The name is absent

Appendix 3 - A logistic transformation

One alternative way to overcome the criticism of assuming that the distance between
two notches is equal for every combination of sequential notches is to apply alternative
transformations besides the usual linear one. For instance, one could use either a logistic
or an exponential transformation.

The idea underlying the use of a logistic transformation is that at the middle of the scale,
ratings can rise rather quickly, as the sovereigns deliver some improvements. Both at
the bottom and top end of the rating scale, however, the increase of an additional notch
is slower, since the requisites of sovereign debt quality are more demanding.

If one assumes that the functional form that describes the relationship between the
creditworthiness rating, Ri, normalized to grade each of the countries on a scale of zero
to one, with zero representing the least creditworthy countries and one representing the
most creditworthy countries, and the set of explanatory variables, X, is the standard
conventional logistic form

(A3.1)

where the vector β includes the parameters of the exogenous variables. The logistic
transformation then becomes

L = ln[R₁ /(1 - R₁ )] = βX ^,

where L1 is the logit of R1.⁸ This equation is not only linear in X, but also linear in the
parameters and can be estimated using ordinary least squares.

Figure A3.1 compares the linear and the logistic transformation and in Table A3.1 we
present the values that we used alternatively in the logistic transformation. Table A3.2
reports the estimation results for the three rating agencies using the respective full panel

⁸Where R₁ = (21 -1) /( 2 × number of categories).

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