Econometric Issues

Fixed Effect versus Random Effect: The individual effect can be controlled using fixed

effect or random effect approaches. A fixed effect model assumes that the individual effect is

correlated with the independent variables in the model, while a random effect model assumes

that there is no correlation between the individual effect and the independent variables. Hausman

(1978) proposes a test to check a more efficient model against a less efficient, but consistent,

model. Under the null hypothesis, both fixed effect and random effect estimates are consistent,

but a random effect estimate is more efficient; whereas, under the alternative hypothesis, the

fixed effect estimate is consistent, but the random effect estimate is not consistent. In this paper

the Hausman test shows that fixed effect model is appropriate for affiliated, non-affiliated and all

companies together.

Endogenous Variable: With the presence of moral hazard, the reinsurance coverage may

make insurer take less precaution controlling risks, which leads to higher incurred losses. Hence

the right-hand side explanatory variable of LR_i,tis endogenous. It is correlated with the error term of the equation (12). To correct the endogeneity, the first stage estimation equation (13) is

used: t i t i D t i t i t i D t i REINS REINS LR DPW LR_, =α₂₀ +β_{21 ,−1}+β_{22 ,−2} +β_{23 ,−1} +β₂₄ln( )_, +ε_, (13) REINS_i,t−1=One lag of reinsurance purchase for the primary insurer iin yeart−1;

t i,

From Wooldridge (2002), the OLS estimators will be biased if the endogenous variables

are included in the estimated model. To conduct an endogeneity test, a set of suitable instrument

variables (IV hereafter) is needed for this potential endogenous variable. The regression-based

approach introduced by Wooldridge (2002) is applied. An appropriate IV needs to be correlated

to the endogenous variable and uncorrelated with the error term in the model. Intuitively, the

direct loss is positively related to the direct written premium by the primary insurers, and the

direct written premium can serve as an IV per se. Therefore, log of direct written premium is

used as one IV for loss incurred. In addition, one and two lags of reinsurance purchase are

included in the model as one instrumental variable. This inclusion can be used to test its effect on

the concurrent losses incurred which may arise due to moral hazard with the reinsurance

coverage.

The reduced form of direct incurred loss is estimated by using all the independent

variables in the estimation (12) and four IVs as the independent variables. After the residual of

this estimate is obtained, the dependent variable, the reinsurance purchase in the equation (12), is

regressed on all the independent variables and the obtained residual, as well. The insignificant

robust t-statistic of estimated coefficient for the error term indicates that the direct incurred loss

is not an endogenous variable in the estimation and the corresponding results are unbiased. The

corresponding p-value is 0.00 which implies that the variable of concurrent loss incurred is

endogenous in the estimation equations, and the OLS estimators are biased. We need to apply

IVs to fix the endogeneity issue.

In addition, equation (13) partly reflects the potential of moral hazard on the part of the

primary insurer. With the presence of moral hazard, the higher level of reinsurance purchase in

primary insurer if other firm characteristics are controlled. Hence, the estimated coefficient of

lag of reinsurance purchase is expected to be positive if moral hazard does exist in the

reinsurance market.

Heteroskedasticity: If the error terms do not have constant variance with each observation,

the heteroskedasticity problem arises. In this case, the OLS estimators are unbiased and

consistent but inefficient because the assumption of the constant variance for error terms is

violated. In the presence of hetoroskedasticity, the variance of the coefficients obtained from

OLS tends to be underestimated, so the OLS standard error is not valid for constructing

confidence intervals and t statistics. To solve this problem, Weighted Least Square (WLS)

estimators or robust standard errors are usually adopted to improve efficiency.

In the estimation, the White test is employed to detect the possible heteroskedasticity

problem. The White test statistics is 2261.24 and corresponding p-value is 0.00. This result

rejects the null hypothesis that the residuals in the model are homoskedasticity. Therefore, the

heteroskedasticity issue occurs when estimating the model, and the estimators are unbiased and

consistent but inefficient. In addition, the normal standard errors are invalid to construct the

confidence intervals and the t-statistics. Therefore, the robust standard errors are used instead to

improve the estimator efficiency in the presence of heteroskedasticity.

Individual Effect versus Pooled OLS: The error term u_i,t in equation (12) can be

decomposed asu_i,t =a_i +ν_i,t, where a_iis called individual effect, ν_i,tis idiosyncratic error and

t i

u_, is composite error. The individual effect is usually unobservable. If the unobserved individual effect is correlated with other independent variables in the model, the pooled OLS estimators are

biased and inconsistent. If the individual effect is a random variable and is uncorrelated with

As a result, the presence of the individual effect to choose the appropriate estimation method

needs to be tested. Breusch and Pagan (1979) derive the Lagrange Multiplier (LM) test to detect

the presence of individual effect. Based on the residuals from the equation (12), the LM test

statistics is 5217.6, which reject the null hypothesis of the absence of individual effect. In the

presence of individual effect, the pooled OLS estimation is not appropriate for our model.

Overidentifying Test: To test the model identification, Anderson Canon and Cragg-

Donald tests are undertaken by using STATA code of “xtivreg2”. The small p-value of these

tests shows the model proposed is identified.