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Standard Errors for Regression-Based Causal Effect Estimates in Economics Using Numerical Derivatives

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Abstract

The aim of nearly all empirical studies in economics is to provide scientific evidence that can be used to assess policy relevant cause-and-effect. In the context of the general potential outcomes framework, we review how a causal effect parameter can be rigorously but tractably specified, identified and estimated along with its asymptotic standard error. For cases in which the analytic and computational requirements for calculation of the ASE are challenging, we suggest the use of numerical derivatives (ND). We detail the specific type of ND software required for this purpose, and note that it is offered as a feature in most statistical packages. As an illustration, we analyze the causal effect of wife's high school graduation on family size using the Stata/Mata deriv command. Code for this example is supplied in an appendix.

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Notes

  1. See any graduate econometrics text under “marginal effects” or “average partial effects.” For more detailed discussions see Terza (2016a, b, 2017), Basu and Rathouz (2005) and Dowd et al. (2014).

  2. Derivations using asymptotic first principles are also offered by Basu and Rathouz (2005) and Wooldridge (2011, pp. 184–186).

  3. Conway and Maxwell (1962).

  4. X is the global symbol replacing the phrase “presumably causal entity of interest” and Y is the global symbol replacing the phrase “outcome of interest.”.

  5. Terza (2020) reviews the GPOF.

  6. Here we use the prefixes “pre-” and “post-” in reference to the change in the X to be hypothetically mandated as part of the relevant counterfactual query.

  7. The DGP is the joint distribution from which the observable data can be sampled.

  8. An entity (e.g., a parameter) is identified if its value would be implied by hypothetical knowledge of relevant aspects of the DGP.

  9. The primitive conditions under which (4) holds are discussed in detail in Terza (2020).

  10. Examples include: the deriv command in Stata/Mata 18® (StataCorp, 2023); the numDeriv package in R (Gilbert and Varadhan, 2022); the NLPFDD package in SAS/IML (SAS Institute, 2023); the hessp and gradp commands in GAUSS (Aptech Systems, 2023); and the DERIVEST suite in Matlab (D’Errico, 2023).

  11. Here we assume that for the pre- scenario of the relevant counterfactual we are setting the education level for individual ω in the population to be equal to the individual’s observable level at the time of the survey – thus, one may write \({\text{X}}^{\text{c}}\left({\upomega}\right)={\text{X}}\left({\upomega}\right)\). We re-emphasize here, however, that \({\text{X}}^{\text{c}}\left({\upomega}\right)\) is not a random variable.

  12. Methodological works related to the CMP include: Conway and Maxwell 1962, Shmueli et al. (2005), Sellers and Shmueli (2010), Daley and Gaunt (2016), Huang (2017), Forthmann et al. (2020), Sellers (2023). Applications of the CMP include: Ghorbani et al. (2023), Shirani-Bidabadi et al. (2020), Yan et al. (2019), and Lord et al. (2008), Bogomolovas et al. (2023), Wimmer et al. (1994), Borle et al. (2007), and Fraser (2020).

  13. We coded this estimator in Stata/Mata 18® (StataCorp, 2023), relying mainly on the Mata moptimize function.

  14. The full Stata/Mata 18(R) code is given in the appendix.

  15. Asymptotic t-statistic was calculated as (AIE Estimate)/ASE, where ASE is the relevant version of the square root of (15) for the CMP.

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Funding

This work was supported by the National Institute on Alcohol Abuse and Alcoholism under subcontract to the Alcohol Research Group (Center Grant #P50 AA05595-41).

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  1. Department of Economics, Indiana University School of Liberal Arts at IUPUI, Indianapolis, IN, 46202, USA

    Joseph V. Terza

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Appendix: Full Stata/Mata Stata/Mata 18® Code

Appendix: Full Stata/Mata Stata/Mata 18® Code

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Terza, J.V. Standard Errors for Regression-Based Causal Effect Estimates in Economics Using Numerical Derivatives. Comput Econ (2024). https://doi.org/10.1007/s10614-024-10565-w

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