FELLOW’S CORNER Foundations of statistical inference based on numerical roots of robust pivot functions

Journal of Econometrics 86 (1998) 387—396

FELLOW’S CORNER Foundations of statistical inference based on numerical roots of robust pivot functions H.D. Vinod* Economics Department, Fordham University, Bronx, NY 10458-5158, USA Received 1 August 1996; final version received 1 November 1997

Abstract Fisher’s pivot functions (PFs) continue to dominate statistical inference and bootstrap literature, despite Efron and Hinkley and Royall’s attempts to inject robustness. Vinod uses Godambe’s pivot functions (GPFs) based on Godambe—Durbin estimating functions (EFs) to develop numerically computed GPF roots. Such GPF roots can fill a long-standing need in the bootstrap literature for robust pivots. Proposition 1 proves that GPFs are more robust than other PFs. Recently, Heyde rigorously explains why confidence intervals (CIs) from GPFs are the shortest and Davison and Hinkley explain why the double bootstrap (d-boot) yields second-order correct CIs. We briefly discuss realistic econometric simulations supporting GPF roots in double bootstraps. ( 1998 Elsevier Science S.A. All rights reserved. JEL classification: C12; C20; C52; C8 Keywords: Bootstrap; Regression; Fisher information; Robustness; Double bootstrap; Heteroscedasticity

1. Introduction As a field of study, econometrics competes with others in the emerging market of ideas for the new millennium. Our ‘buyers’ will be increasingly familiar with computers, seek standardized, platform-independent, credible (numerically

* E-mail: [email protected] 0304-4076/98/$19.00 ( 1998 Published by Elsevier Science S.A. All rights reserved. PII S 0 3 0 4 - 4 0 7 6 ( 9 7 ) 0 0 1 1 9 - X

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accurate) and reproducible inference methods, within the constraints of limited non-experimental economic data. The bootstrap is a computer-intensive inference method that is quickly gaining acceptance and we find greater reliance on out-of-sample forecasting (rather than asymptotic or other theoretical properties) as a criterion for choice among methods. A well-known problem often noted in the vast bootstrap literature (see, e.g., Vinod 1993;Davison and Hinkley 1997) is that reliable inference from bootstraps needs robust pivot functions (PFs), whose distribution does not depend on unknown parameters. Following Vinod (1996b), we argue that Godambe’s (Godambe, 1985) pivot function (GPF), which uses all the information in the sample, can fill that long-standing need. The GPF arose in the growing literature on Godambe and Durbin’s estimating functions (EFs) from 1960, surveyed by Dunlop (1994), Liang and Zeger (1995), Heyde (1997) and Vinod (1998). The EFs are defined as functions g(y, h)"0 of the data y and parameters h. The EF estimators are simply roots of EF"g(y, h)"0 solved for h. This paper uses only quasi-likelihood score functions (QSFs) as optimal EFs. The EF theory proves, Heyde (1997), that desirable properties (e.g., asymptotic normality) of underlying EFs also ensure desirable properties of their roots. Heyde (1997) (p. 2, Chapter 2) provides examples where the EF root is not a ‘sufficient statistic’, while the QSF provides a ‘minimal sufficient partitioning of the sample space’. Vinod and Samanta’s (Vinod and Samanta, 1997) EF example provides superior out-of-sample forecasts compared to the generalized method of moments (GMM). The surveys cited above provide many examples where the EF estimators are distinct from the usual ones. Econometrics has not yet exploited the simplicity with which EFs can be combined and applied to semiparametric and semimartingale models when the QSFs fail to exist Heyde 1997) (Chapter 11). Econometrics texts (e.g., Greene, 1997) (p. 153) define the pivot as a function of h and hK , f (h, hK ) with a known distribution. GPF(y, h) functions include y and p exclude hK from the list of arguments. Proposition 1 will prove that GPFs converge robustly (with minimal assumptions) to the unit normal N(0,1). Let us first emphasize a well-known link between robust inference, non-normality and the ‘information matrix equality’. Let SE denote the standard error and ASE the asymptotic SE. Now Fisher’s PF (FPF) is FPF"z "(hK !h)/ASE which converges in distribution to N(0,1). (1) F The log-likelihood of y &N(k, p2), an independent and identically distributed t (iid) normal variable, is log f (y , k, p2)"!0.5 log 2p!0.5 log p2!0.5(y !k)2/p2. t t

(2)

Since h"(k, p2), the Fisher information I is a 2]2 matrix. Now I , the F 201 matrix of second-order partials, is a diagonal matrix having (p~2, 2~1p~4) and

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the matrix of ‘outer product of gradients’ is

C

D

p~2 2~1c p~3 1 , (3) " 01' 2~1c p~3 4~1(c #2)p~4 1 2 where c "E(y !k)3/p3 and c "E(y !k)4/p4!3 measure skewness and 1 t 2 t kurtosis. Clearly, if c "0 and c "0, the ‘information matrix equality’, 1 2 I "I , holds. Robust methods avoid these assumptions. 201 01' For brevity, our inference uses confidence intervals and indirectly covers its ‘dual’ called significance testing. After all, the tails of confidence intervals are simply the rejection regions of significance tests. The PF-root is defined as a solution of PF"(constant), and a numerical GPF root is a generalization first stated in Vinod (1996b). Now, let us review the well-known link between PF roots and confidence intervals. Two analytic solutions of Fisher’s PF or FPF"z "($1.96) yield the lower and upper limits of the traditional symmeta ric 95% confidence interval (CI95) by the simple inversion I

[hK !1.96ASE, hK #1.96ASE].

(4)

There is no loss of generality in discussing the 95% " 100(1!a)% two-sided CIs, since one can readily modify a ("0.05) to any desired significance level a(1. Also, if a left-hand-tail probability a is different from the right-hand-tail 2 L probability a , one can compute N(0,1) quantiles z and z satisfying U L U Pr[z )z)z ]"1!a !a . (5) L U L U Then the roots solving FPF"z and FPF"z yield the lower and upper limits L U of a more general CI. We shall achieve robustness and generality by using the GPF and by not insisting that its roots be analytic.

2. Need for numerical solutions of Godambe’s pivot functions (GPFs) This section first reviews the evolution of GPF in the statistical literature. Unfortunately, the example used in this literature and simulated in Vinod (1996b) — Cox’s example of estimating h from N(h, p2) with dichotomous random p2 — is unappealing to econometricians. Hence, we use N(h, p2) assuming known rather than dichotomous p2 as our ‘simplest case’. Its log likelihood is ¸"+T ¸ "+log f (y ; h), where +T "+ (throughout below). Now the t/1 t/1 t t t score function is S "(­¸ /­h), and Fisher’s pivot is t t z "(h!hK )M!E­2¸ /­h2N1@2"(h!hK )/SE. (6) F t In our simplest case, hK is the sample mean yN and SE"p/J¹. For robustness against incorrectly known p2, one can use SE"pL /J¹. Efron and Hinkley’s (Efron and Hinkley, 1978) rigorous theory and examples prove the desirability

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of removing the expectation operator E in Eq. (6). Their ‘less structured’ or ‘more robust’ pivot is z "(h!hK )M+!­2¸ /­h2N1@2"(h!hK )/SE. (7) EH t Thus, in our simplest case, z "z , even though E is absent in Eq. (7). This EH F contrasts with Cox’s example, where z Oz . Royall (1986) proves that the EH F asymptotic CIs from Eq. (7) are often invalid, and when the parametric model fails, the variance estimator is inconsistent. Royall’s robust PF is z "(h!hK )M+!­2¸ /­h2NM+[­¸ /­h]2N~1@2"(h!hK )¹1@2M+(y !hK )2N~1@2, R t t t (8) where the last h is replaced by its estimate. The following definition of GPF is simpler and direct: z "+­¸ /­hM+[­¸ /­h]2N~1@2"+(y !h)M+(y !h)2N~1@2"+SI , (9) G t t t t t which defines the notation SI for ‘scaled quasi-scores’. Without the (h!hK ) term, t Eq. (9) does not look like a typical pivot. However, we claim that (i) the very absence of (h!hK ) is an important asset of GPFs for robust inference, and (ii) numerical GPF roots similar to Eq. (4) will yield CIs for the unknown h, associated with EF estimators, while avoiding their sampling distributions. Heyde (1997) (p. 62) proves that CIs from ‘asymptotic normal’ GPFs are shorter than CIs from ‘locally asymptotic mixed normal’ pivots. Mixtures of distinct distributions in the usual bootstrap pivots come from normalizations (e.g., divide by a robust estimate of p2). Hence, we argue that GPFs are preferable for bootstrap-based inference. Let m "%T (1#ijSI ), where i2"!1. Assuming (a) m is uniformly integrT t/1 t T able, (b) E(m )P1 as ¹PR, (c) that RSI 2P1 in probability, as ¹PR, and (d) t T the max of DSI DP0 in probability as ¹PR, McLeish (1974) proved (without t assuming finite second moments) a central limit theorem (CLT) for dependent processes common in econometrics. Proposition 1. Assume that the scaled quasi-scores SI satisfy Mc¸eish+s assumpt tions (a)—(d). ¹he partial sum, z "RSI , ("GPF) converges in distribution to G t N(0,1) as ¹PR. Defining robustness as absence of additional assumptions, z is more robust than z , z and z . G F EH R Proof. The convergence of z follows from McLeish’s CLT. Since z , z and G F EH z pivots contain hK (the root of RS "0), their convergence obviously needs R t further assumptions to ensure asymptotic normality of the (ML) estimator hK . Davidson and MacKinnon (1993) (pp. 136—154) discuss the assumption of ‘weak uniform law of large numbers of a sequence of Hessians’ for asymptotic normality of a hK in Eq. (6). Formal convergence arguments in Efron and Hinkley prove

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that z has ‘less structure’ than z in Eq. (6). Royall’s Eq. (8) reduces to Eq. (7) EH F only if one assumes c "c "0 from Eq. (3), (i.e., only if IK "IK ). Thus, 1 2 201 01' z has less structure than z . Still, z has the (h!hK ) term, which implicitly R EH R forces the assumption that confidence intervals must be symmetric. The absence of (h!hK ) admits more general asymmetric GPF confidence intervals from Eq. (9) using numerical methods. Thus, z is more robust than z ,z and z . G F EH R As in Eq. (4), we can construct a CI from numerical GPF roots h of the following nonlinear equation: z !z "0"Ry !¹h!1.96MR(y !h)2N1@2. (10) G a t t Numerical GPF roots of nonlinear equations like Eq. (10) are impossible without modern computers, and were not mentioned in the EF literature before Vinod (1996b). His simulation shows that Eqs. (9) and (10) yield superior CIs for Cox’s example. He also derives new GPFs for estimating binomial probability, Poisson mean, and Normal distribution standard deviation. In general, pvariate numerical GPF roots can directly yield CIs for almost any function f (h), avoiding a search for pivotal statistics for each f (h).

3. Applications to regressions Econometricians often discuss the usual regression model with ¹ observations and p regressors: y"Xb#e,

E(e)"0, Eee@"p2X.

(11)

The log likelihood function ¸"R¸ contains ¸ "!0.5 log(2np2)!0.5 p~2[e@e ], tt t t and the score S "R(­¸ /­b), only under normality. Even if normality is avoided, t t the quasi-score is often available, and then QSF"0 is the optimal EF. If X"I, its root (solution) is bK "(X@X)~1X@y, the ordinary least-squares (OLS) es0-4 timator. Thus, QSF"0 can be viewed as a set of p ‘normal equations’: (12) X@X~1(y!Xb)"RH@(y !X b)"RH@e "RS "0, tt t t t t where E(ee@)"p2X is assumed to be known, H@ denotes the tth column of t X@X~1, and the score S is p]1. The p roots of Eq. (12) yield the ML estimator of t b under normality. Now the Fisher information matrix is I "[X@X~1X]/pL 2, F where (¹!p)pL 2"(y!XbK )@X~1(y!XbK ). Using the inverse of I as our (ASE)2 F matrix, Fisher’s PF is z "(b!bK )(ASE)~1. Removing the expectation operator F as in Eq. (7), Efron—Hinkley’s PF is z "(b!bK )(ASªE)~1. When X is nonEH stochastic, E[X@X~1X]"[X@X~1X], and z equals z . For the special case of EH F heteroscedasticity, where X"diag(.) is a known diagonal matrix, Royall (1986) suggests the following robust estimator of the covariance matrix: KK "¹[X@X~1X]~1X@X~1 diag (eL 2)X~1X[X@X~1X]~1. t 1

(13)

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If X"I it is noteworthy that Eq. (13) reduces to the Eicker—White heteroscedasticity consistent (HC) covariance matrix estimator often used in econometrics. For unknown X, Royall suggests KK "¹[X@XK ~1X]~1(+X@XK ~1(y !X bK )(y !X bK )@XK ~1X )[X@XK ~1X]~1, t t t t t t 2 (14) using a consistent estimate XK of X. Royall’s PF from Eq. (14) is z "(b!bK )(KK )~1@2. A GPF similar to Eq. (9) would be R 2 [X@X~1Eee@X~1X]~1@2X@X~1e, where Eee@"X, is assumed known. Thus, GPF becomes [X@X~1X]~1@2X@X~1e. For OLS the simplified GPF equals [X@X]~1@2X@e. When X is unknown, but all we know is that it is diagonal, then Vinod (1996b) proposes the following scaled sum of scores as the GPF: (15) z "[X@ diag(e2)X]~1@2X@e. t G Let I denote a p]p identity matrix. If Eee@"diag(e2), the diagonal term cancels t p in the following: (16) Ez z@ "[X@ diag(e2)X]~1@2[X@Eee@X][X@ diag(e2)X]~1@2"I . t p t G G Thus, the GPF of Eq. (15) is robust against arbitrary heteroscedasticity (unknown Eee@). A p-variate version of Proposition 1 holds whenever we can write GPF"RSI , which converges in distribution to N(0, I ) by the CLT. Let ı denote t p a p]1 vector of ones. We obtain a CI95 for a function f (b) by numerically solving for f (b) a system of p equations GPF(y, b)"$1.96(ı)"z . After a rearranging a slightly general version of Eq. (15) we solve (17) z [X@X~1 diag(e2)X~1X]1@2"X@X~1e, t a where the rearrangement avoids inversion of a square root matrix to improve the numerical accuracy of computer implementations. For greater generality one can replace the unknown X with a ‘heteroscedasticity and autocorrelation consistent’ estimate XK in a more general GPF: [X@X~1X]~1@2X@X~1e. Now, we seek further robustness by avoiding McLeish’s assumptions (a)—(d) and by permitting Eee@Odiag(e2). Then the multivariate GPF converges to t a non-normal variable Heyde 1997) (eq. 4.1). Also, typical (small) ¹ values in economic data may not be enough for asymptotic results. Fortunately, the double bootstrap (d-boot) can handle some non-normality (reduce bias, skewness, etc.) according to Vinod (1995) and Davison and Hinkley (1997). Moreover, Booth and Hall (1994) show that the d-boot confidence intervals are second-order correct if 10¹3 resamples are computed. For OLS, X"I in Eq. (12) GPF is Eq. (15), and our d-boot algorithm has following steps. McCullough and Vinod (1998) provide some theory and practical hints on implementing the d-boot.

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1. Compute the OLS estimates bK , a ¹]1 vector of residuals eL , and also a p]1 0-4 vector: z "[X@ diag(eL 2)X]~1@2X@eL . (18) 0-4 We suggest using the singular-value decomposition for better numerical accuracy in inverting the square root matrix. For greater generality this step can use iid recursive residuals, Vinod (1996a, 1997). 2. Let the jth vector of shuffled OLS residual vectors be eL for j"1,2999, using j a bootstrap shuffle. 3. Replace the eL in Eq. (18) by these shuffled vectors eL to yield zL . j j 4. Further shuffling of each zL for k"1,2,K yields zL , which are second stage j jk p]1 vectors. The size of K is determined to minimize the asymptotic mean squared error (MSE) by Booth and Hall (1994). For ¹"25, J"999 the Booth—Hall criterion suggests K"174 for the CI95. Hence we use K"174. 5. We can find any of the JK estimates bK by solving for b the GPF: jk MX@ diag(y!Xb)2XN1@2zL "X@(y!Xb). jk 6. We provide CI95 for a function f (b), which is a function of one or more regression coefficients. Denote its estimate by fK "f (bK ). We need to store jk jk only the counts C "d( fK (fK ) for j"1,2J. That is, we count the j jk 0-4 number of times fK is smaller than the OLS estimate of f (b) among K trials. jk 7. Under ideal conditions (normality of z ), the proportion p "C /K is a wellG j j behaved symmetric random variable in [0,1]. Denote the jth smallest value or ‘order statistic’ for C /K by p . If J"999, we have p "0.025, and j (j) (25) p "0.975, only under ideal conditions. Otherwise, we may, for example, (975) have p "0.030, and p "0.965. Then one simply uses appropriate (25) (975) (30th and 965th) nearby order statistics of fK to yield CI95 "[ fK , fK ]. It (30) (965) is by the right choice of nearby order statistics that the d-boot corrects for the bias, skewness, etc., without assuming (or needing to know) the actual non-normal functional form. The extra cost of d-boot computations is rapidly declining over time. Now we briefly review Vinod’s (1996b) simulations (¹"25,50) using OLS on realistic economic data. The ¹"25 case uses Bell—System data from Vinod (1976). This example is realistic and important, as it was used in my ‘expert’ testimony in the US Justice Dept. versus the Bell System. The Cobb-Douglas production function is: log ½"b #b log C#b log ¸#b D, where ½ is the 0 1 2 3 output, C is capital, ¸ is labor and D is percent telephones using the direct distance dial (DDD) to represent technology. The elasticity of output with respect to capital is (­ log½/­ log C)"b , similar output elasticities are b and 1 2 b and the scale elasticity is (b #b #b ). Their estimates were important in 3 1 2 3 the antitrust case. If the scale elasticity exceeds unity, there are economies of scale and upon an antitrust breakup, the Bell telephone consumers are predicted

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to pay higher prices (in the long run). When D is rescaled to achieve numerical comparability, the OLS coefficients are close to, but not exactly equal to b"(!3, 0.7, 0.3, 0.3)@. For convenience, the simulation fixes this as the true b vector and generates artificial data for log ½"Xb#error, where X has observed data on all regressors and the error vector has (i) unit normal N(0,1) deviates from the GAUSS computer language, or (ii) appropriate Cauchy deviates. For a d-boot, the computational burden of JK (J"999, K"174) solutions in the step (5) is manageable, since the solutions were obtained on a Pentium 133 MHz computer in about 28 hours. Conditional on coverage of the true parameter, the simulation computes w as i ratios of widths of CI95s. The numerator of w has GPF-d-boot interval width i for b and the denominator has a similar width of the usual Student’s t CI95. i Under N(0,1) errors the w are (0.19, 0.27, 0.20, 0.41) for b to b . The average of i 0 3 w or wN +0.27 represents an impressive reductions in the widths achieved by the i GPF-d-boot given that nominal coverage is achieved. Similar width ratios w for i Royall’s PF are (1.14, 0.97, 1.06, 0.88), with wN +1.01. Since the average exceeds unity, no improvement is discerned. However, since Royall’s PF is designed for non normal errors, we use a method in Vinod and Srivastava (1995) to generate Cauchy errors. Although the Cauchy distribution does not possess mean or variance, we still standardize them (subtract the realized mean and divide by the realized standard deviation) to remain comparable to the N(0,1) simulation. Under Cauchy errors Royall’s PF yields w "(0.64, 0.80, 0.73, 0.83), and i wN +0.75. Since this wN represents about 25% width reduction, robustness of Royall’s methods under Cauchy errors is reflected by the simulation. Now GPF-d-boot wN +0.28 under Cauchy errors from w "(0.20, 0.28, 0.21, 0.44) is i even smaller. Thus, conditional on coverage, GPF-d-boot achieves impressive reductions in CI95 widths in this limited simulation, subject to the usual caveats. Vinod’s (1996b) second simulation uses classic data by Nerlove and Waugh (1961) on supply and advertising for oranges (¹"50). Again, GPF-d-boot and Royall’s PF reduce the widths of CI95, especially when the errors are non normal. Moreover, the simulation results are not sensitive to a change from CI95 to a 90% CI. A formal proof of width reductions by Godambe’s PF is conveniently available in Heyde (1997) (pp. 54—66). Similarly, the reduction in coverage error of a CI95 from 1!a#O(¹~1) to 1!a#O(¹~2) by using the d-boot is explained in Davison and Hinkley (1997) (Section. 5.6). In regressions involving time-series data, the usual bootstrap shuffle loses the time identity of regression residuals. Hence, Vinod (1996a) suggests a modified bootstrap based on the so-called recursive residuals, which are always iid by construction. Vinod (1998) uses the modified bootstrap and EFs to estimate a consumption-based capital asset pricing model (C-CAPM), yielding better results and inference than the generalized method of moments (GMM). Vinod (1997) shows how to estimate the C-CAPM with a new hyperbolic utility function using EF-theory and recursive residuals.

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4. Conclusion and final remarks Simplification of econometrics with the help of computer intensive methods is desirable and possible. Efron, Hinkley and Royall have proved that Fisher’s PF is not robust. However, their own pivots in Eqs. (7) and (8) still contain the expression (h!hK ), leading to non-robust (always symmetric) confidence intervals (CIs). For inference on ‘locally almost unidentified’ functions lau(h) (e.g., long-run multiplier) Dufour (1997) proves that CIs obtained by inverting Waldtype statistics [lau(h)!lau(hK )]/SE(laˆu) similar to Eqs. (7) and (8) can have zero coverage probability. Our GPF"RSI , a sum of ¹ items, converges to N(0, I ) t p by the CLT, contains all information in the sample and is certainly not Waldtype. Hence, unlike Wald-type statistics, the GPFs can be used for resampling as their distributions do not depend on unknown parameters. Thus, for lau(h), we can have valid CIs from numerical roots of GPFs, possibly combined with resampling. More such research seems worthwhile, since Dufour shows that lau(h) are ubiquitous. We prove a proposition that the GPF is more robust than other PFs. In p-variate regressions we use GPF numerical roots in a double bootstrap (dboot) to reduce the coverage error of CIs to order 1/¹2. Two small-scale simulations support the theoretical implication (see Heyde (1997); Davison and Hinkley (1997) ) that GPF-d-boot CIs are short (conditional on coverage) and robust. The numerical roots of robust GPFs are tailor-made for bootstraps, and can serve as a foundation for further research in an era of powerful computing. The potential of EFs and GPFs for semiparametric and semimartingale models with nuisance parameters is indicated by Heyde (1997). Our GPF-d-boot CIs can potentially simplify, robustify and improve the currently used asymptotic inference methods.

Acknowledgements The author thanks Professor Godambe for suggesting the project and Bruce McCullough, Matt Morey and two referees for helpful comments.

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