On optimal inference in the linear IV model. (English) Zbl 1434.62160
Summary: This paper considers tests and confidence sets (CSs) concerning the coefficient on the endogenous variable in the linear IV regression model with homoskedastic normal errors and one right-hand side endogenous variable. The paper derives a finite-sample lower bound function for the probability that a CS constructed using a two-sided invariant similar test has infinite length and shows numerically that the conditional likelihood ratio (CLR) CS of M. J. Moreira [Econometrica 71, No. 4, 1027–1048 (2003; Zbl 1151.62367)] is not always “very close”, say 0.005 or less, to this lower bound function. This implies that the CLR test is not always very close to the two-sided asymptotically-efficient (AE) power envelope for invariant similar tests of D. W. K. Andrews et al. [Econometrica 74, No. 3, 715–752 (2006; Zbl 1128.62022)] (AMS).
On the other hand, the paper establishes the finite-sample optimality of the CLR test when the correlation between the structural and reduced-form errors, or between the two reduced-form errors, goes to 1 or \(-1\) and other parameters are held constant, where optimality means achievement of the two-sided AE power envelope of AMS. These results cover the full range of (nonzero) IV strength.
The paper investigates in detail scenarios in which the CLR test is not on the two-sided AE power envelope of AMS. Also, theory and numerical results indicate that the CLR test is close to having the greatest average power, where the average is over a specified grid of concentration parameter values and over a pair of alternative hypothesis values of the parameter of interest, uniformly over all such pairs of alternative hypothesis values and uniformly over the correlation between the structural and reduced-form errors. Here, “close” means 0.015 or less for \(k \leq 20\), where \(k\) denotes the number of IVs, and 0.025 or less for \(0 < k \leq 40\). The paper concludes that, although the CLR test is not always very close to the two-sided AE power envelope of AMS, CLR tests and CSs have very good overall properties.
On the other hand, the paper establishes the finite-sample optimality of the CLR test when the correlation between the structural and reduced-form errors, or between the two reduced-form errors, goes to 1 or \(-1\) and other parameters are held constant, where optimality means achievement of the two-sided AE power envelope of AMS. These results cover the full range of (nonzero) IV strength.
The paper investigates in detail scenarios in which the CLR test is not on the two-sided AE power envelope of AMS. Also, theory and numerical results indicate that the CLR test is close to having the greatest average power, where the average is over a specified grid of concentration parameter values and over a pair of alternative hypothesis values of the parameter of interest, uniformly over all such pairs of alternative hypothesis values and uniformly over the correlation between the structural and reduced-form errors. Here, “close” means 0.015 or less for \(k \leq 20\), where \(k\) denotes the number of IVs, and 0.025 or less for \(0 < k \leq 40\). The paper concludes that, although the CLR test is not always very close to the two-sided AE power envelope of AMS, CLR tests and CSs have very good overall properties.
MSC:
| 62J12 | Generalized linear models (logistic models) |
| 62F25 | Parametric tolerance and confidence regions |
| 62P20 | Applications of statistics to economics |
Keywords:
conditional likelihood ratio test; confidence interval; infinite length; linear instrumental variables; optimal test; weighted average power; similar test; linear IV regressionReferences:
| [1] | Anderson, T. W. (1946), “The non‐central Wishart distribution and certain problems of multivariate statistics.” Annals of Mathematical Statistics, 17, 409-431. · Zbl 0063.00085 |
| [2] | Anderson, T. W. and H.Rubin (1949), “Estimation of the parameters of a single equation in a complete system of stochastic equations.” Annals of Mathematical Statistics, 20, 46-63. · Zbl 0033.08002 |
| [3] | Andrews, D. W., V.Marmer, and Z.Yu (2019), “Supplement to ‘On optimal inference in the linear IV model’.” Quantitative Economics Supplemental Material, 10, https://doi.org/10.3982/QE1082. · Zbl 1434.62160 · doi:10.3982/QE1082 |
| [4] | Andrews, D. W. K. (1998), “Hypothesis testing with a restricted parameter space.” Journal of Econometrics, 84, 155-199. · Zbl 0947.62014 |
| [5] | Andrews, D. W. K. and P.Guggenberger (2018), “Identification‐ and singularity‐robust inference for moment condition models.” Cowles Foundation Discussion Paper No. 1978R, Yale University. |
| [6] | Andrews, D. W. K. and P.Guggenberger (2017), “Asymptotic size of Kleibergen”s LM and conditional LR tests for moment condition models.” Econometric Theory, 33, 1046-1080. · Zbl 1442.62728 |
| [7] | Andrews, D. W. K., M. J.Moreira, and J. H.Stock (2004), “Optimal invariant similar tests for instrumental variables regression with weak instruments.” Cowles Foundation Discussion Paper No. 1476, Yale University. |
| [8] | Andrews, D. W. K., M. J.Moreira, and J. H.Stock (2006), “Optimal two‐sided invariant similar tests for instrumental variables regression.” Econometrica, 74, 715-752. · Zbl 1128.62022 |
| [9] | Andrews, D. W. K., M. J.Moreira, and J. H.Stock (2008), “Efficient two‐sided nonsimilar invariant tests in IV regression with weak instruments.” Journal of Econometrics, 146, 241-254. · Zbl 1429.62650 |
| [10] | Andrews, D. W. K. and W.Ploberger (1994), “Optimal tests when a nuisance parameter is present only under the alternative.” Econometrica, 62, 1383-1414. · Zbl 0815.62033 |
| [11] | Andrews, I. (2016), “Conditional linear combination tests for weakly identified models.” Econometrica, 84, 2155-2182. · Zbl 1410.62193 |
| [12] | Andrews, I. and A.Mikusheva (2016), “Conditional inference with a functional nuisance parameter.” Econometrica, 84, 1571-1612. · Zbl 1410.62054 |
| [13] | Chamberlain, G. (2007), “Decision theory applied to an instrumental variables model.” Econometrica, 75, 609-652. · Zbl 1142.91396 |
| [14] | Chernozhukov, V., C.Hansen, and M.Jansson (2009), “Admissible invariant similar tests for instrumental variables regression.” Econometric Theory, 25, 806-818. · Zbl 1278.62029 |
| [15] | Davidson, R. and J. G.MacKinnon (2008), “Bootstrap inference in a linear equation estimated by instrumental variables.” Econometrics Journal, 11, 443-477. · Zbl 1154.62018 |
| [16] | Dufour, J.‐M. (1997), “Some impossibility theorems in econometrics with applications to structural and dynamic models.” Econometrica, 65, 1365-1387. · Zbl 0886.62116 |
| [17] | Dufour, J.‐M. and M.Taamouti (2005), “Projection‐based statistical inference in linear structural models with possibly weak instruments.” Econometrica, 73, 1351-1365. · Zbl 1155.62486 |
| [18] | Elliott, G., U. K.Müller, and M. W.Watson (2015), “Nearly optimal tests when a nuisance parameter is present under the null hypothesis.” Econometrica, 83, 771-811. · Zbl 1410.62029 |
| [19] | Hillier, G. H. (2009), “Exact properties of the conditional likelihood ratio test in an IV regression model.” Econometric Theory, 25, 915-957. · Zbl 1278.62086 |
| [20] | Kleibergen, F. (2002), “Pivotal statistics for testing structural parameters in instrumental variables regression.” Econometrica, 70, 1781-1803. · Zbl 1141.62361 |
| [21] | Mikusheva, A. (2010), “Robust confidence sets in the presence of weak instruments.” Journal of Econometrics, 157, 236-247. · Zbl 1400.62330 |
| [22] | Moreira, H. and M. J.Moreira (2013), “Contributions to the theory of optimal tests.” Unpublished manuscript, available at Ensaios Economicos, http://hdl.handle.net/10438/11263. |
| [23] | Moreira, H. and M. J.Moreira (2015), “Optimal two‐sided tests for instrumental variables regression with heteroskedastic and autocorrelated errors.” Unpublished manuscript, available at arXiv:1505.06644. · Zbl 1456.62029 |
| [24] | Moreira, M. J. (2003), “A conditional likelihood ratio test for structural models.” Econometrica, 71, 1027-1048. · Zbl 1151.62367 |
| [25] | Moreira, M. J. (2009), “Tests with correct size when instruments can be arbitrarily weak.” Journal of Econometrics, 152, 131-140. · Zbl 1431.62655 |
| [26] | Staiger, D. and J. H.Stock (1997), “Instrumental variables regression with weak instruments.” Econometrica, 65, 557-586. · Zbl 0871.62101 |
| [27] | Wald, A. (1943), “Tests of statistical hypotheses concerning several parameters when the number of observations is large.” Transactions of the American Mathematical Society, 54, 426-482. · Zbl 0063.08120 |
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