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# Bootstrap Methods in Econometrics: Theory and Numerical Performance

## Author

Listed:
• Joel L. Horowitz

(Univ. of Iowa)

## Abstract

The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data. It amounts to treating the data as if they were the population for the purpose of evaluating the distribution of interest. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the approximation obtained from first-order asymptotic theory. Thus, the bootstrap provides a way to substitute computation for mathematical analysis if calculating the asymptotic distribution of an estimator or statistic is difficult. The maximum score estimator Manski (1975, 1985), the statistic developed by Ha..rdle et al. (1991) for testing positive- definiteness of income-effect matrices, and certain functions of time- series data (Blanchard and Quah 1989, Runkle 1987, West 1990) are examples in which evaluating the asymptotic distribution is difficult and bootstrapping has been used as an alternative.1 In fact, the bootstrap is often more accurate in finite samples than first-order asymptotic approximations but does not entail the algebraic complexity of higher-order expansions. Thus, it can provide a practical method for improving upon first-order approximations. First-order asymptotic theory often gives a poor approximation to the distributions of test statistics with the sample sizes available in applications. As a result, the nominal levels of tests based on asymptotic critical values can be very different from the true levels. The information matrix test of White(1982) is a well-known example of a test in which large finite- sample distortions of level can occur when asymptotic critical values are used (Horowitz 1994, Kennan and Neumann 1988, Orme 1990, Taylor 1987). Other illustrations are given later in this chapter. The bootstrap often provides a tractable way to reduce or eliminate finite- sample distortions of the levels of statistical tests.

## Suggested Citation

• Joel L. Horowitz, 1996. "Bootstrap Methods in Econometrics: Theory and Numerical Performance," Econometrics 9602009, University Library of Munich, Germany, revised 05 Mar 1996.
• Handle: RePEc:wpa:wuwpem:9602009
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## References listed on IDEAS

as
1. Blanchard, Olivier Jean & Quah, Danny, 1989. "The Dynamic Effects of Aggregate Demand and Supply Disturbances," American Economic Review, American Economic Association, vol. 79(4), pages 655-673, September.
2. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
3. Nelson, Charles R & Startz, Richard, 1990. "The Distribution of the Instrumental Variables Estimator and Its t-Ratio When the Instrument Is a Poor One," The Journal of Business, University of Chicago Press, vol. 63(1), pages 125-140, January.
4. Härdle, Wolfgang & Hart, Jeffrey D., 1992. "A Bootstrap Test for Positive Definiteness of Income Effect Matrices," Econometric Theory, Cambridge University Press, vol. 8(02), pages 276-292, June.
5. Lancaster, Tony, 1984. "The Covariance Matrix of the Information Matrix Test," Econometrica, Econometric Society, vol. 52(4), pages 1051-1053, July.
6. Nelson, Charles R & Startz, Richard, 1990. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 58(4), pages 967-976, July.
7. Chesher, Andrew & Jewitt, Ian, 1987. "The Bias of a Heteroskedasticity Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 55(5), pages 1217-1222, September.
8. Horowitz, Joel L., 1994. "Bootstrap-based critical values for the information matrix test," Journal of Econometrics, Elsevier, vol. 61(2), pages 395-411, April.
9. Manski, Charles F., 1985. "Semiparametric analysis of discrete response : Asymptotic properties of the maximum score estimator," Journal of Econometrics, Elsevier, vol. 27(3), pages 313-333, March.
10. Hardle, Wolfgang & Hildenbrand, Werner & Jerison, Michael, 1991. "Empirical Evidence on the Law of Demand," Econometrica, Econometric Society, vol. 59(6), pages 1525-1549, November.
11. Hall, Peter & Horowitz, Joel L, 1996. "Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators," Econometrica, Econometric Society, vol. 64(4), pages 891-916, July.
12. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
13. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 60(4), pages 953-966, July.
14. MacKinnon, James G. & White, Halbert, 1985. "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties," Journal of Econometrics, Elsevier, vol. 29(3), pages 305-325, September.
15. repec:cup:etheor:v:8:y:1992:i:2:p:276-90 is not listed on IDEAS
16. Hillier, Grant H., 1985. "On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation," Econometric Theory, Cambridge University Press, vol. 1(01), pages 53-72, April.
17. Manski, Charles F., 1975. "Maximum score estimation of the stochastic utility model of choice," Journal of Econometrics, Elsevier, vol. 3(3), pages 205-228, August.
18. Lafontaine, Francine & White, Kenneth J., 1986. "Obtaining any Wald statistic you want," Economics Letters, Elsevier, vol. 21(1), pages 35-40.
19. Gregory, Allan W & Veall, Michael R, 1985. "Formulating Wald Tests of Nonlinear Restrictions," Econometrica, Econometric Society, vol. 53(6), pages 1465-1468, November.
20. Hansen, Lars Peter & Singleton, Kenneth J, 1982. "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, Econometric Society, vol. 50(5), pages 1269-1286, September.
Full references (including those not matched with items on IDEAS)

### JEL classification:

• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
• C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
• C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
• C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
• C5 - Mathematical and Quantitative Methods - - Econometric Modeling
• C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs

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