Page 363 - Special Topic Session (STS) - Volume 3

STS550 Matteo Mogliani
as the number of predictors increases. To deal with this issue, in this work we
adopt an alternative Empirical Bayes approach that relies on stochastic
approximation algorithms to solve maximization problems when the
likelihood function is intractable, by mimicking standard iterative methods
such as the gradient algorithm. This approach is computationally efficient,
because it requires only a single Monte Carlo run. Using a stochastic
approximation to solve the maximization problem, we get an approximate EM
algorithm, where both E- and M-steps are approximately implemented. Hence,
marginal maximum likelihood estimates of the hyper-parameters and draws
from the posterior distribution of the parameters are both obtained using a
single run of the Gibbs sampler.

3. Results
We evaluate the performance of the proposed models through Monte
Carlo experiments. For this purpose, we use a DGP similar to Equation (1) and
involving = {30, 50} predictors sampled at frequency = 3 and = 200 in-
sample observations. The predictors follow all the same stationary AR(1)
process, but only five are relevant in the model. As for the weighting function
Β (c; ϑ), we choose an exponential Almon lag function. We investigate three
alternative weighting schemes that correspond to fast-decaying weights,
slow-decaying weights, and near-flat weights. For ease of analysis we assume
ℎ = 0 . In this specification, the error terms are assumed i.i.d. normally
distributed, but the design matrix is allowed to present moderate to extremely
high correlation structure.
We compute the average mean squared error (MSE), the average variance
(VAR), and the average squared bias (BIAS2) over R Monte Carlo replications.
Further, we evaluate the selection ability of the models by computing the True
Positive Rate (TPR), the False Positive Rate (FPR), and the Matthews correlation
coefficient (MCC). Simulation results point to a number of interesting features.
First, the models perform overall quite similarly in terms of MSE, although the
BMIDAS-AGL-SS seems to perform somewhat better across DGPs by mainly
providing the smallest bias. This leads to highest TPR and lowest FPR for this
model, entailing better classification of the active and inactive sets across
simulations. Second, the MSE increases substantially with the degree of
correlation in the design matrix, but it tends to decrease with more irrelevant
predictors. It follows that the performance of the models in selecting and
estimating the coefficients of the relevant variables holds the same regardless
the increase in the degree of sparsity. This result is confirmed by the TPR, which
is relatively high and hovers around 80-90% for moderate correlation, and it’s
overall stable across the different values of , suggesting that the models can
select the correct sparsity pattern with a high probability even in finite
samples. However, it is worth noting that the TPR drops to 30-50% for very

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