--- title: "Using Instrumental Variables under Partial Observability of Endogenous Variables for Assessing Effects of Air Pollution on Health" authors_and_venue: "with Tarik Benmarhnia and Prashant Bharadwaj" abstract: "This paper highlights a common issue with instrumental variables (IV) techniques in evaluating the impact of air pollution on health. Since pollutants are almost always produced alongside other pollutants, instrumenting a single endogenous pollutant requires additional assumptions for correct inference and interpretation. We clarify these assumptions and propose that researchers place more structure on the relationship between the instrument and all the relevant pollutants." pdf_url: "https://mauricio-romero.com/pdfs/papers/partialiv.pdf" canonical_url: "https://mauricio-romero.com/pdfs/papers/partialiv.pdf.md" source: research.qmd note: >- Machine-readable Markdown version of the paper, generated for LLMs. --- # Using Instrumental Variables for Assessing the Effects of Air Pollution on Health: A Cautionary Tale^∗ Tarik Benmarhnia† Prashant Bharadwaj‡ Mauricio Romero§ #### Abstract This paper highlights a common issue with instrumental variables (IV) techniques in evaluating the impact of air pollution on health. Since pollutants are almost always produced alongside other pollutants, instrumenting a single endogenous pollutant requires additional assumptions for correct inference and interpretation. We clarify these assumptions and propose that researchers place more structure on the relationship between the instrument and all the relevant pollutants. ^∗Corresponding author: Tarik Benmarhnia. E-mail: [tbenmarhnia@ucsd.edu.](mailto:tbenmarhnia@ucsd.edu) Romero acknowledges financial support from the Asociaci´on Mexicana de Cultura. ^†University of California, San Diego - Department of Family Medicine and Public Health, and Scripps ^‡University of California, San Diego - Department of Economics ITAM - Centro de Investigaci´on Econ´omica ### 1 Introduction Instrumental variable (IV) methods are frequently used to estimate causal effects in economics. In particular, air pollution and its effect on health is an area of research in which the use of IV methods is widespread. In this paper, we search the literature that uses IV methods to assess the impact of atmospheric air pollution on health outcomes and point out an important but largely unemphasized assumption implicit in most of these papers. The basis of this paper is simple: while instruments are often used to create plausibly exogenous variation in single pollutants, pollutants are generally co-produced. Hence, IV models treating a single pollutant as endogenous will yield biased estimates. The direction of bias depends on how co-pollutants interact with each other and with the instrument. In some cases, it is impossible to assess whether biased IV estimates are closer to the true parameters than OLS estimates. We conducted a systematic search to identify all published literature on air pollution and health using IV methods through April 2017. Our goal was to illustrate the variety of studies conducted in this area and to highlight how exclusion criteria violations may occur.[1](#page-1-0) We searched keywords, titles, and abstracts in PubMed, Elsevier, Embase, and Web of Science.[2](#page-1-1) The boolean search used was: ("air pollution" OR "air quality") AND "instrumental variable\*" AND (health OR mortality OR morbidity OR hospital\* OR emergency).[3](#page-1-2) . Twenty-five studies fulfilled the inclusion criteria (see Panel A, Table [1\)](#page-2-0), although only a fraction of them (n=8) were relevant for the issues considered in this paper. Table [2](#page-13-0) lists the non-relevant studies. In Panel B, we show other articles that, based on our knowledge of the field, use IV methods to examine the causal effect of air pollution on health. The papers in Panel B are a selective subsample of a broader set of papers that may exist in this area. Combining the "relevant" papers from Panel A and the papers in Panel B, we examine a total of 20 papers. IV methods are rather recent in this space (the earliest relevant study in our sample is from 2003). We categorize the types of IVs used in three groups: 1) IVs related to regulatory or economic shocks[$Bombardini](#page-7-0) [& Li,](#page-7-0) [2016;](#page-7-0) [Chay & Greenstone,](#page-7-1) [2003b;](#page-7-1) [Deschenes et al.,](#page-8-0) [2012;](#page-8-0) [Lagravinese et al.,](#page-8-1) [2014$](#page-8-1). Some studies used geographical variations in regulation [\(Chay et al.,](#page-7-2) [2003;](#page-7-2) [Chay & Greenstone,](#page-7-3) [2003a;](#page-7-3) [Gutierrez,](#page-8-2) [2015;](#page-8-2) ¹The search was conducted on May 1, 2017. ²We did not use Google Scholar because its search function is limited to either the title or the entire body; it does not allow for abstract searches. ³For PubMed the exact search was: (((instrumental variable\*[Title/Abstract]) AND (Air quality[Title/Abstract] OR Air pollution[Title/Abstract])) AND (health [Title/Abstract] OR mortality[Title/Abstract] OR morbidity OR hospital\*[Title/Abstract] OR emergency[Title/Abstract])). For Eslevier it was: title-abs-key(instrumental variable\*) AND (titleabs-key(Air quality) OR title-abs-key(Air pollution)) AND (title-abs-key(health) OR title-abs-key(mortality) OR title-abskey(morbidity) OR title-abs-key(hospital\*) OR title-abs-key(emergency)). For Embase it was: 'instrumental variable\*':ab,ti AND ('air quality':ab,ti OR 'air pollution':ab,ti) AND ('health':ab,ti OR 'mortality':ab,ti OR 'morbidity':ab,ti OR 'hospital\*':ab,ti OR 'emergency':ab,ti). For Web of Science it was: TS=("instrumental variable\*") AND (TS=("Air quality") OR TS=("Air pollution")) AND TS=(health OR mortality OR morbidity OR hospital\* OR emergency). | Table 1: Summary | table of studies us | ing IV the health | effects of air polli | ution | |------------------|---------------------|-------------------|----------------------|-------| | | | | | | | Study | Journal | Location | Health outc | ome | Air pollutant | IV used | |-------------------------------------------------------------------------|-----------------------------------------------------------|-----------------|--------------------------------------------------------------|--------|----------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------| | Panel A: Relevant artciles fro | | | | - | | | | Ebenstein, Frank, and Reingewertz (2015) | The Israel Medicine Association Journal | Israel | Hospital a | dmis- | PM10 | Sandstorms | | Knittel, Miller, and Sanders (2016) | Review of Economics and
Statistics | USA | Infant mortali | ty | PM10,CO | Fluctuations in traffic and their interaction with local weather conditions | | Lagravinese, Moscone, Tosetti,
and Lee (2014) | Regional Science and Urban Economics | Italy | Hospital a
sions | dmis- | PM10, CO, NO2,
CO and O3 | Climate conditions and humar
activity (traffic congestion and
concentration of manufacturing
industries) | | Moretti and Neidell (2011) | Journal of Human Resources | USA | Hospital a
sions | dmis- | O3, NO2 and CO | Boat traffic into the two major
ports of Los Angeles | | Schlenker and Walker (2016) | Review of Economic Studies | USA | Hospital a
sions | dmis- | CO | Variation in airport congestion
on the East Coast | | Schwartz, Austin, Bind,
Zanobetti, and Koutrakis
(2015) | American Journal of Epidemiology | USA | Total mortality | У | PM 2.5 | Back trajectories of air masses | | Schwartz, Bind, and Koutrakis (2017) | Environmental Health
Perspectives | USA | Total mortality | У | $\begin{array}{ccc} \mathrm{PM2.5}, & \mathrm{BC}, & \mathrm{and} \\ \mathrm{NO2} & & \end{array}$ | Combined height of the plane-
tary boundary layer and wind
speed | | Tonne et al. (2010) | Occupational and Envi-
ronmental Medicine | England | Hospital a | dmis- | NO(x) | Indicator for congestion charging
zone | | | The Economic Journal | Mexico | Infant mortali | ty | PM10 & CO | Meteorological thermal inversions | | Arceo-Gomez, Hanna, and Oliva
(in press)
Bombardini and Li (2016) | The Economic Journal Working Paper | Mexico
China | Infant mortali | | PM10 & CO
SO2 | Meteorological thermal inversions Export shocks | | Chay and Greenstone (2003b) | Quarterly Journal of Economics | USA | Infant mortali | | TSP | Income shocks | | Chay and Greenstone (2003a) | Working Paper | USA | Infant mortali | | TSP | Regulatory nonattainment status | | Chay, Dobkin, and Greenstone (2003) | The Journal of Risk and
Uncertainty | USA | Adult mortalit | | TSP | Regulatory nonattainment sta-
tus | | Deryugina, Heutel, Miller, Molitor, and Reif (2016) | Working Paper | USA | Mortality, hos
ization rate, ar
tal hospital sp
ing | nd to- | PM 2.5 | Changes in the local wind direction | | Deschenes, Greenstone, and
Shapiro (2012) | Working Paper | USA | Mortality and
pital admission | | PM 2.5, O3, SO2,
and CO2 | Emissions cap and trade market | | Gutierrez (2015) | Journal of Population
Economics | Mexico | Infant mortali | ty | Aerosol Optical
Depth (AOD) | Power plants in operation | | He, Fan, and Zhou (2016) | Journal of Environmental
Economics and Manage-
ment | China | Total mortality | У | PM10 | Olympic Games air pollution regulation | | Jayachandran (2009) | Journal of Human Resources | Indonesia | Infant mortali | ty | Satellite based O3 | Wildfire smoke | | Luechinger (2014) | Journal of Health Economics | Germany | Infant mortali | ty | SO2 | Regulatory desulfurization a
power plants, power plants' lo
cation and prevailing wind direc-
tion | | Zhong (2015) | Working Paper | China | Hospital a
sions | dmis- | NO2 | Driving restrictions and superstitions concerning the number 4 | [Luechinger,](#page-8-8) [2014\)](#page-8-8) and one study took advantage of air pollution regulations in Beijing during the 2008 Olympic Games [$He et al.,](#page-8-6) [2016$](#page-8-6). 2) IVs related to variations in road, port, and airport traffic flows [$Knittel](#page-8-4) [et al.,](#page-8-4) [2016;](#page-8-4) [Moretti & Neidell,](#page-9-0) [2011;](#page-9-0) [Schwartz et al.,](#page-9-2) [2015;](#page-9-2) [Zhong,](#page-10-0) [2015$](#page-10-0). 3) IVs related to meteorological or other exogenous environmental events, like thermal inversions [$Arceo-Gomez et al.,](#page-7-4) [in press$](#page-7-4), changes in local wind direction [$Deryugina et al.,](#page-8-5) [2016$](#page-8-5), smoke from wildfires [$Jayachandran,](#page-8-7) [2009$](#page-8-7) or back trajectories of air masses [$Schwartz et al.,](#page-9-2) [2015$](#page-9-2) as IVs. One study used a combination of the three types of IV described above [$Lagravinese et al.,](#page-8-1) [2014$](#page-8-1). As mentioned earlier, we want to highlight a particular violation of the exclusion restriction that is likely to occur in settings where the exposure or treatment is an aggregate variable such as air pollution (e.g., PM10 and CO) and the chosen IV (e.g., thermal inversions) affects this aggregate variable but the researcher only observes a part of what makes up overall air pollution. The converse, where the parameter of interest is the impact of a specific component of an aggregate variable (PM10, rather than overall pollution) but the instrument affects the aggregate variable (pollution), is another form of the same exclusion restriction violation. Some of the papers identified in Table [1](#page-2-0) address the issue we highlight in this paper [$Deryugina](#page-8-5) [et al.,](#page-8-5) [2016;](#page-8-5) [Moretti & Neidell,](#page-9-0) [2011;](#page-9-0) [Zhong,](#page-10-0) [2015$](#page-10-0). [Deschenes et al.](#page-8-0) [$2012$](#page-8-0) even acknowledge the possibility of violating the exclusion restriction (p. 16-17) by including various pollutants affected by a common IV.[4](#page-3-0) ### 2 The basic IV setting To fix ideas, suppose we want to measure the effect of pollution on a health outcome (y) and that the true relationship between the two is: $$y = x_1 \beta_1 + x_2 \beta_2 + \varepsilon, \tag{1}$$ where x¹ is one component of pollution (e.g., PM10), and x² is another component of pollution (e.g., CO2). For ease of exposition and without loss of generality, assume all variables have been standardized such that V (X1) = V (x2) = V (ε) = 1. Additionally, assume cov(x1, ε) 6= 0, so that x¹ is endogenous. Suppose the statistician only observes x1. If the statistician were to estimate the following regression $$y = x_1 \beta_1 + \eta \tag{2}$$ ⁴The authors of two studies claim to have met this exclusion restriction [$Knittel et al.,](#page-8-4) [2016;](#page-8-4) [Schwartz et al.,](#page-9-2) [2015$](#page-9-2). where η = x2β² + ε then $$\widehat{\beta_1^{ols}} = (x_1'x_1)^{-1}x_1'y \to_p V(x_1)^{-1}cov(x_1, y) = \beta_1 + V(x_1)^{-1}cov(x_1, x_2)\beta_2 + V(x_1)^{-1}cov(x_1, \varepsilon) = \beta_1 + \rho_{x_1, x_2}\beta_2 + \rho_{x_1, \varepsilon}$$ If cov(x1, ε) = 0 and cov(x1, x2) = 0 then βols is consistent.[5](#page-4-0) If either cov(x1, x2) 6= 0 or cov(x1, ε) 6= 0 then the OLS estimator of β¹ will be asymptotically biased. Assume we have an instrument z, such that cov(z, ε) = 0 and cov(z, x1) 6= 0. Then the IV estimate of β¹ is $$\widehat{\beta_1^{iv}} = (z'x_1)^{-1}z'y$$ $$\rightarrow_p cov(z, x_1)^{-1}cov(z, y)$$ $$= \beta_1 + cov(z, x_1)^{-1}cov(z, x_2)\beta_2$$ $$= \beta_1 + \rho_{z, x_1}^{-1}\rho_{z, x_2}\beta_2$$ Going from OLS to IV eliminates the bias term caused by the "direct" endogeneity of x¹ (ρx1,εσε) and the "indirect" biased term caused by the correlation of x¹ and x² (β2ρx1,x² ), but introduces another "indirect" biased caused by the correlation between the instrument and x² (β2ρ −1 z,x¹ ρz,x² β2). This is not an omittedvariable problem: If x² is observable, then both OLS and IV may be biased (i.e., ontrolling for pollutants does not solve the problem if the instrument is correlated with these pollutants).[6](#page-4-1) The IV estimate will be consistent as long as cov(z, x2) = 0 and cov(z, x1) 6= 0. The instrument can only be correlated with the component of pollution we observe. If the instrument is correlated with any other component of pollution, then the estimate may be biased. For example, suppose we observe PM10 and want $$\begin{array}{ccc} \widehat{\beta^{ols}} & \rightarrow_p & \beta_1 + \rho_{x_1,\varepsilon} \\ \\ \widehat{\beta^{iv}} & \rightarrow_p & \beta_1 + \rho_{z,x_1}^{-1} \rho_{z,x_2} \beta_2 \end{array}$$ ⁵This is different from the standard measurement error problem. ⁶ In this case to use an instrument (such as thermal inversions, driving restrictions, or environmental regulations) to study the effect of PM10 on health. In that case, our estimate of β¹ will not be consistent if our instrument affects other non-observable components of pollution such as SO2 (i.e., cov(z, x2) = 0) and if that component of pollution affects health (i.e., β² 6= 0). We provide a more general framework in Appendix [A.](#page-10-1) #### 2.1 Results Based on results in the previous section, some (intuitive) observations immediately arise: - 1. If β² = 0 (i.e., if the other pollutant has no effect on health) then IV is consistent . - 2. If cov(z, x2) = 0 (i.e., the instrument only impacts health through PM10) then IV is consistent. One could either assume β² = 0 or cov(z, x2) = 0 while using IV methods to assess the impact of air pollution on health (only one of these assumptions is needed). However, neither assumption is realistic in many settings. For example, cars emit hundreds of volatile organic compounds (VOCs) [$Caplain et al.,](#page-7-5) [2006$](#page-7-5). Thus, using driving restrictions as an instrument for any particular pollutant would lead to biased estimates. In terms of reducing bias, is IV better than OLS? Intuitively, if β² is relatively small, then the "indirect" bias is relatively small for both OLS and IV. Assume without loss of generality that β¹ > 0, ρx1,x² > 0 and ρz,x¹ > 0. Then: Lemma 1. If ρz,x² ≥ 0 and ρx1,ε ≥ 0, then $$\rho_{x_1,x_2} > \rho_{z,x_1}^{-1} \rho_{z,x_2}$$ is a sufficient condition for $$Bias(\widehat{\beta^{iv}}) < Bias(\widehat{\beta^{ols}})$$ where Bias refers to the difference between the probability limit of the estimator and the true value. Proof of Lemma 1 is the Appendix [C](#page-13-1) To assess whether IV "perform better" than OLS, we need assumptions on: 1) the correlation between the endogenous pollutant and the error term; 2) the correlation between the measured pollutant and its unmeasured counterparts; and 3) the ratio of the correlation between the instrument and the various pollutants. For example, if the correlation between the instrument and various pollutants is the same, then it would be unfeasible for the bias in IV to be less than the bias in OLS (since ρx1,x² < 1). If the instrument is mostly correlated with a specific pollutant, then ρ −1 z,x¹ ρz,x² is small, and Lemma 1 is likely to hold, implying that IV perform better than OLS. In general, authors should be specific about their assumptions on the pollution production function and how their chosen instrument affects individual pollutants. #### 2.2 Case study To showcase the problem's extent, we estimate the bias from both OLS and IV in one setting based on [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1). [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1) study the effect of pollution on hospitalization rates for asthma by using flight delays originating in the eastern US to instrument air pollution daily variation near airports in California. They focus on CO and NO2 since airports are a major source of these two pollutants. These two pollutants are highly correlated (see Panel A in Table [3](#page-14-0) in Appendix [C\)](#page-13-1), even after controlling for a rich set of weather and seasonal controls (see Panel B in Table [3](#page-14-0) in Appendix [C\)](#page-13-1). The instrument (taxi time at an airport in the eastern US) is correlated with both pollutants. The authors of the study are aware of this: "Since various pollutants are often correlated with one another, these estimates should be interpreted with caution, as the pollutant of interest will proxy for other correlated air pollutants." The authors overcome this problem by using wind speed and wind direction interacted with taxi time as additional sources of exogenous variation. Using all three sources of exogenous variation, an increase of 1 ppb of CO and NO2 increases asthma rates (per 10 million people) by 0.222 (p-value<0.01) and -2.2 (p-value>0.1) (see Table 5 in [Schlenker and](#page-9-1) [Walker](#page-9-1) [$2016$](#page-9-1)). Had [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1) ignored that these pollutants are correlated, an increase of 1 ppb of CO and NO2 would be (wrongly) associated with increases in asthma rates (per 10 million people) of 0.194 (p-value<0.01) and 12.4 (p-value<0.01) (see Table 4 in [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1)). Considering both pollutants to be endogenous did not change the magnitude of the effect of CO but makes a significant difference for NO2. #### 3 Discussion While instrumental variables are often used to create exogenous variation in single pollutants, pollutants are generally co-produced and any instrument that affects one component of pollution (e.g., PM10) is likely to affect other pollutants not considered in the analysis (e.g., SO2). If pollutants are co-produced, IV models that treat a single pollutant as endogenous will yield biased estimates. 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Retrieved from [http://www.sciencedirect.com/](http://www.sciencedirect.com/science/article/pii/B9780123756787003060) [science/article/pii/B9780123756787003060](http://www.sciencedirect.com/science/article/pii/B9780123756787003060) doi: 10.1016/B978-0-12-375678-7.00306-0 # A A more general case Suppose the true underlying model is the following: $$y = \sum_{k=1}^{N} x_k \beta_k + \varepsilon \tag{3}$$ and that cov(x1, ε) 6= 0, so that x¹ is endogenous. Additionally, suppose we have an instrument z such that cov(z, ε) = 0 and cov(z, x1) 6= 0. Finally, suppose the statistician only observes x1. Then the OLS estimate of β¹ is $$\widehat{\beta^{ols}} = (x_1'x_1)^{-1}x_1'y$$ $$\to_p V(x_1)^{-1}cov(x_1, y)$$ $$= \sum_{k=1}^K \left[ V(x_1)^{-1}cov(x_1, x_k)\beta_k \right] + V(x_1)^{-1}cov(x_1, \varepsilon)$$ $$= \beta_1 + \sum_{k=2}^K \beta_k V(x_1)^{-1}cov(x_1, x_k) + V(x_1)^{-1}cov(x_1, \varepsilon)$$ and the IV estimate of β¹ is $$\widehat{\beta^{iv}} = (z'x_1)^{-1}z'y$$ $$\to_p cov(z, x_1)^{-1}cov(z, y)$$ $$= \sum_{k=1}^K cov(z, x_1)^{-1}cov(z, x_k)\beta_k + cov(z, x_1)^{-1}cov(z, \varepsilon)$$ $$= \beta_1 + \sum_{k=2}^K \beta_k cov(z, x_1)^{-1}cov(z, x_k)$$ Assuming, without loss of generality, that σ^k = 1 for all k and that σ^ε = 1, then $$\begin{split} \widehat{\beta^{ols}} & \to_p & \beta_1 + \sum_{k=2}^K \beta_k \rho_{x_1, x_k} + \rho_{x_1, \varepsilon} \\ \widehat{\beta^{iv}} & \to_p & \beta_1 + \sum_{k=2}^K \beta_k \rho_{z, x_1}^{-1} \rho_{z, x_k} \end{split}$$ Thus, going from OLS to IV one eliminates the bias term caused by the "direct" endogeneity of x¹ (ρx1,ε). But introduces an"indirect" biased term caused by the correlation of x¹ and x^k ( P^K ^k=2 βkρx1,x^k ) and the correlation between the instrument and x^k ( P^K ^k=2 βkρ −1 z,x¹ ρz,x^k ). #### B Proof of Lemma [1](#page-5-0) and some corollaries Proof of Lemma [1](#page-5-0) in the general setting. Suppose the true underlying model is the following: $$y = \sum_{k=1}^{N} x_k \beta_k + \varepsilon \tag{4}$$ and that cov(x1, ε) 6= 0, so that x¹ is endogenous. Additionally, suppose we have an instrument z such that cov(z, ε) = 0 and cov(z, x1) 6= 0. Finally, suppose the statistician only observes x1. Assuming, without loss of generality, that σ^k = 1 for all k, σ^ε = 1, β¹ ≥ 0, ρ^x1,x^k > 0 for all k, and ρz,x¹ > 0. From σ^k = 1 for all k and σ^ε = 1: $$Bias(\beta^{ols}) \rightarrow_{p} \sum_{k=2}^{K} \beta_{k} \rho_{x_{1},x_{k}} + \rho_{x_{1},\varepsilon}$$ $$Bias(\beta^{iv}) \rightarrow_{p} \sum_{k=2}^{K} \beta_{k} \rho_{z,x_{1}}^{-1} \rho_{z,x_{k}}$$ Additionally, assume that β^k > 0 for all k, ρz,x^k ≥ 0 for all k, ρx1,ε ≥ 0, and ρx1,x^k > ρ−¹ z,x¹ ρz,x^k for all k. Then it follows that: $$\rho_{x_{1},x_{k}} > \rho_{z,x_{1}}^{-1}\rho_{z,x_{k}}$$ $$\sum_{k=2}^{K} \beta_{k}\rho_{x_{1},x_{k}} > \sum_{k=2}^{K} \beta_{k}\rho_{z,x_{1}}^{-1\rho_{z,x_{k}}}$$ $$\sum_{k=2}^{K} \beta_{k}\rho_{x_{1},x_{k}} + \rho_{x_{1},\varepsilon} > \sum_{k=2}^{K} \beta_{k}\rho_{z,x_{1}}^{-1}\rho_{z,x_{k}}$$ $$|\sum_{k=2}^{K} \beta_{k}\rho_{x_{1},x_{k}} + \rho_{x_{1},\varepsilon}| > |\sum_{k=2}^{K} \beta_{k}\rho_{z,x_{1}}^{-1}\rho_{z,x_{k}}|$$ $$|Bias(\beta^{ols})| > |Bias(\beta^{iv})|$$ $$Bias(\beta^{ols}) > Bias(\beta^{iv})$$ Two immediate corollaries that can be proved in an almost identical manner are: Corollary 1. If β^k > 0 for all k, ρz,x^k > 0 for all k, ρ^x1,ε < 0, and ρ −1 z,x¹ ρz,x^k > ρ^x1,x^k for all k, then $$\widehat{\beta^{iv}} > \widehat{\beta^{ols}}$$ Corollary 2. If β^k > 0 for all k, ρz,x^k > 0 for all k, ρ^x1,ε > 0, and ρ −1 z,x¹ ρz,x^k < ρ^x1,x^k for all k, then $$\widehat{\beta^{iv}} < \widehat{\beta^{ols}}$$ Table 2: Non-relevant research articles found in the systematic search | Study | Journal | |-----------------------------------------------------------|---------------------------------------------| | Agrawal and Yamamoto (2015) | Indoor Air | | Ambrey and Fleming (2014) | Economics Letters | | Bilger and Carrieri (2013) | Journal of Health Economics | | Brown (2013) | Applied Economics | | Ho and Hite (2009) | Journal of Community Health | | Howley (2017) | Journal of Economic Behavior & Organization | | Jones (2007) | Health economics | | Mullahy and Portney (1990) | Journal of Health Economics | | Nordberg, Filipsson, Gustafsson, Harland, and Roos (2001) | Journal of Sea Research | | Pant (2013) | Respirology | | Saha, Pattanayak, Sills, and Singha (2011) | Health & Place | | Schennach (2013) | Annals of Statistics | | Sim, Suryadarma, and Suryahadi (2017) | World Development | | Strand, Sillau, Grunwald, and Rabinovitch (2015) | Environmetrics | | Weldesilassie, Boelee, Drechsel, and Dabbert (2011) | Environment and Development Economics | | Zaman and el Moemen (2017) | Renewable and Sustainable Energy Reviews | | Zivin and Neidell (2014) | Encyclopedia of Health Economics | # C Extra tables Table 3: Correlation between pollutants and IV in [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1) Panel A: Unconditional correlation | | Taxi time | CO | NO2 | |-----------|-----------|----------|-----| | Taxi time | 1 | | | | CO | 0.105∗∗∗ | 1 | | | NO2 | 0.0305∗∗∗ | 0.798∗∗∗ | 1 | ^∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001 Panel B: Conditional correlation | | Taxi time | CO | NO2 | |-----------|------------|----------|-----| | Taxi time | 1 | | | | CO | -0.0118∗∗∗ | 1 | | | NO2 | -0.0103∗∗∗ | 0.583∗∗∗ | 1 | ^∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001 Correlations are based on the supplementary material data provided by [Schlenker and Walker](#page-9-1) [$2016$](#page-9-1). Panel A shows the raw correlation, while Panel B shows the conditional correlation after controlling for weather and seasonal controls.