6.1 Identification

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 $ G$ and linear index function $ v$:

$\displaystyle G(\bullet ) = \frac{1}{1+\exp\left(-\frac{\bullet -\gamma}{\tau}\right)}\,, \quad
v (x) = \beta_{0}+{\boldsymbol{\beta}}_{1}x.$

(For the sake of simplicity we restrict ourselves to a one-dimensional explanatory variable $ X$.) The previous equations yield

$\displaystyle E(Y\vert X=x) = \frac{1}{1+\exp\left(-\frac{\beta_{0}+\beta_{1}x-\gamma}{\tau}\right)}\,.$ (6.2)

Note that in this model the parameters $ \gamma$ and $ \tau$ control location and scale of the link function, respectively, whereas the parameters $ \beta_{0}$ and $ \beta_{1}$ represent intercept and scale of the index $ v$. We will show now that without further specification $ \gamma$ and $ \tau$ cannot be identified.

First, we see that $ \gamma$ and $ \beta_0$ cannot be estimated separately but the difference $ \beta_0-\gamma$ can. To make this clear, add an arbitrary real constant $ c$ to both the intercept of the index and the location parameter of the link function:

$\displaystyle G^+(\bullet ) = \frac{1}{1+\exp(-\frac{\bullet -(\gamma+c)}{\tau})}\,,
\quad v^+(x) = \beta_{0}+c+\beta_{1}x . $

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 $ \gamma$ and $ \beta_0$. 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 $ c=-\gamma$, respectively the mean of this link to zero, i.e. $ \gamma =0$.

Next, we demonstrate that for arbitrary $ \tau$ the slope coefficient $ \beta_1$ cannot be identified either. Multiplying all coefficients by a non-zero constant $ c$ yields:

$\displaystyle G^{\ast}(\bullet ) = \frac{1}{1+
\exp\left(-\frac{\bullet -\gamma c}{\tau c}\right)}\,,\quad
v^{\ast}(x) = \beta_{0}c+(\beta_{1}c)x . $

Again, the resulting regression $ E(Y\vert X=x)$ will empirically not be distinguishable from the original specification (6.2). $ c=1/\tau$ would normalize the scale of the link function to 1. But since $ c$ can take any non-zero value, we are again faced with the inability to find a unique estimate of the coefficients $ \tau$, $ \beta_0$ and $ \beta_1$. In this case, these parameters are said to be identified up to scale.

The model defined by the normalization $ \gamma =0$ and $ \tau=1$ 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 $ \beta_{0}$ to be part of the nonparametric function $ g(\bullet)$. Moreover, in cases with several explanatory variables one of the following scale normalizations can be applied:

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

$\displaystyle Y^*=v_{\boldsymbol{\beta}}({\boldsymbol{X}}) -\varepsilon, \quad
\varepsilon=\varpi\{v_{\boldsymbol{\beta}}({\boldsymbol{X}})\}\cdotp\zeta$

with $ \varpi(\bullet)$ an unknown function and $ \zeta$ a standard logistic error term independent of $ {\boldsymbol{X}}$. (In Example 1.6 we studied a linear index function $ v$ combined with $ \varpi(u)=\frac{1}{2}\{1+u^2\}$.) The same calculation as for (5.7) (cf. Exercise 5.1) shows now

$\displaystyle P(Y=1 \mid {\boldsymbol{X}}={\boldsymbol{x}}) = E(Y \mid {\boldsy...
...\boldsymbol{x}})}{\varpi\{v_{\boldsymbol{\beta}}({\boldsymbol{X}})\}}\right]\,.$ (6.3)

This means, even if we know the functional form of the link ($ G_{\zeta}$ is the standard logistic cdf) the regression function is unknown because of its unknown component $ \varpi(\bullet)$. As a consequence, the resulting link function

$\displaystyle g(\bullet)=G_\zeta\left[\frac{\bullet}{\varpi\{\bullet\}}\right]\,$

is not necessarily monotone increasing any more. For instance, in Figure 1.9 we plotted a graph of the link function $ g$ as a result of the heteroscedastic error term.

Figure 6.1: Two link functions
\includegraphics[width=1.2\defpicwidth]{SPM-two-link-functions.ps}

For another very simple example, consider Figure 6.1. Here, we have drawn two link functions, $ G(\bullet)$ (upward sloping) and $ G^{-}(\bullet)$ (downward sloping). Note that both functions are symmetric (the symmetry axis is the dashed vertical line through the origin). Thus, $ G(\bullet)$ and $ G^{-}(\bullet)$ are related to each other by

$\displaystyle G^{-}(u)=G(-u).$

Obviously now, two distinct index values, $ u={\boldsymbol{x}}^\top{\boldsymbol{\beta}}$ and $ u^{-}=-{\boldsymbol{x}}^\top{\boldsymbol{\beta}}$ will yield the same values of $ E(Y\vert{\boldsymbol{X}}={\boldsymbol{x}})$ 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.