Individual-level data

The stabsim function simulates individual-level, independent variables. The code below generates data for m=1,000 markets with gpm=2 groups per market and group size ind=5.

## Simulate individual-level, independent variables
library(matchingMarkets)
idata <- stabsim(m=1000, ind=5, seed=123, gpm=2)
head(idata)

##   m.id g.id wst  R
## 1    1    1   0 NA
## 2    1    1   1 NA
## 3    1    1   0 NA
## 4    1    1   1 NA
## 5    1    1   1 NA
## 6    1    2   0 NA

The resulting data contains market and group identifiers m.id and g.id and the independent variable wst\(\sim\) B(1,0.5). The dependent variable R depends on the error terms and is still undefined at this stage.

Group-level data

Next we apply the function stabit that serves three purposes:

• First, it specifies the list of variables to be included in selection and outcome equations and generates group-level variables based on group members’ individual characteristics. For example, the operation ieq="wst" produces the probability that two randomly drawn group members have the same value of wst.
• Second, if simulation="NTU", it draws standard normal, group-level unobservables eta and xi to enter selection and outcome equation and selects equilibrium groups based on the group formation game with non-transferable utility, assuming pairwise aligned preferences as in (Klein 2015a). In the case of two groups per market, this selection rule results in one dominant group with the maximum group valuation and one group comprised of the residual agents.
• Third, the argument method="model.frame" specifies that only the group-level model matrices be generated. Other options are estimators using "NTU" for selection correction using non-transferable utility matching as selection rule or "outcome" for estimation of the outcome equation only.

## Simulate group-level variables (takes a minute to complete...)
mdata <- stabit(x=idata, simulation="NTU", method="model.frame",
                selection = list(ieq="wst"), 
                outcome   = list(ieq="wst"))$model.frame

The resulting object mdata is a list containing data for selection and outcome equations in SEL and OUT, respectively. SEL contains 252,000 rows, one for each of \({5 \choose 10}\) = 252 feasible group in each of the 1,000 markets. A group’s valuation is given by V = +1*wst.ieq + eta. The variable D indicates which groups are observed in equilibrium D=1 and which are not D=0.

head(mdata$SEL, 4)

##   m.id g.id wst.ieq D          V        eta
## 1    1    1     0.4 1 2.97145815  2.5714581
## 2    1    2     0.4 1 1.11284232  0.7128423
## 3    1    3     0.4 0 0.05608277 -0.3439172
## 4    1    4     0.6 0 2.19850877  1.5985088

The outcome data in OUT contains 2,000 rows, one for each of 2 equilibrium groups per market. The group outcome is given by R = -1*wst.ieq + epsilon with epsilon := +0.5*eta + xi.

head(mdata$OUT, 4)

##   m.id g.id intercept wst.ieq           R         xi    epsilon
## 1    1    1         1     0.4 -0.78221286 -1.6679419 -0.3822129
## 2    1    2         1     0.4  0.17095999  0.2145388  0.5709600
## 3    2    1         1     0.6 -0.57408090 -1.3165104  0.0259191
## 4    2    2         1     0.6 -0.08277463  1.4414618  0.5172254

Bias from sorting

The bias in the slope estimate \(\hat\beta-\beta\) = -0.44 - (-1) = 0.56 is illustrated in the left panel of the figure below.

## Naive OLS estimation
lm(R ~ wst.ieq, data=mdata$OUT)$coefficients

## (Intercept)     wst.ieq 
##   0.3961386  -0.4372392

The source of this bias is the positive correlation between epsilon and the exogenous variable wst.ieq (see the right panel below).

## epsilon is correlated with independent variables
summary(lm(epsilon ~ wst.ieq, mdata$OUT))$coefficients

##              Estimate Std. Error  t value     Pr(>|t|)
## (Intercept) 0.3961386 0.08758434 4.522939 6.456494e-06
## wst.ieq     0.5627608 0.15889272 3.541766 4.065683e-04

An analytical example and a formal treatement of this bias is available in (Klein 2015a). We know that epsilon = 0.5*eta + xi. Thus, conditional on eta, the unobservables in the outcome equation are independent of the exogenous variables (because xi does not enter the selection equation).

## xi is uncorrelated with independent variables
summary(lm(xi ~ wst.ieq, mdata$OUT))$coefficients

##                Estimate Std. Error    t value  Pr(>|t|)
## (Intercept) -0.03837509 0.06827652 -0.5620540 0.5741423
## wst.ieq      0.04147217 0.12386508  0.3348173 0.7377980

Correction of sorting bias

The selection problem is resolved when the residual from the selection equation eta is controlled for in the outcome equation.

## 1st stage: obtain fitted value for eta
lm.sel <- lm(V ~ -1 + wst.ieq, data=mdata$SEL); lm.sel$coefficients

##  wst.ieq 
## 1.004501

eta <- lm.sel$resid[mdata$SEL$D==1]
## 2nd stage: control for eta
lm(R ~ wst.ieq + eta, data=mdata$OUT)$coefficients

## (Intercept)     wst.ieq         eta 
## -0.03858257 -0.96578534  0.50366230

The figure below plots the bias from sorting against the independent variable, for the naive OLS and the selection-correction from the structural model.

In most real-world applications, however, the match valuations V are unobserved. The solution is to estimate the selection equation by imposing equilibrium bounds, as derived in (Klein 2015b).