7/31/2019 Pres BayMC2

1/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Bayesian MC2 estimation of the capital-laborsubstitution elasticity in developing countries

Rolando Gonzales Martnez

4th regional meeting economic analysis of public policies with computable

general equilibrium models - ESPAE, Guayaquil

April, 2012

http://find/http://goback/

7/31/2019 Pres BayMC2

2/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Motivation: MAMS Project

Part of the project Strengthening Macroeconomic andSocial Policy Coherence through Integrated Macro-Micro

Modelling1 involved the estimation of the capital-laborsubstitution elasticity for the computable generalequilibrium model MAMS (Maquette for MillenniumDevelopment Goals Simulation, see Logfren andDaz-Bonilla, 2010)

1Organized by the United Nations Development Program (UNDP), theUnited Nations Department of Economic and Social Aairs (UNDESA),

and Unidad de Anlisis de Polticas Sociales y Econmicas de Bolivia(UDAPE).

http://find/http://goback/

7/31/2019 Pres BayMC2

3/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Capital-labor substitution elasticity

CES production function (Arrow et al., 1961):

q = Ah`

1 + (1 )k

1

i 1

, (1)

q is real output, A is factor productivity, 2 [0, 1] is a distribution parameter2 [0,) is the elasticity of substitution between capital andlabor. The rst order condition of the restricted optimizationproblem of the rms is,

pA

1`

1

+ (1 ) k

11

1

`

1

= w,

then,

` =

0

B@pA

`

1 + (1 ) k

11

11

w

1

CA

,

http://find/http://goback/

7/31/2019 Pres BayMC2

4/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Estimable equation

` =

p

w

A

`

1 + (1 ) k

11

1

, (2)

Replacing (1) in (2), the labor demand equals,

` = p

w

A1

q,

then,q

`=

w

p

()A1.

This equation can be log-linearized,

y = 0 + 1x, (3)

for y = ln(q/`), x = ln(w/p), w/p real wages and0 = ln + (1 )lnA, 1 := . See Cicowiez (2011).

http://find/http://goback/

7/31/2019 Pres BayMC2

5/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

A frequentist approach to estimate (3)

Plug up an (additive) error term i.i.d N

0, 2

in (3) to

become it an empirical model:

y = 0 + 1x + (4)

Use any available information to "measure" y = ln(q/l)and x = ln(w/p)2

Estimate (4) with a canned software, as e.g. Eviews orStata.

2For the Bolivian case, yearly data of ouput and output deator fromnational accounts was used to measure q and p, respectively; l and w wereemployed population and earnings from the household survey, respectively.

Since the cut year of the MAMS model is 2006, the total available samplehas 10 observations.

http://find/http://goback/

7/31/2019 Pres BayMC2

6/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Some problems of the frequentist approach

1 Some developing countries have only a short sampleof inaccurate data to estimate (4)3.

2 The usual estimation problems of endogeneity, omittedvariable bias, nonstationarity, serial correlation cannot be

easily corrected with a small sample size.3 If the estimated value of1 is not the expected, it is a

usual frequentist practice to perform an informal "search"process across dierent local-DGPs (most of the time

without the correction to the Type I error) until the model"matches" the theory, the estimations in similar countries,or the subjective perception of the modeler/client.

3But frequentist techniques only make sense in large samples.

Furthermore, an error in x leads to a downward biased estimation of, in amagnitude toward zero. See Hausman (2001) for details.

http://find/http://goback/

7/31/2019 Pres BayMC2

7/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Frequentist results

M1 is equation (4), M2 as in Jabbar (2002), M3 as in Tipper(2011). The frequentist results are not Hendry-congruent4.

4See Hendry, David F (1997). On congruent econometric relations: A

comment. Carnegie-Rochester Conference Series on Public Policy. Number47, 1, 163-190

http://find/http://goback/

7/31/2019 Pres BayMC2

8/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Bayesian approach

With Bayesian methods, it is possible to incorporate previousempirical work and economic theory in (3), together with theexpert criteria of professionals in the eld5.Let X = [1n (x1 , ..., xt)0], = [0 1], y = (y1 , ..., yt)0,

y N(X, s2In ),

then, N(0 , B0), s

2 IG(0/2, 0/2).

js2 , y N(, B1),

B1 = [s2X0X + B10 ]

1,

= B1[s2X0y + B10 0].

5In developing countries the knowledge and the experience of experts in

the eld is a valuable source of information that can be exploited toimprove the estimation of elasticities

http://find/http://goback/

7/31/2019 Pres BayMC2

9/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Markov Chain Monte Carlo (MC2) sampler

(a) Let s2(0) be a starting value of s2.(b) At the g-th iteration,

(g) N((g)

, B(g)1 ),

s2(g) IG(1/2, (g)1 /2),

where,

B(g)1 = [s

2(g1)X0X + B10 ]1

,

(g)

= B(g)1 [s

2(g1)X0y + B10 0],

(g)1 = 0 + (y X(g))0(g)).

with a repetition of the Gibbs sampling (b) until g = B + G itis possible to estimate with (assuming quadratic loss),

minE[L(, )] = min

Z 2 (jy)d.

http://find/http://goback/

7/31/2019 Pres BayMC2

10/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Prior elicitation

From equation 3 it is clear that the prior for 0 would be,

0 = 1ln + (1 1)lnA. (5)

The components of the prior variance-covariance matrix B0,

B0 = Var(0) Cov(0 , 1)

Cov(0 , 1) Var(1) ,

can be elicitated given the fact that,

Var(0) : = Var(1ln + (1 1)lnA)

= [(ln)

2

+ (lnA

)

2

+ 2ln

( +A

)]Var

(1) (6)and,

Cov(0 , 1) = E(01) E(0)E(1)

Thus, 0, 1 and B0 can be fully elicitated assigning values to

the hyperparametes A and , without a previous frequentistestimation

http://find/http://goback/

7/31/2019 Pres BayMC2

11/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Prior elicitation for the Bolivian case

http://find/http://goback/

7/31/2019 Pres BayMC2

12/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Bayesian results

http://find/http://goback/

7/31/2019 Pres BayMC2

13/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

Conclusion

The estimation of in developed countries is close to oneor even larger than one

The estimated value of in the study is congruent withthe empirical fact that in developing countries theestimation of tends to be lower

The Bayesian approach seems an interesting alternative forthe estimation of the capital-labor substitution elasticity indeveloping countries, as with this technique it is possibleto compensate data limitations with a rigorous

inclusion of the experience of professionals in the eld,theoretical concepts or previous empirical work,through prior elicitation.

The combination of the prior criteria with data evidenceproduces an estimator (a Bayesian estimator) that

exploits both sources of information

http://find/http://goback/

7/31/2019 Pres BayMC2

14/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

References

* Arrow, K.J. ,H.B.Chenery, B.S.Minhas, R.M. Solow (1961). Capital-Labor Substitution and Economic

Eciency. The Review of Economics and Statistics, 43, 3, pp.225-250

* Cicowiez, Martn (2011). Un Modelo de Equilibrio General Computado para la Evaluacin de Polticas

Econmicas en Argentina: Construccin y Aplicaciones. Tesis de Doctorado. Universidad Nacional de La

Plata. Captulo 4.

* Hausman, Jerry (2001). Mismeasured Variables in Econometric Analysis :Problems from the Right and

Problems from the Left. The Journal of Economic Perspectives, Vol.15, No.4 (Autumn,2001), pp.57-67.

* Hendry, David F. (1997). On congruent econometric relations: A comment. Carnegie-Rochester

Conference Series on Public Policy, 47(1), 163-190.

* Jabbar Saed, Afaf Abdul (2002). Technological Progress and Capital-Labour Substitution in the Jordanian

Industry: 1985-1997.Journal of Economic and Administrative Sciences, Vol.18, No.2.

* Lofgren, Hans, Carolina Diaz-Bonilla (2010). MAMS: An Economy W ide Model for Analysis of MDG

Country Strategies: Technical Documentation. The World Bank (mimeo).

* Tipper, Adam (2011). One For All? The Capital-Labour Substitution Elasticity In New Zealand.Paper

prepared for the 52nd New Zealand Association of Economists conference, Wellington, NewZealand.

http://find/http://goback/

7/31/2019 Pres BayMC2

15/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

10 iterations, no burn-in, 0 = 1 102, 0 = 1 103

http://find/http://goback/

7/31/2019 Pres BayMC2

16/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

100 iterations, no burn-in, 0 = 1 102, 0 = 1 103

http://find/http://goback/

7/31/2019 Pres BayMC2

17/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

1100 iterations, no burn-in, 0 = 1 102, 0 = 1 103

http://find/http://goback/

7/31/2019 Pres BayMC2

18/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

1100 iterations, no burn-in, 0 = 0 = 1 104

http://find/http://goback/

7/31/2019 Pres BayMC2

19/19

Bayesian MC2

estimation

R. Gonzales

MAMSProject andcapital-laborsubstitutionelasticity

Frequentistapproach

Bayesianapproach

Conclusion

1100 iterations, no burn-in, 0 = 0 = 1 104

http://find/http://goback/