White, H. 1980. A Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica. 48(4) 817-838.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Arny Stromberg, Suojin Wang and David Ruppert, whose help is gratefully acknowledged. 1 1 Introduction The heteroskedasticity consistent covariance matrix estimator is a common tool used for variance estimation of parameter estimates. Originally introduced by Huber (1967) Eicker (1967) and White (1980), the estimate has become popular in the econometric literature. In the last decade the method has also been widely used in the context of generalized estimating equations, see e.g. Liang Zeger (1986) Liang, Zeger Qaqish (1992) and Diggle, Liang Zeger (1994) where it was introduced as the ....

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817-838.

....In that case, the usual OLS estimator of the covariance of the OLS estimates # is in general biased, and so conventional t and F tests do not have their namesake distributions, even asymptotically, under the null hypotheses that they test. The problem was solved by Eicker (1963) and White (1980), who proposed a heteroskedasticity consistent covariance matrix estimator, or HCCME, that permits asymptotically correct inference on # in the presence of heteroskedasticity of unknown form. MacKinnon and White (1985) considered anumber of possible forms of HCCME, and showed that, in nite ....

....samples are drawn from the following DGP,which plays the role of # in the experiments: y # = # u # ; t =1; 10; 29) where # # = x # ## , the t ## component of # # , and the u # are normal white noise. This pattern of heteroskedasticity leads to bias of the OLS covariance matrix; see White (1980). Since # # contains a very high leverage observation for t = 2, the DGP (29) is very strongly heteroskedastic. Note that, since the distributions of all the statistics we consider are independent of the parameters # of the regression, we may,asin (29) set # = 0 without loss of generality. The ....

White, H. (1980). \A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity," ############, 48, 817-838.

....less harmful than underestimation of it, because it results in too conservative inferences. If one uses weights in a typical regression package, the simple covariance matrix of the classical weighted least squares or the so called White heteroskedasticity consistent covariance matrix estimator (White (1980a) White (1980b) are computed. Although these estimators work well under the assumptions behind each of the estimators, the complex survey sampling setup violates the assumptions. A crucial (but not the only) problem in use of these estimators in our current setup is that they completely ignore ....

....than underestimation of it, because it results in too conservative inferences. If one uses weights in a typical regression package, the simple covariance matrix of the classical weighted least squares or the so called White heteroskedasticity consistent covariance matrix estimator (White (1980a) White (1980b) are computed. Although these estimators work well under the assumptions behind each of the estimators, the complex survey sampling setup violates the assumptions. A crucial (but not the only) problem in use of these estimators in our current setup is that they completely ignore the dependence ....

White, H. (1980a): "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48(4), 817--838.

....errors, the usual OLS estimator of the covariance of the OLS estimates is in general asymptotically biased, and so conventional t and F tests do not have their namesake distributions, even asymptotically, under the null hypotheses that they test. The problem was solved by Eicker (1963) and White (1980), who proposed a heteroskedasticity consistent covariance matrix estimator, or HCCME, that permits asymptotically correct inference on in the presence of heteroskedasticity of unknown form. MacKinnon and White (1985) considered a number of possible forms of HCCME, and showed that, in nite ....

....data, we set t = 1 for all t, and for heteroskedastic data, we set t = jx t1 j, the absolute value of the t th component of x 1 . Because of the high leverage observation, this gives rise to very strong heteroskedasticity, which leads to serious bias of the OLS covariance matrix; see White (1980). The v t are independent mean zero variables of unit variance, and in the experiments will be either normal or else drawings from the highly skewed 2 (2) distribution, centred and standardised. In Table 3, we give, for the regression designs considered and the above pattern of ....

White, H. (1980). \A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity," Econometrica, 48, 817-838.

....2 t ) using the OLS regression: y t = # # y t # t (6) If a prediction is unbiased, # =0and# = 1. Table 2 reports the ordinary least squares estimates of (6) and the associated R 2 statistic when applied to variances, standard deviations and logarithmic variances. Standard errors using White s (1980) heteroskedasticity correction are in parentheses. Tab l e 2 : Realized Logarithmic Variance Model Ex Ante Volatility Predictions variances standard deviations log variances # #R 2 # #R 2 # #R 2 FI 0.079 1.238 0.379 0.048 1.086 0.486 0.028 1.024 0.515 (0.058) 0.162) 0.031) ....

....least squares coe#cient estimates for the model defined by equation 4 using the variance, standard deviation and logarithmic variance predictions given by (5) i.e. the ex ante one step ahead volatility predictions coming from the FI, FIX and FIMAX models reported in Table 1. Standard errors using White s (1980) heteroskedasticity correction are in parentheses. For all three models the estimates of # and # are within two standard errors of their hypothesized values. From the R 2 statistics it becomes evident that all three models can 11 As one would expect form the discussion at the beginning of this ....

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White, H. 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a direct Test for Heteroskedasticity." Econometrica 48:817--838.

....the Number of Total Banks, we focus on the arranging banks given their importance in creating and managing the syndicates. These other measures of syndicate structure do not exhibit the same level of significance. With these variables, we estimate OLS regressions and test for significance using White (1980) corrected standard errors. Below, we discuss the results from Table VI in three sections. Section A discusses the loanspecific variables; Section B describes the political risk variables; and Section C discusses the syndicate structure variables, which is the crux or our argument. Before ....

White, H., 1980. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817-838.

....stage of the sampling process, a number of students were drawn from each institution, thus violating the independence assumption. To account for this, the logistic regression models were fit using Huber White estimators of variance, which allows observations that are not independent (Huber, 1967; White, 1980, 1982) The sample weights and sampling stratification schema were also used in the analysis. The logistic models used in this study were fit by sequentially entering the groups of variables in blocks, with each block containing a series of predictor and or control variables. The blocks and ....

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817-829.

....second order expansion in powers of T , and , as well as terms capturing interaction of and with T. 1 2 We then obtain the final response surface specifications by dropping variables with insignificant t ratios. Because the variance of varies with the value of the t ratios are constructed using White s (1980) heteroskedasticity consistent standard errors. The final specifications of the response surfaces are given in Table 1. 5 All computations were done using 386 MATLAB version 3.5k on an IBM compatible 486 machine running at 33 MHz. All N(0,1) deviates were generated using the Forsythe et al. ....

White, H., 1980, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48, 817-838.

....# t i (13) # t i = d 0 d 1 SL t # t i (14) Because SL t is mean zero, the constant term is the sample mean of the dependent variable. The results of this second stage are reported in Table 3. Asymptotic t statistics are corrected for generalized heteroskedasticity using the technique of White (1980). There is a statistically significant negative relation between the slope of the term structure and the subsequent volatility of innovations in consumption growth. The evidence is somewhat stronger for absolute residuals than squared residuals. The relation between the slope and the subsequent ....

....of Table 5 reports the results from (17) Three sample periods are examined: The full 1952:3 through 1999:4 period and the pre experiment (1952:3 1977:4) and postexperiment (1983:1 1999:4) periods. Asymptotic t statistics are corrected for generalized heteroskedasticity using the technique of White (1980). The results are not surprising, given what we have seen in Tables 2 through 4. The average covariance between stock returns and consumption growth is positive and (aside from the short post experiment period) statistically significant. A steeper slope in quarter t reduces this covariance. The ....

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White, Halbert, 1980, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48, 817-838.

....We allow a nonzero var(ut 1) to accommodate the yh. tr# v r# p f r## s# # px# h. xr## . r#. ## # Xr# hyy ## 1 ## # # vpyqr# #h#v h. 4 We report results for the OLS t ratio. In these experiments similar results are obtained when the t ratios are formed as in Hansen (1982) and White (1980). 7 regressors. Stambaugh (1998) studies a special case of our setup, where the errors t # t are perfectly correlated, or equivalently, the analyst observes and uses the correct regressor. He studies finite sample biases that arise due to lagged stochastic regressors. The bias is ....

....heteroskedastic. The analyst in the simulations estimates the regression model (6) He uses the lagged instrument, Z t , which is independent of Z t . The true values of the coefficients in the regression (6) are A0=0, A1=0, B0=1 and B1=0. We form t ratios for the coefficients using the White (1980) Hansen (1982) consistent standard errors. We consider two cases in how we model the market index return. In the first case, the beta and the conditional mean of the market return follow white noise processes. In the second case, there is a common persistent factor driving the movements in both. ....

White, H., 1980, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48, 817-38.

....bill; SKEW t is the conditional skewness, and KURT t the conditional kurtosis. They are given by expression (12) 21 . Table 5 contains the regression results based on the entire sample period and 768 call options, and where the standard error for each coefficient estimate is adjusted by the White (1980) heteroskedasticity consistent estimator. The explanatory variables employed in the regressions tend to be statistically significant. However, there are important differences between the percentage errors associated with either BS or Heston. In particular, a key point of the results is the ....

White, H. (1980). "A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity", Econometrica 48, pp. 817-838.

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White, Halbert [1980]: "A Heteroskedasticity Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, 817-838.

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White, H. 1980. A Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica. 48(4) 817-838.

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White, Halbert, 1980, "A Heteroskedasticity-Consistent Covariance Matrix Estimator and Direct Test of Heteroskedasticity," Econometrica 48: 817-38.

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White, H. (1980), "A heteroskedasticity-consistent covariance matrix estimator and a direst test for heteroskedasticity," Econometrica 48, 817--30. 34

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White, Halbert (1980), "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica 48, p. 817-838.

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White, H. (1980): A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity. Econometrica 48, 817-838. 8 \Deltay t = OE 0 + OE 1 \Deltay t;1 + oeffl t

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White, Halbert, 1980, "A Heteroskedasticity Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, 817838.

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White, H. (1980), "Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct test for Heteroskedasticity", Econometrica, 48, 817-836.

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White, H. 1980. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 817-838.

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White, Halbert, 1980, "A Heteroskedasticity Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, 817838. 36

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White, H. (1980), "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, 817-838.

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White, H., 1980. A Heteroskedasticity-Consistent Covariance Matrix Estimator and Direct Test for Heteroskedasticity. Econometrica, 48: 817-838.

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White, H.,1980, A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity, Econometrica, 48, 817-838.

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White, Halbert (1980). "A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity", Econometrica, 48, 817-838.

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