Standard Errors of Fitted Category Probabilities by the Delta Method for the Nested Logit Model

An Example

I’ll use the following example to illustrate the results in this document: Suppose that we have a three-category response variable \(Y\) with categories \(A\), \(B\), and \(C\), and define the two nested dichotomies \(Y_1\) coded 0 or 1 for categories \(\{A\}\) and \(\{B, C\}\), respectively, and \(Y_2\) coded 0 and 1 for categories \(\{B\}\) and \(\{C\}\). Then \(\psi_1 = \Pr(Y_1 = 1) = \Pr(\{B, C\})\); \(1 - \psi_1 = \Pr(Y_1 = 0) = \Pr(\{A\})\). As well, \(\mathcal{M}_A = \{1\}\) and \(\mathcal{M}_B = \mathcal{M}_C = \{1, 2\}\). Consequently, \[\begin{align*} \phi_A &= \psi_{1, A_1} = 1 - \psi_{1}\\ \phi_B &= \psi_{1, B_1}\psi_{2, B_2} = \psi_{1}(1 - \psi_{2}) \\ \phi_C &= \psi_{1, C_1}\psi_{2, C_2} = \psi_{1}\psi_{2} \end{align*}\] Here, I abuse the notation slightly in the interest of clarity, using letters rather than numbers for the response categories, so the index of response categories, \(k\), takes on the values \(A\), \(B\), and \(C\), rather than 1, 2, and 3.

For the Dichotomous Logit Models

The estimated probability of success \(\psi_{j}\) for the \(j\)th dichotomous logit model is \[\begin{equation*} \psi_{j} = \frac{1}{1 + e^{-\lambda_{j}}} \end{equation*}\] Then \[\begin{align*} \lambda_{j} &= \alpha^{(j)} + \beta_1^{(j)} x_1 + \cdots + \beta_p^{(j)} x_p \\ &= \mathbf{x}^T \boldsymbol{\beta}^{(j)} \end{align*}\] is a function of the regression coefficients, where \(\mathbf{x}^T = [1, x_1, \ldots, x_p]\) (an arbitrary vector of values of the regressors) and \(\boldsymbol{\beta}^{(j)} = [\alpha^{(j)}, \beta_1^{(j)}, \ldots, \beta_p^{(j)}]^T\). The probability of failure is \[\begin{equation*} 1 - \psi_{j} = \frac{1}{1 + e^{\lambda_{j}}} \end{equation*}\] The variance of the logit is \(V(\lambda_{j}) = \mathbf{x}^T V(\boldsymbol{\beta}^{(j)}) \mathbf{x}\),

The derivatives of \(\psi_{j}\) and \(1 - \psi_{j}\) with respect to \(\lambda_{j}\) are \[\begin{align*} \frac{d\psi_{j}}{d\lambda_{j}} &= \frac{e^{-\lambda_{j}}}{\left( 1 + e^{-\lambda_{j}} \right)^2} \\ \frac{d \left(1 - \psi_{j} \right)}{d\lambda_{j}} &= - \frac{e^{\lambda_{j}}}{\left( 1 + e^{\lambda_{j}} \right)^2} \end{align*}\]

By the univariate delta method, \[\begin{align*} V(\psi_{j}) &\approx \left( \frac{d\psi_{j}}{d\lambda_{j}} \right)^2 V(\lambda_{j}) \\ &= \left[ \frac{e^{-\lambda_{j}}}{\left( 1 + e^{-\lambda_{j}} \right)^2} \right]^2 V(\lambda_{j})\\ V(1 - \psi_{j}) &\approx \left[ \frac{d \left(1 - \psi_{j} \right)}{d\lambda_{j}} \right]^2 V(\lambda_{j}) \\ &= \left[ \frac{e^{\lambda_{j}}}{\left( 1 + e^{\lambda_{j}} \right)^2}\right]^2 V(\lambda_{j}) \\ &= V(\psi_{j}) \end{align*}\]

For the Nested Logit Model

The variances of the estimated response-category probabilities for the polytomous response can be obtained similarly by the multivariate delta method, recognizing that these probabilities are products of the dichotomous probabilities. The result is greatly simplified because the dichotomies are independent, and so the covariance matrix of the estimated dichotomous probabilities is diagonal.

The required derivatives are \[\begin{equation*} \frac{\partial \phi_k}{\partial \psi_{j, k_j} } = \prod_{j' \in \mathcal{M}_k - \{j\}} \psi_{j', k_j} \end{equation*}\] for \(j \in \mathcal{M}_k\) and \(k = 1, \ldots, m\). Here, \(-\) denotes set difference. Because \(V(\psi_{j}) = V(1 - \psi_{j})\), it’s always the case that \(V(\psi_{j, k_j}) = V(\psi_{j})\), and so \[\begin{align*} V(\phi_k) &\approx \sum_{j \in \mathcal{M}_k} \left( \frac{\partial \phi_k}{\partial \psi_{j, k_j} } \right)^2 V\left( \psi_{j} \right) \\ &= \sum_{j \in \mathcal{M}_k} \left( \prod_{j' \in \mathcal{M}_k - \{j\}} \psi_{j', k_j} \right)^2 V\left( \psi_{j} \right) \end{align*}\] for \(k = 1, \ldots, m\).

Applying these results to the example, recall, first, that \(\mathcal{M}_A = \{1\}\) and so the set for the product \(\prod_{j' \in \mathcal{M}_{A} - \{j\}} \phi^{(j', A_j)}\), that is, \(j' \in \{1\}-\{1\}\), is empty. In this case, the product is taken = 1, and \(V(\phi_A) = V(\psi_{1})\). That makes intuitive sense, because, as noted previously, \(\phi_A = 1 - \psi_{1}\).

Proceeding with \(B\) and \(C\), \(\mathcal{M}_B = \mathcal{M}_C = \{1, 2\}\). Consequently, each product \(\prod_{j' \in \mathcal{M}_{B} - \{j\}} \psi_{j', B_j}\) and \(\prod_{j' \in \mathcal{M}_{C} - \{j\}} \psi_{j', C_j}\) has only one term, for \(j' = 2\) in the case of \(B\) or \(j' = 1\) in the case of \(C\): \[\begin{align*} V(\phi_B) &= \psi_{2, B_2} V(\psi_{1}) + \psi_{1, B_1} V(\psi_{2}) \\ &= (1 - \psi_{2}) V(\psi_{1}) + \psi_{1} V(\psi_{2}) \\ V(\phi_C) &= \psi_{2, C_2} V(\psi_{1}) + \psi_{1, C_1} V(\psi_{2}) \\ &= \psi_{2} V(\psi_{1}) + \psi_{1} V(\psi_{2}) \\ \end{align*}\]

Yet another application of the delta method produces approximate variances for the individual-category logits. The relevant derivative is \[\begin{equation*} \frac{d\Lambda_k}{d\phi_k} = \frac{1}{\phi_k(1 - \phi_k)} \end{equation*}\] for \(k = 1, \ldots, m\), and so \[\begin{align*} V(\Lambda_k) &\approx \left( \frac{d\Lambda_k}{d\phi_k} \right)^2 V(\phi_k) \\ &= \left[ \frac{1}{\phi_k(1 - \phi_k)} \right]^2 V(\phi_k) \end{align*}\]