# Fitting the Multinomial Probit Model via Markov chain Monte Carlo

### Description

`mnp`

is used to fit (Bayesian) multinomial probit
model via Markov chain Monte Carlo. `mnp`

can also fit the model
with different choice sets for each observation, and complete or
partial ordering of all the available alternatives. The computation
uses the efficient marginal data augmentation algorithm that is
developed by Imai and van Dyk (2005a).

### Usage

1 2 3 4 |

### Arguments

`formula` |
A symbolic description of the model to be fit specifying the response variable and covariates. The formula should not include the choice-specific covariates. Details and specific examples are given below. |

`data` |
An optional data frame in which to interpret the variables
in |

`choiceX` |
An optional list containing a matrix of choice-specific covariates for each category. Details and examples are provided below. |

`cXnames` |
A vector of the names for the choice-specific covariates
specified in |

`base` |
The name of the base category. For the standard multinomial probit model, the default is the lowest level of the response variable. For the multinomial probit model with ordered preferences, the default base category is the last column in the matrix of response variables. |

`latent` |
logical. If |

`invcdf` |
logical. If |

`trace` |
logical. If |

`n.draws` |
A positive integer. The number of MCMC draws. The
default is |

`p.var` |
A positive definite matrix. The prior variance of the
coefficients. A scalar input can set the prior variance to the
diagonal matrix whose diagonal element is equal to that value. The
default is |

`p.df` |
A positive integer greater than |

`p.scale` |
A positive definite matrix. When |

`coef.start` |
A vector. The starting values for the coefficients.
A scalar input sets the starting values for all the coefficients
equal to that value. The default is |

`cov.start` |
A positive definite matrix. When |

`burnin` |
A positive integer. The burnin interval for the Markov
chain; i.e., the number of initial Gibbs draws that should not be
stored. The default is |

`thin` |
A positive integer. The thinning interval for the Markov
chain; i.e., the number of Gibbs draws between the recorded values
that are skipped. The default is |

`verbose` |
logical. If |

### Details

To fit the multinomial probit model when only the most
preferred choice is observed, use the syntax for the formula, ```
y
~ x1 + x2
```

, where `y`

is a factor variable indicating the most
preferred choice and `x1`

and `x2`

are individual-specific
covariates. The interactions of individual-specific variables with each
of the choice indicator variables will be fit.

To specify choice-specific covariates, use the syntax,
```
choiceX=list(A=cbind(z1, z2), B=cbind(z3, z4), C=cbind(z5,
z6))
```

, where `A`

, `B`

, and `C`

represent the choice
names of the response variable, and `z1`

and `z2`

are each
vectors of length *n* that record the values of the two
choice-specific covariates for each individual for choice A, likewise
for `z3`

, *…*, `z6`

. The corresponding variable names via
`cXnames=c("price", "quantity")`

need to be specified, where `price`

refers to the coefficient
name for `z1`

, `z3`

, and `z5`

, and `quantity`

refers to that for `z2`

, `z4`

, and `z6`

.

If the choice set varies from one observation to another, use the
syntax, `cbind(y1, y2, y3) ~ x1 + x2`

, in the case of a
three choice problem, and indicate unavailable alternatives by
`NA`

. If only the most preferred choice is observed, `y1`

,
`y2`

, and `y3`

are
indicator variables that take on the value one for individuals who
prefer that choice and zero otherwise. The last column of the
response matrix, `y3`

in this particular example syntax, is
used as the base category.

To fit the multinomial probit model when the complete
or partial ordering of the available alternatives is recorded, use
the same syntax as when the choice set varies (i.e., ```
cbind(y1, y2,
y3, y4) ~ x1 + x2
```

). For each observation, all the available
alternatives in the response variables should be numerically ordered
in terms of preferences such as `1 2 2 3`

. Ties are allowed. The
missing values in the response variable should be denoted by
`NA`

. The software will impute these missing values using the
specified covariates. The resulting uncertainty estimates of the
parameters will properly reflect the amount of missing data. For
example, we expect the standard errors to be larger when there is more
missing data.

### Value

An object of class `mnp`

containing the following elements:

`param` |
A matrix of the Gibbs draws for each parameter; i.e., the coefficients and covariance matrix. For the covariance matrix, the elements on or above the diagonal are returned. |

`call` |
The matched call. |

`x` |
The matrix of covariates. |

`y` |
The vector or matrix of the response variable. |

`w` |
The three dimensional array of the latent variable, W. The first dimension represents the alternatives, and the second dimension indexes the observations. The third dimension represents the Gibbs draws. Note that the latent variable for the base category is set to 0, and therefore omitted from the output. |

`alt` |
The names of alternatives. |

`n.alt` |
The total number of alternatives. |

`base` |
The base category used for fitting. |

`invcdf` |
The value of |

`p.var` |
The prior variance for the coefficients. |

`p.df` |
The prior degrees of freedom parameter for the covariance matrix. |

`p.scale` |
The prior scale matrix for the covariance matrix. |

`burnin` |
The number of initial burnin draws. |

`thin` |
The thinning interval. |

### Author(s)

Kosuke Imai, Department of Politics, Princeton University kimai@Princeton.Edu, http://imai.princeton.edu; David A. van Dyk, Department of Statistics, University of California, Irvine dvd@uci.edu, http://www.ics.uci.edu/~dvd.

### References

Imai, Kosuke and David A. van Dyk. (2005a) “A Bayesian Analysis
of the Multinomial Probit Model Using the Marginal Data
Augmentation,” *Journal of Econometrics*, Vol. 124, No. 2
(February), pp.311-334.

Imai, Kosuke and David A. van Dyk. (2005b) “MNP: R Package for
Fitting the Multinomial Probit Models,” *Journal of Statistical
Software*, Vol. 14, No. 3 (May), pp.1-32.

Burgette, L.F. and E.V. Nordheim. (2009). “An alternate identifying
restriction for the Bayesian multinomial probit model,” *Technical
report*, Department of Statistics, University of Wisconsin, Madison.

### See Also

`coef.mnp`

, `cov.mnp`

, `predict.mnp`

,
`summary.mnp`

; MNP home page at
http://imai.princeton.edu/research/MNP.html

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ```
###
### NOTE: this example is not fully analyzed. In particular, the
### convergence has not been assessed. A full analysis of these data
### sets appear in Imai and van Dyk (2005b).
###
## load the detergent data
data(detergent)
## run the standard multinomial probit model with intercepts and the price
res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice,
Wisk=WiskPrice, EraPlus=EraPlusPrice,
Solo=SoloPrice, All=AllPrice),
cXnames = "price", data = detergent, n.draws = 500, burnin = 100,
thin = 3, verbose = TRUE)
## summarize the results
summary(res1)
## calculate the quantities of interest for the first 3 observations
pre1 <- predict(res1, newdata = detergent[1:3,])
## load the Japanese election data
data(japan)
## run the multinomial probit model with ordered preferences
res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan,
verbose = TRUE)
## summarize the results
summary(res2)
## calculate the predicted probabilities for the 10th observation
## averaging over 100 additional Monte Carlo draws given each of MCMC draw.
pre2 <- predict(res2, newdata = japan[10,], type = "prob", n.draws = 100,
verbose = TRUE)
``` |