6.4 Random-effects binary Logit

Mathematical representation

$$Prob\left(y_{it}=1\right)=\Lambda \left({\alpha }_i+{\mathbf{x}}_{it}^{\prime }\beta \right),{\alpha }_i\sim \mathrm{N}\left(0,\frac{1}{\omega }\right)$$

(6.7)

• the model is estimated using observations from $N$ groups, each group observed for $T_i$ periods (balanced or unbalanced panels); the total number of observations is ${\sum }_{i=1}^N{T}_i$
• $y_{it}$ is the value of the dependent variable for group $i$, observed in period $t$ and it can take two values: 0 and 1
• ${\mathbf{x}}_{it}$ is a $K\times 1$ vector that stores the values of the $K$ independent variables for group $i$, observed in period $t$
• $\beta$ is a $K\times 1$ vector of parameters
• ${\alpha }_i$ is the group-specific error term for group $i$
• $\omega$ is the precision of the group-specific error term: ${\sigma }_{\alpha }^2=\frac{1}{\omega }$
• $\Lambda \left(\cdot \right)$ is the standard-Logistic cdf

An equivalent representation uses the latent variable $y_{it}^{\ast }$:

$$\begin{array}{cc}\begin{array}{rl}{y}_{it}^{\ast }& ={\alpha }_i+{\mathbf{x}}_{it}^{\prime }\beta +{𝜀}_{it},{𝜀}_{it}\sim \mathrm{Logistic}\left(0,1\right),{\alpha }_i\sim \mathrm{N}\left(0,\frac{1}{\omega }\right),\\ y_{it}& =\left\{\begin{array}{ccc}1\hfill & \hfill \mathrm{if}\hfill & \hfill y_{it}^{\ast }>0\\ 0\hfill & \hfill \mathrm{if}\hfill & \hfill y_{it}^{\ast }\le 0\end{array}\right.\end{array}& \end{array}$$

(6.8)

The mean of the distribution of the ${\alpha }_i$s is restricted to zero and, therefore, these are simply group-specific errors terms. However, including a constant term in the set of independent variables is valid and leads to a specification equivalent to one where the group effects are draws from a normal distribution with mean equal to the parameter associated with the constant term and precision $\omega$.

Priors

Syntax

where:

• y is the dependent variable name, as it appears in the dataset used for estimation
• x1 x2 $\dots$xK is a list of the $K$ independent variable names, as they appear in the dataset used for estimation; 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 two values: 0 and 1 (with 1 indicating “success"). Observations with missing values in y are dropped during estimation, but if a numerical value other than 0 and 1 is encountered, then an error is produced.

BayES automatically drops from the sample used for estimation groups which are observed only once. This is because for these groups the group effect (${\alpha }_i$) cannot be distinguished from the error term (${𝜀}_{it}$).

The optional arguments for the random-effects binary Logit model are:⁴

Reported Parameters

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a random-effects binary 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()
• store()
• mfx()
• predict()

The random-effects binary Logit model uses the store() function to associate the group effects (alpha_i) with specific observations and store their values in the dataset used for estimation. The generic syntax for a statement involving the store() function after estimation of a random-effects binary Logit model is:

The random-effects binary Logit model uses the mfx() function to calculate and report the marginal effects of the independent variables on the probability of success. There are two types of marginal effects which can be requested by setting the "type" argument of the mfx() function equal to 1 or 2:

The generic syntax for a statement involving the mfx() function after estimation of a random-effects binary Logit model is:

and:

for calculating these two types of marginal effects. The default value of the "type" option is 1. See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

The random-effects binary Logit model uses the predict() function to generate predictions of the probability of success. There are two types of predictions which can be requested by setting the "type" argument of the mfx() function equal to 1 or 2:

The generic syntax for a statement involving the predict() function after estimation of a random-effects binary Logit model is:

and:

for generating these two types of predictions effects. The default value of the "type" option is 1. See the general documentation of the predict() function (section B.14) for details on the other optional arguments.

Examples

Example 1

Example 2

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