The Relationship Between Employment and Poverty Using Fuzzy Regression

Besma Belhadj and Fir as Kaabi

Introduction

Poverty has many causes, first among them is unemployment or low-wage employment. Because of their income, or its absence, the poor cannot afford health care, making them more susceptible to disease and less likely to work full- or even part-time.

The eradication of extreme poverty requires addressing its root causes. Therefore, information on the characteristics of the population living in poverty is essential. In particular, data on their employment status (i.e. whether they are employed, unemployed or not part of the labour force) can provide insights into the factors that create poverty. For those who are employed yet still live in poverty, also called the working poor, it is likely that low earnings and, more generally, sub-standard working conditions, are to blame. Conversely, for the poor who are either unemployed or uncounted for in the labour force, poverty may be driven by a lack of employment opportunities and insufficient social protections.

Data on employment broken down by economic class, especially the share of the employed who are poor (the working poverty rate) conveys information on the link between employment and poverty, which is highly relevant for the formulation of effective policies. Employment should be a vector to lift people out of poverty, but this is true only if job quality is sufficient, including adequate earnings, job security and a safe working environment. The relationship between employment and poverty depends greatly on the extent to which decent work is ensured for the labour market.

To express this relationship, we use a classical statistical linear regression binder the poverty rate at the unemployment rate. In this statistical regression, problems can occur in the following situations: the number of observations is inadequate (small data set); difficulties arise in verifying distribution assumptions; the relationship between input and output variables is vague; the events or the degree to which they occur are ambiguous; and linearisation introduces some inaccuracy and distortion.

Thus, this statistical regression is problematic because of vagueness in the relationship between the independent and dependent variables and ambiguity associated with the event. This is the kind of situation that fuzzy regression is meant to address.

This chapter is structured as follows: The second section estimates the poverty rate based on the unemployment rate and explains the results and limitations of this estimation. The third section outlines an alternative estimation method called fuzzy regression. In this section, estimators of a linear fuzzy regression model are constructed, and the consistency of these estimators is established. The last section concludes.

The Poverty Rate as a Function of the Unemployment Rate

We start with the observation that poor people derive most of their income from work. This basic fact means that employment and the access which the poor have to decent earnings opportunities are crucial determinants in the reduction of poverty.

This observation leads us to estimate the relationship between the poverty rate P/ and the unemployment rate Uj in the period 2000-2015 for a group of developing countries for which the unemployment rate and the poverty rate are available, as follows:

This relationship depends on national income: the unemployment rate is generally higher in developing middle-income countries, and the poverty rate is higher in developing low-income countries. These two facts make the relationship negative.

Equation (17.1) is only illustrative, however, because unemployment data generally include not only the long-term unemployed but also the shortterm unemployed, some of whom may be poor. However, the assumption made here is that the majority are long-term unemployed.

Fuzzy Linear Regression Models

Tanaka et al. (1982) were the first to introduce a fuzzy linear regression model. They constructed a linear regression model with fuzzy response data, crisp prediction data and fuzzy parameters as a mathematical programming problem. Their approach was later improved in many other studies, which can be roughly divided into two approaches: linear programming-based methods, called the possibilistic approach (e.g. Tanaka, 1987; Tanaka and Ishibuchi, 1991; Diamond and Tanaka, 1999; Hong and Yi, 2003; Hong et al., 2004), and fuzzy least-squares methods (e.g. Diamond, 1988; D’Urso and Gastaldi, 2000; D’Urso, 2003; Yang and Liu (2003).

Classical Statistical Regression and Fuzzy Regression Analysis

In many situations, the observations cannot be described accurately, for instance, when they depend on individual responses, as in the case of poverty. In this case, we can provide only an approximate description of them or an interval in which to enclose them. Here, we are concerned with a kind of uncertainty which is different from randomness, sometimes referred to as vagueness.

In ordinary regression analysis, deviations between the observed values and the estimates are assumed to be due to random errors. However, these deviations are sometimes due to the indefiniteness of the structure of the system or imprecise observations. The uncertainty in this type of regression model then becomes fuzziness, not randomness.

In fact, randomness and fuzziness are two different kinds of uncertainty (Chang and Ayyub, 2001). In ordinary regression analysis, the performance and validity of the least-squares method are reduced if its assumptions, such as independence and homogeneity of error terms, are violated. Fuzzy linear regression aims to model vague and imprecise phenomena using fuzzy model parameters. Fuzzy regression is different from ordinary least squares in the sense that it is not a statistical method (Kim and Chen, 1997).

The functional form of regression model (1) is as follows:

In some cases, we may need to consider that the relationship expressed in Equation (17.2) may be fuzzy. Indeed, three cases are possible when the predictor variable is fuzzy, but the parameters are crisp,

the predictors are crisp and the parameters are fuzzy,

and when both the predictor and parameters are fuzzy,

where Pj are the fuzzy responses.

Fuzzy Least-Squares Method

In this section, we estimate the parameters of model (17.4) in which the predictor or unknown parameters are trapezoidal fuzzy numbers.

E = [ehem,e,nep^ is a trapezoidal fuzzy number in which e( and ep are the left and right endpoints of the corresponding trapezoid, and em and en are the left and right midpoints (Figure 17.1).

Trapezoidal fuzzy number

Figure 17.1 Trapezoidal fuzzy number.

In model (17.4), if ax and a2 are trapezoidal fuzzy numbers, where ci = and u2 = ^^2/, fl2my&im&ip), then, for U- > 0, ci + д2Uj is

the trapezoidal fuzzy number:

To find an estimate for д, and a2 we minimise the sum of the squared distances (Arabpour and Tata, 2008):

Differentiating with respect to au, aUn, д, a]p and д,(, alm, a2n, a,p, we obtain the estimates of д, and a2 as follows:

In the following sections, we propose two alternative methods to the least- squares method. These are, respectively, the error’s membership function method and the dissemblance index method.

The Error’s Membership Function Method Beginning with model (17.4), we can write:

The term on the right-hand side of Equation (17.7) shows subtraction of fuzzy numbers as follows:

The estimate of л, and a2 in model (17.4) involves solving the following minimisation problem:

Suppose that pr (E) and Ps,+s2u, (<* + a2Uj)are represented as in Figure 17.2; they are expressed, respectively, in Equations (17.10) and (17.11).

Error’s membership function

Figure 17.2 Error’s membership function.

Let

and

Hh+hu, (Д1 + Иг (^) *n Equation (17.8) is represented in Figure 17.2

and expressed in Equation (17.12).

(а)

_g

Now using Equation (17.10), let Va e [0,1] a = —--

Zm ~ Zl

From which Pa =

(zm — Zi) + Zi,Zm, Z,nG. {zn — Zp ] + Zp J

Ь(а) _ i,

Now using Equation (17.11), we obtain a =-£--!L

bjm - bn

From which (аг+а2и)а =(4"),^),^)4“))

= [a {b„„ - bit) + bihbmnb„„a (b,„ - bip) + (?„,]

We must then have

Thus, if we define - Ь^ = zi - a (b„„ -b,i)~ Ьц

Then,

where = д]( + a2jU, j = /, m, n,p i = 1,n.

We note that the graph for Equation (17.12) can be visualised as in Figure 17.2.

We show that to solve Equation (17.9) is to solve the system

This proves that the error’s membership function, pt;tьд2), is also the hatched area of Figure 17.3 given by the difference between the membership function of the observed value p,^ (P,) and the membership function of the estimated value Рщ+-а1и, (a2Uj) of Pr Therefore, the solution to Equation (17.9) is equivalent and it amounts to minimising the hatched areas, hence the following method.

Distance S(A,B) (using the area method)

Figure 17.3 Distance S(A,B) (using the area method)

272 В. Belhadj and F. Kaabi The Dissemblance Index Method

Starting with Equation (17.10), we define A = \_zi,zm,z„,zp~

and starting with Equation (17.11), Ba = jy,i + a (b,m - Ьц ),b,p + a(bin - bip )J where b„ = atl + a2lU, j = l,m,n,p.

Note that A = [г/,г„,,г„,г,,] and В = \_blhb,m,bm,bip~^ intersect at the following two points: zi + a (zm -Zi) = b,i + a (b„„ - b,i) yields

Zp + a(z„ -zp) = bip + a(b„, -bip) yields

i

We then have

and 4“> - = Zp + a (z„ -zp)-bip-a (b,„ - bip)

= zp- bip +a(z„ -zp - b,„ + bip)

If we proceed to integrate from a = 0 to a = 1, we obtain a distance

S(A,B) by summing the distances S(A,B) = S(Aa,Ba)da, which is called

Ja= 0

the dissemblance index of A and B. S(A,B) is a distance between the fuzzy numbers A and B.

The calculation of Equation (17.15) shows that:

where bjj = aj + a2jUj j = l,m,n,p i = l,...,«.

With trapezoidal fuzzy numbers, the distance S(A,B) is the sum of the hatched areas in Figure 17.3.

The estimate of д, and a, in model (17.4) consists of solving the following minimisation problem:

The solution to Equation (17.17) consists of solving Equation (17.18).

The solution to Equation (17.18) for n -» oo is:

Based on Equations (17.19) and (17.20), in developing countries, the relationship between the poverty rate and the unemployment rate is always negative.

One explanation for this finding is that in developing countries, the poor are forced to work without receiving real social assistance. In these countries, it also appears that the long-term unemployed, for the most part, cannot be considered poor. The reason for these two facts is illustrated by the potential for a negative relationship between the unemployment rate and the poverty rate.

The nature of the relationship between poverty and unemployment has recently been unclear. Although this topic has received more extensive scholarly attention worldwide, no studies have examined this negative relationship in developing countries. This is because policymakers approach the general notion of the labour market according to the conditions in a developed market economy. In these economies, which have systems of social insurance, the problem of poverty is rightly captured in the unemployment rate.

Conclusion

In this chapter we use a new method to estimate the parameters of a fuzzy regression model with a crisp predictor and fuzzy parameters. This method is especially attractive because it is based on an integral abstraction of reality while retaining the same process as a classical linear regression. Moreover, in this method, the method of calculation is related to the form of the membership functions and therefore to the reality considered. In a future work, we extend our methodology to the case of a fuzzy predictor and fuzzy parameters.

References

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Diamond P., Tanaka H. (1999), Fuzzy regression analysis. In Roman Slowinski (Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, p. 387. Kluwer Academic.

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