Robust SURE estimates of profitability in the Egyptian insurance market

Abstract

This paper proposes three robust estimators (M-estimation, S-estimation, and MM-estimation) for handling the problem of outlier values in seemingly unrelated regression equations (SURE) models. The SURE model is one of regression multivariate cases, which have especially assumption, i.e., correlation between errors on the multivariate linear models; by considering multiple regression equations that are linked by contemporaneously correlated disturbances. Moreover, the effects of outliers may permeate through the system of equations; the primary aim of SURE which is to achieve efficiency in estimation, but this is questionable. The goal of robust regression is to develop methods that are resistant to the possibility that one or several unknown outliers may occur anywhere in the data. In this paper, we study and compare the performance of robust estimations with the traditional non-robust (ordinary least squares and Zellner) estimations based on a real dataset of the Egyptian insurance market during the financial year from 1999 to 2018. In our study, we selected the three most important insurance companies in Egypt operating in the same field of insurance activity (personal and property insurance). The effect of some important indicators (exogenous variables) issued by insurance corporations on the net profit has been studied. The results showed that robust estimators greatly improved the efficiency of the SURE estimation, and the best robust estimation is MM-estimation. Moreover, the selected exogenous variables in our study have a significant effect on the net profit in the Egyptian insurance market.

Keywords

Contemporaneous correlation Egyptian insurance Generalized Least Squares outliers residual analysis robust estimation Seemingly Unrelated Regressions Equations Model Zellner’s estimator

1. Introduction

The Seemingly Unrelated Regressions (SUR) or Seemingly Unrelated Regression Equations (SURE) model is proposed by Zellner [1, 2], where the main assumption of this model that the errors of the model are related by contemporaneous correlation, see [3, 4]. The generalization of the linear regression model that consists of several regression equations, each has its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and could be estimated separately. The SURE model is a special case of simultaneous equations models; where there are no endogenous variables appear as regresses in any of the equations. That is why the system was called seemingly unrelated, although some authors suggested that the seemingly related term would be more appropriate, since the error terms were assumed to be correlated across the equations. Each equation satisfies the assumptions of the classical linear regression model. The SURE model could be viewed as either the simplification of the general linear model where certain coefficients in matrix B were restricted to be equal to zero, or as the generalization of the general linear model where the regressors on the right-hand-side could be different in each equation, see [5, 6].

Many studies in economics, insurance, and finance are based on regression models which contain more than one equation. Unconsidered factors that influence the error term in one equation are also influence the error terms in other equations. Ignoring this dependence structure of the error terms and estimating these equations separately using Ordinary Least Squares (OLS) estimator leads to inefficient estimates. Therefore, the SURE model has been developed which considers the underlying covariance structure of the error terms across equations. The assumption of SURE model is developed in several econometric applications (or models), such as panel data models and related fields, see [7, 8, 9], and many more.

The Zellner’s estimator of the SURE model depends on the data without any outliers, but in some cases this cannot be achieved. If the dataset contains outliers and influential observations, the Zellner’s estimator is not efficient. The robust estimation methods are considered one of the most important approaches to deal with outliers, which allowed the unequal weight for observations. Robust regression analysis provides good alternative estimators for Zellner’s estimator, when the classical assumptions are not fulfilled, see [10]. The robust estimation methods are discussed in many papers for several regression models, such as count regression model [11], semiparametric partially linear model [12], and others.

The insurance sector in Egypt is one of the most important non-banking financial services activities and the most prominent contributor to the Gross Domestic Product (GDP), as it contributes a large percentage to it. Also, it is closely related to the rest of the economic sectors, and it contributes to managing the risks that may be exposed to economic assets. Kelly et al. [13] investigated the impact of automobile insurance regulation on the size of the involuntary insurance market as well as the level and volatility of auto insurance loss ratios in Canada. They also used SURE model to model this endogeneity, they found that rate reduction orders, product reform, and a pricing “Grid” that established maximum premiums increase the size of the involuntary market, while prior approval does not have any significant effect, unlike U.S. studies, they found that prior approval does not significantly impact loss ratio volatility. Also Tan and Floros [14] investigated the inter-temporal relationship between banking profitability, competition and risk of a sample of Chinese commercial banks by employing several profitability and risk indicators, and using SURE model under a panel data framework. The results support the Structure-Conduct-Performance (SCP) theory which states that there is a negative impact of competition on bank profitability.

This paper is organized as follows; Section 2 presents a background about the SURE model, and some non-robust estimation methods of this model. While in Section 3, we review three robust estimation methods of the SURE model. The results of the application have been presented in Section 4. Finally, Section 5 offers the concluding remarks.

2. SURE model specification and estimations

The SURE model is a system of $m$ multiple regression equations in which each equation has a single dependent and ( $m>1$ ), independent or exogenous variables as in standard regression model. The $m$ equations have no link or relationship with each other except that their disturbances are said to be correlated, this is the simplest version of the linear, constant parameter SURE system for linear regression equations.

The specification of the basic SURE model was proposed by Zellner [1], for the first time as one of his wonderful and successful scientific contribution remains an important option in any modeling exercise using pooled data, under the assumption that the errors of the model are related by contemporaneous correlation. Zellner’s [2] developed the SURE estimator for estimating models with dependent variables that allow for different regressor matrices in each equation, e.g. $X_{i}\neq X_{j}$ , and account for contemporaneous correlation, i.e. $E({u_{i}u_{j}})\neq 0$ . In order to simplify notation, the problem of estimation for a system of regression equations where the random disturbances are correlated with each other was investigated. That is, the regression equations are linked statistically, even though not structurally, through the non-diagonality of the associated variance-covariance matrix. The SURE is used to reflect the fact that the individual equations are in fact related to one another even though, superficially, they may not seem to be, from the background described before. Thronton [15] described SURE specification and estimation. We can assume that if there is a $m$ number of equations that are related each other because the error terms are correlated. The system of $m$ SURE can be stacked in two equivalent compact matrix forms. First, we can express it as a multiple linear regression model:

$\displaystyle\left({{\begin{array}[]{*{20}c}{Y_{1}}\\ {Y_{2}}\\ \vdots\\ {Y_{m}}\\ \end{array}}}\right)_{mn\times 1}=\left({{\begin{array}[]{*{20}c}{X_{1}}&0&% \ldots&0\\ 0&{X_{2}}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&X_{m}\\ \end{array}}}\right)_{mn\times K}$ $\displaystyle\left({{\begin{array}[]{*{20}c}{\beta_{1}}\\ {\beta_{2}}\\ \vdots\\ {\beta_{m}}\\ \end{array}}}\right)_{K\times 1}+\left({{\begin{array}[]{*{20}c}{u_{1}}\\ {u_{2}}\\ \vdots\\ {u_{m}}\\ \end{array}}}\right)_{mn\times 1}$ (1)

This multiple equation can be simply re-written compactly as:

$\displaystyle Y=X\beta+U,$ (2)

where the $Y=(y^{\prime}_{1},\ldots,y^{\prime}_{m})^{\prime}$ is the column vector of observation on the $i^{\text{th}}$ endogenous variable, $X=\text{diag}[{X_{i}}]$ ; with $X_{i}$ (for ${i}=1,2,\ldots,m$ ) is a block diagonal design matrix of the exogenous non-stochastic variables of equation number $i$ with dimension $n\times k_{i}$ , and $\beta=(\beta^{\prime}_{1},\ldots,\beta^{\prime}_{m})^{\prime}$ is the column vector of the stacked coefficient vectors of all equations, the Total number of parameters estimated for all $k$ sub models is $K=\mathop{\sum}\limits_{i=1}^{m}k_{i}$ , while $U=(u^{\prime}_{1},\ldots,u^{\prime}_{m})^{\prime}$ is the column vector of contemporaneous correlated random error.

Second, the SURE model can be rewritten as:

$\displaystyle\begin{array}[]{c}\tilde{Y}\\ n\times m\\ \end{array}=\begin{array}[]{c}\tilde{X}\\ n\times K\\ \end{array}\begin{array}[]{c}{\cal B}\\ K\times m\\ \end{array}+\begin{array}[]{c}{\cal U}\\ n\times m\\ \end{array},$ (3)

where $\tilde{Y}=(y_{1},\ldots,y_{m})$ is the response matrix, $\tilde{X}=(X_{1},\ldots,X_{m})$ is the design matrix, the coefficient matrix here has a constrained structure:

$\displaystyle{\cal B}=\left[{{\begin{array}[]{*{20}c}{{\cal B}_{1}}&0&\ldots&0% \\ 0&{{\cal B}_{2}}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&{{\cal B}_{m}}\\ \end{array}}}\right]$

2.1 SURE assumptions

A1: $E(U)=0$ ,

$\displaystyle\textit{cov}(U)=E({UU^{\prime}})=\left[\begin{array}[]{cccc}% \sigma_{11}&\sigma_{12}&\ldots&\sigma_{1m}\\ \sigma_{21}&\sigma_{22}&\ldots&\sigma_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ \sigma_{m1}&\sigma_{m2}&\ldots&\sigma_{mm}\\ \end{array}\right]$ $\displaystyle\otimes I_{n}=\sum\otimes I_{n}=\Omega,$ (4)

where $I_{n}$ is an $n\times n$ identity matrix and $\otimes$ is the Kronecker product operator.

A2: $X_{i}$ is fixed in repeated samples (non-stochastic matrix)

A3: $X$ is full column rank matrix, i.e., $\textit{rank}(X)=K$ .

Under A1–A3 assumptions, we can apply the OLS on Eq. (2) to estimate $\beta$ as:

$\displaystyle\hat{\beta}_{\text{OLS}}=(X^{\prime}X)^{-1}(X^{\prime}Y).$ (5)

It is well-known that OLS estimator is consistent, but it does not take into account the correlation structure of the disturbances across equations. Consequently, it is generally less efficient, and may yield inefficient estimators. The Generalized Least Squares (GLS) estimator (also known as Zellner’s estimator) is a modification of least squares that can deal with any type of correlation, including contemporaneous correlation, Zellner’s estimator is efficient and also fulfill the maximum likelihood requirement. Because it given best linear unbiased estimators (BLUEs). For the SURE model, the Zellner’s estimator takes:

$\displaystyle\hat{\beta}_{Z}=(X^{\prime}\Omega^{-1}X)^{-1}(X^{\prime}\Omega^{-% 1}Y).$ (6)

The variance-covariance matrix of the Zellner’s estimator is given by

$\displaystyle\textit{cov}(\hat{\beta}_{Z})=(X^{\prime}\Omega^{-1}X)^{-1}.$ (7)

Zellner’s estimator is efficient and fulfill the maximum likelihood requirement. Zellner’s estimator is more efficient than the OLS estimator, but in most situations the covariance ${\sum}$ needed in Zellner’s estimator is unknown. Feasible Generalized Least Squares (FGLS) estimate the elements of ${\sum}$ by $\widehat{{\sum}}$ where $\widehat{{\sum}}$ is calculated based on the OLS residuals for each equation. Note that FGLS can be repeated iteratively, see [16]. Alternatively, the maximum likelihood (ML) estimator can be considered see [17].

3. Robust estimators for the SURE model

3.1 Consequences of outliers in SURE model

In econometrics and statistics, the SURE model has taken an important place. However, since the procedure proposed originally by Zellner is essentially a least squares estimator in a multiple equations model with a particular covariance matrix, it is expected that the estimator is vulnerable to outliers. In practice, data collected in a broad range of applications frequently contains one or more atypical observations called outlier. An exact definition of an outlier often depends on hidden assumptions regarding the data structure and the applied detection method. Yet, some definitions are regarded general enough to cope with various types of data and methods. Hawkins [18] defines an outlier as an observation that deviates so much from other observations as to arouse suspicion that it was generated by a different mechanism. They can occur by chance in a distribution but are mostly indicative of measurement error which one can decide to discard or use statistics that are robust. Moreover, the effects of outliers may permeate through the system of equations; the primary aim of SURE which is to achieve efficiency in estimation is therefore questionable; see [19].

There are two categories of outliers; the first category includes the outliers in Y-dimension (endogenous variable) that are defined as “vertical outlier”, the second category includes the outliers in X-dimension (exogenous variable) these observations are called “leverages points”. Detecting or diagnosing outliers is a very important process in multivariate linear models especially the SURE model which is a special case of these models, and it is statistics that focuses attention on observations having an influence on Zellner’s estimator, which is known to be non-robust, see [20]. Many diagnostic measures have been designed to detect individual case or group of cases that may differ from the bulk of multivariate dataset, such as, graphical analysis, Mahalanobis Distance (MD), studentized residual, Cook’s Squared Distance (D ${}_{\text{i}}$ ), Different Fits Statistic, and other, see [21].

Robust estimators are tried to seek a model which represents the information in the majority of the dataset. Usually we use the properties of efficiency, the Breakdown Point (BDP), and the influence function, see [22] to measure the performance of robust techniques. BDP is the most common method to measure robustness. The BDP is the largest proportion of the contaminations that the data can contain before the estimate fails. Thus the higher the BDP of an estimator, the more robust is. Intuitively, a BDP cannot exceed 0.5 because if there more than half of the data are outliers, it will be impossible to distinguish between the “good” and “bad” distributions. Therefore, the maximum BDP is 0.5 which is the goal of robust estimation, thus 0 $<$ BDP $\leqslant$ 0.5. Although the sample median can achieve the best BDP value, its efficiency is very low. The main purpose of robust estimation is to provide resistant results in the presence of outliers. In order to achieve this stability, robust regression limits the influence of outliers, see [23]. Many robust methods have been proposed to achieve high BDP or high efficiency or both. In the following, we will review and compare some of these methods in SURE model to determine the best robust estimation method.

3.2 M-estimation method

Koenker and Portnoy [24] introduced the M-estimation method of the multivariate linear models; these weighted M-estimates achieve an asymptotic covariance matrix analogous to that of the SURE estimator. They proposed to apply a regression M-estimator, based on a convex loss function, to each coordinate of the response vector. The M-estimation method is a generalization to ML method in context of location models. That is nearly as efficient as traditional methods such as OLS and Zellner as the objective; M-estimation method principle is minimizing the residual function M-estimation is based on residual scale of Zellner’s estimator, see [25]. It can be introduced the M-estimation method for the context of SURE model.

Definition: Let $({X_{j},Y_{j}})\in\mathbb{R}^{n\times({p_{j}+1})}$ for $j=1,2,\linebreak\ldots,m$ with $n\geqslant p+m$ , and let $\rho_{0}$ be a $\rho$ -function with parameter $c$ . Then, the M-estimator of the SURE model $(\hat{\beta},\widehat{\sum})$ are the solutions that minimize $|\sum_{0}|$ of the optimization problem;

$\displaystyle\mathop{\text{min}}\limits_{({\beta,\sum})}\left|{\sum}\right|,% \text{subject to }\frac{1}{n}\sum_{i=1}^{n}\rho_{0}$ $\displaystyle\left\{\left[(\hat{Y}-\hat{X}_{i}B)^{\prime}{\sum{}_{0}^{{-1}}}(% \hat{Y}-\hat{X}_{i}B)\right]^{\frac{1}{2}}\right\}=\Upsilon$ (8)

Where the minimization is over all $B=\text{bdiag}(B_{1},\linebreak\ldots,B_{m})\in\mathbb{R}^{p\times m}$ , and $\sum_{0}\in\text{PDS}(m)$ with $\text{PDS}(m)$ the set of positive definite and symmetric matrices of dimension $m\times m$ , since $B$ and $\sum_{0}$ are initial estimators. The determinant of $\sum$ is denoted by $|\sum|$ , and ${\Upsilon}$ is a positive constant. In order to obtain estimates which can resist outliers $\rho$ should satisfy the following conditions:

Condition 1. $\rho$ is symmetric, twice continuously differentiable and satisfies $\rho(0)=0$ ; Condition 2. $\rho$ is strictly increasing on $[{0,c}]$ and constant on $[{c,\infty}]$ for some $c>0$ .

Here the constant ${\Upsilon}_{0}$ is given by ${\Upsilon}=E_{F}\{{\rho({|e|})}\}$ , to obtain a consistent estimator at an assumed error distribution $F$ . Usually, the errors are assumed to follow a standard normal distribution with mean zero and then we can take $F\sim N_{m}({0,I_{m}})$ . As before, the regression coefficient estimates in the matrix $\hat{\beta}$ can also be collected in the vector $\hat{\beta}=(\hat{\beta}^{\prime}_{1},\ldots,\hat{\beta}^{\prime}_{m})^{\prime}$ . A popular choice is Tukey’s biweight $\rho$ -function:

$\displaystyle\rho(u)=\left\{{{\begin{array}[]{ll}\frac{u^{2}}{2}-\frac{u^{4}}{% 2c^{2}}+\frac{u^{6}}{6c^{4}},&|u|\leqslant c;\\ \\ \frac{c^{2}}{6},&|u|>c.\\ \end{array}}}\right.$

Where $c$ is an appropriate tuning constant. The derivative of this function is known as Tukey’s bisquare function:

$\displaystyle\psi(u)=\rho^{\prime}(u)=\left\{\begin{array}[]{ll}u[1-(\frac{u}{% c})^{2}]^{2},&|u|\leqslant c;\\ \\ \frac{c^{2}}{6},&|u|>c.\\ \end{array}\right.$

For fixed ${\Upsilon}$ , the value of the tuning constant $c$ determines the BDP value. In addition to the minimization condition mentioned above the robust SURE estimators of $\beta$ and ${\sum}$ also satisfy the following equations;

$\displaystyle\hat{\beta}_{M}=\left\{X^{\prime}\left(\widehat{\sum}_{Z}^{-1}% \otimes W_{M}\right)X\right\}^{-1}X^{\prime}$ $\displaystyle\quad\left(\widehat{\sum}_{Z}^{-1}\otimes W_{M}\right)Y,$ (9) $\displaystyle\widehat{\sum}_{Z}=m(\hat{Y}-\hat{X}\hat{{\cal B}})^{\prime}W_{M}% (\hat{Y}-\hat{X}\hat{{\cal B}})$ $\displaystyle\quad\left\{\sum_{i=1}^{n}\upsilon(w_{Mi})\right\}^{-1},$ (10)

where $W_{M}=\text{diag}\{{u({w_{M1}}),\ldots,u({w_{Mn}})}\}$ is a diagonal matrix of weights, $w_{{M}i}^{2}=e_{i}(\hat{{\cal B}})^{\prime}\sum_{Z}^{-1}e_{i}(\hat{{\cal B}})$ , where $e_{i}(\hat{{\cal B}})^{\prime}$ represents the $i^{\text{th}}$ row of the residual matrix $\hat{Y}-\hat{X}\hat{{\cal B}}$ , $u({w_{M}})=\psi({w_{M}})/w_{M}$ ; $\psi({w_{M}})={\rho}^{\prime}(w_{M})$ , and $\upsilon({w_{M}})=\psi({w_{M}})w_{M}-\rho({w_{M}})+{\Upsilon}$ . Note the similarities with the Zellner’s estimator and the ML estimator

In fact, the BDP of M-estimates is $\text{BDP}=1/n\to 0$ , see [26], smaller values of $c$ produce more resistance to outliers, but comes at the price of loss in efficiency under the normal distribution. Usually, the tuning constant is picked to give reasonably high efficiency in the normal case for the Tukey’s bisquare function, generally, $c=$ 4.685 is used to produces 95% efficiency. To compute the M-estimates efficiently, Hubert et al. [27] developed the fast SURE algorithm based on the ideas of Salibian-Barrera and Yohai [28].

3.3 S-estimation method

We introduce S-estimator to deal with outliers for the SURE model as proposed by Bilodeau and Duchesne [29]; S-estimator has been generalized to multivariate estimation of position and dispersion by Davies [30] and Lopuhai [31]. The aim is then to estimate the multivariate location and the scatter matrix of a $p$ -dimension al multivariate population. S-estimator is difficult to calculate. Algorithms usually use resampling methods. S-estimation is based on residual scale of M-estimation. However, Ruppert [32] proposed an improved resampling algorithm. With this algorithm, S-estimator is easier to calculate, S-estimator of regression and S-estimator of position and dispersion can be evaluated. In the next section we will discuss how to adapt that algorithm to the context of SURE model.

Starting from the initial M-estimates, the S-estimates are calculated easily by iterating these estimating equations until convergence. Hence, S-estimator can attain the maximal BDP of 50%. S-estimator with a smaller value of $c$ down weight observations is more heavily and correspond to a higher BDP; see, e.g., [33, 34].

3.4 MM-estimation method

Peremans and Van Aelst [35] proposed the MM-estimator in the context of SURE model, by combining S-estimation with M-estimation. The initial estimate is a high BDP estimate using S-estimator. The second stage computes an M-estimate of the errors scale from the initial high BDP estimate residuals matrix. Also, they showed that MM-estimator is highly efficient, and not sensitive to leverage points compared to an M-estimator, see [36].

Let $\widehat{\sum}_{S}$ denote the S-estimator of variance covariance matrix. Decompose $\widehat{\sum}_{S}$ into a scale component $\hat{\sigma}$ and a shape matrix $\hat{\Gamma}$ such that $\widehat{\sum}_{S}=\hat{\sigma}^{2}\hat{\Gamma}$ with $|\hat{\Gamma}|=$ 1.

The MM-estimation addresses outliers in both the endogenous and the exogenous variables; MM-estimator inherits the BDP of the initial S-estimator. Hence, they can attain the maximal BDP if initial high-BDP S-estimator is used, see [37]. Recently, Abonazel and Rabie [38] studied the efficiency of some robust estimators with application on the Egyptian economy, and they concluded that the best robust estimator is MM-estimator. These estimates have both a high BDP that is 0.5 and high asymptotic efficiency under Gaussian errors. They proved consistency and asymptotic normality assuming errors with an elliptical distribution. Note that while MM-estimator has maximal BDP, there is some loss of robustness because of the bias as a result of contamination that is generally higher as compared to S-estimator.

4. Empirical study

The present research aims to an overview of Egyptian insurance market with a real dataset application. The insurance industry started in the second half of the 19 ${}^{\text{th}}$ century through acting as agents for British and French corporations, with few Egyptian corporations; under Law No. 23 of 1957, insurance corporations working in Egypt were nationalized. Today the Egyptian insurance market is regulated through “Financial Regulatory Authority” (FRA) under Law No. 10 of 1981 and Law No. 118 of 2008, see [39].

In Egypt, the insurance industry plays a vital role in supporting the economy and growing national investments. It provides financial protection for individuals and projects against different risks through its ability to transfer risk through risk pooling and its overall role in risk management. It frees individuals from dependence on families or communities in case of an adverse event. More importantly, poor families or communities may lack means for “self-insurance.” Insurance thus provides an efficient mechanism to protect people from falling into poverty as a consequence of an adverse event. Besides protecting individuals, by providing essential coverage to businesses. Insurance promotes trade and economic activity. Insurers also contribute to capital formation by collecting Premiums from a large number of policyholders and building capital to back risk. Further, it is a main channel for collecting and using national savings in financing national investments and development plans, making available new job opportunities and alleviating the impacts of inflation. Also, the insurance sector is one of the most important non-bank financial services activities and the most prominent contributor to GDP. It is worth noting that the financial results for the year 2018 indicate that the contribution of the insurance sector to the GDP reached 0.9%. The rate of growth in insurance premiums reached 19% during 2018. The number of corporations operating in this sector reached 40, see [40].

4.1 Data

As an empirical application, this paper is concerned with studying the significant impact of the robust estimators above on a real dataset. This data is obtained from the FRA and “Central Agency for Public Mobilization and Statistics” (CAPMAS) in Egypt. Through the annual statistical report on insurance activity during the fiscal yearfrom 1999 to 2018, see [40].

In our study, we have relied on selecting the three largest insurance corporations in Egypt are active in the same field of insurance activity, which is represented by Personal insurance and Property insurance. They are; Misr Insurance Corporation (MINC), Suez Canal Insurance Corporation (SCIC), and Mohandas Insurance Corporation (MOIC). The dataset is limited by the amount of information available for each insurance corporation. One may expect that within the same year the activities of one corporation can affect the others. Hence, the SURE model seems to be appropriate. Unfortunately, the classical and robust estimators of the covariance matrix become singular when all insurance corporations are considered. Therefore, we only focus on the measurements of three insurance corporations and thus their activities can highly influence each other. Since the interest is in modeling dependencies between the corporations within the same year, a SURE model with three blocks is considered. We selected a variety of variables that are the most important indicators issued by insurance corporations; the endogenous variable is the Net Profit for the Year, see, e.g., [41, 42, 43]. While the exogenous variables, in our study, include the Net compensation, the rate of issued reinsurance commissions, the general and administrative expenses rate, commission rate and production costs, and Loss rate.

4.2 Our methodology

In our application, we used the SURE model to examine the impact of some selected variables on the net profit in three insurance corporations in Egypt. Since the Zellner’s estimator for the SURE model is based on the classical covariance matrix, and it is well known that outliers in the data can severely influence classical estimators and their modifications are all very sensitive to outliers in the data. Hence, OLS and Zellner estimators are expected to yield non-robust estimates, and then we should use the robust estimators.

After descriptive statistics, and investigate the correlation among variables through correlation matrix for the dataset, we will during in analysis do the following steps:

Using non-robust (OLS and Zellner) estimators to estimate the SURE model.

Testing the contemporaneous correlation of the model.

Diagnosing (testing) the outliers, normality of the errors, multicollinearity, and heteroscedasticity problems.

Using robust (M-estimation, S-estimation, and MM-estimation) methods to estimate the SURE model.

In the final step, robust and non-robust estimations will be compared to select the best estimation using some goodness-of-fit criteria, see [44].

Table 1
Variable names and definitions

Variable description Abbreviation Variable definitions Measuring unit

Net profit for the year (endogenousvariable) NPY The net profit of a corporation, organization or any individual or entity that does business, is the financial benefit that is achieved when the proportion of earned income exceeds the expenditures, costs and taxes and all other charges including depreciation, interest, then net profit of the year is the difference between a firm’s total revenue and all explicit costs. NPY $=$ Total revenue $-$ Total expenses. 10000 EGP.

Net compensation NEC Net Compensation means refers to all the sums paid by the insurance corporations (the insurer) in accordance with the requirements of the contract, as compensation for the damage caused to the insured. 10000 EGP.

The rate of issued reinsurance commissions RIC It is the commission rate that the reinsurer pays to the assigned insurer in exchange for the premiums assigned to him and is calculated to include the original commission given to the product and an additional commission for part of the expenses of the assigned insurer. %

The general and administrative expenses rate GER General and administrative expenses are based on serving the administrative process in the facilities and are not directly related to the main activity and do not deal with the service of activities such as marketing activity. The typical average for this indicator is less than 10%, and the increase of this indicator for more than 10% from year to year means a decrease in the ability of the insurance company to grow and to issue new insurance products, which represents a constraint on the rights of shareholders. GER $=$ General and administrative expenses $\div$ Total premiums written. %

Commission rate and production costs CPC It is what the insurance company pays to the broker or employee of the marketing apparatus as a commission for each new insurance policy, and the percentage varies from one client to another in some companies, according to the value of the policy and the premiums that the company will receive. The typical average for this indicator is less than 20%. CPC $=$ Commission and production costs $\div$ Total premiums written. %

Loss rate LOR A loss rate is the frequency with which losses are incurred; insurers calculate their loss rates by figuring out what their numbers of losses are for a specific period of time. If loss rates are too high, then insurers may have to either increase premiums or decide not to renew policies that are too risky. The typical average for this indicator is less than 70%, and increasing this index more than 70% from year to year means an increased likelihood that an insurance company will suffer operating losses for that time period, and the insurance company will not be able to operate at a profit. LOR $=$ Net compensation $\div$ Total premiums written. %

Variable description	Abbreviation	Variable definitions	Measuring unit
Net profit for the year (endogenousvariable)	NPY	The net profit of a corporation, organization or any individual or entity that does business, is the financial benefit that is achieved when the proportion of earned income exceeds the expenditures, costs and taxes and all other charges including depreciation, interest, then net profit of the year is the difference between a firm’s total revenue and all explicit costs. NPY $=$ Total revenue $-$ Total expenses.	10000 EGP.
Net compensation	NEC	Net Compensation means refers to all the sums paid by the insurance corporations (the insurer) in accordance with the requirements of the contract, as compensation for the damage caused to the insured.	10000 EGP.
The rate of issued reinsurance commissions	RIC	It is the commission rate that the reinsurer pays to the assigned insurer in exchange for the premiums assigned to him and is calculated to include the original commission given to the product and an additional commission for part of the expenses of the assigned insurer.	%
The general and administrative expenses rate	GER	General and administrative expenses are based on serving the administrative process in the facilities and are not directly related to the main activity and do not deal with the service of activities such as marketing activity. The typical average for this indicator is less than 10%, and the increase of this indicator for more than 10% from year to year means a decrease in the ability of the insurance company to grow and to issue new insurance products, which represents a constraint on the rights of shareholders. GER $=$ General and administrative expenses $\div$ Total premiums written.	%
Commission rate and production costs	CPC	It is what the insurance company pays to the broker or employee of the marketing apparatus as a commission for each new insurance policy, and the percentage varies from one client to another in some companies, according to the value of the policy and the premiums that the company will receive. The typical average for this indicator is less than 20%. CPC $=$ Commission and production costs $\div$ Total premiums written.	%
Loss rate	LOR	A loss rate is the frequency with which losses are incurred; insurers calculate their loss rates by figuring out what their numbers of losses are for a specific period of time. If loss rates are too high, then insurers may have to either increase premiums or decide not to renew policies that are too risky. The typical average for this indicator is less than 70%, and increasing this index more than 70% from year to year means an increased likelihood that an insurance company will suffer operating losses for that time period, and the insurance company will not be able to operate at a profit. LOR $=$ Net compensation $\div$ Total premiums written.	%

Table 2

Some descriptive statistics

Corporation	Variables	Mean	Median	Max.	Min.	Std. Dev.	C.V
MINC
	NPY	64.778	62.400	190.384	8.543	48.431	0.748
	NEC	150.967	151.454	268.456	77.986	53.459	0.354
	RIC	16.840	17.250	20.900	11.900	2.373	0.141
	GER	6.840	7.150	11.300	2.100	3.167	0.463
	CPC	14.990	14.750	20.100	10.900	2.729	0.182
	LOR	81.105	75.000	152.100	38.900	35.000	0.432
SCIC
	NPY	2.785	2.651	6.474	0.865	1.533	0.550
	NEC	14.378	11.853	44.359	0.897	12.929	0.899
	RIC	27.440	26.350	38.900	20.100	5.141	0.187
	GER	10.885	9.850	18.000	4.800	3.777	0.347
	CPC	21.300	20.600	27.800	14.300	3.435	0.161
	LOR	48.045	49.850	64.900	29.700	9.478	0.197
MOIC
	NPY	3.556	2.435	14.588	0.987	3.625	0.921
	NEC	8.011	8.555	15.041	0.975	4.085	0.509
	RIC	37.730	37.750	50.100	24.500	6.936	0.184
	GER	15.120	13.550	29.100	7.700	5.724	0.379
	CPC	30.055	28.800	39.100	25.900	3.890	0.129
	LOR	44.310	40.700	66.900	9.300	16.964	0.383

Table 3

Correlation matrix and VIF values

Corporation	Variables	NPY	NEC	RIC	GER	CPC	LOR
MINC
	NPY	1
	NEC	0.623 (0.003)	1
	RIC	0.299 (0.201)	0.451 (0.046)	1
	GER	0.493 (0.027)	0.650 (0.002)	0.321 (0.167)	1
	CPC	0.714 (0.001)	0.574 (0.008)	0.268 (0.254)	0.798 (0.000)	1
	LOR	$-$ 0.025 (0.916)	0.216 (0.360)	0.598 (0.005)	$-$ 0.097 (0.684)	$-$ 0.255 (0.277)	1
	VIF	–	2.155	2.226	3.254	3.527	2.396
SCIC
	NPY	1
	NEC	0.870 (0.000)	1
	RIC	$-$ 0.445 (0.049)	$-$ 0.455 (0.044)	1
	GER	0.525 (0.017)	0.685 (0.001)	$-$ 0.229 (0.330)	1
	CPC	0.495 (0.027)	0.446 (0.049)	0.053 (0.826)	0.439 (0.053)	1
	LOR	0.473 (0.035)	0.369 (0.110)	$-$ 0.723 (0.000)	0.306 (0.190)	0.245 (0.297)	1
	VIF	–	2.742	3.244	2.028	1.748	2.697
MOIC
	NPY	1
	NEC	0.639 (0.002)	1
	RIC	$-$ 0.581 (0.007)	$-$ 0.909 (0.000)	1.00
	GER	$-$ 0.282 (0.228)	$-$ 0.659 (0.001)	0.629 (0.003)	1
	CPC	$-$ 0.291 (0.199)	$-$ 0.679 (0.001)	0.657 (0.001)	0.865 (0.000)	1
	LOR	$-$ 0.037 (0.877)	0.602 (0.005)	$-$ 0.499 (0.024)	$-$ 0.521 (0.018)	$-$ 0.521 (0.018)	1
	VIF	–	7.184	6.075	4.188	4.406	1.695
Overall
	NPY	1
	NEC	0.837 (0.000)	1
	RIC	$-$ 0.529 (0.000)	$-$ 0.691 (0.000)	1
	GER	$-$ 0.280 (0.030)	$-$ 0.392 (0.002)	0.671 (0.000)	1
	CPC	$-$ 0.414 (0.001)	$-$ 0.611 (0.000)	0.849 (0.000)	0.813 (0.000)	1
	LOR	0.417 (0.009)	$-$ 0.612 (0.000)	$-$ 0.515 (0.000)	$-$ 0.438 (0.001)	$-$ 0.534 (0.000)	1
	VIF	–	2.464	4.328	3.150	6.063	1.736

4.3 Results

In our application, R version 3.6.1 (“vglm” and “systemfit” packages) was used to perform the analysis, see [45]. Table 1 displays the description of the selected variables in our study, and some descriptive statistics of these variables have been presented in Table 2. In general, it can note that all variables not have large variation, because the coefficient of variation (C.V) of all variables less than one.

Table 3 presents the pairwise correlation coefficients between all variables associated with two-tailed significant t-test in parentheses. It can note that the correlation between GER and CPC is the higher correlation, while the smallest correlation is between NPY and LOR that’s in MINC. In addition, it can note that the correlation between NPY and NEC is the higher correlation, while the smallest correlation is between RIC and CPC in corporation SCIC. As in MOIC the correlation between NEC and RIC is the higher correlation, while the smallest correlation is between NPY and LOR.

As in overall corporations it is possible to notice that all variables are highly significantly correlated with each other and all less than 0.90. Moreover, the results of Table 3 indicate that the data not have multicollinearity problem1 because all the values of the Variance Inflation Factor (VIF) are less than 10 for all correlation matrices in insurance corporations.

4.3.1 Non-robust estimations

The estimation results using non-robust (OLS and Zellner) estimators have been presented in Table 4 it is present the values of estimated coefficients, the standard errors, and the significance of each variable. These results indicate that Zellner’s estimator have smallest standard errors Moreover, we find that there are some not significant coefficients.

Table 4
Non-robust estimators results of the SURE model

Corporation Variables OLS Zellner

Coefficient Std. error Coefficient Std. error

MINC

Intercept $-$ 174. 8837 ${}^{}$ 73. 6447 $-$ 63. 6492 ${}^{}$ 48. 7581

NEC 0. 3761 0. 2128 0. 0017 0. 1061

RIC $-$ 0. 2920 4. 8727 9. 3586 ${}^{}$ 2. 4897

GER $-$ 6. 8160 4. 4145 3. 0270 2. 7887

CPC 15. 1283 ${}^{}$ 5. 3323 $-$ 1. 0531 3. 0421

LOR 0. 0943 0. 3427 $-$ 0. 4235 ${}^{}$ 0. 1392

SCIC

Intercept $-$ 1. 4633 2. 5441 $-$ 0. 3631 1. 5856

NEC 0. 1075 ${}^{}$ 0. 0226 0. 0542 ${}^{*}$ 0. 0168

RIC 0. 0246 0. 0617 $-$ 0. 0017 0. 0360

GER $-$ 0. 0806 0. 0665 $-$ 0. 0341 0. 0429

CPC 0. 0520 0. 0678 0. 0367 0. 0472

LOR 0. 0374 0. 0305 0. 0417 ${}^{}$ 0. 0193

MOIC

Intercept $-$ 4. 6822 11. 2482 $-$ 6. 1783 5. 0187

NEC 1. 1152 ${}^{}$ 0. 3485 0. 8956 ${}^{***}$ 0. 1860

RIC 0. 0856 0. 1887 0. 1567 ${}^{}$ 0. 0887

GER 0. 0308 0. 1899 0. 1000 0. 1091

CPC 0. 0597 0. 2866 $-$ 0. 0346 0. 1306

LOR $-$ 0. 1396 ${}^{}$ 0. 0408 $-$ 0. 0863 ${}^{}$ 0. 0217

Corporation	Variables	OLS	Zellner
		Coefficient	Std. error	Coefficient	Std. error
MINC
	Intercept	$-$ 174.	8837 ${}^{*}$	73.	6447	$-$ 63.	6492 ${}^{**}$	48.	7581
	NEC	0.	3761	0.	2128	0.	0017	0.	1061
	RIC	$-$ 0.	2920	4.	8727	9.	3586 ${}^{***}$	2.	4897
	GER	$-$ 6.	8160	4.	4145	3.	0270	2.	7887
	CPC	15.	1283 ${}^{**}$	5.	3323	$-$ 1.	0531	3.	0421
	LOR	0.	0943	0.	3427	$-$ 0.	4235 ${}^{***}$	0.	1392
SCIC
	Intercept	$-$ 1.	4633	2.	5441	$-$ 0.	3631	1.	5856
	NEC	0.	1075 ${}^{***}$	0.	0226	0.	0542 ${}^{***}$	0.	0168
	RIC	0.	0246	0.	0617	$-$ 0.	0017	0.	0360
	GER	$-$ 0.	0806	0.	0665	$-$ 0.	0341	0.	0429
	CPC	0.	0520	0.	0678	0.	0367	0.	0472
	LOR	0.	0374	0.	0305	0.	0417 ${}^{**}$	0.	0193
MOIC
	Intercept	$-$ 4.	6822	11.	2482	$-$ 6.	1783	5.	0187
	NEC	1.	1152 ${}^{**}$	0.	3485	0.	8956 ${}^{***}$	0.	1860
	RIC	0.	0856	0.	1887	0.	1567 ${}^{*}$	0.	0887
	GER	0.	0308	0.	1899	0.	1000	0.	1091
	CPC	0.	0597	0.	2866	$-$ 0.	0346	0.	1306
	LOR	$-$ 0.	1396 ${}^{**}$	0.	0408	$-$ 0.	0863 ${}^{***}$	0.	0217

Note: The superscripts ${}^{***}$ , ${}^{**}$ , and ${}^{*}$ indicate statistical significance at the 0.001, 0.01 and 0.05 level, respectively.

4.3.2 Testing contemporaneous correlation

In SURE model the errors for different individual equations are contemporaneously correlated, we have seen that in the absence of contemporaneous correlation, applying the OLS method to each equation separately gives the most efficient estimators behind it, and there will be no need to use the Zellner’s estimator. For the above reason, it is important to test that contemporaneous covariance is equal to zero:

$H_{0}$ : $\sigma_{21}=\sigma_{31}=\sigma_{32}=0$ $H_{1}$ : At least one of the covariances is different from zero.

The appropriate test statistic for accepting or rejecting a null hypothesis ( ${H}_{0}$ ) “Lagrange Multiplier Test Statistics (LM)” see [47], and this statistic takes the following form in the cases of the three equations: ${LM}=n({{r}_{21}^{2}+{r}_{31}^{2}+{r}_{32}^{2}})$ ; where ${r}_{{ij}}^{2}$ denotes the square of the correlation coefficient between the residuals of the OLS estimator of ${i},{j}$ . Then ${LM}\sim\chi_{({m({m-1})/2})}^{2}$ under the validity of the ${H}_{0}$ , see [48]. We will start with the calculation of the covariance matrix $\widehat{\sum}_{\text{OLS}}$ , which depends on the vectors of the residuals of the restricted OLS estimator as follows:

$\displaystyle\widehat{\sum}_{\text{OLS}}=\left(\begin{array}[]{lll}1141.104&13% .359&34.683\\ 13.359&0.591&0.7329\\ 34.683&0.733&5.359\\ \end{array}\right)$

Depending on the values of $\widehat{\sum}_{\text{OLS}}$ , then ${LM}=n({{r}_{21}^{2}+{r}_{31}^{2}+{r}_{32}^{2}})=20({0.265+0.197+0.1695})=12.625$ .

Since ${LM}=12.625>\chi_{({3,5\%})}^{2}=7.815$ , then we reject the null hypothesis of no correlation between the errors in favor of the alternative hypothesis that there is a contemporaneous correlation between the errors of the equations, and conclude that there are potential efficiency gains from estimating the three insurance corporation’s equations jointly using Zellner’s estimator.

Based on the results in Table 5, we concluded that Zellner’s estimator is better than OLS estimator. It can note that the Zellner’s estimator has the smallest values of all goodness-of-fit measures, and higher $R^{2}$ and $R^{2}$ (Adj.) values. And the model does not have heteroscedasticity problem because the $p$ -value (0.999) of BPG test [49] is greater than 0.05. In addition, it can note that the OLS residuals are distributed not normally, because the $p$ -value of Shapiro-Wilk test [50] less than 0.05. Moreover, it can note that the model has outliers problem because the $p$ -value of Bonferroni test [51] is less than 0.05.

Table 5
Goodness-of-fit measures of non-robust estimation methods

Measure OLS Zellner

MSE: Mean squared error 382.3513 42.8342

RMSE: Root of mean squared error 12.2877 4.0460

MAE: Mean absolute error 217.7329 96.9229

$R^{2}$ : R-Squared 0.7187 0.8490

$R^{2}$ (Adj.): Adjusted R-Squared 0.6182 0.8088

Residuals diagnostic tests

I. Heteroscedasticity test: Breusch-Pagan-Godfrey (BPG)

GQ-Statistic 0.005 $p$ -value 0.999

II. Normality test: Shapiro-Wilk

W-Statistic 0.847 $p$ -value 2.471 $\times$ 10 ${}^{-6}$

III. Outlier test: Bonferroni

Bonferroni-Statistic 6.1896 $\times$ 10 ${}^{-9}$ $p$ -value 1.032 $\times$ 10 ${}^{-10}$

Measure	OLS	Zellner
MSE: Mean squared error	382.3513	42.8342
RMSE: Root of mean squared error	12.2877	4.0460
MAE: Mean absolute error	217.7329	96.9229
$R^{2}$ : R-Squared	0.7187	0.8490
$R^{2}$ (Adj.): Adjusted R-Squared	0.6182	0.8088

Residuals diagnostic tests
GQ-Statistic	0.005	$p$ -value	0.999
II. Normality test: Shapiro-Wilk
W-Statistic	0.847	$p$ -value	2.471 $\times$ 10 ${}^{-6}$
III. Outlier test: Bonferroni
Bonferroni-Statistic	6.1896 $\times$ 10 ${}^{-9}$	$p$ -value	1.032 $\times$ 10 ${}^{-10}$

4.3.3 Diagnosing of outliers

To check the cause of the violation SURE assumptions about normality, we do a multivariate diagnostic plots corresponding to our analysis of the dataset for analyzing the residuals of the SURE model. The method used is the residual graph vs. fitted, normal Q-Q plot, scale location, cook’s distance, MD, and boxplot. Figure 1 shows that the some observations may give you problems with the SURE model, residual vs. fitted, model valid if the dots spread around 0, and the data is not entirely spread around 0, so that the model is not valid. Scale-location, some points had a great residual value, this is indicated with a point away from the line. Normal Q-Q plot shows the residual spread not normally because there is a point not spread around that line some points are most likely an outlier of data. Moreover, the distribution of the residuals is conformed the decision of Shapiro-Wilk test; this indicates that there are outlier values in the residuals.

Figure 1.

Residuals of OLS results.

In Fig. 2, the values are plotted for identifying outlier points, the outlier points are identified by MD appeared to be same as it was observed in the leverage values ( $h_{ii}$ ). Though there are different methods for detecting outlier points, but it has been found that the maximum outlier can be detected by cook’s distance, cook’s distance indicates the difference between the value of the regression coefficient by incorporating the i-th observation and the SURE coefficients without the i-th observation. Figure 2 shows that some observations very large impact on the regression line. Also the boxplot indicates that there are some outlier values in the residuals. So it can be concluded that the outlier values that have a major influence on the SURE model. Therefore, we will apply the robust estimation methods to get a better estimation.

Figure 2.

Cooks, Mahalanobis distances, and Boxplot of OLS results.

Table 6

Robust estimators results of the SURE model

Corporation	Variables	M-estimator				S-estimator				MM-estimator
		Coefficient		Std. error		Coefficient		Std. error		Coefficient		Std. error
MINC
	Intercept	$-$ 66.	8626 ${}^{*}$	33.	7501	$-$ 42.	0081 ${}^{*}$	30.	9878	$-$ 46.	8418 ${}^{*}$	23.	9751
	NEC	0.	2150 ${}^{**}$	0.	0861	0.	1033	0.	8312	0.	0537 ${}^{*}$	0.	0271
	RIC	6.	6131 ${}^{***}$	1.	8568	6.	4259 ${}^{***}$	1.	5906	7.	0078 ${}^{***}$	1.	0916
	GER	1.	0886	2.	5101	0.	9618	2.	0719	2.	1938 ${}^{***}$	0.	0132
	CPC	0.	3484	1.	5045	$-$ 1.	4988 ${}^{*}$	1.	2179	$-$ 2.	8720 ${}^{***}$	0.	3248
	LOR	$-$ 0.	5463 ${}^{***}$	0.	0612	$-$ 0.	3916 ${}^{***}$	0.	0409	$-$ 0.	3708 ${}^{***}$	0.	0393
SCIC
	Intercept	22.	9143 ${}^{***}$	0.	9963	23.	1747 ${}^{***}$	0.	9096	23.	0655 ${}^{***}$	0.	6995
	NEC	0.	0855 ${}^{***}$	0.	0109	0.	0647 ${}^{***}$	0.	0106	0.	0516 ${}^{*}$	0.	0262
	RIC	0.	1156 ${}^{**}$	0.	0356	0.	0341	0.	0350	0.	0122	0.	0155
	GER	$-$ 0.	0977 ${}^{***}$	0.	0201	$-$ 0.	0925 ${}^{**}$	0.	0294	$-$ 0.	0698 ${}^{***}$	0.	0120
	CPC	$-$ 0.	0748 ${}^{*}$	0.	0383	$-$ 0.	0093	0.	0152	0.	0146 ${}^{*}$	0.	0118
	LOR	0.	0389 ${}^{***}$	0.	0089	0.	0484 ${}^{***}$	0.	0135	0.	0461	0.	1265
MOIC
	Intercept	12.	7740 ${}^{**}$	3.	9821	15.	2022 ${}^{***}$	2.	9789	16.	0524 ${}^{***}$	1.	0917
	NEC	0.	9394 ${}^{*}$	0.	7714	0.	6778 ${}^{***}$	0.	1032	0.	6093 ${}^{***}$	0.	0498
	RIC	0.	0300	0.	0815	0.	0588	0.	0609	0.	0851 ${}^{*}$	0.	0584
	GER	$-$ 0.	0061 ${}^{*}$	0.	0205	0.	1288 ${}^{*}$	0.	0726	0.	1286 ${}^{*}$	0.	0690
	CPC	0.	0507	0.	0657	$-$ 0.	0547 ${}^{*}$	0.	0397	$-$ 0.	0659 ${}^{**}$	0.	0307
	LOR	$-$ 0.	0821 ${}^{***}$	0.	0135	$-$ 0.	0859 ${}^{***}$	0.	0214	$-$ 0.	0778 ${}^{***}$	0.	0104

Note: The superscripts ${}^{***}$ , ${}^{**}$ , and ${}^{*}$ indicate statistical significance at the 0.001, 0.01 and 0.05 level, respectively.

Table 7

Goodness-of-fit measures of robust SURE estimators

Measure	M-estimator	S-estimator	MM-estimator
MSE: Mean squared error	4.2856	3.1595	2.9252
RMSE: Root of mean squared error	2.0702	1.7775	1.1703
MAE: Mean absolute error	22.6838	19.4812	17.6854
$R^{2}$ : R-Squared	0.8908	0.9095	0.9875
$R^{2}$ (Adj.): Adjusted R-Squared	0.8616	0.8854	0.9842

4.3.4 Robust estimations

Table 6 shows that, for all robust methods, most exogenous variables are significant because the $p$ -values of their variables less than 0.05, these results indicate that the robust estimation methods have smallest standard errors compared with non-robust (OLS and Zellner) estimators, and the three models are significant overall. While the results in Table 7 show that the three models have the smallest values of all goodness-of-fit measures, and higher $R^{2}$ and $R^{2}$ (Adj.) values than Zellner and OLS models. This means that the robust estimation methods improved the efficiency and the significant of the SURE model compared with the Zellner and OLS methods. Since MM-estimation has the minimum MSE, RMSE, MAE and higher values of $R^{2}$ and $R^{2}$ (Adj.), so we can say that MM-estimate is the best estimation method for this dataset.

5. Concluding remarks

In this paper, we have discussed some robust estimation methods for the SURE model with an empirical study on the Egyptian insurance market during the fiscal year from 1999 to 2018. The use of the robust estimation methods in the presence of outliers tends to improve the efficiency and reduce the bias compared with the non-robust (OLS and Zellner) estimation methods. The results indicate that the robust estimation methods have the smaller values of MSE, RMSE, and MAE, and higher R-squared than OLS and Zellner methods, although the data contains few outliers. The best estimation for this dataset is obtained by MM-estimation, and the selected exogenous variables in our study have a significant effect on the net profit in the Egyptian insurance market. Therefore, we recommend that non-robust methods residuals be examined, if they have outliers, than a robust estimation method should be used to get an efficient estimation In future work, the Monte Carlo simulation study can be performed to compare the different robust estimation methods for this model in different situations (different samples sizes, different number of exogenous variables, and so on), see [52, 53].

Footnotes

This problem arises when the exogenous variables are highly inter-correlated. Then it becomes difficult to disentangle the separate effects of each of the exogenous variables on the endogenous variables. As a result, the estimated regression parameters may be statistically insignificant and/or have, unexpectedly, different signs. Thus, conducting a meaningful statistical inference would be difficult for the researcher, see e.g. [] for handling and solving this problem in SURE models.

References

Zellner

. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American statistical Association, 1962; 57(298): 348-368.

Zellner

. Estimators for seemingly unrelated regression equations: some exact finite sample results. Journal of the American Statistical Association, 1963; 58(304): 977-992.

Stewart Gilbert

. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM Journal on Numerical Analysis, 1980; 17(3): 403-409.

Parks Richard

. Efficient estimation of a system of regression equations when disturbances are both serially and contemporaneously correlated. Journal of the American Statistical Association, 1967; 62(318): 500-509.

Youssef

. The statistical curvature of seemingly unrelated unrestricted regression equations. The Egyptian Statistical Journal, 1997; 41: 43-50.

Youssef

. A New Distribution Form for SURE Estimates. http//citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.9215&rep=rep1&type=pdf.

Abonazel

. Different estimators for stochastic parameter panel data models with serially correlated errors. Journal of Statistics Applications and Probability, 2018; 7: 423-434. doi: 10.18576/jsap/070303.

Kotakou Christina

. Panel data estimation methods on supply and demand elasticities: the case of cotton in Greece. Journal of Agricultural and Applied Economics, 43.1379-2016-113710; 2011: 111-129.

Abonazel

. Generalized estimators of stationary random-coefficients panel data models: asymptotic and small sample properties. REVSTAT-Statistical Journal, 2019; 17(4): 493-521.

10.

Yogyakarta

. Brief introduction seemingly unrelated regression equations (SUR) models application and development. Aldon mhp & sinaga; 2015.

11.

Abonazel

Saber

. A comparative study of robust estimators for poisson regression model with outliers. Journal of Statistics Applications and Probability, 2020; 9: 279-286. doi: 10.18576/jsap/090208.

12.

Abonazel

Gad

. Robust partial residuals estimation in semiparametric partially linear model. Communications in Statistics-Simulation and Computation, 2020; 49: 1223-1236. doi: 10.1080/03610918.2018.1494279.

13.

Kelly

Anne

, Si Li. The Impact of Regulation on the Availability and Profitability of Auto Insurance in Canada. Available at SSRN 1698042; 2013.

14.

Tan, Yong, Christos Floros. Risk, profitability, and competition: evidence from the Chinese banking industry. The Journal of Developing Areas, 2014: 303-319.

15.

Thronton, James. Econometrics – SUR Models Text Files. University of Eastern Michigan, USA; 2013. Downloaded from http://people.emich.edu/jthornton/text-filesEcon515_out_sur_model.doc. July 7th, 2015.

16.

Schmidt

. A note on the estimation of seemingly unrelated regression systems. Journal of Econometrics, 1978; 7(2): 259-261.

17.

Srivastava

Giles

. Seemingly unrelated regression equations models: Estimation and inference (Vol. 80). CRC press, 2020.

18.

Hawkins.

,Identification of Outliers, Chapman and Hall, London, 1980. doi: 10.1080/00401706.1984.10487956.

19.

Adepoju

Akinwumi

. Effects of atypical observations on the estimation of seemingly unrelated regression model. Journal of Mathematical Sciences, 2017; 5(2): 30-35.

20.

Chen, Jengrong James, Testing for outliers in linear models. Retrospective Theses and Dissertations; 1978: 6448. https//lib.dr.iastate.edu/rtd/6448.

21.

Kannan

Manoj

. Outlier detection in multivariate data. Applied Mathematical Sciences, 2015; 9(47): 2317-2324.

22.

Hampel Frank

. The influence curve and its role in robust estimation. Journal of the American Statistical Association, 1974; 69(346): 383-393.

23.

Rousseeuw

, et al. Robust multivariate regression. Technometrics, 2004; 46(3): 293-305.

24.

Koenker

Portnoy

. M-estimation of multivariate regressions. Journal of the American Statistical Association, 1990; 85(412): 1060-1068.

25.

Kmenta, Jan Roy

. Small sample properties of alternative estimators of seemingly unrelated regressions. Journal of the American Statistical Association, 1968; 63(324): 1180-1200.

26.

Rousseeuw

Leroy

. Robust regression and outlier detection, John Willey & Sons. Inc., New York; 1987.

27.

Hubert

Tim

Özlem

. Fast robust SUR with economical and actuarial applications. Statistical Analysis and Data Mining: The ASA Data Science Journal, 2017; 10(2): 77-88.

28.

Salibian-Barrera

Yohai

. A fast algorithm for S-regression estimates. Journal of Computational and Graphical Statistics, 2006; 15(2): 414-427.

29.

Bilodeau

Duchesne

. Robust estimation of the SUR model. Canadian Journal of Statistics, 2000; 28(2): 277-288.

30.

Davies

. Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices. The Annals of Statistics, 1987; 15(3): 1269-1292.

31.

Lopuhaa Hendrik

. On the relation between S-estimators and M-estimators of multivariate location and covariance. The Annals of Statistics, 1989: 1662-1683.

32.

Ruppert

. Computing S-estimators for regression and multivariate location/dispersion. Journal of Computational and Graphical Statistics, 1992; 1(3): 253-270.

33.

Rocke

. Robustness properties of S-estimators of multivariate location and shape in high dimension. The Annals of Statistics, 1996: 1327-1345.

34.

Van Aelst

Gert

. Multivariate regression S-estimators for robust estimation and inference. Statistica Sinica, 2005: 981-1001.

35.

Peremans

Van Aelst

. Robust inference for seemingly unrelated regression models. Journal of Multivariate Analysis, 2018; 167: 212-224. doi: 10.1016/j.jmva.2018.05.002.

36.

Kudraszow

Ricardo

. Estimates of MM type for the multivariate linear model. Journal of Multivariate Analysis, 2011; 102(9): 1280-1292.

37.

Berrendero

Beatriz

David

. On the maximum bias functions of MM-estimates and constrained M-estimates of regression. The Annals of Statistics, 2007; 35(1): 13-40.

38.

Abonazel

Rabie

. The impact of using robust estimations in regression models: an application on the Egyptian economy. Journal of Advanced Research in Applied Mathematics and Statistics, 2019; 4(2): 8-16.

39.

Wagdi

. Egyptian Insurance Market: History and Structure. Available at SSRN, 2014: 2385329.

40.

Financial Regulatory Authority (FRA). Annual Statistical report of Egyptian Insurance Market; 2019. http://wwwfra.gov.eg/content/efsa_en/eisa_pages_en/report_eisa_en.htm.

41.

Farid

. Using segmental time series to design the best model to explain the relationship between: early warning indicators and profit or loss for the year, an indicator for evaluating the performance of public insurance corporation in the Egyptian insurance market. Egyptian Journal of Insurance and Actuarial Sciences, 2014; 4: 47-111.

42.

Tzung-Ming

Chaang-Yung

. Business performance assessment of insurance company via Grey relational analysis. Journal of Grey System, 2011; 23(1): 83-90.

43.

Rahmani

, et al. Non-parametric frontier analysis models for efficiency evaluation in insurance industry: a case study of Iranian insurance market. Neural Computing and Applications, 2014; 24(5): 1153-1161.

44.

McElroy

. Goodness of fit for seemingly unrelated regressions: Glahn’s r2y.x and hooper’s r2. Journal of Econometrics, 1977; 6(3): 381-387.

45.

Henningsen

Jeff

. Systemfit: a package for estimating systems of simultaneous equations in R. Journal of Statistical Software, 2007; 23(4): 1-40.

46.

Abonazel

. New ridge estimators of SUR model when the errors are serially correlated. International Journal of Mathematical Archive, 2019; 10(7): 53-62.

47.

Breusch

Adrian

. The lagrange multiplier test and its applications to model specification in econometrics. The Review of Economic Studies 1980; 47(1): 239-253.

48.

Dufour

Lynda

. Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions. Journal of Econometrics, 2002; 106(1): 143-170.

49.

Breusch

Pagan

. A simple test for heteroscedasticity and random coefficient variation. Econometrica: Journal of the Econometric Society, 1979: 1287-1294.

50.

Shapiro, Samuel Sanford Martin

. An analysis of variance test for normality (complete samples). Biometrika, 1965; 52(3/4): 591-611.

51.

Galambos

Simonelli

. Bonferroni-type inequalities with applications. New York: SpringerVerlag. New York, Inc. 1996.

52.

Abonazel

. A practical guide for creating monte carlo simulation studies using R. International Journal of Mathematics and Computational Science 2018; 4(1): 18-33.

53.

Abonazel

. Handling outliers and missing data in regression models using R: simulation examples. Academic Journal of Applied Mathematical Sciences, 2020; 6(8): 187-203. doi: 10.32861/ajams.68.187.203.

Residuals diagnostic tests
I. Heteroscedasticity test: Breusch-Pagan-Godfrey (BPG)
GQ-Statistic	0.005	$p$ -value	0.999
II. Normality test: Shapiro-Wilk
W-Statistic	0.847	$p$ -value	2.471 $\times$ 10 ${}^{-6}$
III. Outlier test: Bonferroni
Bonferroni-Statistic	6.1896 $\times$ 10 ${}^{-9}$	$p$ -value	1.032 $\times$ 10 ${}^{-10}$