Bayesian Estimation of InputOutput Tables for Russia Oleg Lugovoy - - PowerPoint PPT Presentation

Bayesian Estimation of Input‐Output Tables for Russia

Oleg Lugovoy (EDF, RANE) Andrey Polbin (RANE) Vladimir Potashnikov (RANE) WIOD Conference April 24, 2012 Groningen

Outline

• Motivation
• Objectives
• Bayesian methods: a short intro
• Experiments:

– Bayesian vs. RAS & Entropy: MC experiment – Updating IOT for Russia

• Some conclusions
• Further steps

Objectives

• Methodological:

– Incorporate uncertainties into IOT estimates – Apply Bayesian framework for IOT updating

• Practical:

– Full density profile estimates with covariates for Russian IOT (OKONH, OKVED 2001‐2010, 15, 23 and 79 activities)

Motivation

• Unsatisfied demand for Russian IOT forces users of the data

to estimate, update, disaggregate, get their best estimate. Each procedure involves assumptions, which draw results. However, there are a number of ways to do that, and it is not always straightforward to prefer one assumption to another, based on available information.

• The assumptions made on the (IOT) estimation stage might

be crucial for an analysis that applies the estimated IOT as an input data.

• Bayesian inference suggests a natural way to accommodate

uncertainties into estimation process. Assigning probability distributions for unknown parameters it allows to trace the link from assumptions to economic analysis results.

‐ unknown parameters ‐ data ‐ prior distribution of parameters ‐ likelihood function ‐ Posterior distribution (combination of information from prior and data)

Bayesian inference

Bayesian Approach to Statistics

• Since we are uncertain about the true value of

parameters, we will consider them to be random variables.

• The rules of probability are used directly to make

inferences about the parameters.

• Known information about parameters can be naturally

incorporated to the estimates using priors.

• The result of the estimate is a posterior distribution of

uncertain parameters which comes from two sources: the prior distribution and the observed data.

How it works?

• For simple (1‐parameter) models closed form solution

can be derived

• For complicated models, were it is difficult to derive

posterior distribution, sampling methods applied

• The most efficient sampling algorithms for now is

Monte Carlo Markov Chains (MCMC) method with Gibbs or Metropolis‐Hasting algorithm

Bayesian perspective on IOT estimation

• Problem:
• Solution:

– application of MCMC to sample elements of A‐

• matrix. Each version of A should satisfy all the

constrains.

, ,

, ,

i j j i j i

Y AX a a a = = ≥

∑

Estimating IOT: Bayesian perspective

• Transforming problem from

Y = AX (A – matrix is unknown) to Bz = Y* (z – vector is unknown, standard problem in linear algebra) where z – vectorized matrix A Y* ‐ combined Y and a_hat vectors therefore we have to sample z in form: where z‐tilde is a particular solution, F – fundamental matrix, psi – stochastic component.

(1) (1)

z z F ξ = + 

Experimental estimates

Estimates

• Performance: Bayesian vs. RAS vs. Max.Entropy

– 1. Artificial data: MC experiment – 2. Historical data: IOT‐2003 (OKONH 23x23)

• Updating with limited information:

– 3. Historical USE‐2006 (OKVED 15x15) – 4. Forecasting USE 2007‐2010 (OKVED 15x15)

• 1. Monte‐Carlo experiment: Bayesian
• vs. RAS, vs. Cross‐Entropy
• Generate arbitrary

A‐matrices 4x4 for six years

• Assume we don’t know the last matrix:
• Estimate A6 with RAS and Minimal Cross‐

Entropy, assuming we know A5, Y6 and X6

• Estimate A6 with MCMC, assuming we know

A1 …A5, Y6 and X6 (estimating standard deviation for A elements based on A1 …A4 , and assigning the information to priors)

1 2 6

, ,.., A A A

6

A

MC experiment (cont.)

• Three cases, 10 000 experiments each:

– i.i.d. – Stationary AR(1) – Random walk

2 , , ,

( , ), 3

t i j i j i j

a N m i σ ≤ ฀

1 2 , , , , , ,

(1 ) , (0, ), 3

t t t t i j i j i j i j i j i j

a m a N i ρ ρ ε ε σ

−

= − + + ≤ ฀

1 2 , , , , ,

, (0, ), 3

t t t t i j i j i j i j i j

a a N i ε ε σ

−

= + ≤ ฀

• Share of experiments where Bayesian methodology provided results

closer to true matrix

• Distance criteria:
• RMSE: root of mean squared error
• MAE: mean absolute error
• MAPE: mean absolute relative error

69.0% 67.1% 74.1% 72.4% 78.1% 76.2% MAPE 67.7% 66.1% 73.1% 71.2% 77.8% 76.0% MAE 63.0% 62.0% 67.8% 67.3% 73.2% 72.2% RMSE RAS Entropy RAS Entropy RAS Entropy Random-Walk AR(1) process Independent process

Monte‐Carlo experiment: results

( )

2 4 4 1 1

ˆ 1/16

ij ij i j

RMSE a a

= =

= −

∑ ∑

4 4 1 1

ˆ 1/16

ij ij i j

MAE a a

= =

= −

∑ ∑

4 4 1 1

ˆ 1/16

ij ij i j ij

a a MAPE a

= =

− =

∑ ∑

Monte‐Carlo experiment: results

0.01 0.02 0.03 0.04 0.05 0.06 0.01 0.02 0.03 0.04 0.05 0.06 RMSE in RAS Method RMSE in Bayesian Method

• 2. Experiment: Updating IOT for Russia
• Symmetric IOT 23x23 for 2003, OKONH (Soviet

type) definition of activities

• Assuming IOT 1998‐2002 are known, 2003 –

unknown

• Prior mean: IOT‐2002
• Prior distribution – truncated normal
• Prior st.d. – estimated in 1998‐2002
• Comparison of the results with “RAS” and “Cross‐

Entropy”

Experiment: Updating IOT for Russia (cont)

• Comparison of the results

where RMSPE - root of mean squared percentage error

RMSE MAE MAPE RMSPE Bayes 0.0074 0.0029 0.1844 0.4502 RAS 0.0067 0.0026 0.1728 0.4604 Entropy 0.0065 0.0026 0.1797 0.4552 2 1 1

ˆ 1/ ( * )

m n ij ij i j ij

a a RMSPE m n a

= =

  − =      

∑ ∑

Experiment: Updating IOT for Russia (cont)

• 3. Experimental estimates of

Russian IOT for 2006

Estimation of USE matrix in OKVED (NACE) definition of activities

Standard problem

X A Y ⋅ =

A – unknown input‐output matrix Y – known intermediate demand vector X – known output vector What inference can we make toward A, when no any other information is available?

What inference can we make toward A, when X and Y are known

Let’s sample N variants of matrix A, satisfying to the constrains: Y = AX, given X и Y, using MCMC, non‐informative priors:

aij ~ uniform(0,1)

Sampling USE‐2006 table; N = 45000

Estimated А‐matrix (blue) vs. true value (red)

Analysis

• Sparse matrix: number of values are close to

zero – for activities with relatively small

• utput.
• Posterior distributions are asymmetric toward
• zero. This is a result of constrain on the

columns sum (<1). If one of a value is relatively large, others should tend to zero.

Distribution of pair‐wise correlation coefficients

Constrains on the coefficients

• Substantial correlation between the sampled

A‐matrix elements means that constraining

• ne or some of the estimated parameters will

affect other.

• Let’s impose a constrain on one of the

coefficients: A(D,D), specifying ”tight” prior for it.

Sampling with “tight” prior for A(D,D)

Sampling with “tight” prior for A(D,D)

Sampling with “tight” prior for A(D,D)

Comparison of results with true values: no constrains

Comparison of results with true values: constrained А(D,D)

• 4. Experiment: Forecasting USE with

unknown Y

• A (t‐1) is known
• Y (t) is unknown:

b < Y < c

Forecasting USE‐2006 with unknown Y

Forecasting USE with unknown Y

prior year 2007 year 2008 year 2009 year 2010

Some conclusions

• Bayesian approach is a flexible and natural tool to

incorporate uncertainties in data into estimation process

• The experimental estimates demonstrate a way
• f application of Bayesian inference for updating

and estimating IOT

• The results of the estimates – multidimensional

distribution of the estimated parameters might be used as an input information for sensitivity analysis on a stage of implementation of the analysis.

Further steps:

• Extending the information for the estimates,

involving data from National Accounts and

• ther available.
• Joint estimates of multiple accounts (in

current and constant prices).

• Disaggregation of OKVED‐15 tables to OKVED‐

79.

Thank you for your attention!

• lugovoy@gmail.com

apolbin@gmail.com potashnikov.vu@gmail.com