`stabit2.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 two-sided matching game; i.e., a matching of students to colleges.

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 (\(D=0\)).
$$ \begin{array}{lcl}
D &= & 1[V \in \Gamma] \\
V &= & W\beta + \eta
\end{array}
$$
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 Sorensen, 2007).
The match valuation \(V\) is a linear function of \(W\), a matrix of characteristics for
*all feasible* matches, and \(\eta\), a vector of random errors. \(\beta\) 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 \(Y\).
$$ \begin{array}{lcl}
R &= & 1[Y > c] \\
Y &= & X\alpha + \epsilon
\end{array}
$$
Here, \(Y\) is a linear function of \(X\), a matrix of characteristics for *observed*
matches, and \(\epsilon\), a vector of random errors. \(\alpha\) is a paramter vector to
be estimated.

The structural model imposes a linear relationship between the error terms of both equations as \(\epsilon = \kappa\eta + \nu\), where \(\nu\) is a vector of random errors and \(\kappa\) is the covariance paramter to be estimated. If \(\kappa\) 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.

stabit2(OUT = NULL, SEL = NULL, colleges = NULL, students = NULL, outcome = NULL, selection, binary = FALSE, niter, gPrior = FALSE, censored = 1, thin = 1, nCores = max(1, detectCores() - 1), ...)

OUT | data frame with characteristics of all observed matches, including
market identifier |
---|---|

SEL | optional: data frame with characteristics of all observed and unobserved matches, including
market identifier |

colleges | character vector of variable names for college characteristics. These variables carry the same value for any college. |

students | character vector of variable names for student characteristics. These variables carry the same value for any student. |

outcome | formula for match outcomes. |

selection | formula for match valuations. |

binary | logical: if |

niter | number of iterations to use for the Gibbs sampler. |

gPrior | logical: if |

censored | draws of the |

thin | integer indicating the level of thinning in the MCMC draws. The default |

nCores | number of cores to be used in parallel Gibbs sampling. |

... | . |

Sorensen, M. (2007). How Smart is Smart Money? A Two-Sided Matching Model of Venture Capital.
*Journal of Finance*, 62 (6): 2725-2762.

# NOT RUN { ## --- SIMULATED EXAMPLE --- ## 1. Simulate two-sided matching data for 20 markets (m=20) with 100 students ## (nStudents=100) per market and 20 colleges with quotas of 5 students, each ## (nSlots=rep(5,20)). True parameters in selection and outcome equations are ## all equal to 1. xdata <- stabsim2(m=20, nStudents=100, nSlots=rep(5,20), verbose=FALSE, colleges = "c1", students = "s1", outcome = ~ c1:s1 + eta + nu, selection = ~ -1 + c1:s1 + eta ) head(xdata$OUT) ## 2. Correction for sorting bias when match valuations V are observed ## 2-a. Bias from sorting lm1 <- lm(y ~ c1:s1, data=xdata$OUT) summary(lm1) ## 2-b. Cause of the bias with(xdata$OUT, cor(c1*s1, eta)) ## 2-c. Correction for sorting bias lm2a <- lm(V ~ -1 + c1:s1, data=xdata$SEL); summary(lm2a) etahat <- lm2a$residuals[xdata$SEL$D==1] lm2b <- lm(y ~ c1:s1 + etahat, data=xdata$OUT) summary(lm2b) ## 3. Correction for sorting bias when match valuations V are unobserved ## 3-a. Run Gibbs sampler (when SEL is given) fit2 <- stabit2(OUT = xdata$OUT, SEL = xdata$SEL, outcome = y ~ c1:s1, selection = ~ -1 + c1:s1, niter=1000 ) ## 3-b. Alternatively: Run Gibbs sampler (when SEL is not given) fit2 <- stabit2(OUT = xdata$OUT, colleges = "c1", students = "s1", outcome = y ~ c1:s1, selection = ~ -1 + c1:s1, niter=1000 ) ## 4. Implemented methods ## 4-a. Get coefficients fit2 ## 4-b. Coefficient table summary(fit2) ## 4-c. Get marginal effects summary(fit2, mfx=TRUE) ## 4-d. Also try the following functions #coef(fit2) #fitted(fit2) #residuals(fit2) #predict(fit2, newdata=NULL) ## 5. Plot MCMC draws for coefficients plot(fit2) # }