1. Econometrics
Chengyuan Yin
School of Mathematics

2. 24. Simulation Based Estimation
Econometrics
24. Simulation Based Estimation

3. Settings Conditional and unconditional log likelihoods
Likelihood function to be maximized contains unobservables
Integration techniques
Bayesian estimation
Prior times likelihood is intractible
How to obtain posterior means, which are open form integrals
The problem in both cases is “…how to do the integration?”

4. A Conditional Log Likelihood

5. Application - Innovation
Sample = 1,270 German Manufacturing Firms
Panel, 5 years,
Response: Process or product innovation in the survey year? (yes or no)
Inputs:
Imports of products in the industry
Pressure from foreign direct investment
Other covariates
Model: Probit with common firm effects
(Irene Bertschuk, doctoral thesis, Journal of Econometrics, 1998)

6. Likelihood Function Joint conditional (on ui) density for obs. i.
Unconditional likelihood for observation i
How do we do the integration to get rid of the heterogeneity in the conditional likelihood?

7. Obtaining the Unconditional Likelihood
The Butler and Moffitt (1982) method is used by most current software
Quadrature
Works for normally distributed heterogeneity

8. Hermite Quadrature

9. Example: 8 Point Quadrature
Nodes for 8 point Hermite Quadrature
Use both signs, + and - ,
Weights for 8 point Hermite Quadrature , , ,

10. Butler and Moffitt’s Approach Random Effects Log Likelihood Function

11. Monte Carlo Integration

12. The Simulated Log Likelihood

13. Quasi-Monte Carlo Integration Based on Halton Sequences
For example, using base p=5, the integer r=37 has b0 = 2, b1 = 2, and b3 = 1. Then H37(5) = 2  5-3 =

14. Panel Data Estimation A Random Effects Probit Model

15. Log Likelihood

16. ( / ( ) = 0.578)

17. Quadrature vs. Simulation
Computationally, comparably difficult
Numerically, essentially the same answer. MSL is consistent in R
Advantages of simulation
Can integrate over any distribution, not just normal
Can integrate over multiple random variables. Quadrature is largely unable to do this.
Models based on simulation are being extended in many directions.
Simulation based estimator allows estimation of conditional means  essentially the same as Bayesian posterior means

18. A Random Parameters Model

19. Estimates of a Random Parameters Model

20. RPM

21. RPM

23. Movie Model

24. Parameter Heterogeneity

25. Bayesian Estimators “Random Parameters”
Models of Individual Heterogeneity
Random Effects: Consumer Brand Choice
Fixed Effects: Hospital Costs

26. Bayesian Estimation
Specification of conditional likelihood: f(data | parameters)
Specification of priors: g(parameters)
Posterior density of parameters:
Posterior mean = E[parameters|data]

27. The Marginal Density for the Data is Irrelevant

28. Computing Bayesian Estimators
First generation: Do the integration (math)
Contemporary - Simulation:
(1) Deduce the posterior
(2) Draw random samples of draws from the posterior and compute the sample means and variances of the samples. (Relies on the law of large numbers.)

29. Modeling Issues
As N , the likelihood dominates and the prior disappears  Bayesian and Classical MLE converge. (Needs the mode of the posterior to converge to the mean.)
Priors
Diffuse  large variances imply little prior information. (NONINFORMATIVE)
INFORMATIVE priors – finite variances that appear in the posterior. “Taints” any final results.

30. A Random Effects Approach
Allenby and Rossi, “Marketing Models of Consumer Heterogeneity”
Discrete Choice Model – Brand Choice
“Hierarchical Bayes”
Multinomial Probit
Panel Data: Purchases of 4 brands of Ketchup

31. Structure

32. Bayesian Priors

33. Bayesian Estimator Joint Posterior=
Integral does not exist in closed form.
Estimate by random samples from the joint posterior.
Full joint posterior is not known, so not possible to sample from the joint posterior.

34. Gibbs Sampling: Target: Sample from f(x1, x2) = joint distribution
Joint distribution is unknown or it is not possible to sample from the joint distribution.
Assumed: f(x1|x2) and f(x2|x1) both known and samples can be drawn from both.
Gibbs sampling: Obtain one draw from x1,x2 by many cycles between x1|x2 and x2|x1.
Start x1,0 anywhere in the right range.
Draw x2,0 from x2|x1,0.
Return to x1,1 from x1|x2,0 and so on.
Several thousand cycles produces a draw
Repeat several thousand times to produce a sample
Average the draws to estimate the marginal means.

35. Gibbs Cycles for the MNP Model
Samples from the marginal posteriors

36. Results Individual parameter vectors and disturbance variances
Individual estimates of choice probabilities
The same as the “random parameters model” with slightly different weights.
Allenby and Rossi call the classical method an “approximate Bayesian” approach.
(Greene calls the Bayesian estimator an “approximate random parameters model”)
Who’s right?
Bayesian layers on implausible priors and calls the results “exact.”
Classical is strongly parametric.
Neither is right – Both are right.

37. Comparison of Maximum Simulated Likelihood and Hierarchical Bayes
Ken Train: “A Comparison of Hierarchical Bayes and Maximum Simulated Likelihood for Mixed Logit”
Mixed Logit

38. Stochastic Structure – Conditional Likelihood
Note individual specific parameter vector, i

39. Classical Approach

40. Bayesian Approach – Gibbs Sampling and Metropolis-Hastings

41. Gibbs Sampling from Posteriors: b

42. Gibbs Sampling from Posteriors: Ω

43. Gibbs Sampling from Posteriors: i

44. Metropolis – Hastings Method

45. Metropolis Hastings: A Draw of i

46. Application: Energy Suppliers
N=361 individuals, 2 to 12 hypothetical suppliers
X=[(1) fixed rates, (2) contract length, (3) local (0,1),(4) well known company (0,1), (5) offer TOD rates (0,1), (6) offer seasonal rates]

47. Estimates: Mean of Individual i
MSL Estimate
Bayes Posterior Mean
Price
-1.04 (0.396)
-1.04 (0.0374)
Contract
(0.0240)
(0.0224)
Local
2.40 (0.127)
2.41 (0.140)
Well Known
1.74 (0.0927)
1.71 (0.100)
TOD
-9.94 (0.337)
-10.0 (0.315)
Seasonal
-10.2 (0.333)
-10.2 (0.310)

48. Reconciliation: A Theorem (Bernstein-Von Mises)
The posterior distribution converges to normal with covariance matrix equal to 1/N times the information matrix (same as classical MLE). (The distribution that is converging is the posterior, not the sampling distribution of the estimator of the posterior mean.)
The posterior mean (empirical) converges to the mode of the likelihood function. Same as the MLE. A proper prior disappears asymptotically.
Asymptotic sampling distribution of the posterior mean is the same as that of the MLE.