AN ANALYSIS OF POVERTY IN ITALY

THROUGH A FUZZY REGRESSION MODEL

S. Montrone, F. Campobasso, P. Perchinunno, A. FanizziUniversità degli Studi di Bari - Dipartimento di Scienze Statistiche

ICCSSA 2011 GEOG-AN-MOD 11

Santander, 20-23 June 2011

ICCSSA 2011 GEOG-AN-MOD 11

Santander, 20-23 June 2011

Over recent years, and related in particular to the significant

contemporary international economic crisis, an increasingly worrying rise

in poverty levels has been observed both in Italy, as well as in other

countries.

INTRODUCTIONINTRODUCTION

The present work elaborates data revealed by the EU-SILC survey (2006)

regarding the perception of poverty by Italian families,

through a fuzzy regression model.

1. Different approaches to the poverty (absolute, relative, subjective)

INDEX INDEX

2. Techniques of the Fuzzy Set (A Fuzzy Regression Model)

3. The application of the Fuzzy Approach: construction of Eu-Silc indicators

and definition of fuzzy numbers

4. Results of the Fuzzy Regression Model

Traditional distinction between absolute and relative poverty:

•the first is understood as the incapacity to reach an objective level of wellbeing;

•the second is based on the assumption that the social condition of an individual,

cannot be well defined without taking in to account his context of living.

1. DIFFERENT APPROACHES TO THE POVERTY1. DIFFERENT APPROACHES TO THE POVERTY

A transversal approach is considered as subjective, through which the poor is

defined as his perception in comparison with the rest of society (in terms of

perceived wellbeing).

2. A FUZZY REGRESSION MODEL2. A FUZZY REGRESSION MODEL

Fuzzy regression techniques can be used to fit fuzzy data into a

regression model.

Diamond (1988) treated the simple Fuzzy regression model introducing

a metrics into the space of triangular fuzzy numbers.

In this work we explicit the expression of the parameters of the model

with fuzzy asymmetric intercept in the multiple case, starting from the

simple model handled by Diamond.

A FUZZY NUMBERSA FUZZY NUMBERS

Modalities of quantitative variables are commonly given as

exact single values, although sometimes they cannot be

precise (the imprecision of measuring instruments and the

continuous nature of some observations).

On the other hand qualitative variables are commonly

expressed using common linguistic terms (which also

represent verbal labels of sets with uncertain borders).

The appropriate way to manage such an uncertainty of

observations is provided by fuzzy numbers.

A triangular fuzzy number for the variable X is characterized by this function

that expresses the membership degree of any possible value of X to

TRL )x,x,x(X~

Diamond (1988) introduced the following metrics onto the space of triangular

fuzzy numbers

:

2TRLTRL

2 )y,y(y,,)x,x,x(d)Y~

,X~

(d 2RR

2LL

2 )yx()yx()yx(

A FUZZY NUMBERSA FUZZY NUMBERS

0,1X:μX~

X~

The same Author derived the expression of the estimated coefficients in

a fuzzy regression model of a dependent variable on a single

independent variable .

We generalize this estimation procedure to the case of several

independent variables with a fuzzy asymmetric intercept.

X~

Y~

A FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPTA FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPT

Assuming to regress a dependent variable on k independent variables in a set of n units,

the linear regression model with a fuzzy intercept is given by this formula:

where

TRiLiii )y,y,(yY~

TRijLijijij )x,x,x(X~

ijj* X

~bA

~~iY

TRL )a,a (a,A~

aa L

aa R

A FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPTA FUZZY REGRESSION MODEL WITH FUZZY ASYMMETRIC INTERCEPT

The estimates of the fuzzy regression coefficients are determined by

minimizing the sum of the Diamond’s squared distances, between

theoretical and empirical values of the dependent variable respect to a,

b1, .., bk, ,

The function to minimize assumes different expressions according to the

signs of the regression coefficients b1, .., bk.

2

ijji )X~

bA~

,Y~

d(

ESTIMATION OF THE FUZZY REGRESSION MODELESTIMATION OF THE FUZZY REGRESSION MODEL

γ γ

The estimates of the fuzzy regression coefficients are so given by

this formula

= [ X' X + (XL' XL + XR' XR) ] -1 [ X'y + (XL'yL +XR'yR) ]

where:

y is the n-dimensional vector of cores of the dependent variable;

yL and yR are the n-dimensional vectors of the lower extremes and the upper

extremes respectively of the dependent variable;

X is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent

variables and two vectors 0;

XL is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent

variables and two vectors -1 and 0;

XR is the n×(k+3) matrix formed by vectors 1, k vectors of cores of the independent

variables and two vectors 1 and 0;

is the vector (a, b1, .., bk, γL, γR)'.

ESTIMATION OF THE FUZZY REGRESSION MODELESTIMATION OF THE FUZZY REGRESSION MODEL

A fuzzy version of the index R2, which may be called Fuzzy Fit Index (FFI),

will be used in order to evaluate how the model fits data:

The more this index is next to one, the better the model fits the observed

data.

Finally we propose a stepwise procedure in order to simplify,

in computational terms, the identification of the most significant

independent variables.

2

2*

),~

(

),~

(

YYd

YYdFFI

i

i

STEPWISE PROCEDURESTEPWISE PROCEDURE

In the present study data are elaborated arising from EU-SILC survey

regarding the perception of the Italian families in “getting through to the end

of the month”.

3. THE APPLICATION OF THE FUZZY APPROACH3. THE APPLICATION OF THE FUZZY APPROACH

It emerges, in particular, that the majority of households surveyed declared

themselves to be in a state of hardship (either in great hardship 13.4%, in

hardship 19.4% or in some degree of hardship, 40.2%). There are, however,

few families (6.0%) declaring that they get through to the end of the month

with absolute confidence.

3. THE APPLICATION OF A FUZZY REGRESSION MODEL 3. THE APPLICATION OF A FUZZY REGRESSION MODEL

Table 1. Distribution of households surveyed by level of hardship in terms of “getting through to the end of the month”.

Getting through to the end of the month… Absolute values %

1 with great difficulty 2,884 13.4%

2 with difficulty 4,181 19.4%

3 with some difficulty 8,643 40.2%

4 fairly easily 4,506 21.0%

5 easily 1,115 5.2%

6 very easily 170 0.8%

Total 21,499

100.0% Source: EU-Silc, 2006.

The present work aims to identify the relationship between several

independent variable Xi (expenses for rent or mortgage payments, for the

running of the household and for other debts) and a single dependent

variable Y (the difficulty of “getting through to the end of month”).

3. DEFINITION OF FUZZY NUMBERS3. DEFINITION OF FUZZY NUMBERS

Furthermore, in order to normalize the data collected, the explanatory

variables have been quantified with the same criteria.

In particular, the response categories in terms of mortgage payments, rent

and household costs are centred on 1, 3 and 5, whilst the response categories

in terms of expenses for other debts are centred on 0, 1, 3 and 5.

3. DEFINITION OF FUZZY NUMBERS3. DEFINITION OF FUZZY NUMBERS

The verification of those expenses which determine the degree of difficulty (in

terms of getting through to the end of the month) is conducted:

• firstly, comparing families in rental accommodation against homeowners

with a mortgage

• secondly, according to geographical area (north of Italy, centre and south

and islands).

4.RESULTS OF THE REGRESSION MODEL4.RESULTS OF THE REGRESSION MODEL

The estimated regression coefficients for families in rented houses are reported

below:

The most relevant expenses (in terms of difficulty in getting through to the end

of month) results those relative to rental payments in all geographic area. The

“expenses for other debt” are relevant only in the North area.

4.RESULTS OF THE REGRESSION MODEL4.RESULTS OF THE REGRESSION MODEL

Families in rented

North Centre South and

Islands

Intercept 2.72 2.25 2.48

Spread left 0.32 0.36 0.47

Spread right 0.28 0.30 0.27

Rental expenses 0.22 0.19 0.20

Household expenses

0.14 0.14 0.16

Other debts 0.12

FFI 0.52 0.52 0.53

The estimated regression coefficients for families with mortgage rates are

reported below:

The most relevant expenses results those relative to the payment of mortgage

rates and differently from the previous model, above all in the south of Italy.

Besides, as we can see by the spread values, the variability is higher in this

case.

4.RESULTS OF THE REGRESSION MODEL4.RESULTS OF THE REGRESSION MODEL

Familes with mortgage

North Centre South and

Islands

Intercept 2.19 2.26 2.94

Spread left 0.33 0.49 0.30

Spread right 0.67 0.22 0.70

Mortgage expenses

0.20 0.22 0.23

Household expenses

0.13 0.13 0.14

Other debts 0.15

FFI 0.56 0.52 0.54

In this work we propose a Fuzzy Regression Model in order to identify the

factors that most influence the perception of poverty by Italian families.

5. CONCLUDING REMARKS5. CONCLUDING REMARKS

A subjective approach to poverty suggests the adoption of a fuzzy regression

model, made possible by an initial transformation of data into triangular fuzzy

numbers

The results of the analysis of poverty levels has showed at what degree the

most relevant expenses (in terms of getting through to the end of the month),

for Italian families, are those for rent and mortgage in the different

geographical areas.