stabit.Rd
The function provides a Gibbs sampler for a structural matching model that estimates preferences and corrects for sample selection bias when the selection process is a onesided matching game; that is, group/coalition formation.
The input is individuallevel data of all group members from onesided matching marktes; that is, from group/coalition formation games.
In a first step, the function generates a model matrix with characteristics of all feasible groups of the same size as the observed groups in the market.
For example, in the stable roommates problem with \(n=4\) students \(\{1,2,3,4\}\) sorting into groups of 2, we have \( {4 \choose 2}=6 \) feasible groups: (1,2)(3,4) (1,3)(2,4) (1,4)(2,3).
In the group formation problem with \(n=6\) students \(\{1,2,3,4,5,6\}\) sorting into groups of 3, we have \( {6 \choose 3} =20\) feasible groups. For the same students sorting into groups of sizes 2 and 4, we have \( {6 \choose 2} + {6 \choose 4}=30\) feasible groups.
The structural model consists of a selection and an outcome equation. The Selection Equation
determines which matches are observed (\(D=1\)) and which are not (\(D=0\)).
$$ \begin{array}{lcl}
D &= & 1[V \in \Gamma] \\
V &= & W\alpha + \eta
\end{array}
$$
Here, \(V\) is a vector of latent valuations of all feasible matches, ie observed and
unobserved, and \(1[.]\) is the Iverson bracket.
A match is observed if its match valuation is in the set of valuations \(\Gamma\)
that satisfy the equilibrium condition (see Klein, 2015a). This condition differs for matching
games with transferable and nontransferable utility and can be specified using the method
argument.
The match valuation \(V\) is a linear function of \(W\), a matrix of characteristics for
all feasible groups, and \(\eta\), a vector of random errors. \(\alpha\) is a paramter
vector to be estimated.
The Outcome Equation determines the outcome for observed matches. The dependent
variable can either be continuous or binary, dependent on the value of the binary
argument. In the binary case, the dependent variable \(R\) is determined by a threshold
rule for the latent variable \(Y\).
$$ \begin{array}{lcl}
R &= & 1[Y > c] \\
Y &= & X\beta + \epsilon
\end{array}
$$
Here, \(Y\) is a linear function of \(X\), a matrix of characteristics for observed
matches, and \(\epsilon\), a vector of random errors. \(\beta\) is a paramter vector to
be estimated.
The structural model imposes a linear relationship between the error terms of both equations as \(\epsilon = \delta\eta + \xi\), where \(\xi\) is a vector of random errors and \(\delta\) is the covariance paramter to be estimated. If \(\delta\) were zero, the marginal distributions of \(\epsilon\) and \(\eta\) would be independent and the selection problem would vanish. That is, the observed outcomes would be a random sample from the population of interest.
stabit(x, m.id = "m.id", g.id = "g.id", R = "R", selection = NULL, outcome = NULL, simulation = "none", seed = 123, max.combs = Inf, method = "NTU", binary = FALSE, offsetOut = 0, offsetSel = 0, marketFE = FALSE, censored = 0, gPrior = FALSE, dropOnes = FALSE, interOut = 0, interSel = 0, standardize = 0, niter = 10, verbose = FALSE)
x  data frame with individuallevel characteristics of all group members including market and groupidentifiers. 

m.id  character string giving the name of the market identifier variable. Defaults to 
g.id  character string giving the name of the group identifier variable. Defaults to 
R  dependent variable in outcome equation. Defaults to 
selection  list containing variables and pertaining operators in the selection equation. The format is

outcome  list containing variables and pertaining operators in the outcome equation. The format is

simulation  should the values of dependent variables in selection and outcome equations be simulated? Options are 
seed  integer setting the state for random number generation if 
max.combs  integer (divisible by two) giving the maximum number of feasible groups to be used for generating grouplevel characteristics. 
method  estimation method to be used. Either 
binary  logical: if 
offsetOut  vector of integers indicating the indices of columns in 
offsetSel  vector of integers indicating the indices of columns in 
marketFE  logical: if 
censored  draws of the 
gPrior  logical: if 
dropOnes  logical: if 
interOut  twocolum matrix indicating the indices of columns in 
interSel  twocolum matrix indicating the indices of columns in 
standardize  numeric: if 
niter  number of iterations to use for the Gibbs sampler. 
verbose  . 
Operators for variable transformations in selection
and outcome
arguments.
add
sum over all group members and divide by group size.
int
sum over all possible twoway interactions \(x*y\) of group members and divide by the number of those, given by choose(n,2)
.
ieq
sum over all possible twoway equality assertions \(1[x=y]\) and divide by the number of those.
ive
sum over all possible twoway interactions of vectors of variables of group members and divide by number of those.
inv
...
dst
sum over all possible twoway distances between players and divide by number of those, where distance is defined as \(e^{xy}\).
Klein, T. (2015a). Does AntiDiversification Pay? A OneSided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.
Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with gprior distributions, volume 6, pages 233243. NorthHolland, Amsterdam.
# NOT RUN { ##  SIMULATED EXAMPLE  ## 1. Simulate onesided matching data for 200 markets (m=200) with 2 groups ## per market (gpm=2) and 5 individuals per group (ind=5). True parameters ## in selection equation is wst=1, in outcome equation wst=0. ## 1a. Simulate individuallevel, independent variables idata < stabsim(m=200, ind=5, seed=123, gpm=2) head(idata) ## 1b. Simulate grouplevel variables mdata < stabit(x=idata, simulation="NTU", method="model.frame", selection = list(add="wst"), outcome = list(add="wst"), verbose=FALSE) head(mdata$OUT) head(mdata$SEL) ## 2. Bias from sorting ## 2a. Naive OLS estimation lm(R ~ wst.add, data=mdata$OUT)$coefficients ## 2b. epsilon is correlated with independent variables with(mdata$OUT, cor(epsilon, wst.add)) ## 2c. but xi is uncorrelated with independent variables with(mdata$OUT, cor(xi, wst.add)) ## 3. Correction of sorting bias when valuations V are observed ## 3a. 1st stage: obtain fitted value for eta lm.sel < lm(V ~ 1 + wst.add, data=mdata$SEL) lm.sel$coefficients eta < lm.sel$resid[mdata$SEL$D==1] ## 3b. 2nd stage: control for eta lm(R ~ wst.add + eta, data=mdata$OUT)$coefficients ## 4. Run Gibbs sampler fit1 < stabit(x=idata, method="NTU", simulation="NTU", censored=1, selection = list(add="wst"), outcome = list(add="wst"), niter=2000, verbose=FALSE) ## 5. Coefficient table summary(fit1) ## 6. Plot MCMC draws for coefficients plot(fit1) ##  REPLICATION, Klein (2015a)  ## 1. Load data data(baac00); head(baac00) ## 2. Run Gibbs sampler klein15a < stabit(x=baac00, selection = list(inv="pi",ieq="wst"), outcome = list(add="pi",inv="pi",ieq="wst", add=c("loan_size","loan_size2","lngroup_agei")), offsetOut=1, method="NTU", binary=TRUE, gPrior=TRUE, marketFE=TRUE, niter=800000) ## 3. Marginal effects summary(klein15a, mfx=TRUE) ## 4. Plot MCMC draws for coefficients plot(klein15a) # }