Bayesian Estimation of Input Output Tables for Russia

Transcription

1 Bayesian Estimation of Input Output Tables for Russia Oleg Lugovoy (EDF, RANE) Andrey Polbin (RANE) Vladimir Potashnikov (RANE) WIOD Conference April 24, 2012 Groningen

2 Outline Motivation Objectives Bayesian methods: a short intro Experiments: Bayesian vs. RAS & Entropy: MC experiment Updating IOT for Russia Some conclusions Further steps

3 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 , 15, 23 and 79 activities)

4 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.

5 Bayesian inference - unknown parameters - data - prior distribution of parameters - likelihood function - Posterior distribution (combination of information from prior and data)

6 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.

7 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

8 Bayesian perspective on IOT estimation Problem: Y = AX, i a = a, a 0 i, j j i, j Solution: application of MCMC to sample elements of A- matrix. Each version of A should satisfy all the constrains.

9 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 z = z + F ξ (1) (1) therefore we have to sample z in form: where z-tilde is a particular solution, F fundamental matrix, psi stochastic component.

10 Experimental estimates

11 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 (OKVED 15x15)

12 1. Monte-Carlo experiment: Bayesian vs. RAS, vs. Cross-Entropy Generate arbitrary A A A-matrices 4x4 for six years Assume we don t know the last matrix: Estimate A 6 with RAS and Minimal Cross- Entropy, assuming we know A 5, Y 6 and X 6 Estimate A 6 with MCMC, assuming we know A 1 A 5, Y 6 and X 6 (estimating standard deviation for A elements based on A 1 A 4, and assigning the information to priors),,.., A A 6

13 MC experiment (cont.) Three cases, experiments each: i.i.d. a Nm (, σ ), i 3 t 2 i, j i, j i, j Stationary AR(1) a = (1 ρ) m + ρa + ε, ε N(0, σ ), i 3 t t 1 t t 2 i, j i, j i, j i, j i, j i, j Random walk a = a + ε, ε N(0, σ ), i 3 t t 1 t t 2 i, j i, j i, j i, j i, j

14 Monte-Carlo experiment: results Share of experiments where Bayesian methodology provided results closer to true matrix Independent process AR(1) process Random-Walk Entropy RAS Entropy RAS Entropy RAS RMSE 72.2% 73.2% 67.3% 67.8% 62.0% 63.0% MAE 76.0% 77.8% 71.2% 73.1% 66.1% 67.7% MAPE 76.2% 78.1% 72.4% 74.1% 67.1% 69.0% Distance criteria: RMSE: root of mean squared error MAE: mean absolute error ( ˆ ) /16 i= 1 j= 1 ij ij RMSE = a a 4 4 1/16 ˆ i= 1 j= 1 ij ij MAE = a a MAPE: mean absolute relative error MAPE 1/ = i= 1 j= 1 a ij aˆ a ij ij

15 Monte-Carlo experiment: results 0.06 RMSE in Bayesian Method RMSE in RAS Method

16 2. Experiment: Updating IOT for Russia Symmetric IOT 23x23 for 2003, OKONH (Soviet type) definition of activities Assuming IOT are known, 2003 unknown Prior mean: IOT-2002 Prior distribution truncated normal Prior st.d. estimated in Comparison of the results with RAS and Cross- Entropy

17 Experiment: Updating IOT for Russia (cont) Comparison of the results RMSE MAE MAPE RMSPE Bayes RAS Entropy where RMSPE - root of mean squared percentage error RMSPE m n a ˆ ij a ij = 1/( m* n) i= 1 j= 1 a ij 2

18 Experiment: Updating IOT for Russia (cont)

19 3. Experimental estimates of Russian IOT for 2006 Estimation of USE matrix in OKVED (NACE) definition of activities

20 Standard problem Y = A X 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?

21 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: a ij ~ uniform(0,1) Sampling USE-2006 table; N = 45000

22 Estimated А-matrix (blue) vs. true value (red)

23 Analysis Sparse matrix: number of values are close to zero for activities with relatively small output. 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.

24 Distribution of pair-wise correlation coefficients

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26 Constrains on the coefficients Substantial correlation between the sampled A-matrix elements means that constraining one 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.

27 Sampling with tight prior for A(D,D)

28 Sampling with tight prior for A(D,D)

29 Sampling with tight prior for A(D,D)

30 Comparison of results with true values: no constrains

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

32 4. Experiment: Forecasting USE with A (t-1) is known unknown Y Y (t) is unknown: b < Y < c

33 Forecasting USE-2006 with unknown Y

34 Forecasting USE with unknown Y prior year 2007 year 2008 year 2009 year 2010

35 Some conclusions Bayesian approach is a flexible and natural tool to incorporate uncertainties in data into estimation process The experimental estimates demonstrate a way of 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.

36 Further steps: Extending the information for the estimates, involving data from National Accounts and other available. Joint estimates of multiple accounts (in current and constant prices). Disaggregation of OKVED-15 tables to OKVED- 79.

37 Thank you for your attention!