The probit model is a flexible and widely-used tool for the analysis of such discrete choice behavior. Choosing between alternatives is omnipresent in everyday life, from the choice of a vehicle for traveling to work over different brands in a supermarket to companies deciding among production plans. Many scientific areas apply the probit model for studying the driving factors behind decision makers’ choices, for example transportation (Bolduc 1999; Shin et al. 2015) and marketing (Allenby and Rossi 1998; Haaijer et al. 1998; Paap and Franses 2000). Estimating the probit model’s parameters traditionally is performed via maximizing the likelihood function numerically. With rising model complexity however, this approach becomes both computationally expensive and does not guarantee convergence to the global optimum.

We briefly formulate the probit model and its estimation and refer to Train (2009) and Bhat (2011) for further details. Say that \(N\) deciders choose among \(J \geq 2\) alternatives at each of \(T\) choice occasions. The values for \(J\) and \(T\) can be decider-specific, though we do not show this dependence in our notation. Let \(y_{nt} \in \{1,\dots,J\}\) label the choice of decider \(n\) at occasion \(t\). Assume that the choice was rational in the sense that \(y_{nt}\) yields the highest utility \(U_{nt}\) for \(n\) at \(t\). The probit model defines \[U_{nt} = X_{nt} \beta + \epsilon_{nt}\], where \(X_{nt}\) is a \(J\times P\)-matrix of \(P\) characteristics for each alternative, \(\beta\) is a coefficient vector of length \(P\) and \(\epsilon_{nt} \sim N(0,\Sigma)\) denotes the vector of jointly normal distributed unobserved influences. We ensure identifiability by taking utility differences and fixing one error-term variance. This implies that instead of \(\Sigma\), we estimate \(J(J-1)/2-1\) parameters of a transformed covariance matrix.

The researcher aims to estimate the values for \(b\) and \(\Sigma\), most commonly by the maximum likelihood method. Let \(\theta\) denote the vector of the \(P\) coefficients of \(b\) and \(J(J-1)/2-1\) identified parameters of \(\Sigma\). Note that the length of \(\theta\) rises quadratically with \(J\). The maximum likelihood estimate \(\hat{\theta}\) is obtained by solving \[\begin{equation} \label{eq:ll} \arg \max_\theta \log \sum_{n,t,j} 1(y_{nt} = j) \int 1(j = \arg \max U_{nt}) \phi(\epsilon_{nt}) d \epsilon_{nt}, \end{equation}\] where \(1(\cdot)\) denotes the indicator function and \(\phi(\cdot)\) the normal density. The integral part of does not have a closed-form expression and hence must be approximated numerically.

The {ino} package provides the function `sim_mnp()`

to
simulate data from a probit model. We simulate 10 data sets.

```
<- 100
N <- 10
T <- 3
J <- 3
P <- c(1,-1,0.5)
b <- diag(J)
Sigma <- function() {
X <- sample(0:1, 1)
class <- ifelse(class, 2, -2)
mean matrix(stats::rnorm(J*P, mean = mean), nrow = J, ncol = P)
}<- replicate(10, sim_mnp(
probit_data N = N, T = T, J = J, P = P, b = b, Sigma = Sigma, X = X
simplify = FALSE) ),
```

The following lines specify the `ino`

object. The
likelihood is computed via `f_ll_mnp()`

which is provided via
{ino}. Via the `global`

argument, we can specify the true
parameter vector thats leads to the global optimum. The
`mpvs = "data"`

input specifies that we want to loop over the
ten provided data sets.

```
<- attr(probit_data[[1]], "true")[-1]
true <- setup_ino(
probit_ino f = f_ll_mnp,
npar = 5,
global = true,
data = probit_data,
neg = TRUE,
mpvs = "data",
opt = set_optimizer_nlm(iterlim = 1000)
)
```

We initialize `runs = 100`

times randomly.

`<- random_initialization(probit_ino, runs = 100) probit_ino `

We initialize on a subset of proportion 20% and 50%, which was selected randomly and using kmeans, respectively.

```
for(how in c("random", "kmeans")) for(prop in c(0.2,0.5)) {
<- subset_initialization(
probit_ino arg = "data", how = how, prop = prop,
probit_ino, ind_ign = 1:3, initialization = random_initialization(runs = 100)
) }
```

3 optimization runs reached the iteration limit of 1000 iterations:

```
library("dplyr", warn.conflicts = FALSE)
summary(probit_ino, "iterations" = "iterations") %>% filter(iterations >= 1000)
#> # A tibble: 3 × 5
#> .strategy .time .optimum .optimizer iterations
#> <chr> <drtn> <dbl> <chr> <int>
#> 1 random 5290.284 secs 507. stats::nlm 1000
#> 2 subset(kmeans,0.2) 9330.853 secs 1151. stats::nlm 1000
#> 3 subset(kmeans,0.2) 9328.964 secs 975. stats::nlm 1000
```

We exclude them from further analysis:

```
<- which(summary(probit_ino, "iterations" = "iterations")$iterations >= 1000)
ind <- clear_ino(probit_ino, which = ind) probit_ino
```

`plot(probit_ino, by = ".strategy", time_unit = "mins", nrow = 1)`

We see that the subset initialization strategies reduce the computation time significantly, in comparison to the random initialization on the full data set.

Allenby, Greg M., and Peter E. Rossi. 1998. “Marketing Models of
Consumer Heterogeneity.” *Journal of Econometrics* 89 (1):
57–78.

Bhat, Chandra. 2011. “The Maximum Approximate Composite Marginal
Likelihood (MACML) Estimation of Multinomial Probit-Based Unordered
Response Choice Models.” *Transportation Research Part B:
Methodological* 45.

Bolduc, Denis. 1999. “A Practical Technique to Estimate
Multinomial Probit Models in Transportation.” *Transportation
Research Part B: Methodological* 33 (1): 63–79.

Haaijer, Rinus, Michel Wedel, Marco Vriens, and Tom Wansbeek. 1998.
“Utility Covariances and Context Effects in Conjoint MNP
Models.” *Marketing Science* 17 (3): 236–52.

Paap, Richard, and Philip Hans Franses. 2000. “A Dynamic
Multinomial Probit Model for Brand Choice with Different Long-Run and
Short-Run Effects of Marketing-Mix Variables.” *Journal of
Applied Econometrics* 15 (6): 717–44.

Shin, Jungwoo, Chandra R. Bhat, Daehyun You, Venu M. Garikapati, and Ram
M. Pendyala. 2015. “Consumer Preferences and Willingness to Pay
for Advanced Vehicle Technology Options and Fuel Types.”
*Transportation Research Part C: Emerging Technologies* 60.

Train, Kenneth. 2009. *Discrete Choice Methods with Simulation*.
2. ed. Cambridge Univ. Press.