estimate.affinity.matrix {affinitymatrix}

Estimate Dupuy and Galichon's model

Description

This function estimates the affinity matrix of the matching model of Dupuy and Galichon (2014), performs the saliency analysis and the rank tests. The user must supply a matched sample that is treated as the equilibrium matching of a bipartite one-to-one matching model without frictions and with Transferable Utility. For the sake of clarity, in the documentation we take the example of the marriage market and refer to "men" as the observations on one side of the market and to "women" as the observations on the other side. Other applications may include matching between CEOs and firms, firms and workers, buyers and sellers, etc.

Usage

estimate.affinity.matrix(
  X,
  Y,
  w = rep(1, N),
  A0 = matrix(0, nrow = Kx, ncol = Ky),
  lb = matrix(-Inf, nrow = Kx, ncol = Ky),
  ub = matrix(Inf, nrow = Kx, ncol = Ky),
  pr = 0.05,
  max_iter = 10000,
  tol_level = 1e-06,
  scale = 1,
  nB = 2000,
  verbose = TRUE
)

Arguments

X

The matrix of men's traits. Its rows must be ordered so that the i-th man is matched with the i-th woman: this means that nrow(X) must be equal to nrow(Y). Its columns correspond to the different matching variables: ncol(X) can be different from ncol(Y). For the sake of clarity of exposition when using descriptive tools such as show.correlations, it is recommended assigning the same matching variable to the k-th column of X and to the k-th column of Y, whenever possible. If X has more matching variables than Y, then those variables that appear in X but no in Y should be found in the last columns of X (and vice versa). The matrix is demeaned and rescaled before the start of the estimation algorithm.

Y

The matrix of women's traits. Its rows must be ordered so that the i-th woman is matched with the i-th man: this means that nrow(Y) must be equal to nrow(X). Its columns correspond to the different matching variables: ncol(Y) can be different from ncol(X). The matrix is demeaned and rescaled before the start of the estimation algorithm.

w

A vector of sample weights with length nrow(X). Defaults to uniform weights.

A0

A vector or matrix with ncol(X)*ncol(Y) elements corresponding to the initial values of the affinity matrix to be fed to the estimation algorithm. Optional. Defaults to matrix of zeros.

lb

A vector or matrix with ncol(X)*ncol(Y) elements corresponding to the lower bounds of the elements of the affinity matrix. Defaults to -Inf for all parameters.

ub

A vector or matrix with ncol(X)*ncol(Y) elements corresponding to the upper bounds of the elements of the affinity matrix. Defaults to Inf for all parameters.

pr

A probability indicating the significance level used to compute bootstrap two-sided confidence intervals for U, V and lambda. Defaults to 0.05.

max_iter

An integer indicating the maximum number of iterations in the Maximum Likelihood Estimation. See optim for the "L-BFGS-B" method. Defaults to 10000.

tol_level

A positive real number indicating the tolerance level in the Maximum Likelihood Estimation. See optim for the "L-BFGS-B" method. Defaults to 1e-6.

scale

A positive real number indicating the scale of the model. Defaults to 1.

nB

An integer indicating the number of bootstrap replications used to compute the confidence intervals of U, V and lambda. Defaults to 2000.

verbose

If TRUE, the function displays messages to keep track of its progress. Defaults to TRUE.

Value

The function returns a list with elements: X, the demeaned and rescaled matrix of men's traits; Y, the demeaned and rescaled matrix of men's traits; fx, the empirical marginal distribution of men; fy, the empirical marginal distribution of women; Aopt, the estimated affinity matrix; sdA, the standard errors of Aopt; tA, the Z-test statistics of Aopt; VarCovA, the full variance-covariance matrix of Aopt; rank.tests, a list with all the summaries of the rank tests on Aopt; U, whose columns are the left-singular vectors of Aopt; V, whose columns are the right-singular vectors of Aopt; lambda, whose elements are the singular values of Aopt; UCI, whose columns are the lower and the upper bounds of the confidence intervals of U; VCI, whose columns are the lower and the upper bounds of the confidence intervals of V; lambdaCI, whose columns are the lower and the upper bounds of the confidence intervals of lambda; df.bootstrap, a data frame resulting from the nB bootstrap replications and used to infer the empirical distribution of the estimated objects.

See Also

Dupuy, Arnaud, and Alfred Galichon. "Personality traits and the marriage market." Journal of Political Economy 122, no. 6 (2014): 1271-1319.

Examples

# Parameters
Kx = 4; Ky = 4; # number of matching variables on both sides of the market
N = 200 # sample size
mu = rep(0, Kx+Ky) # means of the data generating process
Sigma = matrix(c(1, 0.326, 0.1446, -0.0668, 0.5712, 0.4277, 0.1847, -0.2883,
                 0.326, 1, -0.0372, 0.0215, 0.2795, 0.8471, 0.1211, -0.0902,
                 0.1446, -0.0372, 1, -0.0244, 0.2186, 0.0636, 0.1489,
                 -0.1301, -0.0668, 0.0215, -0.0244, 1, 0.0192, 0.0452,
                 -0.0553, 0.2717, 0.5712, 0.2795, 0.2186, 0.0192, 1, 0.3309,
                 0.1324, -0.1896, 0.4277, 0.8471, 0.0636, 0.0452, 0.3309, 1,
                 0.0915, -0.1299, 0.1847, 0.1211, 0.1489, -0.0553, 0.1324,
                 0.0915, 1, -0.1959, -0.2883, -0.0902, -0.1301, 0.2717,
                 -0.1896, -0.1299, -0.1959, 1),
               nrow=Kx+Ky) # (normalized) variance-covariance matrix of the
               # data generating process
labels_x = c("Educ.", "Age", "Height", "BMI") # labels for men's matching variables
labels_y = c("Educ.", "Age", "Height", "BMI") # labels for women's matching variables

# Sample
data = MASS::mvrnorm(N, mu, Sigma) # generating sample
X = data[,1:Kx]; Y = data[,Kx+1:Ky] # men's and women's sample data
w = sort(runif(N-1)); w = c(w,1) - c(0,w) # sample weights

# Main estimation
res = estimate.affinity.matrix(X, Y, w = w, nB = 500)

# Summarize results
show.affinity.matrix(res, labels_x = labels_x, labels_y = labels_y)
show.diagonal(res, labels = labels_x)
show.test(res)
show.saliency(res, labels_x = labels_x, labels_y = labels_y,
              ncol_x = 2, ncol_y = 2)
show.correlations(res, labels_x = labels_x, labels_y = labels_y,
                  label_x_axis = "Husband", label_y_axis = "Wife", ndims = 2)