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Bayesian Estimation
In Bayesian statistics, the parameters to be estimated are treated as random variables associated with a subjective probability distribution that describes the state of knowledge about the parameters. The knowledge either may exist before observing any sample information or might be derived from both prior and sample information. In the former case, the associated probability distribution is a prior distribution. In the latter, that distribution is a posterior distribution. Thus, different from the classical statistics that concentrates on point estimates of a (set of) parameter(s), the objective of Bayesian statistics is usually the achievement of the posterior distribution of a (set of) parameter(s).
Using the Bayesian Theorem, it can be shown that the joint posterior distribution of a set of parameters can be obtained from the combination of sample information and the joint prior distribution of the parameters (see Judge et al., 1988). That is,
EMBED Equation.3 (A3.1)
where ( is the vector of parameters of interest, ( is their variance-covariance matrix, and y is the matrix of sample observations. Expression (A.3.1) states that the posterior joint density function of ( and ( (i.e., EMBED Equation.3 ) is proportional to (() the likelihood function EMBED Equation.3 (which contains all the sample information) times the prior density function EMBED Equation.3 . Intuitively, the prior information about the parameters is modified by the available sample information (through the likelihood function) to obtain the posterior information about the parameters.
Attainment of the posterior distribution requires specification of the prior distribution and the likelihood function. In the present study, the prior distribution must incorporate the restrictions implied by the economic theory. That is, the assumptions required for the adopted economic theory to be true are included as prior beliefs. The prior distribution must then assign probability zero to regions of the parameter space that do not meet the restrictions and positive probability otherwise. Following ODonnell et al. (1999), the following non-informative joint prior distribution was adopted
EMBED Equation.3 (A3.2)
where ( is the subspace of parameter vectors that satisfy the restrictions, N is the number of equations, and I(( ( () is an indicator function that takes the value 1 when a given vector of parameters ( belongs to ( and takes the value 0 otherwise.
The SUR model to be estimated is EMBED Equation.3 , where the N equations have been stacked. To specify the likelihood function, then, a distribution for ( must be assumed. Following Judge et al. (1985), it is assumed that the error vector has a multivariate normal distribution, i.e. EMBED Equation.3 . The likelihood function is then
EMBED Equation.3 (A3.3)
where A is the N x N symmetric matrix with the (i,j)th element equal to aij = (yi-g(Xi,())(yj-g(Xj,()).
Finally, having specified both the prior distribution and the likelihood function, the joint posterior distribution is
EMBED Equation.3 (A3.4)
As in ODonnell et al. (1999), the interest is on the characteristics of the marginal distribution of ( and, then, ( is considered a nuisance parameter that can be integrated out of equation (A3.4). This procedure yields
EMBED Equation.3 (A3.5)
Additional integration of this joint posterior distribution would eventually give the marginal distribution of (. However, this is not analytically possible. The way this is overcome is by adopting numerical methods. In particular, computer-intensive algorithms can be implemented in the estimation of the marginal distribution.
Markov Chain Monte Carlo (MCMC) algorithms, like the Gibbs sampler and the Metropolis-Hastings (MH) algorithm, have recently become very popular in applied economics. They constitute a technique for generating random variables from a marginal distribution without need of analytically calculating the density (Cassella and George, 1992). That is, instead of computing or approximating the posterior distribution EMBED Equation.3 directly, those algorithms allow generating a sample (1,, (m ~ EMBED Equation.3 without knowing EMBED Equation.3 . The characteristics of the marginal density can then be calculated with a large enough sample. For example, the mean of EMBED Equation.3 is calculated using the sample mean (Cassella and George, 1992)
EMBED Equation.3 (A3.6)
Hence, for a large enough sample size m, any population characteristic can be obtained from the generated observations.
The Bayesian results shown in this paper are based on the Metropolis-Hastings algorithm presented in ODonnell et al. (1999). This algorithm consists of the following steps
An arbitrary starting value of the parameter vector (i, i = 0, is specified such that the constraints are satisfied; that is, (0 ( (.
Given (i, a candidate value for (i+1, (C, is generated from a symmetric transition density q((i,(C).
(C is used to evaluate the constraints; if any constraint is violated, then the function (((i,(C) is set equal to zero and the algorithm jumps to step 5).
If the constraints hold, then (((i,(C) is set equal to min[g((C)/ g((i), 1], where EMBED Equation.3 is the kernel of EMBED Equation.3 ; that is, EMBED Equation.3 .
An independent uniform random variable (U) is generated from the interval [0,1].
The next value in the sequence, (i+1, is generated from the rule
EMBED Equation.3
The value of i is set equal to i+1 (i = i+1), and the procedure continues in step 2.
Note that the symmetric transition density q((i,(C) in step 2) must be specified. As in ODonnell et al., the following multivariate normal distribution is adopted
EMBED Equation.3 (A3.7)
where EMBED Equation.3 is the estimated covariance matrix obtained in the SUR estimation, and h is a scalar used to control the size of the step in the iteration (i.e., h gives the rate at which the parameter space B is investigated).
Iteration of this procedure m times gives a sequence of parameter vectors (1,, (m ( B. For some s < m, the following holds: (s+1, , (m ~ EMBED Equation.3 . In words, after a large enough sequence of size s (the burn-in period), the m-s final drawings converge, in the sense that they are drawn from the distribution EMBED Equation.3 . Finally, the estimated ( is simply the sample mean of (s+1, , (m.
For estimation purposes, the starting values, the value of the scalar h, and the sizes of the burn-in period (s) and the whole sample (m) had been established. The value of h was set equal to 0.025 after trying different values. The selection was based on the maximum rate at which candidate vectors were accepted as the next value in the sequence. For many alternative values of h, there was no candidate accepted as next value in the sequence.
The total number of iterations was m = 110,000. The first 10,000 were used for the burn-in period (s). In terms of the starting values, although the SUR estimates do not satisfy the constraints (i.e., the estimated vector b does not belong to B), those estimates were used to draw an initial value (0. However, after trying 100,000 draws, no vector satisfied the restrictions, implying that the SUR estimates are far from the required parameter space B.
An arbitrary vector (0 was then chosen such that these starting values satisfy the constraints. The existence of candidate vectors (C such that (C ( B and (((i,(C) > U was, however, very sensitive to changes in the starting values. That is, similar to what happened with alternative values of h, the case B = {(0} has been the result in many runs that tried different starting values. Additionally, when some restrictions on the set B were relaxed, only 34 out of 100,000 iterations satisfy both (C ( B and (((i,(C) > U, implying that (i+1 = (i in almost all the iterations. Given this available data set (the sample information), it seems that the set B is very restricted and narrow. Consequently, it is difficult to get a reliable posterior distribution.
Finally, convergence of the distribution was checked by taking two sub-samples, s1 and s2, of the last m-s iterations and comparing their means. The sub-sample s1 is composed by the first 10,000 observations, while s2 is composed by the last 50,000 observations. A likelihood ratio test to compare the mean vectors of the two sub-samples was performed. Results indicate that equality of the mean vectors is rejected, meaning that the iterations have not converged. Increasing the size of the burn-in period and the total number of iterations did not change the result of the convergence test. The MH parameter estimates, then, cannot be said to characterize the marginal distribution EMBED Equation.3 .
A non-informative prior distribution is a distribution that does not contain specific information about the parameter. Equation (A3.2), form example, does not specify any exact distribution p((,(). If p((,() is said to be a normal distribution, in contrast, then that would be an informative prior distribution. As a consequence of adopting a non-informative prior, when this prior is combined with the likelihood function in equation (A3.1), the posterior distribution is dominated by the sample information.
Detailed theoretical explanation of the Gibbs sampler and the MH algorithm is provided in Cassella and George (1992), Chib and Greenberg (1996), Gelfand et al. (1990), Gelfand et al. (1992), and Gelfand and Smith (1990). For empirical implementation, see Atkinson and Dorfman (2001), Griffiths et al. (1999), ODonnell et al. (1999), and Terrel (1996).
The burn-in period guarantees two characteristics of the sub-sample of size m-s used to estimate
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