ANNEX
If, we assume that a number of variables observed (latent variables), then what the method
does is to extract a group of indicators to characterize a variable whose behavior has not been
observed (Informality Index). Defining occupation following positions jobs:
Informal Sector
Informal employment in formal
X1 = Employees
X1 = Self-employment in agriculture or subsistence
X2= Self-employed
X2 = unpaid workers in different units Informal Sector
X3= Employers
X3 = Paid Domestic Service
X4= Unpaid Workers
X4 = Salaried workers unprotected sectors working to formal
X5= Other Workers
X5 = Unprotected workers without remuneration fixed in formal
With this these, the index is obtained from informality relative proportions in which each
position represents the employment of the total, thereby reducing the dimensionality of the
original set of variables.
ANNEX
Consider that:
TA  Salaried workers
X ij
TCP  Self-Employed workers
i  Job position 1, ,5
Emp  Employers
UW  Unpaid Workers
where :
j  Economic Activity Subsector 1, ,56
OT  Other workers
X1= Vector of weights TA NAICS subsector with order of 56x1:












x
1
x
1
x
1
X 11   56 1  , X 12   56 1  ,..., X 156   56 1 






x
1
x
1
x
1
 
 
 
 x11 
 x11 
 x11 
Thus :
 X 11 


X
1


X1   2 



 X1 
 56 
Thus :
 X 21 


X
2


X2  2 



X2 
 56 
X2=Vector of weights TCP NAICS subsector with order 56x1:












x
2
x
2
x
2
X 21   56 1  , X 2 2   56 1  ,..., X 2 56   56 1 






x
2
x
2
x
2
 
 
 
 x 21 
 x 21 
 x 21 
ANNEX
X3=Vector of weights Emp NAICS subsector with order56x1:












x
3
x
3
x
3
X 31   56 1  , X 3 2   56 1  ,..., X 356   56 1 






  x3 
  x3 
  x3 
 x 31 
 x 31 
 x 31 
Thus :
 X 31 


 X3 
X3   2 



X3 
 56 
X4=Vector of weights UW NAICS subsector with order of 56x1:












x
4
x
4
x
4
X 41   56 1  , X 4 2   56 1  ,..., X 4 56   56 1 






  x4 
  x4 
  x4 
 x 41 
 x 41 
 x 41 
Thus :
 X 41 


X
4


X4  2 



X4 
 56 
ANNEX
X5=Vector of weights OT NAICS subsector with order of 56x1:












x
5
x
5
x
5
X 51   56 1  , X 5 2   56 1  ,..., X 5 56   56 1 






  x5 
  x5 
  x5 
 x 51 
 x 51 
 x 51 
Thus :
Group X 1, X 2, X 3, X 4, X 5, order of 56 x5:
 X 11

 X 11
Group  


 X1
 56
X 21
X 31
X 41
X 21
X 31
X 41



X 256
X 356
X 456
X 51 

X 51 
 

X 556 
 X 51 


X
5


X5   2 



X5 
 56 
ANNEX
Econometric Estimation in with Software
Year of study: 2008
Principal Components of correlation:
Variable
Vector 1
X1
q1
X2
q2
X3
q3
X4
q4
X5
q5
Eigenvalue
Variance Prop.
Cumulative Prop.
Yields the following vector characteristic:
Vector 1 order 5x1=Vector Characteristic (VC)
 q6

 q7
VC   q 8

q9
q
 10








ANNEX
Squares of the elements of Vector Characteristic(CVC)
 q 11 


 q 12 
CVC   q 13 


 q 14 
q 
 15 
Principal Component (CP) = (Vector Characteristic)*(square root of Eigen value)
CP  VC  EIGENVALUE
 q 16

 q 17
CP   q 18

 q 19
q
 20








Weighting and Scoring=(Elements of Vector Characteristic)/(Principal Component)
Weighting 
VC
CP
 q 21 


q
 22 
Weighting   q 23 


q
24


q 
 25 
ANNEX
Proportion of Variance Explained
PROP.VARIANCE  (q 26) * (100)  %
Informality Index by Economic Activity Subsector, vector order 56x1
(Matrix X1, X2, X3, X4,X5 of order 56x5)*(Vector of Weights order 5x1)
 X 11 X 21 X 31 X 41

 X 11 X 21 X 31 X 41
Informality Index  





 X1
 56 X 2 56 X 356 X 4 56
 q 21 
X 51 
 

 q 22 
X 51 
*  q 23 

 


 q 24 

X 5 56  56 X 5  
 q 25  5 X 1
ANNEX
Adding Weights Vector, vector order 56x1 = S by subsector of X1, X2, X3, X4 y X5
Average Weights Vector, vector order 56x1=
by x
subsector de X1, X2, X3, X4 y X5
Yields some measures of central tendency: mean, standard deviation and variance of the Index of
Informality by sector of Economic activity:
Mean ( x )
Standard Deviation (σ)
Variance (σ2)
Index of Informality Standardized by Subsector of Economic Activity (IIE)
vector order 56x1 = Index of Informality (56x1)