Before discussing how single index models can be estimated,
we first have to mention an important caveat you should always keep
in mind when interpreting estimation results. Consider the following
binary response model with logit link
and linear
index function
:
(For the sake of simplicity we restrict ourselves to a one-dimensional
explanatory variable
.)
The previous equations yield
 |
(6.2) |
Note that in this model the parameters
and
control
location and scale of the link function, respectively,
whereas the parameters
and
represent
intercept and scale of the index
. We will show now that
without further specification
and
cannot be
identified.
First, we see that
and
cannot be estimated
separately but the difference
can. To make this
clear, add an arbitrary real constant
to both the intercept of
the index and the location parameter of the link function:
It is easily verified that this specification leads to the
identical model as in (6.2). As a result, we will
clearly not be able to empirically distinguish between the two
specifications, i.e. between the intercept parameters
and
. We conclude that neither for the intercept of the
index nor for the location of the link function can a
unique estimate be found. For further identification we set
, respectively the mean of this link to zero, i.e.
.
Next, we demonstrate that for arbitrary
the slope
coefficient
cannot be identified either. Multiplying all
coefficients by a non-zero constant
yields:
Again, the resulting regression
will empirically not
be distinguishable from the original specification
(6.2).
would normalize the scale
of the link function to 1. But since
can take any non-zero value,
we are again faced with the inability to find a unique
estimate of the coefficients
,
and
. In this case, these parameters
are said to be identified up to scale.
The model defined by the normalization
and
is labeled standard logit model. This is the typical normalization
used for logit analysis. Other normalizations are
conceivable. Note that in treatments of the SIM (unknown link)
many authors assume
to be part of the
nonparametric function
. Moreover, in cases with several
explanatory variables one of the following scale normalizations
can be applied:
- set one of the slope coefficients equal to 1, or
- set the length of the coefficient vector to 1.
Hence, normalization in the standard
logit model usually differs from that made for the SIM.
Consequently, estimated coefficients for
logit/probit and SIM can be only compared if the
same type of normalization is applied to both
coefficient vectors.
An additional identification problem arises
when the distribution of the error term in the
latent-variable model (5.6) depends on the
value of the index. We have already seen some implications of this
type of heteroscedasticity in Example 1.6.
To make this point clear, consider the latent variable
with
an unknown function and
a standard logistic error term independent of
.
(In Example 1.6 we studied a linear index function
combined with
.)
The same calculation as for (5.7) (cf. Exercise 5.1)
shows now
![$\displaystyle P(Y=1 \mid {\boldsymbol{X}}={\boldsymbol{x}}) = E(Y \mid {\boldsy...
...\boldsymbol{x}})}{\varpi\{v_{\boldsymbol{\beta}}({\boldsymbol{X}})\}}\right]\,.$](spmhtmlimg1794.gif) |
(6.3) |
This means, even if we know the functional form of the link
(
is the standard logistic cdf) the regression function
is unknown because of its unknown component
.
As a consequence, the resulting link function
is not necessarily monotone increasing any more. For
instance, in Figure 1.9 we plotted a graph of the link function
as a result of the heteroscedastic error term.
Figure 6.1:
Two link functions
|
For another very simple example, consider Figure 6.1.
Here, we have drawn two link functions,
(upward sloping) and
(downward sloping).
Note that both functions are symmetric (the
symmetry axis is the dashed vertical line through the origin).
Thus,
and
are related to each
other by
Obviously now, two distinct
index values,
and
will yield the same values
of
if the link functions are chosen accordingly.
Since the two indices differ only in that the coefficients
have different sign, we conclude that the coefficients are
identified only up to scale and sign
in the most general case.
In summary, to compare the effects of two particular
explanatory variables, we can only use the ratio of their coefficients.