In order to expeditiously deal with these challenges, a technique referred to as Bayesian Averaging of Classical Estimates (BACE) is employed. As explained by Sala-i-Martin (1997) and Doppelhofer et al. (2000), rather than assuming there is a single correct model, as in a simplification search, they assign weights corresponding to a “level of confidence” for each model. This then allows them to calculate a weighted OLS average for each of the potential variables. The only prior assumption required is the expected model size, e.g. out of _K_ variables proposed, perhaps only _k_ would be expected to be significant.<sup>41</sup> The main statistics generated by BACE for each regressor include the posterior inclusion probability (PIP), the posterior mean coefficient, and the posterior mean standard deviation. The latter two are calculated unconditionally and conditional on inclusion. The PIP indicates the probability that a regressor drawn from set _K_ should be included in a robust model of size _k_ . The posterior means are the weighted averages of all the coefficients and their standard deviations for all models. The posterior means that are conditional upon inclusion use weights based on the assumption that each regressor _must_ be included in any model of size _k_ . For this purpose, the weights are revised to sum to one only for those models containing the regressor. The ratio of the posterior mean coefficients to the posterior mean standard deviations is analogous to the _t-_ statistic. Doppelhofer _et al._ (2000) caution that this ratio is _not_ the same as a _t-_ statistic but they go on to suggest that it can be used in much the same way. A ratio with an absolute value of 2 or higher indicates that approximately 95% of the probability density associated with the posterior mean for a regression coefficient will exclude zero. In addition, in the language of Doppelhofer _et al._ (2000), a variable is referred to as being “robust” when PIP > _k_ / _K and_ it is well estimated so that the absolute value of the mixed _t_ -ratio is greater than two.

The author’s implementation of BACE will allow testing of up to 19 variables without exceeding the memory and capacity of a standard laptop computer.<sup>42</sup> This is not a problem for the regression equations for _fb_ which requires only 16 variables. The regression equation for aggregate deviations _e_ requires 26 variables while the regression for _r_ requires 28 variables. For these two cases, a stepwise procedure will be employed to eliminate the least significant variables before the remaining 19 variables are tested using the BACE procedure. The variable with the highest probability of being rejected as insignificant is removed. The variable with the next highest _p_ -value, given the removal of the first variable, is also removed. Next both of the removed variables are checked for re-inclusion into the model: any variable whose _p_ -value is lower than 5 percent is added back in to the model. Once the adding back step has been performed, the next variable is removed. This process is repeated until 19 variables remain. In order to cast the net fairly wide, the expected model size for _r_ and _e_ is set at 16 out of 19 variables. The expected model size for _fb_ is set at 14 out of 17 variables.

It should be noted that the procedure above rests on the assumption that the error term in a regression equation of interest is normally distributed. This is a fundamental requirement for hypothesis testing. Therefore, before any variables are eliminated through the procedures above, the distribution of the regression residual is examined using the Jarque-Bera test. If the null hypothesis of a normal distribution is rejected, then dummy variables may be added to obtain a normally distributed error term. The dummy variables will be selected from the largest positive and negative residuals in the error term. The dummy variables can be retained as useful explanatory variables if they can be linked to relevant circumstances.

> 41 Magnus _et al_ (2010) propose an alternative technique that is less computationally demanding.  The author did not have access to software using this methodology.

> 42 Using EVIEWS software.

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