Embed Size (px)
Citation preview
Luis Daniel Torres Gonzalez and Jangho Yang
The Persistent Statistical Structure of the US Input-Output Coefficient Matrices: 1963-2007
April 2018 Working Paper 04/2018 Department of Economics The New School for Social Research
The views expressed herein are those of the author(s) and do not necessarily reflect the views of the New School for Social Research. © 2018 by Luis Daniel Torres Gonzalez and Jangho Yang. All rights reserved. Short sections of text may be quoted without explicit permission provided that full credit is given to the source.
The Persistent Statistical Structure of the US Input-OutputCoefficient Matrices: 1963-2007∗
Luis Daniel Torres Gonzalez† and Jangho Yang‡
April 26, 2018
Abstract
The paper finds evidence for the existence of a statistical structure in the US input-output (I-O)coefficient matricesA = {ai j} for 1963-2007. For various aspects of matricesAwe find smooth andunimodal empirical frequency distributions (EFD) with a remarkable stability in their functionalform for most of the samples. The EFD of all entries, diagonal entries, row sums, and the (left andright) Perron-Frobenius eigenvectors are well described by fat-tailed distributions while the EFDof column sums and eigenvalue moduli are well explained by the normal distribution and the Betadistribution, respectively. The paper provides several economic interpretations of these statisticalresults based on the recent developments in the I-O analysis and the price of production literature.Our findings question some probabilistic assumptions conventionally adopted in the stochastic I-Oanalysis literature and call for a statistical approach to the discussion of the structure of I-Omatrices.
Keywords: Input-Outputmatrices, Stochastic Input-OutputAnalysis, US Input-Output Table, EconomicStructure, Bayesian estimationJEL Class: C11, C46, C51, C52, C67, D57, O51.
∗Authors by alphabetical order.†Department of Economics, New School for Social Research, 6 East 16th Street, New York, NY, USA 10011-8872, +52 1
5529208389, [email protected].‡University of Oxford, Oxford Martin School, 34 Broad St, Oxford, Oxfordshire, UK OX1 3BD, +44 7380 580815,
1
1 Introduction
The modern era in the study of multisectoral linear economic models started with the work of WassilyLeontief. In his groundbreaking book The Structure of the American Economy, Leontief (1951) isinterested in studying the structure of the economy, that is, the study of the general interdependence ofthe different elements of the economic system. To do so, Leontief develops a theoretical Input-Output(I-O) model and constructs empirical measurements for it.
The fundamental element of the I-O model is the I-O flow matrix Zn×n= {zi j}, that is, the matrix that
records the industry-to-industry money flows and tracks the intermediate inputs a specific industry needsfrom other industries for the production of its own output. In the money flow zi j , the i-th industry sells anamount of commodities i valuated at zi j to the j-th industry for the production of commodity j. From theperspective of the i-th industry, this sale represents production for intermediate consumption. From theperspective of the j-th industry, the amount zi j represents the purchase of commodities as inputs for theirproduction process. Consider now the vector containing the gross output of all industries x
1×n= {xj}
and the diagonal matrix xn×n= diag{x1, ..., xn}. Assuming a technology with fixed proportions between
input i and outputs j, that is zi j = ai j · xj , we can represent the technology of the economy with the I-Ocoefficient matrix A
n×n=
{ai j
}, where each ai j represents the amount of the i-th commodity required to
produce one unit of the j-th output.With this representation of the complex network of industries’ interdependences through matrices Z
and A, Leontief originated the branch in economics called I-O analysis. Ever since then, there has beenincreasing interest in the study of the properties of I-O matrices that has led to a substantial progress inthe theoretical sophistication of the model and its increasing empirical validity as well. For an extensiveliterature review on I-O analyses, see Miller and Blair (2009) and ten Raa (2005).
In Leontief’s original conception, the observedZ andA represent the structure of the actual economy.They provide descriptions of the interdependences in the different sectors. However, if we want toanalyze the actual economy by looking at properties from the complex network of industries’ interactionsrepresented in the observed matrices Z and A, then we must study the structure of these matrices. Todo so, we have to detect regularities in the data, i.e. ‘stylized facts’ or remarkable constancies in theempirical evidence (Kaldor, 1957, p. 591), that can guide us to identify the constraints determining thematrices. The information gained from the identification of the regularities and constraints would allowus to make inferences about actual economies.
Despite the substantial interest in the study of structural properties of I-O matrices, there has notbeen enough attention paid to the statistical properties and regularities of the observedZ andAmatrices.This gap in the literature is rather puzzling considering the importance of ‘stylized facts’ of empiricalevidence in the development of proper theories. Most of the I-O studies that adopt statistical andprobabilistic approaches only pertain to the precision of I-O modeling and the estimation/updating ofthe I-O entries (West, 1986; Gurgul, 2007). For most of the these studies, the origin of probabilisticstructures of entries zi j and ai j are predicated on the idea of ‘measurement errors’ that might emerge
2
from data collection, confidentiality, reporting, and sampling, among others (ten Raa, 2005; Ruedaet al., 2013). However, some of the assumptions made in those studies are not empirically verified.Furthermore, understanding the randomness in the data solely as the outcome of human errors leavesout the possibility that the observed regularities are the result of the economic process.
This paper attempts to fill this gap in the literature by 1) evaluating if there is a statistical structure inmatrices A and 2) characterizing its statistical properties based on several representative probabilisticmodels. We will look into the statistical structure of the observed I-O matrices and establish importantstylized facts regarding matrices A. These stylized facts would guide us to further develop existingmodels to properly explain the empirical regularities. For instance, we show that the distribution ofcolumn sums of matrix A, κj =
∑ni=1 ai j , for i, j = 1, ..., n, is well explained by a symmetric distribution
such as the Normal distribution, while their row sums ρi =∑n
j=1 ai j , is explained by a rapidly decayingdistribution with a fat tail such as the Weibull distribution with its shape parameter being smaller than1. Since the column sums distribution represents the industry dispersion of the ratio of the circulatingcapital to gross output (whose inverse is the productivity of intermediate input), the symmetric patternof the distribution of column sums would require some theoretical model to explain why there is across-sectional convergence of productivities. In contrast, the row sums of A matrix, ρi, represent therelative importance of industry i’s product in the production of the rest of the industries. The fact thatwe have a fat-tail pattern in the row sums distribution calls for an explanation for endogenous formationof concentration of supply chains on a small number of industries.
Our understanding of the I-O matrices and approach to studying them is consistent with the holisticapproach advocated in Jensen et al. (1988) and Jensen et al. (1991).1 Identifying the statistical structureof the I-O matrices understands any subset of the n × n cells in relation to the rest of the elements ofthe matrix. This is because we understand that the statistical structure persistent in all the empirical I-Omatrices has emerged from some fundamental economic processes that govern the interdependence ofproduction in an economy.
We believe that the identified stylized facts of the I-O matrices will provide a fresh new look atpreceding works on I-O analyses. First, our findings would place some studies that assume certainprobabilistic features of these matrices with no empirical validation on shaky foundations. Moreover,the existence and characteristics of a statistical structure in the I-O matrices would question the origin ofrandomness as measurement errors and the notion of randomness as an i.i.d. processes. The empiricalregularities would call for a richer understanding that can capture the salient patterns of the systememerging from some economic forces in the production process.
1‘The holistic approach views the cells of the input-output table not simply as observations of individual categories ofeconomic activity, but as elements of an entity, which collectively present a "portrait" of an economy in terms of generalstructural characteristics. This view suggests new approaches to analysis and description of economic structure, both in termsof whole-economy concepts and in terms of the contribution of parts to the whole as a micro-macro interface.’ (Jensen et al.,1991, p. 229) It is worthwhile to note that there is an accounting reason that justifies the holistic approach. The accountingconstraints on Z and A are: 1) both the flow and coefficient entries are semi-positive: zi j ≥ 0 and ai j ≥ 0; 2) the individualflows cannot be greater than its gross output, zi j < (xi, xj ), as well as industries’ row and column sums :
∑nj=1 zi j < xi and∑n
i=1 zi j < xj ; and 3) the column sums of matrix A must fulfill the corresponding normalization constraint∑ni=1 ai j < 1.
3
The rest of the paper is composed of the following sections. Section 2 reviews the statisticalregularities and properties of actual I-O matrices already identified by the literature. The objective ofthis section however is not to review all their probabilistic models and statistical exercises, but only topresent the main interest of their research and report on the statistical regularities and properties foundin their I-O data.
Section 3 presents empirical evidence on the existence of a statistical structure in the observedmatrices A and statistical results on its characterization. We study 10 matrices A, constructed from theUS Benchmark I-O accounts, covering the 44-year period from 1963-2007. Matrices A are constructedin the highest detail (highest level of disaggregation) —the number of sectors considered ranges from351 to 478. Appendix A presents the details regarding the construction of our database.
In this section we compute the empirical frequency distributions of coefficients ai j , as well as fortheir following functions: the diagonal elements aii, their column sums κj , and their row sums ρi, fori, j = 1, ..., n. We also study some aspects of the spectrum of the matrices, such as the absolute valueof its eigenvalues |λt | and the coefficients of the left and right Perron-Frobenius (P-F) eigenvectorsqL
11×n= {q1, j} and qR
1n×1= {qi,1}—the left and right eigenvectors of A associated to the P-F eigenvalue
λ1, i.e. the maximal eigenvalue. Next, we fit probabilistic models to the different empirical frequencydistributions to understand the statistical structure of the matrices. Last, we provide an economicinterpretation of the results.
In Section 4 we contrast our empirical results with different probabilistic and statistical aspectscontained in the literature. We question the notion and origin of randomness in the I-O matricesadopted in the literature. In addition, we show how the usual probabilistic assumptions adopted byliterature are sometimes against the actual empirical findings of the US I-O matrices. Section 5concludes the paper by summarizing the motivation, objective, and results and suggesting future linesof research.
2 A Review of the Statistical Regularities Identified in the I-O Matrices
The economics of input-output and the multisectoral economic models are well established in the eco-nomics discipline. However, probabilistic and statistical aspects have only been considered marginally(West, 1986; Gurgul, 2007). Our literature review identifies three major lines of research on this issue:1) the stochastic I-O analysis literature, 2) the Bródy’s Conjecture debate, and 3) recent price of pro-duction models. The objective of this section however is not to give an exhaustive review of all theirprobabilistic and statistical exercises, but to focus on the statistical regularities found in their studies.
2.1 Stochastic I-O Analysis Literature
The most substantial literature involving random aspects in multi-sectoral models is the so-called‘stochastic I-O analysis’. The motivation of most of the stochastic I-O analysis literature is to study the
4
precision of I-O modeling and the estimation/updating of the I-O entries. Nevertheless, these studies donot attempt to find the statistical regularities of the I-O matrices. As far as we are concerned, Jackson(1986) and Wibe (1982) are the only papers that look into some statistical characteristics of actual I-Omatrices. Even in these papers, however, their focus is on the input and output properties of only thefirms and establishments within each industry in matrix A .
For most of this literature, the justification for the use of random features in the I-O modeling comesfrom measurement errors generated while constructing matrices Z =
[zi j
]or the commodity-industry
oriented Use U =[uk j
]and Make V =
[v jk
]tables.2 In more detail, the sources of randomness in
entries zi j , ui j , and vi j are errors in data collection, confidentiality, reporting, and sampling amongother things (ten Raa, 2005; Rueda et al., 2013). Only a few studies consider that randomness in theI-O flows or coefficient entries could come from sources related to economic factors.
According to Gurgul (2007), Rueda et al. (2013), and West (1986) the objective of the literature isto study:
• Biases in Leontief’s inverse matrix Ln×n= (I − A)−1 = {Li j} and different multipliers.
• Moments, confidence intervals, density functions in L.• Moments and confidence intervals for ai j .• Statistical estimation of ai j and the multipliers.
For these purposes, the literaturemakes various assumptions about the errors regarding themultiplicativeor additive structure of the errors, e.g. the symmetric nature of variance, the (in)dependence of theentries, and the specific type of probability distribution of the coefficients. Given that the constructedentries zi j , uk j , and vjk are ‘plagued’ with all sort of errors, these random elements will be transferedto matrices A and L and to the multipliers.
2.2 Bródy’s Conjecture
By setting a probabilistic model for the I-O coefficient matrix A and estimating the first two eigenvalues,λ1 and λ2, Brody (1997) puts forward the hypothesis that the spectral gap λ1
|λ2 |will tend to zero as the
size of the matrix n increases. In this probabilistic model, each ai j ∼ i.i.d.(µ, σ2) .3 This conjecture
implies that economic systems with large A matrices, like the ones representing the actual working ofeveryday market economies, require fewer iterations to reach an output equilibrium, i.e. the proportionsgiven by the P-F eigenvector qR
1 , which is later known as Bródy’s Conjecture.Bródy arrives at this result purely on the basis of algebraic and probabilistic formalism—he neither
uses statistical properties of actual I-O matrices to corroborate his probabilistic assumptions nor studiesthe empirical eigenvalues of thesematrices. Therefore, Bródy’s suggestion assumes that his probabilistic
2SeeAppendixA for a description of the construction of the industry-by-industry I-O coefficientsA from these commodity-industry tables.
3Bródy proposes the following probabilistic model: Each column of A is a sample of {x} of size n, where x is a positiveand continuous random variable with unknown probability distribution {x} but with a given mean µ and variance σ2. Bródyimplicitly assumes that elements within each sample are independent, hence each input coefficient ai j ∼ i.i.d.
(µ, σ2
).
5
model represents the statistical characteristics of actual economies. In the succeeding literature, Bidardand Schatteman (2001) provide a proof of Bródy’s conjecture while Sun (2008) and Schefold (2013)introduce theorems to further develop Bródy’s Conjecture based on Goldberg et al. (2000) and Goldbergand Neumann (2003) papers on the mathematical random matrix literature. However, none of theseworks consider the statistical characteristics of observed ai j and λt .
Bródy’s Conjecture has also spurred a number of empirical works whose purpose is to empiricallyvalidate the conjecture. The path taken by these empirical studies is to compute the empirical spectrumgap at different values of n, the level of aggregation. These studies have successfully identified somestatistical regularities in the spectrum of the observed I-Omatrices that are persistent for all the countries,years, and aggregation levels considered in their study.4 Given the aggregation level, their main findingsshow that:
1. When looking at the complex plane, eigenvalues are clustered around zero.2. Only a few subdominant eigenvalues have a magnitude different from zero, but this number is
important —i.e. we cannot say that all the eigenvalues are close to 0.3. The absolute value of the eigenvalues exhibit a remarkable and persistent organization by having
smooth and unimodal empirical frequency distributions, which seem to follow a pattern of expo-nential decay—There is no evidence of a degenerate distribution of the subdominant eigenvaluesaround 0.5
2.3 Recent Price of Production Models
Empirical computations of price of productionmodels for different countries and years have consistentlyproduced a nearly linear price and wage curves, i.e. the prices of production and the wage rate (orshare) as a function of the rate of profit.6 This is considered a puzzle in the price of productionliterature because the constraints on the key exogenous variables in these models are not strong enoughto persistently produce near linear price and wage curves. These constraints are the positivity (ai j ≥ 0)and productivity (λ < 1) of the I-O coefficient matrix A, and the positivity (lj, xj > 0) of the laborcoefficient vector l and gross output vector x. In order to explain the observed near-linearity in theprice and wage curves, recent work in the literature has advanced hypotheses based on the statistical
4See Mariolis and Tsoulfidis (2014), Mariolis and Tsoulfidis (2016b), Nassif and Shaikh (2015), and Schefold (2013).5Some authors have been interested in the behavior of the spectral gap and the corresponding eigenvalue distribution
when changing the aggregation level. Mariolis and Tsoulfidis (2014), Gurgul and Wojtowicz (2015), and Nassif and Shaikh(2015) present interesting empirical regularities: the spectral gap λ1
|λ2 |tends to increase as the disaggregation level increases.
In other words, the magnitude of the highest eigenvalues tends to increase with the disaggregation level and, therefore, thenumber of eigenvalues with an important magnitude increases, although less proportionally with the numbers of sectors inthe disaggregation procedure. Nassif and Shaikh (2015) shows that the empirical frequency distribution of |λt | tends to be‘L’-shaped as n increases.
6Prices of production are theoretical prices that equalizes the rates of profit taken as given the technology and the wage orprofit rate. See Pasinetti (1977, 1988) for an expostion of this model and Shaikh (2016) and Mariolis and Tsoulfidis (2016b)for a modern review of this literature.
6
regularities in the structure of technology and demand. This structure consists in (Torres, 2017a):7
(1) the properties of matrix A and(2) the relationship that matrix A has with vectors l and x
In the study of (1), Iliadi et al. (2014), Mariolis and Tsoulfidis (2011, 2015, 2016a), and Torres(2017a,b) find the same characteristics in the eigenvalues of observed I-O matrices as the ones in theBródy’s Conjecture literature. Torres (2017a; 2017b) finds in addition that its column sums κj =
∑ni=1 ai j
display a smooth, unimodal, and highly symmetric empirical frequency distributions. Regarding (2),Torres (2017a) finds evidence of a statistical tendency for proportionality (∝) between vectors l andx and the P-F eigenvectors of matrix A: l ∝ qL
1 and x ∝ qR1 . The deviations between lj − l∗1 qL
1, j andxi − x∗1qR
i,1, have a smooth, unimodal, highly peaked, and symmetric empirical frequency distributions,centered around 0.8 These results are based on the construction of empirical frequency distributionsand their visual inspection, without any attempt to characterize these distributions.
3 A Study of the Statistical Structure of the US I-O coefficient matrices
The regularities in the statistical behavior of the eigenvalues, column sums, and the relationship of thelabor and output vectors with the P-F eigenvectors summarized in the previous section, indicate thatthe I-O coefficient matrices A might have persistent structural properties of a statistical nature. Theobjective of this section is to present empirical evidence for the existence of a statistical structure inmatrices A and to shed some light on its statistical characterization.
To do so, we present statistical information of different variables involving some of or all thecoefficients ai j . These variables represent relevant economic and mathematical aspects of matrices Afrom different points of view. Based on the US Benchmark I-O accounts, compiled by BEA (2015),we construct ten A matrices, covering 44 years from 1963-2007.9 This database presents industryinformation at the most detailed level possible: between 351 to 478 industries. All the matrices refer tototals, that is, domestic plus imported inputs. We construct non-negative, indecomposable I-O matricesfor non-fictitious industries with strictly positive wage bill and value-added.10
7Iliadi et al. (2014) and Mariolis and Tsoulfidis (2011, 2015, 2016a) consider matrix J = 1−λ1λ1
A(I−A)−1 whereas Torres(2017b,a) considers matrix A. Both matrices share the same eigenvectors.
8When matrix A is diagonalizable, then it has n linearly independent (left and right) eigenvectors, so vectors l and x canbe represented as l = l∗1qL
1 + ... + l∗nqLn and x = x∗1qR
1 + ... + x∗nqRn , where the l∗t and x∗t are the coordinates of l and x in the
vector space given by the left and right eigenvectors, respectively. Hence, if l ∝ qL1 and x ∝ qR
1 , then 0 = l− l∗1qL1 =
∑nt=2 l∗t qL
t
and 0 = x − x∗1qR1 =
∑nt=2 x∗t qR
t (Torres, 2017a).91963, 1967, 1972, 1977, 1982, 1987, 1992, 1997, 2002 and 2007.10The manipulations were minimal. Due to the aggregate nature of the following statistical exercises, it is unlikely that the
results would be affected by a change in a small portion of the information. The rest of the details in the construction of thedatabase are given in Appendix A.
7
3.1 The Set of Variables to Study
We study seven variables that show important aspects of matrices A. The first four variables are relatedto the configurations of the coefficients ai j , whereas the rest 3 variables are related to the eigenvaluesand eigenvectors. The following are the variables we consider:
1. Pooled ai j . Leontief (1986, Ch.1,2,8) considers that there is a fundamental relationship betweenoutputs xj and inputs Zi j , measured by the ai j coefficients, which is ‘relatively invariable’, ‘in-flexible’, and that changes slowly through time. This input-output relationship is determinedmainly by the technology —customs and institutional arrangements are complementary determi-nants of the ai j . Hence, the technological structure of an economy, regarding the commodityinputs, is represented by the I-O coefficient matrix A.11 Due to the accounting bound given by0 ≤ ai j ≤ κj < 1, coefficients ai j are constrained by their sum. This means that their values arenot independent —e.g. if one coefficient has a relatively large value, then the others must havesmall values. Because we have between 351 to 478 industries, we decided to pool all the entriesfor each year.
2. Diagonal entries aii. In the literature on linear economic models there is an interest indominant diagonal matrices, i.e. matrices for which positive scalars d1, ..., dn exist such thatdj |aii | >
∑i,j
dj |ai j |, for j = 1, ..., n (Takayama 1974, Ch. 4.C). On the other hand, they have a
connection with mathematical aspects of the matrices, such as the trace and the sum of eigenval-ues, Tr(A) =
∑ni=1 aii =
∑ni=1 λt .
3. Column sums κj =∑n
i=1 ai j . These sums are related to the commodity inputs per dollar ofgross output and value-added per dollar of gross output for each industry, (vaj), κj = 1 − vaj .In models with only circulating capital, κj = 1
1+u jrepresents the capital-gross output κj , which
is directly related to u j , the capital-value added ratio, i.e. the inverse of capital productivity.Hence, regularities in industries’ column sums indicate an organization on industries’ capitalproductivities and value added created per dollar of production.
4. Row sums ρi =∑n
j=1 ai j . Each ai j represents the proportion of the i-th input outlay on the valueof production of industry j and shows the relative importance of each input in terms of eachindustries’ value of production. By taking the row sums of A, the comparisons of the different ρiindicate the relative importance of each input in the productive structure of an economy: A highρi implies that commodity i is an indispensable input in the production network. 12
11The labor coefficient vector l and the matrix of capital stock coefficients B complement the picture. We will not considerthese aspects of technology in this paper.
12From amathematical point of view, both column and row sums are popular matrix norms and, according to Gerschgorin’scircles, they could constitute relevant constraints for the eigenvalues. See chapter 5.2 and example 7.1.4 in Meyer (2000).
8
5. Eigenvalues’ absolute value |λt |. Studying the eigenvalues λt of matrix A can help us to identifyhidden economic and mathematical properties of the matrix that otherwise would be difficult tosee. To show this, assumematrixA has n-distinct eigenvalues λ1, ..., λn.13 Then, we can representit as (Meyer, 2000, p. 517-8):
A =λ1
qL1 qR
1qR
1 qL1 + ... +
λn
qLnqR
n
qRnqL
n (1)
= λ1A1 + ... + λnAn, (2)
where At =1
qLt qR
tqRt qL
t . That is, the actual matrix A is a linear combination of rank-one matricesAt , constructed from the left and right eigenvectors. Therefore, the observed strong concentrationof the λt around zero reported in Section 2 suggests that, if the λt are ordered by their absolutevalue, then A ≈
∑ct λtAt , for a small integer c < n.14 In multisectoral price and growth models,
the P-F eigenvalue λ1 is directly associated with the maximum rates of return and growth aneconomy can achieve.15 More generally, eigenvalue λs is the capital-gross output ratio of theeconomywhen, in a given economy, the vector of labor coefficients l is proportional to eigenvectorqLs , l ∝ qL
s , and/or when the output vector x is proportional to eigenvector qRs , x ∝ qR
s .16 Giventhat eigenvalues belong to the complex field, one way to represent their statistical information isto take their magnitude or absolute value: |λt |.
6. Entries of the Perron-Frobenius eigenvectors qL1, j and qR
i,1. In the same models where λ1 is relatedto the maximum profit and growth rates, eigenvectors qL
1 and qR1 correspond to the proportions for
the corresponding price and output vectors. These vectors are the only positive eigenvectors, sothey constitute the economically meaningful directions of self-expansion. On the other hand, theleft and right P-F eigenvectors define A1, the first layer of matrix A in Equation (2). This couldbe seen as the core of the structure of technology, as is in the tiered approach to the fundamentaleconomic structure in Jensen et al. (1991). Finally, the empirical evidence on the statisticaltendency towards the proportionality between l ∝ qL
1 and x ∝ qR1 reviewed in Section 2 makes
relevant the study of the statistical properties of the P-F eigenvectors.
13‘Any defective square matrix A [a matrix with repeated eigenvalues] can be made to have all distinct eigenvalues bychoosing an appropriate perturbation direction F for a sufficiently small ε , 0’ (Aruka, 2011, p. 46)
14See Schefold (2013) and Mariolis and Tsoulfidis (2016b).15See Pasinetti (1977, Ch. 5,7) and Takayama (1974, Ch. 6,7) for a detailed discussion.16Goodwin (1974; 1976) introduces the notion of Normalized General Coordinates, that is, the change of basis from
the commodity space from matrix A to the space given by the eigenvectors, which represent composite commodities of theoriginal system in A.
9
3.2 Descriptive results
Wefirst report in Figure 1 the heat map of thematrices to see their general patterns. This plot presents theabove 95% quantile components in a black box and the remaining components in a white box. This heatmap effectively displays the most salient pattern in the matrix. The result shows that almost all diagonalentries in each matrix contain large technical coefficients, which indicates that intra-industry trade is animportant component in the signal of the matrix. Also, large values are distributed disproportionallyamong rows. This means that the inputs of some industries are more important than others in theproduction network.17
Figure 1: Heat map of the US input coefficient matrices A = {ai j}: 1963-2007, between 351-478 industries.The black boxes represent the signal component while the white boxes represent the noise component. Visualinspection reveals two persistent regularities: the diagonal elements ai j and certain rows ρi have a prominentrole in the signal of matrices A.
17It is worthwhile to mention that the IO matrix is a sparse matrix with a great number of zeros. The following is theproportion of zero entry in each matrix.1963: 74.4%, 1967: 88.2%, 1972: 38.8%, 1977: 30.9%, 1982: 20.6%, 1987: 27.2%,1992: 31.8%, 1997: 32.6%, 2002: 24.7%, 2007: 21.3%
10
Statistic N Mean St. Dev. Min Max
ai j 1,903,407 0.001 0.010 0.000 0.824aii 4,343 0.037 0.056 0.000 0.479ρi 4,343 0.558 1.323 0.000∗ 22.872κj 4,343 0.558 0.150 0.020 0.979|λt | 4,343 0.054 0.068 0.000∗ 0.532qL
1, j 4,343 0.015 0.046 0.000∗ 0.753qRi,1 4,343 0.081 0.085 0.000∗ 0.716
Table 1: Summary statistics of 7 variables of the US I-O coefficient matrices A = {ai j}: pooled information ofeach variable for all the 10 matrices for the period 1963-2007 (between 351-478 industries). The symbols refer toall the ai j coefficients, the diagonal elements of the matrix aii , the column and row sums of the matrix κj and ρi ,the absolute value of the eigenvalues |λt |, and the coefficients of the left and right Perron-Frobenius eigenvectorsqL
1, j and qRi,1. Zeros with an asterisk are not actually a zero but a very small number rounded off to 3 decimal
places.
Table 1 shows the summary statistics of the seven variables (ai j , aii, κj , ρi, |λt |, qL1, j , and qR
i,1) forall years —Appendix B presents the summary statistics for individual years. Figures 2 to 8 show theempirical frequency distribution (EFD), cumulative distribution function (CDF), and complementarycumulative distribution function (CCDF) for each variable and year.
Figures 2 to 8 present evidence for the existence of a statistical structure in matrices A. The plots forall the variables show smooth, unimodal, and highly peaked EFD in most cases, suggesting that thereis a central tendency in their behavior.18 Furthermore, the shapes of the EFDs are remarkably constantfor almost all the samples. The relevance of this time invariance is outstanding given the tremendouschanges in the technology, the institutions, and the organization of production that the US economy hasgone through in the past decades.19
This persistent pattern in A matrices raises a subtle point about the ‘structural change’ of productionprocess. For Leontief (1986) and Carter (1970), the structure of the economy is represented by theI-O matrices so that the structural change is understood as changes in the ai j coefficients resulting indifferent A. The observed persistent pattern A implies that, even though every coefficient ai j changesover time, the statistical structure of matrix A, might stay highly stable.
18The same general patterns are present for years 1947 and 1958. However, because for these years the level of detail isonly for 78 sectors, the EFDs are somewhat noisy, although they have the same shapes and present the same central tendencies.
19It is worthwhile to note that these regular patterns of A are persistent in spite of major changes in the methodology forthe construction of the I-O accounts experienced in 1972.
11
aij
log_
dens
ity
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.00
10.
011
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
CDF
log(x)
log[
P(X
> x
)]
0.00 0.02 0.04 0.06
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
● ● ● ● ●●●●●●●●●●●●●
●●
●●
●
●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−16 −12 −8 −6 −4
−5
−4
−3
−2
−1
0
● ●●
●●
●●
●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
Figure 2: Pooled coefficients ai jaii
log_
dens
ity
0.0 0.1 0.2 0.3 0.4
0.02
0.05
0.10
0.20
0.50
1.00
2.00
5.00
10.0
020
.00
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ● ● ●
CDF
log(x)
log[
P(X
> x
)]
0.0 0.1 0.2 0.3
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ● ●
● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−14 −10 −6 −2
−5
−4
−3
−2
−1
0 ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
Figure 3: Diagonal coefficients aii
ρi
log_
dens
ity
0 2 4 6 8 10
0.00
10.
005
0.01
00.
050
0.10
00.
500
1.00
0
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ●● ●
CDF
log(x)
log[
P(X
> x
)]
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−6 −4 −2 0 2
−5
−4
−3
−2
−1
0 ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
Figure 4: Row sums ρi =∑n
j=1 ai j
κj
dens
ity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1963196719721977198219871992199720022007
● ● ● ●● ● ● ●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ●●● ●● ● ●
CDF
log(x)
log[
P(X
> x
)]
0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
1.0
● ● ● ● ● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ●
● ● ● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ●
● ● ● ●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ●
● ● ● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ●●●● ●
● ● ●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●● ●
● ●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●
● ● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ●
● ● ● ●● ● ● ●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●●●
●
●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−1.5 −1.0 −0.5 0.0
−5
−4
−3
−2
−1
0 ● ● ● ● ● ●●● ●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
●
● ● ● ● ● ●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●
●
●
●
●
●
●
● ● ● ●● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
●
● ● ● ●● ● ● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●
●
●
●
●
● ● ● ● ●●● ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●
●
●
●
●
● ● ● ● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
●
Figure 5: Column sums κj =∑n
i=1 ai j
Empirical frequency distributions (EFD), cumulative density functions (CDF), and complementary cumulativedensity functions (CCDF) of different elements of the US input coefficient matrices A = {ai j}: 1963-2007,between 351-478 industries. Each one of the figures shows smooth and highly peaked EFD. Whereas there is asharp rate of decay in the EFD of the pooled elements ai j , the diagonal elements aii , and the ρi , panels (a) to(c), respectively, the EFD of the column sums κj is highly symmetric. The functional form of the EFD, CDF, andCCDF shows a remarkable persistence in spite of the series of technological and organizational changes in theUS economy.
12
|λt|
log_
dens
ity
0.1 0.2 0.3 0.4 0.5
0.05
0.10
0.20
0.50
1.00
2.00
5.00
10.0
0
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●● ●● ● ● ●
CDF
log(x)
log[
P(X
> x
)]
0.0 0.1 0.2 0.3 0.4
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ● ●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−7 −5 −3 −1
−5
−4
−3
−2
−1
0 ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●
●
●
● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●
●
●
● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
Figure 6: Eigenvalues’ modulus |λt |
qi1R
log_
dens
ity
0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.02
0.05
0.10
0.20
0.50
1.00
2.00
5.00
10.0
0
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●● ● ● ● ● ●
CDF
log(x)
log[
P(X
> x
)]
0.00 0.10 0.20 0.30
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ●
● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−10 −8 −6 −4 −2
−5
−4
−3
−2
−1
0● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
● ● ● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ●● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●
●
●
●
● ●● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
Figure 7: Right Perron-Frobenius eigenvector’scoefficient qR
i,1
q1jL
dens
ity
0.0 0.1 0.2 0.3 0.4 0.5
05
1015
1963196719721977198219871992199720022007
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●● ●
CDF
log(x)
log[
P(X
> x
)]
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ●● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●● ● ● ● ● ● ●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●● ●●●● ●●● ●●●●●● ●● ●● ● ● ● ●●● ● ● ● ● ● ●
● ● ● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●
●
●
●
●
●
CCDF:log−log
log(x)
log[
P(X
> x
)]
−6 −5 −4 −3 −2 −1
−5
−4
−3
−2
−1
0● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●
●
●
●
●
●
● ● ● ●●● ● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
●
● ● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
● ● ● ●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●
●
●
●
●
●● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●
●
●
●
●
●
Figure 8: Left Perron-Frobenius eigenvector’s co-efficient qL
1, j
Empirical frequency distributions (EFD), cumulative density functions (CDF), and complementary cumulativedensity functions (CCDF) of the absolute value of eigenvalues |λt | and the coefficients of the right qR
i,1 and left qL1, j
Perron-Frobenius eigenvectors of the US input coefficient matrices A: 1963-2007, between 351-478 industries.Except for the left eigenvectors in 1997, 2002, and 2007, each one of the figures in the panels shows smooth andhighly peaked EFD. Whereas there is a sharp rate of decay in the EFD of the |λt | and qR
i,1, panels (a) and (b),respectively, the EFD of qL
1, j shows sharp increase and then a somehow slower decrease.
13
3.3 The Models
In this section we introduce several probabilistic models for our target variables: pooled ai j , diagonalentries aii, column sums κj , row sums ρi, eigenvalues’ absolute value |λt |, and entries of the Perron-Frobenius eigenvectors, qL
1, j and qRi,1.
For each variable we estimate between 3-6 probabilistic models, based on the support of the variable—some variables have support [0, 1) and others [0,∞). The probabilistic models used for the estimationare Gamma (G), Beta (B), Weibull (W), Exponential (E), Pareto (P), Normal (N) and Log-normaldistributions (LN), whose functional forms are in the following:
fG(x;α, β) =βαxα−1e−βx
Γ(α)for x ∈ (0,∞) (3)
fB(x;α, β) =xα−1(1 − x)β−1
B(α, β)for x ∈ [0, 1] (4)
fW(x;α, β) =α
β
(xβ
)α−1e−(x/β)
αfor x ∈ [0,∞] (5)
fE(x;α) = αe−αx for x ∈ [0,∞] (6)
fP(x; xm, α) =αxαmxα=1 for x ∈ [xm,∞] (7)
fLN(x; µ, σ) =1
xσ√
2πe−(lnx−µ)2
2σ2 for x ∈ (0,∞] (8)
fN(x; µ, σ) =1
σ√
2πe−(x−µ)2
2σ2 for x ∈ (0,∞]. (9)
The following are the probabilistic models we fit for each variable:
• For the distributions of the pooled ai j = [0, 1) , f (ai j) and the diagonal entries aii = [0, 1), f (aii),which have a rapidly decaying pattern, we exclude zero values and fit six models: Exponential,Weibull, Beta, Gamma, Pareto, and Log-normal distributions.
• For the distributions of the column sums κj = (0, 1), f (κj), which have a symmetrical pattern, wefit three models: Beta, Gamma, and (truncated) Normal distributions.
• For the distributions of the row sums ρi = (0,∞), f (ρi), which have a rapidly decaying pattern,we fit five models: Exponential, Weibull, Gamma, Pareto, and Log-normal distributions.
• For the distributions of Eigenvalues’ absolute value |λt | = [0, 1), f (|λt |) and the right P-F eigen-vector qR
i,1 = [0, 1) , f (qRi,1), which have a rapidly decaying pattern in our data, we fit 6 models:
Exponential, Weibull, Beta, Gamma, Pareto, and Log-normal distributions.
14
• For the distribution of the left P-F eigenvector qL1, j = [0, 1) , f (qL
1, j), which have a right skewedpattern, we fit four models: Weibull, Beta, Gamma, and Log-normal distributions.
3.4 Model Estimation Results
The different parameters of the distributions are estimated using Bayesian methods with a uniform prior.The Hamiltonian Monte Carlo (HMC) is used for the posterior simulation with 1,000 iterations and 3chains. Then, we compute the accuracy of the model estimation using leave-one-out cross-validation(LOO-CV) and Widely Applicable Information Criteria (WAIC), which takes the log-likelihood of themodel corrected for model complexity. WAIC is asymptotically equal to LOO.20 See Appendices Cand D for detailed discussions on the estimation and model comparison method.
Table 2 presents the results from the WAIC index, showing the rank of models in a decreasing order.The smaller the WAIC is, the more preferred the model is. The number in parenthesis is the standarderror of WAIC. For example, the distribution of all entries, f (ai j) for year 1963 prefers the log-normaldistribution whose WAIC index is -2912 with the standard error 95.
The most noticeable result we can see from Table 1 is that the probability density function bestfitting the EFD of the variable is relatively persistent for different samples. This temporal persistenceof the functional form of the EFDs provides some evidence for the existence of a statistical structureof the matrices, indicating that structural change in the US matrices A is characterized by changes inthe parameters determining the shape, scale, and/or location of the density function, not by changes inthe underlying data generating process. Detailed estimation results for all the models are available inAppendix D.
Figures 9 to 17 present the frequency distribution, the fit of different distributions, and the Q-Q plotsfor one representative example, which typifies the patterns observed for other years.
All entries, f (ai j): The distributions of all entries are well described by the log-normal distribution,ln(ai j) ∼ N(µ, σ2). As is well known, the log-normal distribution is one of the fat tailed distributions,meaning that some of the technical coefficients in A have relatively large values while most of themare very small. As the heat map of A shows, this rapidly decaying pattern with a fat-tail results fromthe relative importance of the diagonal entries and some entries in a small number of rows, furthersuggesting that, without the diagonal part, the I-O network is highly concentrated on a few sectors.
20For a detailed discussion on the HMC and its application see Gelman et al. (2013). See Vehtari et al. (2015) forleave-one-out cross-validation and Widely Applicable Information Criteria.
15
f(.)
1963
1967
1972
1977
1982
1987
1992
1997
2002
2007
f(a
ij)
LN:-2912
(95)
LN:-2377
(72)
LN:-5575
(102)
LN:-5742
(104)
LN:-6117
(108)
LN:-5804
(109)
LN:-5785
(109)
LN:-4848
(126)
LN:-4970
(118)
LN:-4812
(125)
W:-2873
(95)
W:-2321
(80)
W:-5502
(103)
W:-5671
(107)
W:-6023
(108)
W:-5723
(110)
W:-5695
(108)
W:-4814
(124)
W:-4915
(117)
W:-4770
(121)
G:-2765
(107)
G:-2237
(91)
G:-5308
(114)
G:-5433
(140)
G:-5757
(156)
G:-5464
(153)
G:-5453
(136)
G:-4697
(125)
G:-4779
(118)
G:-4669
(118)
B:-2748
(111)
B:-2224
(94)
B:-5301
(116)
B:-5418
(147)
B:-5727
(178)
B:-5434
(173)
B:-5436
(146)
B:-4695
(125)
B:-4777
(118)
B:-4668
(118)
E:-2015
(226)
E:-1904
(137)
P:-4293
(110)
P:-4474
(112)
P:-4855
(114)
P:-4507
(115)
P:-4481
(113)
P:-3577
(130)
P:-3670
(122)
P:-3524
(127)
P:-1373
(89)
P:-690(70)
E:-3239
(275)
E:-3099
(471)
E:-3060
(642)
E:-3026
(614)
E:-3091
(483)
E:-3035
(214)
E:-3087
(202)
E:-3120
(166)
f(a
ii)
W:-1537
(64)
W:-1635
(51)
W:-2044
(67)
G:-2224
(69)
W:-2141
(66)
W:-2224
(67)
B:-2186
(70)
G:-2290
(73)
B:-1933
(68)
B:-1605
(59)
G:-1536
(63)
LN:-1629
(54)
G:-2042
(67)
W:-2223
(68)
G:-2135
(66)
G:-2219
(67)
G:-2185
(70)
B:-2288
(73)
G:-1933
(68)
G:-1603
(59)
B:-1535
(62)
G:-1626
(51)
B:-2036
(68)
B:-2219
(69)
B:-2131
(66)
B:-2216
(67)
W:-2173
(68)
W:-2284
(71)
W:-1920
(67)
W:-1592
(59)
LN:-1493
(70)
B:-1617
(52)
LN:-1943
(68)
LN:-2094
(69)
LN:-2053
(70)
LN:-2125
(70)
E:-2009
(58)
LN:-2104
(74)
E:-1755
(55)
LN:-1479
(63)
E:-1340
(55)
E:-1602
(53)
E:-1870
(63)
E:-2054
(64)
E:-1980
(63)
E:-2057
(63)
LN:-1988
(70)
E:-2089
(62)
LN:-1740
(69)
E:-1471
(48)
P:1
340(79)
P:1
920(56)
P:1
595(86)
P:1
662(90)
P:1
758(84)
P:1
656(86)
P:1
748(98)
P:1
590(99)
P:1
412(97)
P:1
451(82)
f(ρ
i)
LN:2
31(56)
LN:2
13(67)
LN:7
7(73)
LN:9
3(75)
W:1
47(77)
LN:1
8(76)
LN:4
7(77)
W:1
94(75)
W:2
15(68)
W:1
93(62)
W:2
62(57)
W:2
61(68)
W:1
16(74)
W:1
34(76)
LN:1
68(84)
W:3
4(76)
W:5
5(78)
LN:2
30(74)
G:2
56(77)
G:2
13(69)
G:2
87(61)
G:3
09(74)
G:1
77(78)
G:2
00(84)
G:2
07(85)
G:8
9(82)
G:1
16(86)
G:2
44(84)
LN:2
62(73)
LN:2
77(57)
E:3
25(71)
E:3
92(91)
E:3
65(102)
E:3
90(112)
E:4
18(113)
E:3
30(108)
E:3
66(116)
E:4
30(111)
E:3
75(100)
E:3
11(82)
P:2
527(53)
P:3
036(62)
P:2
848(73)
P:3
047(74)
P:2
996(81)
P:2
746(80)
P:2
826(81)
P:2
988(80)
P:2
721(72)
P:2
481(73)
f(κ
j)
N:-326(27)
N:-412(30)
N:-468(31)
N:-483(32)
B:-435(29)
N:-515(33)
N:-490(33)
N:-369(30)
B:-388(25)
B:-323(23)
B:-317(27)
B:-404(29)
B:-457(31)
B:-468(33)
N:-434(30)
B:-495(34)
B:-473(33)
B:-365(32)
N:-379(26)
N:-312(24)
G:-286(32)
G:-373(33)
G:-438(34)
G:-439(35)
G:-387(34)
G:-495(38)
G:-450(40)
G:-290(54)
G:-344(29)
G:-277(26)
f(|λ
i|)
E:-1066
(39)
B:-1492
(63)
B:-1999
(106)
B:-2368
(164)
B:-2196
(140)
B:-2461
(189)
B:-2724
(227)
B:-2502
(196)
B:-1779
(92)
B:-2049
(207)
W:-1065
(39)
G:-1486
(60)
G:-1995
(104)
G:-2360
(163)
G:-2188
(138)
G:-2452
(188)
G:-2711
(226)
G:-2494
(194)
G:-1778
(90)
G:-2038
(206)
G:-1065
(39)
W:-1469
(50)
W:-1942
(86)
W:-2246
(131)
W:-2100
(112)
W:-2317
(157)
W:-2535
(194)
W:-2351
(161)
W:-1744
(74)
W:-1885
(175)
B:-1060
(39)
E:-1453
(45)
E:-1774
(55)
E:-1923
(57)
E:-1839
(55)
E:-1838
(57)
E:-1892
(58)
E:-1882
(59)
E:-1632
(52)
E:-1415
(48)
LN:-1016
(44)
LN:-680(94)
LN:-1283
(73)
LN:-1462
(143)
LN:-1353
(108)
LN:-1620
(175)
LN:-1829
(225)
LN:-1620
(184)
LN:-1094
(63)
LN:-1299
(209)
P:1
337(48)
P:1
284(152)
P:5
97(182)
P:2
32(259)
P:4
32(229)
P:-92
(277)
P:-394(321)
P:-106(289)
P:6
13(169)
P:-181(296)
f( q
L 1,j
)LN
:-2410
(77)
LN:-3318
(89)
LN:-3499
(90)
W:-3962
(97)
W:-4399
(103)
W:-3501
(94)
W:-3882
(100)
W:-3603
(96)
W:-2900
(87)
W:-2855
(86)
W:-2364
(73)
W:-3263
(87)
W:-3477
(91)
LN:-3937
(93)
LN:-4345
(103)
LN:-3490
(93)
LN:-3859
(100)
G:-3533
(101)
G:-2849
(89)
G:-2801
(92)
G:-2305
(72)
G:-3177
(86)
G:-3384
(92)
G:-3855
(105)
G:-4262
(123)
G:-3423
(94)
G:-3794
(104)
LN:-3525
(92)
B:-2836
(90)
LN:-2798
(83)
B:-2294
(72)
B:-3162
(87)
B:-3366
(94)
B:-3823
(111)
B:-4203
(141)
B:-3408
(95)
B:-3768
(110)
B:-3513
(104)
LN:-2801
(94)
B:-2770
(101)
E:-2004
(89)
E:-2698
(114)
E:-2852
(127)
E:-3235
(162)
E:-3485
(236)
E:-2844
(119)
E:-3078
(149)
E:-2943
(131)
E:-2413
(101)
E:-2330
(127)
P:6
34(73)
P:4
27(86)
P:2
95(92)
P:2
2(102)
P:-518(110)
P:2
83(101)
P:-69
(109)
P:2
11(110)
P:4
96(101)
P:2
15(98)
f( q
R i,1)
G:-1380
(31)
W:-1523
(34)
G:-1968
(40)
W:-1689
(31)
W:-2040
(36)
G:-1808
(40)
G:-1921
(40)
G:-1024
(39)
B:-782(34)
G:-605(31)
B:-1380
(31)
B:-1522
(34)
B:-1965
(40)
B:-1686
(32)
B:-2039
(36)
B:-1803
(41)
B:-1919
(39)
W:-1020
(39)
G:-781(36)
B:-604(29)
W:-1377
(32)
G:-1520
(34)
W:-1964
(40)
G:-1682
(32)
G:-2038
(36)
W:-1798
(41)
W:-1918
(39)
B:-1011
(41)
W:-780(35)
W:-603(30)
LN:-368(20)
LN:-387(26)
LN:-723(30)
LN:-416(26)
LN:-740(31)
LN:-558(26)
LN:-715(32)
LN:-35
(24)
LN:-10
(26)
LN:1
28(18)
Table2:
WidelyAp
plicable
Inform
ationCriteriaindexof
theestim
ated
probabilistic
modelsfordifferent
componentsof
theUSinputc
oefficientm
atrices
A={a i
j}:1963-2007,
between351-478industries.T
hecomponentso
fAare:
thepooled
entriesa
ij,the
diagonal
elem
ents
a ii,therowsumsρi,thecolumn
sumsκj,theabsolute
valueof
theeigenvalues|λ
i|,andtheentriesof
theleftandrightP
erron-Frobeniuseigenvectors
qL 1,jand
qR i,1.
Theinitialsforeach
probabilitymodelareG=Gam
ma,B=Be
ta,W
=Weibull,
E=Ex
ponential,P=Pa
reto,L
N=Lo
gNo
rmal,and
N=No
rmal.T
henumbersin
theparenthesis
arethestandard
erroro
fthe
estim
ated
WAICindex.
Thelowe
rthe
WAICindexis,the
morethemodelispreferred.
16
log(aij)
freq
uenc
y
−10 −8 −6 −4 −2 0
0.00
0.05
0.10
0.15
0.20
0.25
Normal
● ●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●
●●
●
Q−Q Plot: Log−Normal
log(y)
chec
k
−10 −6 −2
−10
−6
−2
Figure 9: All coefficients, ln(aii): 1967aii
freq
unec
y
0.0 0.1 0.2 0.3 0.4
05
1015
2025
30 BetaGamma
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●
●●●●●
●●● ●
●
●
●
●
●
Q−Q Plot: Beta
temp
chec
k
0.0 0.2 0.4
0.0
0.1
0.2
0.3
0.4
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●● ●●
● ●●●
●
●
●
Q−Q Plot: Gamma
temp
chec
k
0.0 0.2 0.4
0.0
0.2
0.4
Figure 10: Diagonal coefficients aii: 1967
ρi
freq
uenc
y
−10 −8 −6 −4 −2 0 2
0.00
0.05
0.10
0.15
0.20
0.25
0.30 Normal
●
●●
●●●●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●
●
●
Q−Q Plot: Log−Normal
log(y)
chec
k
−8 −4 0 2
−6
−2
02
4
Figure 11: Row Sums ρi: Log-normal Fit, 1967
ρi
freq
uenc
y
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Weibull
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●● ●●
●●●
●
●
Q−Q Plot: Weibull
y
chec
k0 5 10 15 20
05
1015
Figure 12: Row Sums ρi: Weibull Fit, 1997
κj
freq
uenc
y
0.2 0.4 0.6 0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
NormalBeta
●
●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●
●
Q−Q Plot: Normal
col_sum[[4]]
chec
k
0.2 0.4 0.6 0.8
0.0
0.4
0.8
Figure 13: Column Sums κj: 1967
Histograms, fitted probability distributions models, and Q-Q plots for different aspects of the US input coefficientmatrices A = {ai j}: different years, between 351-478 industries.
17
|λk|
freq
uenc
y
0.0 0.1 0.2 0.3 0.4 0.5
02
46
810
BetaGamma
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●●● ●●●
●●● ●●
●●● ●
●
●
●
●
●
Q−Q Plot: Beta
temp
chec
k
0.0 0.2 0.4
0.0
0.2
0.4
0.6
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●● ●●●●● ● ●●
●●
●●●
●
●
●
●
Q−Q Plot: Gamma
temp
chec
k
0.0 0.2 0.4
0.0
0.2
0.4
0.6
Figure 14: Eigenvalues’ modulus |λt |:1967
q1jL
freq
uenc
y
0.00 0.05 0.10 0.15 0.20 0.25
02
46
810
BetaGamma
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●
● ●●●●● ●● ● ●●
●●● ●
●●●●●
●
●
Q−Q Plot: Beta
temp
chec
k
0.00 0.10 0.20
0.0
0.2
0.4
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●● ● ●● ●●●
●●●●●●
●
●
Q−Q Plot: Gamma
temp
chec
k
0.00 0.10 0.20
0.0
0.2
0.4
0.6
Figure 15: Left Perron-Frobenius eigenvector’s co-efficient qL
1, j:1967
qi1R
freq
uenc
y
−10 −8 −6 −4 −2
0.00
0.05
0.10
0.15
0.20
Normal
●
●●
●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●
●●
●
Q−Q Plot: Log−Normal
log(y)
chec
k
−10 −6 −2
−15
−10
−5
0
Figure 16: Right Perron-Frobenius eigenvector’scoefficient qR
i,1:1967
qi1R
freq
uenc
y
0.0 0.2 0.4 0.6
010
2030
40 Weibull
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●
●●●●●
●●
●
●
●
●
Q−Q Plot: Weibull
right_eigen_list[[12]]
chec
k
0.0 0.2 0.4 0.6
0.0
0.2
0.4
0.6
Figure 17: Right Perron-Frobenius eigenvector’scoefficient qR
i,1: 2007
Histograms, fitted probability distributions models, and Q-Q plots for different aspects of the US input coefficientmatrices A = {ai j}: different years, between 351-478 industries.
Diagonal entries, f (aii): The distributions of diagonal entries are well explained by the Weibull,Gamma, and Beta distributions. The Weibull distribution with its shape parameter is a subexponentialdistribution and is considered a heavy tailed distribution. Our estimation results show that αW is farbelow 1 in all cases. The Gamma distribution can also have significant kurtosis and have a fat tail as well.The kurtosis of the Gamma distribution is calculated by 6/αG , the shape parameter. In our estimationresult, αG for the diagonal distribution is normally around 0.5, implying a high kurtosis. Two scaleparameters of the Beta distribution αB and βB are estimated around 0.5 and 12, respectively, so that the
18
expected value is around 0.04 while having a high degree of right skweness and kurtosis, confirmingits fat-tail behavior. Considering the fact that the diagonal entries are among the most pronounced partof A, the fat tail behavior in their distribution suggests that even among the relatively important entries,the relative difference between the aii entries is significant, with a few aii having very high values. Itis worthwhile to note, however, that the tail of diagonal entries distribution is less obese than that of allentries distribution whose kurtosis is normally 3 times higher than the diagonal counterpart.
Column sums, f (κj): The distributions of column sums are well described by the normal distribution inalmost all samples. Some years have the Beta distribution as the best performing distribution, whose twoscale parameters α and β being very close, implying the symmetry of the distribution. The economicinterpretation of this result is intriguing. Since the column sums distribution represents the industrydispersion of the ratio of the circulating capital to gross output (whose inverse is the productivity ofintermediate input), the symmetric pattern of the distribution suggests that there is a cross-sectionalconvergence of the productivity of the circulating capital. Also, in the linear production models withthe uniform wage w and profit (or interest) rates r , this observed symmetry suggest a statistical ten-dency towards uniformity in the capital-labor ratios or compositions of capital k j , because κj = 1
1+wk j+r.
Row sums, f (ρi): The distributions of row sums are well described by the log-normal distribution andthe Weibull distribution with the shape parameter α being far below 1. As we discussed before thesetwo distributions capture the fat-tail behavior of the row sums. Since the row sums of A matrix, ρirepresents the relative importance of industry i’s product in the production of others, the result showsthe production structure of the US economy, in which the supply chain network concentrates on a smallnumber of industries.
Eigenvalues , f (|λt |): The distributions of eigenvalue moduli are well described by the Beta distri-bution. The Gamma and Weibull distributions are also a good model chasing the Beta with a smallmargin. Two scale parameters of the Beta distribution αB and βB are estimated around 0.5 and 9,respectively, so that the expected value is around 0.05 while having a high degree of right skewnessand kurtosis. The behavior of eigenvalues’ modulus suggests a statistical tendency to a certain degreeof proportionality among the columns of A, i.e. among industries’ intermediate input technologies.21For each linearly dependent column in A there is a corresponding zero eigenvalue. The fact that theexpected value of eigenvalue moduli is close to zero with a relatively heavy tail suggests that matrix Acan be well approximated as a linear combination of a small number of matrices At =
1qLt qR
tqRt qL
t , thatis, as A ≈
∑ct λtAt for a small c � n. However, it is worthwhile to not that the relatively heavy tail
behavior implies that there are still several important eigenvalues with a significant magnitude.
21This tendency toward proportionality in the columns of A is used by Mariolis and Tsoulfidis (2011) and Schefold (2013)as a hypothesis to explain certain regularities in empirical computation of price of production models.
19
Perron-Frobenius eigenvectors, f (qL1, j) and f (qR
i,1): The distributions of the right eigenvectors prefereither the Weibull or the Log-normal distributions as the best fitting model, while the distributions ofthe left eigenvectors prefer either Gamma or Weibull distributions but with a small margin with theBeta distribution. All estimated Gamm, Weibull, Beta consistently have a high degree of kurtosis andexhibit a fat-tail behavior. In linear production models, the left P-F eigenvectors is associated with the(row) vectors of labor coefficient l and prices p whereas the right P-F eigenvector is associated with thevectors of net y and gross x output. Under the assumption that A is diagonalizable, vectors l and x canbe represented as l = l∗1qL
1 + ... + l∗nqLn and x = x∗1qR
1 + ... + x∗nqRn . Hence, the empirical evidence of
previous studies on the statistical tendency towards the proportionality between if l ∝ qL1 and x ∝ qR
1means that there is a one-and-one correspondence between the probability distribution of (qL
1 , qL1 ) and
the probability distribution of (l and x). The fact that eigenvector distributions have a fat-tail behaviorallows us to have a more systematic approach to studying labor and output vector distributions since thelatter could also have a fat tail behavior.
4 Discussion
In the last sectionwe reported the existence of smooth, unimodal, and highly peaked empirical frequencydistributions (EFD) for all the aspects of matrix A and showed that the functional form is time invariantfor almost all samples. The use of the frequency and cumulative distributions gives a collective view ofthe matrices, an account of their macro-state, consistent with the holistic approach advocated by Jensenet al. (1988) and Jensen et al. (1991). Based on these results we conclude that our findings provideevidence for the existence of a statistical structure in the I-O coefficient matrices A
There are two theoretical aspects that we can now address based on the contrast between the em-pirical evidence obtained in Section 3 and some stochastic and statistical literature reviewed in Section 2.
The Origin of Randomness: Section 2.1 showed that, with few exceptions, for the stochastic I-O analysis literature, randomness of the elements in the tables
(ui j, vi j
)and matrices
(ai j
)comes
from errors generated in the construction of the I-O accounts —data randomness is originated due toimprecisions from data collection to aggregation of the information. However, the persistent patternsobserved in many aspects of I-O matrices over 40 years suggest that a more fundamental force than justhuman error might be in place, which requires further research looking into the economic forces behindthis pattern.
Understanding the probabilistic structure solely based on the existence of human errors leaves asidethe possibility that the regularities in the observed frequency distributions are the result of the economicprocess. It is important to note that the probabilistic assumptions used in the stochastic models lackany economic interpretation other than misrepresentation of the ‘true’ value of the parameters —noisearound a true parameter. It is hard to interpret the empirical regularities in the data as the outcome ofsystematic measurement errors —the construction of the I-O accounts in the US went through a drastic
20
change since 1972 and the statistical structure remained the same.
Probabilistic Assumptions vs. Empirical Regularities: The surveys of the stochastic I-O analysisliterature by West (1986) and Gurgul (2007) show that this literature relies frequently on probabilis-tic assumptions of the ai j components such as additive or multiplicative random structure, normal-ity, symmetry, independence, identity in distribution, among others. For instance, while Goldberget al. (2000) assume ai j ∼i.i.d, E
(ai j
)= 1
n and Var(ai j) ≤ c1n2 for c1 > 0, Goldberg and Neumann
(2003) allow within column interdependence and only assumes that E(ai j
)= 0, Var(ai j) ≤ c
n2 and|Cov (aik, aim)| ≤ c
n3 , i, k,m = 1, ..., n, k , m. These studies do not make probabilistic assumptionsbased on the information of actual I-O matrices but rely solely on mathematical convenience. For in-stance, West mentions that ‘[n]ormality was assumed because this is the distribution commonly invokedwith sampling theory, and is probably more amenable to mathematical manipulation ... many of theirassumptions appear too restrictive to be of practical use’ (West 1986, p.364).
The following are some contrasts between the empirical statistical characteristics found in the dataand the probabilistic assumptions in the literature:
• ai j coefficients are far from being normally distributed. They are asymmetrically distributed witha particularly sharp decay in their values.
• The accounting constraint on the coefficients in each column, 0 <∑n
i=1 ai j < 1 for all j = 1, ..., n,implies that their elements cannot be independent of each other. If few elements have high values,the remaining ai j would be constrained to have low values. The sharp decay in the frequencydistribution of the ai j gives evidence for this.
• Neither column nor row sums are equal. The column and row dimensions show a completelydifferent statistical behavior —column sums are symmetrically distributed whereas row sums areasymmetrically distributed. It is not possible to consider a priori that all columns or rows havethe same mean.
5 Conclusion
In this paper we showed that there is a statistical structure in the observed I-O coefficient matrices Aby carrying out a statistical exploratory data analysis and model estimation on the US A matrices forthe period 1963-2007. We found a series of regularities. First, there are smooth, unimodal, and highlypeaked empirical frequency distributions for several functions of ai j —the pooled ai j , the diagonalelements aii, the column sums and row sums of A, eigenvalues’ modulus |λi |, and the elements of theleft and right Perron-Frobenius eigenvectors, the eigenvectors associated to the maximal eigenvalue, qL
1, jand qR
i,1. Second, the distributions of all entries, diagonal entries, row sums, and the P-F eigenvectors arewell described by a fat-tailed distribution such as the Log-normal, Weibull with the below unity shapeparameter, and the Gamma distribution with a high kurtosis. In contrast, the distributions of columnsums are well explained by the normal distribution due to its symmetry while the distributions of the
21
eigenvalue moduli prefers the Beta distribution. Finally, the general characteristics of these frequencydistributions are maintained for almost all samples. This result is informative considering the fact thatthe US economy experienced tremendous changes in its technology, institutions, and the organizationof production. The general patterns are also invariant to the methodologies in the construction of theI-0 accounts and matrices. The evidence of the existence of statistical stylized facts of A calls for futureresearch on potential economic theories that explain such regularities.
Acknowledgements
We want to thank Duncan Foley, Geoffrey Hewings, Michael Lahr, and Anwar Shaikh for stimulatingconversations and detailed comments on early drafts of this paper. We would also like to thank FidelAroche, Henryk Gurgul, Theodore Mariolis, and Lefteris Tsoulfidis for their comments. The firstversion was presented at the International Input-Output Conference at Seoul, South Korea, 4-8 July2016. We thank all our commentators on that occasion. The usual disclaimer applies. Daniel Torresacknowledges financial support by CONACYT and DGRI-SEP.
References
Aruka, Y. (2011). Complexities of Production and Interacting Human Behaviour. Physica-Verlag.
BEA (2009). Concepts and Methods of the U.S. Input-Output Accounts. Bureau of Economic Analysis,U.S. Department of Commerce, updated edition.
BEA (2015). Benchmark io accounts.
Bidard, C. and Schatteman, T. (2001). The spectrum of random matrices. Economic Systems Research,13(3):289–298.
Brody, A. (1997). The second eigenvalue of the leontief matrix. Economic Systems Research, 9(3):253–258.
Carter, A. P. (1970). Structural Change in the American Economy. Harvard University Press.
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). BayesianData Analysis. Third Edition. CRC Press.
Gelmann, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria forbayesian models. Statistics and Computing, 24(6).
Goldberg, G. and Neumann, M. (2003). Distribution of subdomiant eigenvalues of matrices withrandom rows. SIAM Journal on Matrix Analysis and Applications, 24(3):747–761.
22
Goldberg, G., Okunov, P., Neumann, M., and Schneider, H. (2000). Distribution of subdominanteigenvalues of random matrices. Methodology and Computing in Applied Probability, 2(2):137–51.
Goodwin, R. M. (1974). The use of normalized general coordinates in economic analysis. In Mitra,A., editor, Economic Theory and Planning: Essays in Honour of A. K. Das Gupta, chapter 3, pages26–38. Calcutta: Oxford University Press.
Goodwin, R. M. (1976). Use of normalized general coordinates in linear value and distribution theory.In Polenske, K. and Skolka, J., editors, Advances in Input-Output Economics, chapter 29, pages581–602. Cambridge, Mass.: Ballinger Publ. Co.
Gurgul, H. (2007). Stochastic input-output modeling. Ekonomia Menedzerska, (2):57–70.
Gurgul, H. and Wojtowicz, T. (2015). On the economic interpretation of the brody’s conjecture.Economic Systems Research, 27(1):122–131.
Iliadi, F., Mariolis, T., Soklis, G., and Tsoulfidis, L. (2014). Bienenfeld’s approximation of productionprices and eigenvalue distribution: further evidence from five european economies. Contributions toPolitical Economy, 33(1):35–54. Advance Access published May 25, 2014.
Jackson, R. W. (1986). The full-distribution approach to aggregate representation in the input-outputmodeling framework. Journal of Regional Science, 26(3):515–531.
Jensen, R., Dewhurst, J., West, G., and Hewings, G. (1991). On the concept of fundamental economicstructure. In Dewhurst, J., Jensen, R., and Hewings, G., editors, Regional Input-Output Modelling,chapter 12. Avebury.
Jensen, R., West, G., and Hewings, G. (1988). The study of regional economic structure using input-output tables. Regional Studies, 22(3):209–220.
Kaldor, N. (1957). A model of economic growth. The Economic Journal, 67(268):591–624.
Leontief, W. (1951). The Structure of American Economy, 1919-1939: An Empirical Application ofEquilibrium Analysis. 2nd Editon, Harvard University Press.
Leontief, W. (1986). Input-Output Economics. 2nd Editon, Oxford University Press.
Mariolis, T. and Tsoulfidis, L. (2011). Eigenvalue distribution and the production price-profit raterelationship in linear single-product systems: theory and empirical evidence. Evolutionary andInstitutional Economics Review, 8(1):87–122.
Mariolis, T. and Tsoulfidis, L. (2014). On brody’s conjecture: theory, facts and figures about instabilityof the us economy. Economic Systems Research, 26(2):209–223.
23
Mariolis, T. and Tsoulfidis, L. (2015). Capital theory paradoxes and paradoxical results: Resolved orcontinued? Department of PublicAdministration, PanteionUniversity andDepartment of Economics,University of Macedonia.
Mariolis, T. and Tsoulfidis, L. (2016a). Capital theory: Less is more. Dep. of Public Administration,Panteion University and Dep. of Economics, University of Macedonia.
Mariolis, T. and Tsoulfidis, L. (2016b). Modern Classical Economics and Reality. A Spectral Analysisof the Theory of Value and Distribution. Springer.
Meyer, C. D. (2000). Matrix analysis and applied linear algebra. SIAM: Society for Industrial andApplied Mathematics.
Miller, R. E. and Blair, P. D. (2009). Input-Output Analysis. Foundations and Extentions. CambridgeUniversity Press.
Nassif, L. and Shaikh, A. (2015). Eigenvalue distribution, matrix size and the linearity of wage-profitcurves. 23rd IIOA conference, June 22-26 2015, Mexico City.
Pasinetti, L. L. (1977). Lectures on the theory of production. Palgrave Macmillan.
Pasinetti, L. L. (1988). Sraffa on income distribution. Cambridge Journal of Economics, 12(1):135–138.
Rueda, J. M., Dietzenbacher, E., Fernandez, E., and Amores, A. (2013). The bias of the multipliermatrix when supply and use tables are stochastic. Economic System, 25(4):435–448.
Schefold, B. (2013). Approximate surrogate production functions. Cambridge Journal of Economics,37(5):1161–1184.
Shaikh, A. (2016). Capitalism: Competition, Conflict, Crises. Oxford: University Press.
Sun, G.-Z. (2008). The first two eigenvalues of large random matrices and brody’s hypothesis on thestability of large input-output systems. Economic Systems Research, 20(4):429–432.
Takayama, A. (1974). Mathematical Economics. The Driden Press.
Team, S. D. (2018). RStan: the R interface to Stan, Version 2.17.3.
ten Raa, T. (2005). The Economics of Input-Output Analysis. Cambridge University Press.
Torres, L. D. (2017a). The near-linearity in the empirical relation between wages and the rate of profits:an explanation based on the statistical structure of technology and demand. Mimeo.
Torres, L. D. (2017b). Regularities in prices of production and the concentration of compositions ofcapitals. Working Paper 09/2017, Department of Economics, The New School for Social Research.
24
Vehtari, A., Gelmann, A., and Gabry, J. (2015). Efficient implementation of leave-one-out cross-validation and waic for evaluating fitted bayesian models. Unpublished:http://www.stat.columbia.edu/ gelman/research/unpublished/.
Watanabe, S. (2010). Asymptotic equivalence of bayes cross validation and widely application infor-mation criterion in singular learning theory. Journal of Machine Learning Research, 11:3571–3594.
Watanabe, S. (2013). A widely applicable bayesian information criterion. Journal of Machine LearningResearch, 14(1):867–897.
West, G. R. (1986). A stochastic analysis of an input-output model. Econometrica, 54(2):363–374.
Wibe, S. (1982). The distribution of input coefficients. Economics of Planning, 18(2):65–70.
Appendices
A Construction of the Industry-by-Industry I-O Coefficient Matrices
This paper takes the U.S. Benchmark Input-Output Accounts (BIOA), assembled by the Bureau ofEconomic Analysis (BEA), and constructs 10 input matrices for the years 1963, 1967, 1972, 1977,1982, 1987, 1992, 1997, 2002, and 2007.22 We constructed matrices A
n×n=
{ai j
}, where n is the
number or industries, considering the two methodologies BEA adopted to construct its BIOA. For years1963 to 1967, BEA published an industry-by-industry monetary transaction flow table Z
n×n= {zi j}
and each industries’ gross output, which we gather in the gross output vector x1×n= {xj}. With this
information, matrix A is derived by normalizing each columns of matrix Z by its corresponding grossoutput value: A = Zx−1, where x
n×n= diag{x1, ..., xn}.
From 1972 onwards, the BIOA are constructed from a commodity-by-industry perspective andthey are gathered in the Use U
c×nand the Make V
n×ctables, where c and n stand respectively for the
number of commodities and industries. The Use table U contains the monetary value of the use ofcommodities as intermediate consumption by industries and as final demand —it has commodity-by-industry dimensions. The Make V table indicates the commodity monetary value of production byindustries—it has industry-by-commodity dimensions. There are two types of Use and theMake tables:before and after redefinitions.23 The database is constructed using the latter. Following Miller and Blair(2009, ch. 5), the industry-by-industry I-O coefficient matrices are constructed under the industry-basedtechnology assumption24 as A = D
n×c· Bc×n
, where D = Vz−1, B = Ux−1, and z = diag{z1, ..., zc} is thediagonal matrix with total commodities’ output.
22The full database and the list of eliminated industries is available upon request.23The tables after redefinitions are obtained when the‘the secondary products and their associated inputs [are] excluded
from the industry that produced them and [are] included in the industry in which they were primary’ (BEA, 2009, p. 1-9).24When ‘[a]ll commodities produced by an industry are assumed to have the same input structure’ (Miller and Blair, 2009,
p. 192).
25
There are several modifications made in the original matrices and tables Z, U and V. First, weeliminated the industries with zero intermediate input consumption and negative value-added andwages.Most of these industries are ‘fictitious industries’ according to the BEA. We also eliminated a set offictitious commodities.25 Second, in few occasions, the raw information contained negative entries.Their magnitude was typically very small. We decided to replace those entries with zeros. Their numberof cases varied from 0 to 18 and in the majority of the cases they were concentrated in few industries.Finally, after constructing matrices A, we decided to eliminate the non-basic industries. The rows orcolumns associated with non-basic industries produce zero entries in the left and right Perron-Frobenius(P-F) eigenvectors. Because we are interested in studying the P-F eigenvectors and because ‘non-basicindustries’, at the level of disaggregation with which are working, are somehow arbitrarily determinedby the BEA, we eliminated these industries and made matrix A indecomposable.
Based on these modifications, we constructed the desired database: non-negative, indecomposableI-O matrices with strictly positive value-added and wages, and non-fictitious industries. The finaldimensions (year) of the matrices are: 351 (1963), 438 (1967), 450 (1972), 478 (1977), 472 (1982),456 (1987), 466 (1992), 461 (1997), 406 (2002), and 371 (2007).
B Summary Statistics
ai j
Year N Mean St. Dev. Min Max1963 123,201 0.002 0.013 0.000 0.7771967 191,844 0.001 0.011 0.000 0.7451972 202,500 0.001 0.010 0.000 0.7711977 228,484 0.001 0.010 0.000 0.7591982 222,784 0.001 0.010 0.000 0.7561987 207,936 0.001 0.010 0.000 0.8241992 217,156 0.001 0.010 0.000 0.7921997 207,025 0.001 0.009 0.000 0.6782002 164,836 0.001 0.009 0.000 0.6702007 137,641 0.001 0.009 0.000 0.665
aii
Year N Mean St. Dev. Min Max1963 351 0.047 0.072 0.000 0.4311967 438 0.039 0.059 0.000 0.4211972 450 0.036 0.058 0.000 0.3671977 478 0.033 0.053 0.000 0.3531982 472 0.037 0.057 0.000 0.4061987 456 0.034 0.052 0.000 0.4261992 466 0.038 0.053 0.000 0.3291997 406 0.036 0.051 0.000 0.3442002 455 0.034 0.051 0.000 0.4792007 371 0.042 0.056 0.000 0.305
ρi
Year N Mean St. Dev. Min Max1963 351 0.581 1.078 0.001 11.0881967 438 0.572 1.223 0.0001 12.7791972 450 0.548 1.292 0.0001 15.4751977 478 0.549 1.375 0.0001 19.9721982 472 0.568 1.440 0.00000 21.8671987 456 0.524 1.282 0.0001 19.6551992 466 0.541 1.417 0.0001 22.8721997 406 0.579 1.389 0.00000 17.6662002 455 0.571 1.451 0.00000 21.4672007 371 0.555 1.158 0.00001 15.411
κ j
Year N Mean St. Dev. Min Max1963 351 0.581 0.151 0.141 0.9791967 438 0.572 0.151 0.154 0.9491973 450 0.548 0.143 0.123 0.9281977 478 0.549 0.146 0.153 0.9541982 472 0.568 0.152 0.143 0.9631987 456 0.524 0.137 0.099 0.9321992 466 0.541 0.143 0.086 0.9421997 406 0.579 0.151 0.173 0.9492002 455 0.571 0.163 0.020 0.9592007 371 0.555 0.158 0.128 0.944
Table 3: Summary statistics of the pooled ai j coefficients, the diagonal aii , column sums κj , and row sums ρj foreach US I-O coefficient matrices A = {ai j}.
25The set of industries and commodities eliminated consisted in royalties; Commodity Credit Corporation; directly,transferred, and noncomparable imports; scrap, used and secondhand goods, rest of the world industry, Inventory valuationadjustment, household industry, and sometimes diverse federal, state and local government industries.
26
|λi |
Year N Mean St. Dev. Min Max1963 351 0.080 0.083 0.0001 0.5221967 438 0.070 0.075 0.000 0.5321972 450 0.051 0.066 0.000 0.4921977 478 0.049 0.064 0.000 0.5051982 472 0.052 0.066 0.000 0.5001987 456 0.049 0.065 0.000 0.4901992 466 0.048 0.064 0.000 0.4861997 406 0.049 0.063 0.000 0.4812002 455 0.047 0.064 0.000 0.5282007 371 0.054 0.068 0.000 0.511
qL1, j
Year N Mean St. Dev. Min Max1963 351 0.021 0.049 0.00002 0.4171967 438 0.017 0.045 0.00001 0.4011972 450 0.015 0.045 0.00000 0.4921977 478 0.012 0.044 0.000 0.6401982 472 0.009 0.045 0.000 0.7531987 456 0.016 0.044 0.000 0.4261992 466 0.013 0.044 0.000 0.6601997 406 0.019 0.046 0.000 0.4162002 455 0.015 0.044 0.000 0.5652007 371 0.016 0.050 0.000 0.723
qRi,1
Year N Mean St. Dev. Min Max1963 351 0.058 0.040 0.001 0.3081967 438 0.069 0.052 0.0005 0.2631972 450 0.043 0.036 0.00001 0.2541977 478 0.068 0.048 0.001 0.2661982 472 0.044 0.035 0.0003 0.1871987 456 0.054 0.044 0.0003 0.3251992 466 0.048 0.042 0.0003 0.2161997 406 0.142 0.125 0.0001 0.5552002 455 0.127 0.106 0.0004 0.7162007 371 0.175 0.130 0.003 0.607
Table 4: Summary statistics of the absolute value of the eigenvalues |λt | and the left and right Perron-Frobeniuseigenvectors qL
1, j and qRi,1 for each US I-O coefficient matrices A = {ai j}
C HMC and WAIC&LOO-CV
Hamiltonian Monte Carlo (HMC) algorithm, which is also known as Hybrid Monte Carlo (Duane1987), is a Markov chain Monte Carlo (MCMC) method, a fundamental simulation technique inBayesian Statistics. HMC is known to be more efficient than other random walk MCMC algorithmssuch as the Metropolis-Hastings due to the gradient information of a dynamical system constructedfrom the target density. This paper uses RStan software that implements HMC algorithm along with aparticular sampler, called No U-turn sampler (Team, 2018). For detailed technical discussion on HMCwith No U-turn sampler see Gelman et al. (2013).
For a model evaluation and comparison, we use a fully Bayesian method called Widely ApplicableInformation Criteria (WAIC) and Leave-one-out cross-validation (LOO-CV) (Watanabe, 2010, 2013).They are fully Bayesian because they utilize the entire posterior distribution rather than a point estima-tion. WAIC is asymptotically equal to LOO. For detailed survey on different model selection criteria,see Gelmann et al. (2014).
Here, we briefly summarize WAIC calculation. First, the model fit or the posterior predictiveaccuracy is assessed using the log pointwise predictive density (LPPD). For each data point i = 1, . . . , n,the expected log-likelihood of the data is calculated given the posterior distribution and is summed overthe entire data points:
LPPD =n∑i=1
Log[∫
p[yi |θ]p[θ |y]dθ]. (10)
27
where y is data and θ is a set of parameters. The LPPD can be evaluated by drawing from the posteriorsimulations. Denoting the simulated θ to be θs, s = 1, . . . , S, the computed LPPD, which I denote withan asterisk, LPPD*, is written as:
LPPD* =
n∑i=1
Log
[1S
S∑s=1
p[yi |θs]
](11)
The higher the LPPD is, the more accurately the model fit the data.Since the LPPD of observed data y is solely based on the log-likelihood, it does not account for
the number of parameters in the model and therefore is prone to an overestimation of the predictiveaccuracy. One way to deal with this problem is to augment the LPPD with a measure of bias correction.In WAIC, the correction is made by subtracting the ‘effective number of parameters’ of the modelepWAIC, which is a measure of ‘model complexity’. It is defined as the difference between the modeldeviance given the posterior distribution, 2 Log[E[p[yi |θ]]], and the expectation of the model deviance,E[2 Log[p[yi |θ]] :
epWAIC =
n∑i=1(2 Log[E[p[yi |θ]]] − E[2 Log[p[yi |θ]]]) , (12)
which can be computed by:
ep*WAIC = 2
n∑i=1
(Log
[1S
S∑i=1
p[yi |θs]
]−
1S
S∑s=1
Log[p[yi |θss]]
). (13)
Finally, WAIC is obtained by subtracting epWAIC from the model fit, LPPD. In deviation scale, themeasure is multiplied by -2 so that WAIC is written as follows:
WAIC = −2(lppd− epWAIC). (14)
The lower WAIC, the more accurate the posterior prediction is for new data and the more parsimoniousthe model is.
D Parameter Estimation
The following table summarizes the results of parameter estimation for the top two performing modelsfor each variable of A– all entries ai j , the diagonal elements aii, the column sums and row sums ρi andκj , eigenvalues’ modulus |λi |, and the elements of the left and right Perron-Frobenius eigenvectors, theeigenvectors associated to the maximal eigenvalue, qL
1, j and qRi,1:
28
f(.)
Parameter
1963
1967
1972
1977
1982
1987
1992
1997
2002
2007
f(a
ij)
αE
154.43
(0.85)
90.62(0.63)
502.28
(1.31)
602.32
(1.55)
659.67
(1.38)
633.95
(1.55)
587.88
(1.45)
534.42
(1.52)
528.21
(1.62)
526.56
(1.49)
αW
0.42
(0)
0.61
(0)
0.32
(0)
0.31
(0)
0.29
(0)
0.29
(0)
0.28
(0)
0.3(0)
0.31
(0)
0.31
(0)
βW
0.00
17(0)
0.00
66(0)
0.00
02(0)
0.00
01(0)
0.00
01(0)
0.00
01(0)
0.00
01(0)
0.00
02(0)
0.00
02(0)
0.00
02(0)
αB
0.26
(0)
0.46
(0)
0.18
(0)
0.17
(0)
0.16
(0)
0.16
(0)
0.16
(0)
0.18
(0)
0.18
(0)
0.18
(0)
βB
38.58(0.48)
39.11(0.51)
85.5
(0.64)
99.21(0.7)
98.62(0.59)
96.09(0.67)
88.9
(0.68)
91.58(0.57)
93.92(0.62)
92.45(0.76)
αG
0.27
(0)
0.47
(0)
0.18
(0)
0.18
(0)
0.16
(0)
0.16
(0)
0.16
(0)
0.18
(0)
0.18
(0)
0.18
(0)
βG
41.66(0.55)
43.05(0.54)
90.84(0.67)
105.8(0.67)
104.48
(0.74)
101.51
(0.72)
93.69(0.67)
95.76(0.62)
97.79(0.71)
96.02(0.78)
αP
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.01
(0.01)
0.02
(0)
µLN
-7.63(0.01)
-5.86(0.01)
-10.35
(0.01)
-10.67
(0.01)
-11.3(0.01)
-11.2(0.01)
-11.16
(0.01)
-10.46
(0.01)
-10.29
(0.01)
-10.35
(0.01)
σLN
2.65
(0.01)
1.7(0.01)
3.13
(0.01)
3.18
(0.01)
3.67
(0.01)
3.52
(0.01)
3.62
(0.01)
3.6(0.01)
3.54
(0.01)
3.61
(0.01)
f(a
ii)
αE
20.25(1.08)
22.29(1.18)
25.9
(1.3)
27.77(1.3)
25.61(1.18)
28.15(1.4)
25.64(1.16)
28.38(1.34)
26.66(1.4)
22.6
(1.25)
αW
0.58
(0.03)
0.81
(0.03)
0.64
(0.02)
0.65
(0.02)
0.65
(0.02)
0.65
(0.02)
0.65
(0.03)
0.63
(0.02)
0.62
(0.02)
0.65
(0.03)
βW
0.03
(0)
0.04
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.03
(0)
αB
0.44
(0.03)
0.71
(0.04)
0.49
(0.03)
0.5(0.03)
0.51
(0.03)
0.51
(0.03)
0.5(0.03)
0.48
(0.03)
0.48
(0.03)
0.51
(0.03)
βB
8.49
(0.83)
14.78(1.24)
12.07(1.07)
13.35(1.12)
12.61(1.08)
13.66(1.13)
12.29(1.01)
13.13(1.12)
12.31(1.11)
11(1.06)
αG
0.46
(0.03)
0.74
(0.05)
0.51
(0.03)
0.52
(0.03)
0.53
(0.03)
0.52
(0.03)
0.51
(0.03)
0.5(0.03)
0.49
(0.03)
0.52
(0.03)
βG
9.3(0.96)
16.57(1.44)
13.25(1.22)
14.49(1.21)
13.72(1.23)
14.78(1.28)
13.2
(1.12)
14.13(1.17)
13.05(1.22)
11.79(1.11)
αP
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
µLN
-4.42(0.12)
-3.91(0.08)
-4.5
(0.1)
-4.56(0.1)
-4.42(0.09)
-4.54(0.1)
-4.47(0.11)
-4.64(0.1)
-4.6
(0.12)
-4.33(0.11)
σLN
2.17
(0.08)
1.43
(0.05)
2.1(0.07)
2.14
(0.08)
1.99
(0.06)
2.04
(0.07)
2.31
(0.07)
2.34
(0.08)
2.45
(0.08)
2.21
(0.09)
f(κ
j)
αE
1.73
(0.09)
1.75
(0.08)
1.83
(0.09)
1.83
(0.09)
1.77
(0.08)
1.91
(0.09)
1.86
(0.08)
1.72
(0.08)
1.74
(0.09)
1.81
(0.09)
αW
0.76
(0.03)
0.71
(0.02)
0.63
(0.02)
0.64
(0.02)
0.62
(0.02)
0.6(0.02)
0.6(0.02)
0.64
(0.02)
0.67
(0.02)
0.69
(0.03)
βW
0.48
(0.04)
0.43
(0.03)
0.36
(0.03)
0.36
(0.03)
0.37
(0.03)
0.33
(0.03)
0.33
(0.03)
0.4(0.03)
0.42
(0.03)
0.42
(0.03)
αG
0.69
(0.04)
0.62
(0.03)
0.51
(0.03)
0.52
(0.03)
0.5(0.03)
0.47
(0.03)
0.47
(0.03)
0.51
(0.03)
0.56
(0.03)
0.57
(0.04)
βG
1.19
(0.1)
1.08
(0.09)
0.93
(0.08)
0.94
(0.08)
0.88
(0.07)
0.9(0.08)
0.88
(0.08)
0.89
(0.07)
0.97
(0.09)
1.04
(0.09)
αP
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
µLN
-1.43(0.07)
-1.56(0.07)
-1.85(0.08)
-1.84(0.08)
-1.84(0.08)
-2(0.09)
-1.98(0.08)
-1.78(0.09)
-1.67(0.09)
-1.68(0.09)
σLN
1.4(0.05)
1.47
(0.05)
1.69
(0.06)
1.67
(0.05)
1.82
(0.06)
1.84
(0.06)
1.85
(0.06)
1.84
(0.06)
1.77
(0.06)
1.88
(0.07)
f(κ
j)
αB
5.33
(0.4)
5.42
(0.38)
5.85
(0.39)
5.6(0.36)
5.36
(0.34)
6.09
(0.39)
5.81
(0.38)
4.66
(0.3)
5.63
(0.4)
4.99
(0.36)
βB
3.84
(0.29)
4.05
(0.28)
4.81
(0.32)
4.6(0.29)
4.08
(0.26)
5.51
(0.35)
4.92
(0.32)
3.37
(0.22)
4.1(0.28)
4.02
(0.29)
αG
12.52(0.99)
12.61(0.85)
13.09(0.88)
12.33(0.77)
11.98(0.73)
13.35(0.9)
12.71(0.82)
10.34(0.62)
12.89(0.9)
10.6
(0.78)
βG
21.54(1.75)
22.05(1.52)
23.89(1.63)
22.48(1.43)
21.09(1.31)
25.51(1.75)
23.51(1.56)
17.79(1.09)
22.28(1.6)
19.13(1.46)
µN
0.58
(0.01)
0.57
(0.01)
0.55
(0.01)
0.55
(0.01)
0.57
(0.01)
0.52
(0.01)
0.54
(0.01)
0.58
(0.01)
0.58
(0.01)
0.55
(0.01)
σL
0.15
(0.01)
0.15
(0.01)
0.14
(0)
0.15
(0)
0.15
(0)
0.14
(0)
0.14
(0)
0.16
(0.01)
0.15
(0.01)
0.16
(0.01)
Table5:
Summarystatisticso
fthe
estim
ated
parametersfor
allentries
a ij,thediagonal
elem
ents
a ii,thecolumnsumsa
ndrowsumsρiandκj.Th
esubscripts
E,B,
G,W
,N,L
NrepresentE
xponentia
l,Be
ta,G
amma,Weibull,
Norm
al,L
og-Normal
distributio
ns,respectively.
29
f(.)
Parameter
1963
1967
1972
1977
1982
1987
1992
1997
2002
2007
f(|λ
i|)
αE
12.45(0.64)
14.37(0.67)
19.71(0.91)
20.45(0.9)
19.16(0.87)
20.34(0.94)
20.77(0.99)
21.1
(1.05)
20.49(1)
18.38(0.96)
αW
0.99
(0.04)
0.84
(0.03)
0.63
(0.03)
0.54
(0.02)
0.57
(0.02)
0.47
(0.02)
0.43
(0.02)
0.48
(0.02)
0.68
(0.03)
0.43
(0.02)
βW
0.08
(0)
0.07
(0)
0.04
(0)
0.04
(0)
0.04
(0)
0.03
(0)
0.03
(0)
0.03
(0)
0.04
(0)
0.04
(0)
αB
0.94
(0.07)
0.67
(0.04)
0.47
(0.03)
0.38
(0.02)
0.41
(0.02)
0.33
(0.02)
0.29
(0.01)
0.33
(0.02)
0.51
(0.03)
0.3(0.02)
βB
10.71(0.96)
8.92
(0.72)
8.73
(0.7)
7.46
(0.68)
7.55
(0.63)
6.55
(0.59)
5.99
(0.56)
6.76
(0.65)
9.89
(0.9)
5.4(0.58)
αG
1.02
(0.07)
0.7(0.04)
0.48
(0.03)
0.39
(0.02)
0.42
(0.02)
0.33
(0.02)
0.29
(0.02)
0.34
(0.02)
0.53
(0.03)
0.3(0.02)
βG
12.81(1.11)
10.11(0.85)
9.41
(0.83)
8(0.72)
8.11
(0.69)
6.85
(0.67)
6.16
(0.61)
7.11
(0.68)
10.77(1)
5.58
(0.6)
αP
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
0.02
(0)
µLN
-3.09(0.07)
-3.53(0.17)
-4.32(0.2)
-4.73(0.27)
-4.5
(0.23)
-5.06(0.31)
-5.4
(0.34)
-5.06(0.32)
-4.22(0.2)
-5.2
(0.4)
σLN
1.25
(0.05)
3.57
(0.12)
4.23
(0.14)
5.83
(0.19)
5.18
(0.17)
6.4(0.21)
7.31
(0.25)
6.62
(0.23)
4.12
(0.15)
7.57
(0.27)
f( q
L 1,j
)αE
47.6
(2.4)
59.95(2.79)
65.48(3.2)
81.42(3.79)
112.46
(5.33)
62.34(2.75)
75.13(3.46)
66.88(3.26)
53.62(2.58)
63.91(3.25)
αW
0.54
(0.02)
0.51
(0.02)
0.5(0.02)
0.49
(0.02)
0.47
(0.02)
0.49
(0.02)
0.46
(0.02)
0.49
(0.02)
0.51
(0.02)
0.49
(0.02)
βW
0.01
(0)
0.01
(0)
0.01
(0)
0.01
(0)
0(0)
0.01
(0)
0.01
(0)
0.01
(0)
0.01
(0)
0.01
(0)
αB
0.39
(0.02)
0.35
(0.02)
0.34
(0.02)
0.33
(0.02)
0.29
(0.01)
0.33
(0.02)
0.3(0.02)
0.33
(0.02)
0.36
(0.02)
0.33
(0.02)
βB
17.4
(1.77)
19.96(1.96)
20.97(1.94)
23.87(2.3)
26.9
(2.65)
19.5
(1.78)
20.86(2.09)
20.69(2.05)
18.07(1.78)
18.64(2)
αG
0.4(0.02)
0.37
(0.02)
0.35
(0.02)
0.34
(0.02)
0.31
(0.02)
0.34
(0.02)
0.32
(0.02)
0.34
(0.02)
0.37
(0.02)
0.34
(0.02)
βG
19.45(2.04)
21.92(2.13)
23.27(2.25)
27.9
(2.57)
34.63(3.4)
21.43(2.04)
23.88(2.24)
22.98(2.05)
20.02(1.94)
22.19(2.32)
αP
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
0(0)
µLN
-5.52(0.1)
-5.92(0.1)
-6.09(0.11)
-6.39(0.11)
-6.96(0.12)
-6.1
(0.11)
-6.49(0.11)
-6.18(0.12)
-5.81(0.12)
-6.12(0.13)
σLN
1.94
(0.07)
2.05
(0.07)
2.17
(0.07)
2.34
(0.08)
2.53
(0.08)
2.35
(0.08)
2.52
(0.08)
2.56
(0.08)
2.51
(0.09)
2.54
(0.09)
f( q
R i,1)
αW
1.51
(0.06)
1.35
(0.05)
1.24
(0.05)
1.43
(0.05)
1.27
(0.04)
1.31
(0.05)
1.15
(0.04)
1.29
(0.05)
1.13
(0.05)
1.39
(0.06)
βW
0.06
(0)
0.08
(0)
0.05
(0)
0.08
(0)
0.05
(0)
0.06
(0)
0.05
(0)
0.14
(0.01)
0.15
(0.01)
0.19
(0.01)
αB
1.94
(0.13)
1.55
(0.09)
1.42
(0.09)
1.66
(0.09)
1.42
(0.08)
1.57
(0.1)
1.25
(0.07)
1.36
(0.08)
1.09
(0.06)
1.46
(0.09)
βB
31.71(2.4)
21.02(1.47)
31.7
(2.34)
22.69(1.48)
30.93(2.16)
27.56(1.96)
24.93(1.8)
9.23
(0.62)
6.54
(0.46)
6.83
(0.52)
αG
2.04
(0.14)
1.63
(0.11)
1.48
(0.08)
1.75
(0.1)
1.47
(0.09)
1.67
(0.1)
1.3(0.07)
1.57
(0.09)
1.23
(0.08)
1.77
(0.12)
βG
35.3
(2.78)
23.78(1.82)
34.7
(2.26)
25.75(1.81)
33.45(2.32)
31.08(2.08)
27.4
(1.87)
12.36(0.88)
8.71
(0.68)
10.11(0.79)
µLN
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
0.01
(0.01)
σLN
3.24
(0.12)
3.18
(0.1)
3.69
(0.13)
3.15
(0.1)
3.66
(0.12)
3.39
(0.11)
3.65
(0.12)
2.61
(0.09)
2.67
(0.1)
2.24
(0.08)
Table6:
Summarystatisticsof
theestim
ated
parametersforeigenvalues’modulus|λ
i|,andtheelem
entsof
theleftandrightP
erron-Frobeniuseigenvectors,
theeigenvectorsassociated
tothemaximaleigenvalue,f
qL 1,jand
qR i,1.
ThesubscriptsE,
B,G,W
,N,L
NrepresentE
xponentia
l,Be
ta,G
amma,Weibull,
Norm
al,
Log-No
rmal
distributio
ns,respectively.
30
