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

Tests for heteroscedasticity after fit
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        ^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
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^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^)
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^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^)
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^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
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    . ^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^)
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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^)
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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
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References
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    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
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^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
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 Manual:  [R] fit
On-line:  help for @fit@, @regmh@, @probith@, @_tscore@