6.8 Conditional Logit

Mathematical representation

$$\begin{array}{cc}{p}_{mi}=Prob\left(y_i=m|{\mathbf{z}}_{mi}^{\prime }\delta +{\mathbf{x}}_i^{\prime }{\beta }_m\right),\hfill & m=1,\dots ,M\hfill \\ p_{0i}=1-\sum _{m=1}^M{p}_{mi},\hfill & \text{with additional structure on each }{p}_{mi}\hfill \end{array}$$

(6.15)

• the model is estimated using $N$ observations
• $y_i$ is the value of the dependent variable for observation $i$ and it can take $M+1$ values: 0, 1, …, M
• ${\mathbf{z}}_{mi}$ is a $K\times 1$ vector that stores the values of the $K$ independent variables for observation $i$, which are specific to alternative $m$
• ${\mathbf{x}}_i$ is an $L\times 1$ vector that stores the values of the $L$ independent variables for observation $i$, which are common to all alternatives
• $\delta$ is a $K\times 1$ vector of parameters, common to all alternatives
• ${\beta }_m$ is an $L\times 1$ vector of parameters for alternative $m$, $m=1,2,\dots ,M$
• there are $J=K+M\cdot L$ parameters to be estimated

The latent-variable representation of the conditional Logit is:

$$\begin{array}{ccccc}\hfill y_{1i}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{z}}_{1i}^{\prime }\delta +{\mathbf{x}}_i^{\prime }{\beta }_1\hfill & \hfill +\hfill & \hfill {𝜀}_{1i}\hfill \\ \hfill y_{2i}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{z}}_{2i}^{\prime }\delta +{\mathbf{x}}_i^{\prime }{\beta }_2\hfill & \hfill +\hfill & \hfill {𝜀}_{2i}\hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill y_{Mi}^{\ast }\hfill & \hfill =\hfill & \hfill {\mathbf{z}}_{Mi}^{\prime }\delta +{\mathbf{x}}_i^{\prime }{\beta }_M\hfill & \hfill +\hfill & \hfill {𝜀}_{Mi}\hfill \\ \hfill \hfill \end{array}{y}_i=\left\{\begin{array}{ccc}\hfill 0\hfill & \hfill \text{if}\hfill & \underset{j}{max}\left\{y_{ji}^{\ast }\right\}\le 0\hfill \\ \hfill 1\hfill & \hfill \text{if}\hfill & \underset{j}{max}\left\{y_{ji}^{\ast }\right\}=y_{1i}^{\ast }\hfill & \hfill \text{and}\hfill & \hfill y_{1i}^{\ast }>0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill M\hfill & \hfill \text{if}\hfill & \underset{j}{max}\left\{y_{ji}^{\ast }\right\}=y_{Mi}^{\ast }\hfill & \hfill \text{and}\hfill & \hfill y_{Mi}^{\ast }>0\hfill \end{array}\right.$$

(6.16)

Let:

The ${𝜀}_{mi}$s are assumed to follow a multivariate Logistic distribution (Malik & Abraham, 1973) with mean zero.

Priors

Syntax

where:

• y is the dependent variable name, as it appears in the dataset used for estimation
• z1 z2 $\dots$zK is a list of the names, as they appear in the dataset used for estimation except for the alternative index, of the independent variables which are associated with variables that vary by alternative; for each ‘zk’ variable, the dataset must contain $M+1$ variables whose names start by zk and followed by an underscore and the index of the alternative to which the variable corresponds (counting starting at zero)
• x1 x2 $\dots$xL is a list of the names, as they appear in the dataset used for estimation, of the independent variables which are common to all alternatives; when a constant term is to be included in the model, this must be requested explicitly

The dependent variable, y, in the dataset used for estimation must contain only consecutive integer values, with the numbering starting at 0 (base category). Observations with missing values in y are dropped during estimation, but if a non-integer numerical value is encountered or if the integer values are not consecutive (for example there are no observations for which $y_i=1$), then an error is produced.

The optional arguments for the conditional Logit model are:⁸

Reported Parameters

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a conditional Logit model is created, then the following results are saved in the model item and are accessible via the ‘.’ operator:

Additionally, the following functions are available for post-estimation analysis (see section B.14):

• diagnostics()
• test()
• pmp()
• mfx()
• predict()

The conditional Logit model uses the mfx() function to calculate and report the marginal effects of the independent variables on the probability of each outcome, $m$, occurring. Because the model calculates only one type of marginal effects, the only valid value for the "type" option is 1. The generic syntax for a statement involving the mfx() function after estimation of a conditional Logit model is:

See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

The conditional Logit model uses the predict() function to generate predictions of the probability each of the $M+1$ outcomes occuring. Because the model generates only one type of predictions, the only valid value for the "type" option is 1. The generic syntax for a statement involving the predict() function after estimation of a conditional Logit model is:

See the general documentation of the predict() function (section B.14) for details on the other optional arguments.

Examples

Example 1

Example 2

Share this content:

© 2016–20 Grigorios Emvalomatis