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7.2 Ordered Logit model

Mathematical representation

yi = x iβ + 𝜀 i,𝜀i Logistic 0,1 yi = 1 if γ0 < yi γ1 2 if γ1 < yi γ2 M if γ M1 < yi γ M (7.2)

Priors




Parameter Probability density function Default hyperparameters



β p β = |Pβ|12 2πK2 exp 1 2 β mβ P β β mβ mβ = 0K, Pβ = 0.001 IK
δ p δ = |Pδ|12 2πM2 2 exp 1 2 δ mδ P δ δ mδ mδ = 0M2, Pδ = 0.001 IM2



Syntax

[<model name> = ] ologit( y ~ x1 x2  xK [, <options> ] );

where:

PIC The dependent variable, y, in the dataset used for estimation must contain only consecutive integer values, with the numbering starting at 1. 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 yi = 2), then an error is produced.

The optional arguments for the ordered Logit model are:2

Gibbs parameters


"chains"

number of chains to run in parallel (positive integer); the default value is 1

"burnin"

number of burn-in draws per chain (positive integer); the default value is 10000

"draws"

number of retained draws per chain (positive integer); the default value is 20000

"thin"

value of the thinning parameter (positive integer); the default value is 1

"seed"

value of the seed for the random-number generator (positive integer); the default value is 42

Hyperparameters


"m_beta"

mean vector of the prior for β (K ×1 vector); the default value is 0K

"P_beta"

precision matrix of the prior for β (K ×K symmetric and positive-definite matrix); the default value is 0.001 IK

"m_delta"

mean vector of the prior for δ ( M 2 ×1 vector); the default value is 0M2

"P_delta"

precision matrix of the prior for δ ( M 2 ×M 2 symmetric and positive-definite matrix); the default value is 0.001 IM2

Dataset and log-marginal likelihood


"dataset"

the id value of the dataset that will be used for estimation; the default value is the first dataset in memory (in alphabetical order)

"logML_CJ"

boolean indicating whether the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood should be calculated (true|false); the default value is false

Reported Parameters




β

variable_name

vector of parameters associated with the independent variables

γ

gamma_m

vector of cutoff points (M 2)




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

Samples

a matrix containing the draws from the posterior of β and γ

x1,,xK

vectors containing the draws from the posterior of the parameters associated with variables x1,,xK (the names of these vectors are the names of the variables that were included in the right-hand side of the model)

gamma_2,,
  gamma_{M-1}

vectors containing the draws from the posterior of the cutoff parameters, for m = 2,,M 1

logML

the Lewis & Raftery (1997) approximation of the log-marginal likelihood

logML_CJ

the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood; this is available only if the model was estimated with the "logML_CJ"=true option

nchains

the number of chains that were used to estimate the model

nburnin

the number of burn-in draws per chain that were used when estimating the model

ndraws

the total number of retained draws from the posterior ( =chains draws)

nthin

value of the thinning parameter that was used when estimating the model

nseed

value of the seed for the random-number generator that was used when estimating the model

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

The ordered Logit model uses the mfx() function to calculate and report the marginal effects of the independent variables on the probability of the response variable being in each one of the M categories: Prob y = m|x, for m = 1,2,,M. 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 an ordered Logit model is:

mfx( ["type"=1] [, "point"=<point of calculation>] [, "model"=<model name>] );

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

Examples

Example 1

myData = import("$BayESHOME/Datasets/dataset10.csv"); 
myData.constant = ones(rows(myData), 1); 
 
ologit( y ~ constant x1 x2 x3 x4 );

Example 2

myData = import("$BayESHOME/Datasets/dataset10.csv"); 
myData.constant = ones(rows(myData), 1); 
 
myModel = ologit( y ~ constant x1 x2 x3 x4, 
    "m_beta"=zeros(5,1), "P_beta" = 0.01*eye(5,5), 
    "m_delta"=zeros(3,1), "P_delta" = 0.1*eye(3,3), 
    "burnin"=10000, "draws"=40000, "thin"=4, "chains"=2, 
    "logML_CJ" = true, "dataset"=myData); 
 
diagnostics("model"=myModel); 
 
mfx("point"="mean","model"=myModel); 
mfx("point"="median","model"=myModel);

2Optional arguments are always given in option-value pairs (eg. "chains"=3).

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© 2016–20 Grigorios Emvalomatis