SUMM/Review diagnostic and specification tests

[josef-pkt]

I wrote the diagnostic tests initially mainly base on Green and a few articles, mostly with translating formulas to code.
This was a long time ago, and it needs revisiting to see what those tests are actually doing, and to extend them beyond OLS.

3 kinds of test in terms of using different versions and options

• identical results (up to numerical precision) but different ways of computing them
• asymptotically equivalent results under some given assumptions, that might agree numerically in small samples only in special cases
• results that differ because different assumptions on maintained or auxiliary assumptions, i.e. robust to other specification issues.

Most of the, especially older, econometrics text books and articles are not very specific on this, e.g.
"We can estimate Sigma by ..."
"An asymptotically equivalent way of estimating Sigma is ..."

issues:

• centered versus uncentered rsquared
  Score/LM conditional moment tests #2096 (comment)

• projection + rsquared versus wald
  Green describes for projecting on some variables/exog that are unconstrained and then use uncetered rsquared for the OLS with projected variables. (My guess is now that if constant is the only absorbed variable, then this is identical to using centered rsquared, as in het_xxx.

• non-OPG versions
  all the auxiliary regression versions assume OPG. Davidson and MacKinnon argue that those don't have good small sample properties

• mix observed Hessian and orthogonal/blockdiagnonal property of expected Hessian.
  AFAIU in LEF we use the property that asymptotically or in expected terms variance or second moments have zero off diagonal block with respect to mean parameters. We often impose this even when we use the observed Hessian within blocks. E,g, het test don't take into account that mean parameter are estimated , at least not in the moment conditions. mean tests don't take variance second moment assumptions into account. I think in small samples, the off-diagonal block wouldn't be zero, and I think it wouldn't be zero even in expected terms if the LEF is not the true distribution (LEF property of block diagonality is still a maintained assumption).
  Related: Green 5th edition on nonlinear models explains that mean and variance don't have the blockdiagonal structure of the (asymptotic) hessian.