# f.influence: Field of Influence In ioanalysis: Input Output Analysis

## Description

Calculates the field of influence. Can handle first to nth order field of influence. Uses the method as Sonis & Hewings 1992. This is a recursive technique, so computation time depends on the size of the data and order of field of influence.

NOTE: If you want to examine a % productivity shock to a specific region-sector, see inverse.important.

## Usage

 1 f.influence(io, i , j) 

## Arguments

 io An InputOutput class object from as.inputoutput i Numeric. The row component(s) of the coefficient(s) of interest j Numeric. The column component(S) of the coefficient(s) of interest

## Details

First Order Field of Influence - This is simply the product of the jth column of the Leontief inverse multiplied by the ith row of the Leontief inverse. In matrix notation:

F_1[i, j] = L_{.j} L_{i.}

where F denotes the field of influence, and i and j are scalars

Nth Order Field of Influence - This is a recursive function used to calculate higher order fields of influence. The order cannot exceed the size of the Intermediate Transaction Matrix (Z). I.e. if Z is 20x20, you can only calculate up to the 19th order. The formula is as follows:

F_k[(i_1,...,i_k), (j_1,...,j_k)] = \frac{1}{k-1} ∑_{s=1}^k∑_{r=1}^k (-1)^{s+r+1} l_{i_s,j_r} F_{k-1}[i_{-s}, j_{-r}]

where F is the field of influence, k is order of influence, l_ij is the ith row and jth column element of the Leontief Inverse and -s indicates the sth element has been removed.

## Value

Returns a matrix of the Field of Influence

## Author(s)

John J. P. Wade, Ignacio Sarmiento-Barbieri

## References

Sonis, Michael & Hewings, Geoffrey J.D. (1992), "Coefficient Chang in Input-Output Models: Theory and Applications," Economic Systems Research, 4:2, 143-158 (https://doi.org/10.1080/09535319200000013)

Blair, P.D. and Miller, R.E. (2009). "Input-Output Analysis: Foundations and Extensions". Cambridge University Press

Nazara, Suahasil & Guo, Dong & Hewings, Geoffrey J.D., & Dridi, Chokri, 2003. "PyIO. Input-Output Analysis with Python". REAL Discussion Paper 03-t-23. University of Illinois at Urbana-Champaign. (http://www.real.illinois.edu/d-paper/03/03-t-23.pdf)

inverse.important
  1 2 3 4 5 6 7 8 9 10 11 data(toy.IO) class(toy.IO) # First order field of influence on L[3,2] i <- 3 j <- 2 f.influence(toy.IO, i, j) # Second order field of influence on L[3,2], L[4,5], L[6, 3], and L[1,10] i <- c(3, 4, 6, 1) j <- c(2, 5, 3, 10) f.influence(toy.IO, i, j)