inverse of a sum of standard normal cdfs

[jburto2]

I'm trying to derive an estimate of delta[sub:2sypu7vi]t[/sub:2sypu7vi] in a logistic transition model. Here is what is known:

E(Y[sub:2sypu7vi]it[/sub:2sypu7vi]| y[sub:2sypu7vi]it-1[/sub:2sypu7vi]) = {[e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi]+alpha*y[sub:2sypu7vi]it-1[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi]+alpha*y[sub:2sypu7vi]it-1[/sub:2sypu7vi][/sup:2sypu7vi]]} + {[e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]}

and

E(Y[sub:2sypu7vi]it[/sub:2sypu7vi]) = [e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]

I know that:
[e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]] = E{[e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi]+alpha*y[sub:2sypu7vi]it-1[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi]+alpha*y[sub:2sypu7vi]it-1[/sub:2sypu7vi][/sup:2sypu7vi]]} + E{[e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]delta[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]}

Using a standard normal approximation to the standard logist cdf, I get:
[e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]] = PHI[(16*sqrt(3)/15*pi)*delta[sub:2sypu7vi]t[/sub:2sypu7vi]]/sqrt[1+(16*sqrt(3)/15*pi)[sup:2sypu7vi]2[/sup:2sypu7vi]*sigma[sup:2sypu7vi]2[/sup:2sypu7vi]] + PHI[(16*sqrt(3)/15*pi)*delta[sub:2sypu7vi]t[/sub:2sypu7vi]]

Where PHI is the standard normal cdf and pi=3.14...

If this is correct, is it possible to take an inverse of the sum of the two standard normal cdfs?

In other words, can I say that:
[(16*sqrt(3)/15*pi)*delta[sub:2sypu7vi]t[/sub:2sypu7vi]]/sqrt[1+(16*sqrt(3)/15*pi)[sup:2sypu7vi]2[/sup:2sypu7vi]*sigma[sup:2sypu7vi]2[/sup:2sypu7vi]] + [(16*sqrt(3)/15*pi)*delta[sub:2sypu7vi]t[/sub:2sypu7vi]] = PHI[sup:2sypu7vi]-1[/sup:2sypu7vi]{[e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]/[1+e[sup:2sypu7vi]mu[sub:2sypu7vi]t[/sub:2sypu7vi][/sup:2sypu7vi]]}

???

THANKS!

 

[royhaas]

The problem is that \(\displaystyle \Phi^{-1}\) is not a linear function, so your last statement is not correct.