In this model, I assume that the regulator does not “forbear” from closing down a bank. But the researcher observes \( X \), which is the regulators signal, \( Z \), plus noise (i.e., due to opacity). I assume the researcher can observe “opacity'' for each firm, which is simply the standard deviation of the noise component.
N <- 1000
mean_Z <- -3
Z_star <- 0.2
sigma_min <- 0.2
sigma_max <- 1.8
The regulator observes \( Z \), which is the probability of failure, and simply shuts the bank down if \( Z > Z^{*} = 0.2 \). There is no "forbearance” in this model. Some banks that are not shut down, fail anyway. We don't observe the distinction between a regulator-ordered closing and a failure.
# Regulator-observed score
z <- rnorm(N, mean_Z)
Z <- exp(z)/(1 + exp(z))
closed <- Z >= Z_star
# Measurment error in earnings
sigma <- runif(N, min = sigma_min, max = sigma_max)
# Earnings is true signal plus noise
x <- z + rnorm(N, mean = 0, sd = sigma)
X <- exp(x)/(1 + exp(x))
# Banks can be closed or can fail
failed <- rbinom(N, size = 1, prob = Z) | closed
table(failed, closed)
## closed
## failed FALSE TRUE
## FALSE 892 0
## TRUE 50 58
We can measure forbearance as in Gallemore (2013) and then estimate whether this measure is associated with opacity.
# Measure of forbearance
forbear <- ifelse(failed, 0, X)
# Regression of
summary(lm(forbear ~ sigma))
##
## Call:
## lm(formula = forbear ~ sigma)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.1068 -0.0649 -0.0312 0.0220 0.8132
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.03530 0.00848 4.16 3.4e-05 ***
## sigma 0.03981 0.00757 5.26 1.8e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.112 on 998 degrees of freedom
## Multiple R-squared: 0.027, Adjusted R-squared: 0.026
## F-statistic: 27.6 on 1 and 998 DF, p-value: 1.78e-07