Introduction
Bayesian econometrics offers a powerful framework that combines classical statistical methods with prior beliefs, enhancing parameter estimation and inference. When applied to GMM (Generalized Method of Moments), Bayesian GMM becomes a comprehensive tool for addressing complex econometric models. This post covers the theory, benefits, and EViews implementation of Bayesian GMM, complete with references for further reading.
1. A Quick Overview of GMM
GMM is an econometric estimation technique widely used for models where moment conditions can be defined based on the data. For a dataset \(\{y_t, x_t\}_{t=1}^T\), the GMM estimation objective is:
\[E[g(y_t, x_t, \theta)] = 0\]
where \(g(\cdot)\) is a function involving observed data and unknown parameters \(\theta\). The GMM estimator \(\hat{\theta}_{GMM}\) minimizes:
\[\hat{\theta}_{GMM} = \arg \min_\theta \left[g(\theta)' W g(\theta)\right] \]
with \(W\) being the weighting matrix. Hansen (1982) introduced GMM, laying the foundation for this widely applicable method [Hansen, 1982].
2. Bayesian Perspective on GMM
Bayesian GMM incorporates prior knowledge with sample information to update beliefs about model parameters:
\[p(\theta | y, x) \propto p(y | \theta, x) p(\theta) \]
Likelihood Function \(p(y | \theta, x)\): In Bayesian GMM, a pseudo-likelihood is constructed using the GMM objective:
\[p(y | \theta, x) \propto \exp\left(-\frac{1}{2} g(\theta)' W g(\theta)\right)\]
Prior Distribution \(p(\theta)\): Encodes prior beliefs about parameters, which could be informed by expert opinion or previous research.
Posterior Distribution: Combines the likelihood and prior:
\[ p(\theta | y, x) \propto \exp\left(-\frac{1}{2} g(\theta)' W g(\theta)\right) p(\theta)\]
3. Why Use Bayesian GMM?
Advantages of Bayesian GMM
- Incorporation of Prior Information: Enables the use of external knowledge, making it ideal for small sample sizes or specific econometric contexts [Gelman et al., 2013].
- Full Posterior Analysis: Unlike traditional GMM that offers point estimates, Bayesian GMM produces full posterior distributions, allowing for credible intervals and uncertainty analysis [Robert, 2001].
- Flexibility: Adapts to complex models such as hierarchical structures and models with parameter uncertainty [Greenberg, 2012].
- Robust Inference: Useful for models where asymptotic normality assumptions of traditional GMM may not hold.
4. MCMC for Bayesian GMM
Markov Chain Monte Carlo (MCMC) is essential for sampling from the posterior distribution. The Metropolis-Hastings algorithm is often used for Bayesian GMM:
- Start with an initial parameter vector \(\theta^{(0)}\).
- Propose a new parameter \(\theta'\) from a proposal distribution.
- Calculate the acceptance ratio: \[\alpha = \min\left(1, \frac{p(\theta' | y, x)}{p(\theta^{(i)} | y, x)} \cdot \frac{q(\theta^{(i)} | \theta')}{q(\theta' | \theta^{(i)})}\right)\]
- Accept \(\theta'\) with probability \(\alpha\); otherwise, retain \(\theta^{(i)}\) [Chib & Greenberg, 1995].
5. Practical Implementation in EViews
Step-by-Step EViews Code for Bayesian GMM
' Step 1: Load Data
' Load your dataset into EViews
series y = _exch
series x1 = _infl
series x2 = _opr
' Step 2: Set Up Moment Conditions
' Create moment conditions based on the residuals
equation reseq.ls y c x1 x2
reseq.makeresid res
' Define moment conditions (e.g., using instruments)
series m1 = res
series m2 = res * x1(-1)
series m3 = res * x2(-1)
' Add more moment conditions if needed
' Step 3: Calculate the GMM Objective Function
' Create a function to calculate the GMM objective function
scalar obs = @obs(y)
' Step 4: Initialize Weighting Matrix
matrix(3, 1) moments = 0
matrix(obs, 3) sample_moments = 0
' Calculate initial sample moments for weighting matrix
for !t = 1 to obs
sample_moments(!t, 1) = m1(!t)
sample_moments(!t, 2) = m2(!t)
sample_moments(!t, 3) = m3(!t)
next
matrix covariance_matrix = @cov(sample_moments)
weight = @inverse(covariance_matrix)
' Step 5: Compute Initial Posterior Density
vector initial_coefs = cc
for !t = 1 to obs
moments(1, 1) = moments(1, 1) + m1(!t)
moments(2, 1) = moments(2, 1) + m2(!t)
moments(3, 1) = moments(3, 1) + m3(!t)
next
moments = moments / obs
scalar gmm_obj_value = @t(moments) * weight * moments
' Define initial priors
vector(3) priors
priors(1) = @dnorm((cc(1)-3.0)/0.5)*(1/0.5)
priors(2) = @dnorm((cc(2)-0.1)/0.5)*(1/0.5)
priors(3) = @dnorm((cc(3)+0.2)/0.5)*(1/0.5)
' Compute initial posterior
scalar pseudo_logl = -0.5 * gmm_obj_value
scalar posterior = exp(pseudo_logl) * @prod(priors)
'scalar posterior = @recode(posterior=na,0.00001,posterior)
' Step 6: Implement MCMC Using Metropolis-Hastings
' Initialize the parameter vector for MCMC
vector initial_coefs = cc
' Run MCMC for 10,000 iterations
!nburn_in=50000
!niterations = 100000
matrix(!niterations-!nburn_in,3) iteration_results=na
for !iteration = 1 to !niterations
' Propose new parameter values using a random walk
vector(3) proposed_coefs
for !i = 1 to 3
proposed_coefs(!i) = initial_coefs(!i) + 0.1*@nrnd ' Adjust the proposal standard deviation as needed
next
matrix pre_initial_coefs=initial_coefs
' Temporarily update coefficients
cc(1) = proposed_coefs(1)
cc(2) = proposed_coefs(2)
cc(3) = proposed_coefs(3)
' Recalculate the moment conditions and objective function with proposed coefficients
for !t = 1 to obs
if !t = 1 then
m1(!t) = y(!t) - cc(1) - cc(2) * x1(!t) - cc(3) * x2(!t)
m2(!t) = m1(!t)
m3(!t) = m1(!t)
else
m1(!t) = y(!t) - cc(1) - cc(2) * x1(!t) - cc(3) * x2(!t)
m2(!t) = m1(!t) * x1(!t-1)
m3(!t) = m1(!t) * x2(!t-1)
endif
next
moments(1, 1) = @mean(m1)
moments(2, 1) = @mean(m2)
moments(3, 1) = @mean(m3)
scalar gmm_obj_new = @t(moments) * weight * moments
scalar pseudo_logl_new = -0.5 * gmm_obj_new
' Update the weighting matrix at specific intervals (e.g., every 100 iterations)
if @mod(!iteration, 5) = 0 then
for !t = 1 to obs
sample_moments(!t, 1) = m1(!t)
sample_moments(!t, 2) = m2(!t)
sample_moments(!t, 3) = m3(!t)
next
covariance_matrix = @cov(sample_moments)
weight = @inverse(covariance_matrix)
endif
' Calculate priors for proposed coefficients
vector(3) new_priors
new_priors(1) = @dnorm((cc(1)-3.0)/0.5)*(1/0.5) 'c1 ~ N(3.0,0.5^2)
new_priors(2) = @dnorm((cc(2)-0.1)/0.5)*(1/0.5) 'c2 ~ N(-0.1,0.5^2)
new_priors(3) = @dnorm((cc(3)+0.2)/0.5)*(1/0.5) 'c3 ~ N(0.2,0.5^2)
scalar posterior_new = exp(pseudo_logl_new) * @prod(new_priors)
' Calculate acceptance probability
scalar alpha = @recode(posterior_new / posterior<1,posterior_new / posterior,1)
' Accept or reject the proposal
if @rnd < alpha then
posterior = posterior_new
initial_coefs = proposed_coefs ' Update the accepted coefficients
else
' Revert to old coefficients if rejected
initial_coefs = pre_initial_coefs
endif
' Save the iteration results
if !iteration>!nburn_in then
for !i = 1 to 3
iteration_results(!iteration-!nburn_in, !i) = initial_coefs(!i)
next
endif
next
' Step 7: Save the iteration results to a file or inspect them in EViews
' You can view or expo rt `iteration_results` as needed
Toy Model
We estimate the following model \[y_t=\alpha+\beta x_{1,t} +\gamma x_{2,t} +\epsilon_t\] and use constant, \(x_{1,t-1}\) and \(x_{2,t-1}\) as the instruments. Figure 1 presents the simulated estimates. The estimated densities for intercept, \(\beta\) and \(\gamma\) are displayed in Figures 2-4 respectively. Each of the densities is based on 50,000 samples after accounting for 50,000 burn-ins.
6. Use Cases and Practical Applications
Bayesian GMM is well-suited for:
- Small Sample Analysis: Useful when traditional GMM may not provide reliable estimates [Gelman et al., 2013].
- Policy Evaluation: Incorporates prior beliefs, offering more informed policy insights [Sims & Zha, 1998].
- Complex Econometric Models: Handles models with parameter uncertainty or hierarchical structures efficiently [Greenberg, 2012].
7. Conclusion
Bayesian GMM enriches traditional GMM by incorporating prior information and providing a full posterior distribution. This approach allows for robust inference, especially in cases where classical assumptions do not hold. With EViews' built-in GMM estimation and MCMC routines, implementing Bayesian GMM becomes accessible and efficient, providing researchers with a powerful tool for econometric analysis.
References
- Hansen, L. P. (1982). Large Sample Properties of Generalized Method of Moments Estimators. Econometrica, 50(4), 1029–1054.
- Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). Chapman & Hall/CRC.
- Chib, S., & Greenberg, E. (1995). Understanding the Metropolis-Hastings Algorithm. The American Statistician, 49(4), 327–335.
- Greenberg, E. (2012). Introduction to Bayesian Econometrics (2nd ed.). Cambridge University Press.
- Robert, C. P. (2001). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation (2nd ed.). Springer-Verlag.
- Sims, C. A., & Zha, T. (1998). Bayesian Methods for Dynamic Multivariate Models. International Economic Review, 39(4), 949–968.
© 2021 Olayeni Olaolu Richard. Originally published 11/18/2024
