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Analysis of multiple linear regression of the total income of some insurance variables which are Fire insurance, Accident insurance, Vehicles insurance, liabilities insurance, and Marine insurance

Analysis of multiple linear regression of the total income of some insurance variables which are Fire insurance, Accident insurance, Vehicles insurance, liabilities insurance, and Marine insurance

 Analysis of multiple linear regression of the total income of some insurance variables which are Fire insurance, Accident insurance, Vehicles insurance, liabilities insurance, and Marine insurance. 



1.1     Background of the study

In some sense, we can say that insurance appears simultaneously with the appearance of human societies: money economic (with markets, money, financial instruments and so on) and non-money or natural economies (without money, markets, financial instruments and so on. The second type is a more ancient form than the first. In such an economy and community, we can see insurance in the form of people helping each other; for example, if a house burns down, the members of the community help build a new one. Should the same thing happen to one's neighbour, the other neighbours must help; otherwise neighbours will not receive help in the future. This type of insurance has survived to the present day in some countries where modem money economy with its financial instrument is not widespread.

Turning to insurance in the modern sense (i.e. insurance in a modern money economy, in which insurance is part of the financial sphere), early methods of transferring or distributing risk were practiced by Chinese and Babylonia trades as long ago as the 3"1 and 2nd millennial BC, respectively. Chinese merchants traveling treacherous river rapids would redistribute their wares across many vessels to limit the loss due to any single vessels capsizing. The Babylonians developed a system which was recorded in the famous code of Hammurabi, C. 1750 BC, and practiced by early Mediterranean sailing merchants. If a merchant received a loan to fund his shipment, he would pay the lender an additional sum in exchange for the lender's guarantee to cancel the loan should the shipment be stolen or lost at sea.

Achaemenian monarchs of Ancient Persia were the first to insure their people and made it official by registering the insuring process in government notary offices. The insurance tradition was performed each year in Norouz (beginning of the Iranian New Year); the heads of different ethnic groups as well as others willing to take part, presented gift to the monarch.

The most important gift was presented during a special ceremony. When a gift was worth more than 10,000 Derrik (Achaemenian gold coin) the issue was registered in a special office. This was advantageous to those who presented such special gifts. For others, the present were fairly assessed by the confidents of the court. Then the assessment was registered in special offices.

The purpose of registering was that whenever the person who presented the gift registered by the court was in trouble, the monarch and the court would help him. Janez, a historian and writer, writes in one of his books on ancient Iran; "whenever the owner of the present is in trouble or wants to construct a building, set up a feast, have his children married, etc. the one in charge of this in the court would check the registration. If the registered amount exceeded 10,000 Derrik, he or she would receive an amount of twice as much".

The Greeks and Romans introduced the origins of health of life insurance. 600 AD when they organized guilds called "benevolent societies" which cared for the families and paid funeral expenses of members upon death. Guilds in the middle ages served a similar purpose. The Talmud deals with several aspects of insuring foods. Before insurance was established in the late 17th century, "friendly societies" existed in England in which people donated amounts of money to a general sum that could be used for emergencies.

Some forms of insurance had developed in London by the early decades of the seventeenth century. For example, the will of the English colonist Rober Hayma mentions two "policies of insurance" taken out with the diocesan chancellor of London. Arthur duck. Of the value of E10 each, one relates to the sage arrival of Hayman's ship in Ganyana and the other is in regard to "one hundred pounds assured by the said Doctor Arthur Ducke on my life" Hayman's will was signed and sealed on 17 November 1628 but not proved until 1633. Toward the end of the seventeenth century London's growing importance as a centre for trade increased demand for marine insurance. In the late 1680s, Edward Lloyd opened a coffee house that became a popular haunt of ship owners, merchants, and ships captains, and thereby a reliable source of the latest shipping news. It became the meeting place for parties wishing to insure cargoes and ships and those willing to underwrite such ventures. Today Lloyd' of London remains the leading market (note that it is not an insurance company) for example and other specialist types of insurance, but it worlcs rather differently than the more familiar kinds of insurance.

Insurance as we know it today can be traced to the great fire of London devoured more than 13,000 houses. The devastating effects of the fire development of insurance "from a matter of convenience into one of urgency, a change of opinion reflected in sir Christopher Wren's inclusion of a site for the insurance office' in his new plan for London in 1667" a number of attempted fire insurance schemes came to nothing but in 1681 Nicholas Barbon, and eleven associates established England's first fir insurance company, the insurance office for houses' at the back of the Royal exchange. Initially, 5,000 homes were insured by Barbon's insurance office.

The first insurance company in the United States underwrote fire insurance and was formed in Charles town (modern-day Charleston, South Carolina in 1732.

1.3     Aims and Objectives

The study aimed to determine whether any relationship exist among the variables (i.e. accident, motor, liabilities, and marine insurance) by fitting appropriate model for data collected and analyse.

The specific objectives of the study are:

To formulate an equation showing the relationship among the variables

 To estimate the rate at which sources of income depends on variables (i.e. fire insurance, accident insurance, motor insurance, liabilities insurance and marine insurance)

 To determine the significant of the estimated regression co-efficient

 To know which department of insurance contribute most to the insurance company

1.3   Scope and Limitation

Expectedly, the study ought to cover the whole of insurance (life and non-life insurance) companies in Nigeria, but because of time, the voluminous nature of this approach and other, the study is therefore designed to have an analytical overview of the total income, fire insurance, Accident insurance, liabilities insurance and marine insurance during the period in review and specifically limiting the efforts of the statistical evaluation to non-life insurance companies in Nigeria.

The study however, covered period of thirty years of sources of income of non-life insurance companies form (1976-2005).

 In the research work, the sources of data were mainly secondary form central Bank of Nigeria Annual survey and National insurance commission (NAICOM) annual report


 MATERIAL AND METHODS

2.1 Methodology

The regression technique is a statistical tool that cut across almost all disciplines-science, medical science, social science, technology etc. in solving their everyday problems, especially in decision making process where decisions are reached with a degree of accuracy.

In order to predict a variable from other variables, it is necessary to construct a mathematical model whose line or curve, as the nature of the relationship permits, is close as possible to the actual line of the variable of interest.

2.1.2     Types of Regression

There are many regression models as much as that of mathematical equations. In fact, it put it as straight as possible; there is limitless number of regression models. This is because a unique model may be adopted or developed to fit a single set of variables so one can imagine the complexity of trying to identify each one as a unique type if regression model

However, regression models can be broadly classified into two broad types, the linear and no-linear regression models. As the name implies, the linear regression models refers to a regression model that is linear in parameters; and not necessarily in variables. While the nonlinear regression model implies any model otherwise, there are quite a number of generally accepted non-linear regression models, which were developed to suit uniquely characterized categories of data i.e. set of variables.

They include

The Exponential Regression Model

The Quadratic Model

The Cubic Model

The Logarithmic Model, Just to mention a few

In this write up, however, we shall place our emphasis on the linear which is the model that will be adopted for analytical purpose.


2.2 Statistical tools 

2.2.1 The Linear Regression Model

The linear regression model is the most widely used model, since most practical regression cases are linearly related. It involves building a model that is linear in form, such that when fits, it forms a linear line across the actual dependent line. The regression equation under the linear regression model is given as




where

Y = dependent variable

 = independent variable

=         the parameter of the model; and

=      the disturbance or error term

However, there are basically three techniques that can be used to estimate a linear regression model and they are:

The least square regression technique

The moving average technique and

The semi-average technique

Our interest, however, focuses on the least square regression technique which shall be employed for analytical purpose in this project. However, before discussion on this technique (i.e. the least squares), let us first take a look at the propositions of the regression model as well as the assumptions following the linear regression model.

2.2.2     The Proposition of Regression Model

This term regression model was introduced by Francis Gallon its major proponent. He found that although there was tendency for tall parents to have tall children and for short parent to have short children, the average height of children born to parents of a given height tend to more or "regress" towards the average height of the population as a whole.

This law was later confirmed by Karl Pearson who collected more than 1000 records of height of members of family groups. He found that the average height of sons of a group of tall fathers was greater than their father's height.

2.2.3     Assumption of the Classical Linear Regression Model

Like other regression models, the linear regression also has its own assumption guiding the efficiency of the model. These assumptions are briefly highlighted below:

The model is linear in parameters (and not the variables)

The A' values are fixed in repeated sampling, i.e. non-random. 

Zero mean value of the error term. That is given the value of X, the mean or the expected value of the error term is zero,  i.e.  E (ui/Xi ) = 0

Homoscedasticity of the disturbance term ui . That is the ui’s have equal variances

No autocorrelation between the disturbance terms (i.e. the correlation between any ui, and uj  given any Xi and Xt  is zero).

)


vi  There is zero covariance between   and   (i.e e( = 0


= E[ (-E(Xl))],  Since  = 0 

= E-E()E(Xi)

= 0, since E()E(Xl) = 0

The number of observations, «, must be greater than the number of parameters

to be estimated.

  Variability in A'-values in a given sample must not all be the same. 

The regression model must be correctly specified; .i.e. there must be no specification bias of the error in the model used in empirical analysis.

There is no perfect multicollinearity, i.e. the correlation between all  ,and  is zeoro: E(ui,uj) = 0

2.2.4    The Least Square Regression Techniques

The least squares regression technique is the most efficient of all the techniques used for estimating a linear regression model. This is because of all the regression lines that could be drawn to represent the dependent variable; the least squares regression line of Y on X is the line for which the sum of squares of the vertical deviations of all the points from the actual line is least

2.2.5 Dispersion of the Estimators

It is important to note that the term "dispersion of  simply implies the variance of the estimator  Since we are dealing with them in matrix form, it is not mathematically correct to term their dispersion as variances.

Now, recall from equation 

X'Y

Symbolizing the dispersion by D, this implies that the dispersion of  will be given as:

D(

But since by definition,

 =  X'Y = L'Y,  where L' = X', therefore,


D(X'D(Y)X


At this junction, it is important to note that one of the assumptions of the classical linear regression model is that the disturbance terms are normally and independently distributed with zero mean and homoscedastic variances, i.e. ui   ̴   NID ( 0, )

Note that the variance of the distribution term is the same as that of Y. Therefore, incorporating this assumption into equation above give rise to

D(β) =X' (X'X X

=  X'X

Also, recall that =  X'X = I , where I implies an identity matrix. Hence, the dispersion of β in (3.6) is given as

D(β) = I 

=  

However, the violation of the assumptions underlying above about the method of the least squares (or the least squares regression techniques) leads to the problems called the multicollinearity problem, the heteroscedasticity problem and lastly the autocorrelation problem; which are the regression problems.

2.2.6    The Multicollinearity Assumption

The term "multicollinearity" or collinearity was derived by Ragnar Frisch, who used it to refer to mean the existence of a "perfect", or "exact" linear relationship among some or all explanatory variables of a regression model.

It is used to refer to a situation where a perfect or exact linear relationship exists in some or all the explanatory variables i.e. an explanatory variable expresses in linear form to another.

However, the linear regression model (classical) assumes no perfect multicollinearity relationship exist among the explanatory variables because, if multicollinearity is perfect, the regression coefficient of the X variables (i.e. explanatory variables) indeterminate and their standard errors are infinite. However, if multicollinearity is less than perfect, the regression coefficient although determinate, possess large standard errors, which means the coefficient cannot be estimated with great precision or accuracy.

In other words, an independent a variable must not be completely expressed in linear form to another. If this is so, the assumption is over ruled; hence the regression model generated is no efficient enough.

There are various situations that may give rise to a multicollinearity problem within a linear regression model. As Montgomery and Peck put it, the problem may be due to the following factors and these include:

The data collection method employed, i.e. sampling over a limited range of the values taken by the regression in the population,

Constraints on the model or in the population being sampled. For example in theregression of electricity or pipe-borne water, on income and house size, there is a physical constraint in the population that families whit higher income generally have homes than families with lower income 

Model Specification, e.g. adding polynomial terms to a regression model, especially when the range of the A'variable is small

An over determined model: This happens when the model has more explanatory variables than the number of observations i.e. when the number of explanatory variable is large or more than the number of observations,

When  an  explanatory variables  is  a proportion  or per cent  of another

explanatory variable.

The practical consequences of the problem of multicollinearity with a linear regression model include:

Although BLUE (Best Linear Unbiased Estimators), the OLS (Ordinary Least Squares) estimators have large variances and covariance; making precise estimation difficult or impossible.

Because if the consequence (1) above, the confidence interval tend to be much wider, leading to the acceptance of the null hypotheses (i.e the true population confidence is zero) implies the β’s are not confidence

 Also because of the consequence (1) above, the t-statistic tend to be statistically insignificant.

Although the t-statistic is statistically insignificant the overall goodness of fit, the confidence of determination, R2  can be very high.

The OLS estimators and their standard errors can be sensitive to small changes in the data

 2.2.7     Test for Detecting Multicollinearity

These methods are based on Frisch's confluence analysis. As highlighted above, the problem of multicollinearity in a regression model gives rise to some consequence, thereby making the linear regression model less efficient as desired. To this end, it is necessary to detect and if possible rectify the problem. The seriousness of the effects of multicollinearity seems to depend on the degree of intercorrelation,  as well as on the overall correlation coefficient, ,  Thus one might suggest that the standard errors, the partial correlation coefficient ( ) and the total R2 may be employed for detecting or testing for multicollinearity. Yet, none of these criteria by itself is a satisfactory indicator of multicollinearity because; large standard errors don't always appear with multicollinearity. However, large standard errors may be as a result of various other reasons and not only because of the presence of multicollinearity.

Thus, since multicollinearity is essentially a sample phenomenon, we therefore, do not have the unique method of detecting it or measuring its strength; what we have are some RULES OF THUMBS. Some of these techniques to detect the existence of multicollinearity in a linear regression model, of which some formal and some informal mclude:

Estimation of auxiliary regression, Kleins Multicollinearity test

High coefficient determination, R2 but few significant t-ratio. If R2 is high, f-test will reject the null hypathoses.

High pairwise correlation among regression especially when p>0.8  

Examination of partial correlation

Eigen values and conditional index

Tolerance and variance inflation factor (VIT)

2.2.8     The Klein's MulticoUinearity Test

Since multicollinearity arises because one or more of the regression are exact or approximately linear combination of the other regressors, one way to find out which X variable is related to other X variables and compute the corresponding R2 i.e. Rf each one of these regression is auxiliary to the main regression on the other A"s, where

̴ 

where  n = The sample size

k = The number of explanatory variable including the intecept term 

 = the coefficient of determination in the regression of Xt variable on the remaining variables, X1, . . . Xk  follows the F distribution with k - 1 and n- k + 1 degrees of freedom

If the calculated F is greater than the tabulated F, the particular X, is collinear with the other X 's , instead of testing all auxiliary R* we can adopt Klein's rule of thumb which suggest that multicollinearity is a troublesome problem only if the R2 obtained from an auxiliary regression is greater than the overall R2 , i.e. that obtained from the regression of Y on all the regressors.

The test of hypothesis for this purpose is given below as:

H0 :  The model is not affected by a problem of multicollinearity  

H1:  The model is affected by a problem of multicollinearity

Decision Rule: Reject the null hypothesis, H0 , if   otherwise do not reject H0.


2.2.9    High *2 but Few Significant /-ratio

This is the "classic" symptom of multicollinearity. If R2  high the F-test in most cases will reject the null hypothesis that the partial slope coefficients are simultaneously equal to zero, but the  individual t-ratio will show that none or very few of the partial slope coefficient are statistically different from zero.

Its disadvantage is that multicollinearity is considered harmful only when all of the influence of the explanatory variables on y cannot be disentangled.

2.2.10    High Pairwise Correlation among Regression

Another rule of thumb is that if the pairwise of zero order correlation coefficient between two regressors is high (i.e. p > 0.8), then multicollinearity is a serious problem (sufficient but not a necessary condition).


Decomposition of sum of square of ANOVA

Having obtained known the least square estimator β, follows from a model that the vector of fitted values is

Y = X β

=LY

The vector of residuals is

U = Y - X β

= Y-LY

= (1-L)Y

The total sum of square of the responses variable is

SST =    = 


Adding and subtracting the quantity we have 

SST =  ( + ()

       = SSE + SSR

Therefore, the first pair of parentheses on the right equation constitutes the residual sum of squares, SSE, while the second is the sum of square due to regression, SSR. The total sum of squares thus breakdown, into the fundamental analysis of variance (ANOVA) identity

SST = SSE + SSR.

However, there are k regressors in the model, the number of degree of freedom for regressior is k-1, for the total is n-1, and for the residual n-k. the features of ANOVA table can be summerised as follows: 

Sources of variation

Degree of freedom

Sum of Square

Mean Square

F calculated


Regression




   






Residual









Total







2.2.11   Significant Test for Overall Regression Coefficients

Inferences about the regression parameters can be drawn through analysis of variance (ANOVA) table. The individual test reveals the contribution of each prediction to significance of the regression equation. 

Significant test for combine regression coefficient can also be examined through the (ANOVA) above

Hypotheses


   is not true

Decision Rule

If 

Otherwise 

Significant test for Separate Regression Coefficient 

Hypotheses


  

 

ANALYSIS OF DATA

3.1 Result and interpretation

This chapter feature the detailed analysis of multiple linear regression of the total income of some insurance variables which are Fire insurance, Accident insurance, Vehicles insurance, liabilities insurance, and Marine insurance.   

The parameters will be estimated and their significance will be tested using different statistical tools. The true regression model is of the form:


Where

X1,= F.I =  Fire Insurance

X2 = A.I = Accident Insurance

X3= V.I = Vehicle Insurance

X4 - L.I =   Liability Insurance

X5= M.I = Marine Insurance

Y = T.I =   Total Insurance


= 1,2, 3, 4, 5.

Table 3.1: Model Summary


Model Summary

Model

R

R-Sqaure

Adjusted R-Sqaure

Std. Error of the estimate


1

.998a

.996

.995

1244069.80



From Table 3.1 above, the multiple correlation R = 0.996 which means there is strong correlation in the dependent variable and the joint independent variables or dependent variables. Also, the coefficient of determinations  R2 = 0.995 means the goodness of fit showed that the five departments used explained 99.6% ,  of the variations in the Total Income,


 Table 3.2:  ANOVA

ANOVA

Model

Sum of square

df

Mean Square

F


sig


1     Regression

Residual

Total

9.0E+015

3.7E+013

9.0E+015

5

24

29

1.800E+0.15

1.548E+012

1162.825


.000a


a. predictors: (constant), marine, liabilities, accident, vehicles, fire.

b. Dependent variables: Total income

from table 3.2, the overall test of ANOVA showed that all departments studied contribute significantly to the income of insurance company.


Table 3.3: coefficients

Coefficients



Model 

Unstandardized 

coefficients

Standardized coefficient



t



sign



B

Std. Error

Beta




1    constant

      Fire

     accident

     Vehicle

      Liabilities

     marine

-17389

1.608

0.830

1.217

11.293

.955

303268.8

.597

.734

.215

2.461

.100


.253

.128

.344

.194

.186

-.573

2.694

1.131

5.668

4.590

9.576

.572

.013

.269

.000

.000

.000


a. dependent variable: total income

From table 3.3 the individual contribution using t-test showed that four departments marine insurance, liabilities insurance, vehicles insurance and fire insurance – contributed significantly to the incomes of insurance company while accident insurance is not significantly contributed to the income of insurance company.

 The percentage contribution of each department to total income is given by

Table 3.4: Percentage contributed 

Department

Percentage contributed


Fire  insurance  

25.30%


Accident insurance

12.30%


Vehicle insurance 

34.40%


Liabilities insurance

14.90%


Marine insurance

18.60%


We observe that Fire department contribute 25.30o/0 to the dependent variable, and Fire, Accident, veh.cle, Liabilities, and Marine Departments contribute 34.40%, 14.90%, and 18.60%, respectively.

Table 3.5: Model Summary


Model



R



R Square


Adjusted R

Square


Std. Error of the Estimate



1


.998"


.996


.995


1.25098E6



a. Predictors: (Constant), X5, X4, X1, X3

After removing the non-significant variable, the multiple correlation R = 0.998 which means there is strong correlation between the dependent and independent variables: Fire, Vehicle, Liabilities, and Marine Departments and also R = 0.996 which measure the goodness of fit so that the four department used explained 99.6% to the income of insurance companies.

From Table 3-6 below, the overall test of ANOVA shows that four departments studied contribute significantly to the income of insurance companies.

From Table 4-6, the individual using t-test shows that four departments (i.e. Fire, Vehicle, Liabilities, and Marine Departments) contributed significantly to the income of durance companies.

Table 3.6: ANOVA


ANOVA


Model

Sum of Square

df

Mean Square

F

Sig


1      Regression

Residual

Total

8.997E15 3.912E13 9.036E15                  

4                25           29          

2.249E15 1.565E12

1437.191

.000a
















a. predictors (constant) x5, x4, x1, x3

b. Dependent variable: y



Table 3.7: Coefficients'

Coefficients'





Standardized









Unstandardized Coefficients


Coefficients







Model


B


Std. Error


Beta


t


Sig.



1           (Constant)


-267803.279


293296.730




-.913


.370



x1


2.199


.290


.346


7.584


.000



x3


1.369


.168


.387


8.154


.000



x4


10.561


2.387


.139


4.424


.000



x5


.962


.100


.188


9.608


.000



a. Dependent variable: y

The percentage contribution of each department to total income, after removing one variable that is not significant to the model, is given by

Fire insurance 34.60%

Vehicle insurance 38.70%

Liabilities insurance 13.90%

Marine insurance 18.80%


3.2 Discussion

This project work is concerned with how various classifications of non-life insurance by department contribute to the income of non-life insurance companies in Nigeria.

The data used was extracted from Central Bank of Nigeria Annual Survey and National Insurance Commission (NAICOM) Annual Report covers period of 30 years from 1976-2005 for the following sources r=Total income, with X,,i = 1,2,3,4,5 corresponding to the contributions from fire insurance, accident insurance, vehicle insurance, liabilities insurance and marine insurance.

The multiple linear regression has been employed in this project. The regression model fitted for this work is

Y = -173891.4 + 1.608X1, + 0.830X2 + 1.217 X3 + 11.293X4, + 0.955 X5 (s.e.) (0.597) (0.734) (0.215) (2.461) (0.100) R2 = 0.996, N = 30

The regression coefficient contribute positively to the model. The high positive value of R2 shows that the model adequately fit the data.

A coefficient of determination of 99.6% shows that the five department indicators considered explained 99.6% of the insurance companies.

The percentage contribution of each of the department is also obtained as follows: Fire insurance 25.3%, accident insurance 12.3%, vehicle insurance 34.4%, liabilities insurance 14.9%, and marine insurance 18.6%.

Durbin-Watson test was used in testing for autocorrelation. It is therefore recommended that government should put in place relevant policies which will afford better participation of the populace in insurance business in Nigeria, i.e. cutting down on their taxes, minimizing losses resulting from pure risk (i.e. making road safer from motorists to reduce motor accidents) and the society should encourage education on insurance business to improve the development of the economy and the nation as a whole.

    

CONCLUSION

In this project work, the linear regression model is appropriate for the insurance department data in which liabilities insurance department is contributing most to the total income of the insurance companies that is 11.293 contributing to the total income of insurance companies, Fire Insurance is contributing next to the total income of insurance companies that is 1 .608 contributing to the total income of insurance companies.

The individual contribution using t-test shows that four departments have significant association with total income of insurance companies - Fire Insurance Department, Vehicle Insurance Department, Liabilities Insurance Department, and Marine Insurance Department. While Accident Insurance Department does not have significant association with Total Income of insurance companies.

Also, the joint test using F-test shows that all departments of insurance contribute significantly to the income of insurance company.

Test for autocorrelation shows that there is negative autocorrelation.

 

REFERENCES

Adebowale S.A. (2006): Statistics for Engineers, Managers and Scientists, Alfredo

Alaba O. O. (2010): Unpublished Lecture Note on Regression Analysis and Analysis Of  Variance II.

Ayoola, J.F. (2009): Unpublished Lecture Note on Regression Analysis and Analysis of Variance I.

John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman (1999): "Applied 

Linear Statistical Models," 4th ed., WCB/McGraw-Hill Companies Inc., New York

Moses Yode Omotosho (2007): "Econometrics: A Practical Approach," Revised Edition. Yosod Book Publisher.

Murray R. Spiegel, Larry J. Stephens (1999): "Schaum's Outline of Theory and Problems of Statistics," 3rd ed., McGraw-Hill Book Company.

 Ojo J. F. (2010): Unpublished Lecture Note on Acturial Statistics.


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