentification come to mind. If no signs of underidentification show up, Anderson/Rubin statistics can be left aside as well. However, Angrist/Pischke statistics for the individual endogenous regressors will be reported. Some brief notes and thoughts on those statistics: Cragg/Donald F-stat: - tests the null of weak identification, so we want to reject - equivalent to the regular first stage F-stat in the case of just one endogenous variable - refers to all endogenous regressors together, doesn’t indicate which one is weakly identified. Failure to reject might be due to all endogenous variables or just one. - Stock/Yogo (2005) provide critical values for relative and size bias for up to 3 endogenous regressors and 30 instruments. Since those values all seem to be in a certain ballpark, being a bit “hand-wavey” as Mark termed it will probably be considered appropriate. - only a viable statistic unless errors are heteroskedastic or serially correlated; not an issue in my case Kleibergen/Paap F-Stat: - tests the null of underidentification. If we already reject weak identification according to C/D, K/P is unnecessary. - again, with a single endogenous variable, simply the regular first stage F-stat - doesn’t necessarily rely on i.i.d. assumption, but not an issue here anyway - also, refers to all endogenous variables together. So failure to reject doesn’t tell me which regressor(s) are underidentified. In my case, these F-stats pretty much implode from just one endogenous variable to extended sets following the approaches [1] or [2]. I’ve come across some other empirical papers using a similar or the same procedure, and the ones that actually do report F-statistics (C/D or K/P) on the extended cases also show sharp declines in the F-values, indicating weak or underidentification of some form. I’m just wondering how come. The way I understand the null of the C/D F-stat, it tests whether the instruments are _jointly_ only weakly correlated to the endogenous regressors. Could failure to reject (as indicated by F-stats around 8) be due to the little correlation between say ex1_z to ex2_en, considering ex1 and ex2 are completely distinct constructs, and there’s a couple of these cases? At the same time, (ex2)^2_z does little to explain ex2_en which ex_z doesn’t already do, so some of my instruments barely differ using the main approach [1]/[2]. Completely dropping everything related to either ex1 or ex2 didn’t really help any, either, and my research has to incorporate both ex1 and ex2. Does anyone have any intuition on how the C/D F-stat *should* be expected to behave going from [3] to [2]? Again, other papers with similar approaches I found all document sharp declines in the C/D F-stat. But with my limited understanding of the matter, I’d rather not base my reasoning on my own econometric intuition and simple reference to other papers that simply took the statistics at face value. The Sargan/Hansen statistic of overidentification also tests whether _any_ of the instruments fail the orthogonality criterion. In the base case I can confidently fail (interesting wording) to reject the null. Since I know ex1 and ex2 are highly correlated to the dependent variable, so are by construction the newly generated instruments such as ex1_z. Hence, does this seem plausible as cause for the sudden rejection of the null? Comparing results of different sets of instruments, as suggested by Mark, did not help me so far. The Angrist/Pischke statistic for weak- and underidentification of individual endogenous regressors, also provides mixed evidence. Extending the model to incorporate en, ex1_en, ex2_en as endogenous terms instrumented accordingly with a set of (interacted) instruments found valid for “en” actually found “en” to be identified the worst, as indicated by an weakid AP F-stat of around 2.5, with weakid AP F-stats of 5.7 and 19.6 for ex1_en and ex2_en respectively. The AP underid F-stats were quite high, with p-values between 0 and 0.01. According to the fully extended case, i.e. [2] but with z1, z2, and z3, the AP F-stats for underidentification exhibit p-values between 0 and 0.001, whereas the F-stats for weak identification range between 3 and 8. Naively assuming the instruments do their job quite well (after all, they did fairly well in explaining “en” as sole endogenous variable) and the poor results of the aforementioned could be explained, would any other statistics come to mind, void of those problems (if there are “problems” in the calculations of those statistics at all)? In particular I’d think about a comparison of the partial R² to Shea’s partial R², or Shea’s partial R² as such. c) generating instruments of the interacted exogenous variables with fitted values of the endogenous variable: another approach we already discussed generates fitted values of the endogenous variable first, let’s call them “enhat”. After this preliminary step, in the main 2SLS procedure, instruments are constructed by interacting this fitted value with ex1, ex2, (ex1)^2, and (ex2)^2. So the Stata command is ivreg2 y ex1 ex2 (ex1)^2 (ex2)^2 (en ex1_en ex2_en (ex1)^2_en (ex2)^2_en = enhat enhat_ex1 enhat_ex2 enhat_(ex1)^2 enhat_(ex2)^2 ) If I’m not mistaken, this procedure is also recommended by Wooldridge, and I’ve seen it applied in two empirical papers. One only showed some A/P F-stats for an extended case with multiple endogenous interaction terms, the other paper showed C/D and K/P F-stats that were greatly reduced from the base case to their version of extended case. d) A variation of c): since a preliminary regression already predicted fitted values for the endogenous variable, would it be viable to include those in the second stage directly instead of using them as an instrument? At the same time, I’d think of excluding the endogenous variable from the set of instrumented variable. In essence, instead of ivreg2 ex1 ex2 (en ex1_en ex2_en = enhat enhat_ex1 enhat_ex2) I would run ivreg2 ex1 ex2 enhat (ex1_en ex2_en = enhat_ex1 enhat_ex2) So enhat would be an included instrument instead of an excluded instrument. Therefore, it would still contribute to instrument the endogenous interaction terms. And in my humble/naïve opinion, it would be sufficiently predicted already, since it’s the fitted value. Are there any upsides or downsides to this approach in contrast to c) ? e) Using separate sets of instruments to predict separate endogenous interaction terms: in one of the papers by an influential author I found, the following approach seems to have been employed. They start with getting fitted values of the endogenous variable, enhat. In a second step, they run multiple first stage regressions to instrument the endogenous interaction terms. In particular, they run ex1_en = enhat_ex1 + controls + original uninteracted instruments ex2_en = enhat_ex2 + controls + original uninteracted instruments Thus, they use selective instruments for each endogenous interaction term, e.g. not using enhat_ex2 to predict ex1_en. This seems to pretty much resemble the recommendation from Wooldridge according to approach “c)” as well, albeit using not each generated instrument for each endogenous interaction term, but only the corresponding ones. I had already thought about a similar procedure somewhere along approach “b)”, since I’m afraid some of my instruments in that case are either redundant or too much/too little correlated to other endogenous variables or the dependent variable for the aforementioned test statistics to show the results they do. Therefore, this might attenuate my problem of too many unnecessary (in some cases) instruments. On a side note, I do not know whether they include “enhat” as such in those first stage regressions of the interaction terms. If this approach was valid, would enhat need to be included as instrument for the interaction terms, or would the interactions such as enhat_ex2 for ex2_en suffice? Unfortunately, the authors do not show any multivariate statistics, but only statistics for the prediction of the single endogenous regressor en/enhat. f) control function approach: Wooldridge suggests predicting (by all the exogenous regressors and the to-be-excluded instruments) the endogenous variable in a first step and then including the residual from this regression in the main equations in which the endogenous variable is used. He explains (taken from some lecture slides) “If we believe y2 [the dependent variable] has a linear RF with additive normal error independent of z [an exogenous variable], the addition of v2_hat [the predicted residual] solves the endogeneity problem regardless of how y2 appears.” Unfortunately, even in his books I didn’t find too many explanations on this procedure, or at least not too many explanations I could make sense of. Following the base case [3], the results of the control function approach are equivalent. Toying around with my extended case (but only non-quadratic interactions), the results between “c)” and this control function approach clearly differ when the predicted residual is included. Unless I made some stupid mistake, coefficients of the endogenous terms are rather similar when also including interactions of ex1/ex2 and the predicted residual, and standard errors differ quite a bit as well. I’d greatly appreciate if anyone could chime in and provide some insights on how the control function approach is supposed to be incorporated with multiple endogenous (interaction) variables, i.e. whether and how interactions of the residual need to be incorporated as well. Any feedback - no matter how extensive - is highly appreciated, Jason * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/faqs/resources/statalist-faq/ * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:

**References**:**RE: st: RE: Testing for instrument relevance and overidentification when the endogeneous variable is used in interaction terms***From:*"Schaffer, Mark E" <[email protected]>

**Re: st: RE: Testing for instrument relevance and overidentification when the endogeneous variable is used in interaction terms***From:*Jason Wichert <[email protected]>

**RE: st: RE: Testing for instrument relevance and overidentification when the endogeneous variable is used in interaction terms***From:*"Schaffer, Mark E" <[email protected]>

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