Misleading BarnettZehnwirth Documentation

[cdietrich215]

I think there could be some slightly misleading aspects of the BarnettZehnwirth documentation. Namely, I think that the documentation's inclusion of the valuation predictor does not actually impact results. LDFs do not differ by origin period.

As shown in the code below (run on chain ladder 0.8.24), the valuation predictor shown in the documentation does not impact the resulting LDFs, because there is aliasing between valuation, development, and origin. I believe this would happen whenever the valuation parameter is linear. To impact the model I think we'd need to make it slightly more complex (either stepwise or allowing more degrees of freedom).

import chainladder as cl

abc = cl.load_sample('abc')

# Linear coefficients for origin, development, and valuation
print('LDFs without valuation parameter')
display(cl.BarnettZehnwirth(formula='origin+development').fit(abc).ldf_)
print('LDFs with valuation parameter (original)')
display(cl.BarnettZehnwirth(formula='origin+development+valuation').fit(abc).ldf_)
print('LDFs with valuation parameter (fixed)')
# Possible fix?
display(cl.BarnettZehnwirth(formula='origin + development + valuation + I(valuation**2)').fit(abc).ldf_)

Additionally I think there is some LaTeX formatting issues.