...- 
help for ^htest^, ^szroeter^, and ^white^           (Jeroen Weesie/ICS 
Dec 10, 1996) ..-

Tests for heteroscedasticity after fit - --------------------------------------

^htest^ [varlist] [^, r^hs ^u^nivar ]

^szroeter^ varname

^white^ [^, p^reserve ]

^htest^, ^szroeter^, and ^white^ are for use after ^fit^; see help ^fit^.

See the notes below for a discussion of the interpretation of test results in case of weighted regression.

Description - -----------

^htest^, ^szroeter^, and ^white^ provide tests for the assumption of the linear regression model that the residuals e are homoscedastic, i.e., have constant variance. The tests differ with respect to the specification of residual variances under the alternative hypothesis.

^white^ considers the general (unrestricted) alternative hypothesis in which no assumptions is made on the residual variances (White 1980).

^szroeter^ considers the alternative hypothesis that the residual variances are monotonically increasing in some variable (Szroeter 1978).

^htest^ performs standard score tests for H: b=0 for two parametric forms of heteroscedasticity:

^multiplicative heteroscedastcity^ (Cook & Weinberg 1983)

var(y) = s^^2 * exp( b1.z1 + b2.z2 + .. + bk.zk)

^random coefficient heteroscedasticity^ (REF)

var(y) = s^^2 + b1.z1^^2 + b2.z2^^2 + .. + bk.zk^^2

If no varlist is specified, ^htest^ uses the fitted values of the regression.

^szoeter^ and ^htest^ test homoscedasticity against specific models (monotonicity; multiplicative and random coefficient heteroscedasticty) under the alternative hypothesis. Due to the general structure of the alternative, ^white^ is not very powerfull against any particular form of heteroscedasticity. Options (^htest^) - -------

^rhs^ specifies that the independent variables of the ^fit^ted regression should be used.

^univar^ specifies that univariate score tests for each of the variables is displayed in addition to the "combined" test.

If ^htest^ suggest multiplicative forms of heteroscedasticity, regression models can be estimated using ^regmh^ or ^regmhv^ . The random coefficient model is currently not supported.

Options (^white^) - -------

^preserve^ specifies that the ^white^ may ^preserve^ the data in memory and drop all variables and cases that are not needed in the calculations. This is costly for large data sets. However, ^white^ has to generate k(k+1)/2 temporary variables, where k = #rhs-vars in the ^fit^ted model.

Examples - -------- . ^fit income edu sex exp exp2 industry^ (fit a regression model)

. ^white^

. ^szroeter edu^ (variance increases in edu?)

. ^htest^ (homoscedasticity in fitted values) . ^htest edu sex exp exp2 industry^ (homoscedasticity in rhs-vars) . ^htest, rhs^ (shorthand for the previous command) . ^htest edu exp age, univar^ (display univariate tests as well)

Details (^white^) - -------

White (1980) noted that the classical estimate

Vc = s^2 inv(X'X)

of the variance V of OLS regression estimates and the Huber/White/robust estimate

Vr = X' inv(sum e_i^2 x_i x_i') X

of V are likely to be close under homoscedasticity. White (1980) proposed a Hausmann-type test of the matrix difference Vc-Vr as a test for homoscedasticy against general forms of heteroscedasticity. (Actually, White showed that Vc and Vr may also be close under particular forms of hetero-scedasticty as well. See White (1980) for details.)

Details (^szroeter^) - -------

Szoeter's class of tests for homoscedasticity against the alternative that the residual variance increases in some variable x are defined in terms of

sum_i h(x(i)) e(i)^^2 H = -------------------- i=1..n sum_i e(i)^^2

where h(x) is some weight function that increases in x. Note that H can be interpreted as a weighted avarege of the h(x), weighted by the squares residuals. Under homoscedasticty, H should be approximately equal to the unweighted average of h(x). Large values of H suggest that e(i)^^2 tends to be large where h(x) is large, i.e., the variance indeed increases in x, while small values of H suggest that the variance actually decreases in x.

King (1982) suggest to use h(x(i)) = rank(x(i) in x(1)..x(n)). The Q-statistic displayed by ^szroeter^ is a transformation of H that is asymptotically standard normal distributed under homoscedasticity.

^szroeter^ displays a Q-statistic (Judge 1985:452). Under homoscedasticity, Q is approximately N(0,1) distributed. Q>2 suggests that the residual variance increases in varname, while Q<-2 suggests that it decreases in varname.

Homoscedastity and weighted regression - --------------------------------------

References - ----------

Cook, R.D., and S. Weisberg (1983) Diagnostics for heteroscedasticity in regression. Biometrica 70: 1-10.

Judge, G.G. et al. (1985) The Theory and Practice of Econometrics. 2nd Ed. New York: Wiley.

Szroeter, J. (1978) A Class of Parametric Tests for Heteroscedasticity in Linear Econometric Models. Econometrica 46, 1311-28.

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

Saved results - -------------

^htest^ saves results for the final (multivariate) score tests in

scalar ^S_3^ #df for the asymptotic chi-square distribution under Ho. scalar ^S_4^ score statistic scalar ^S_5^ #df for the asymptotic chi-square distribution under Ho. scalar ^S_6^ score statistic

^white^ saves results in

scalar ^S_1^ test statistic scalar ^S_2^ #df for the asymptotic chi-square distribution under Ho.

^szroeter ^ saves results in

scalar ^S_1^ test statistic

Also See - --------

Manual: [R] fit On-line: help for @fit@, @regmh@, @probith@, @_tscore@