占位，持续更新。。所有内容都在2楼。

研究动机

经济增长的影响因素非常多，这使得计量模型具有很多形式。哪种对？作者考虑了模型的不确定性，使用Bayesian Averaging of Classical Estimates选择变量，计算估计系数的加权平均值。

主要结论

BayesianAveraging of Classical Estimates (BACE) constructs estimates by averaging OLScoefficients across models. The weights given to individual regressions have aBayesian justification similar to the Schwarz model selection criterion.

Of 67explanatory variables we find 18 to be significantly and robustly partiallycorrelated with long-term growth and another three variables to be marginallyrelated. The strongest evidence is for the relative price of investment,primary school of enrollment, and the initial level of real GDP per capita.

BACE的优势

Our BACE approachhas several important advantages.

(1) incontrast to a standard Bayesian approach that requires the specification of aprior distribution for all parameters, BACE requires the specification of onlyone prior hyper-parameter: the expected model size.

(2) theinterpretation of the estimates is straightforward for economists not trainedin Bayesian inference: the weights applied to different models are proportionalto the logarithm of the likelihood function corrected for degrees of freedom (analogousto the Schwarz model selection criterion).

(3) use onlyOLS

(4) weconsider models of all sizes and no variables are held fixed and thereforeuntested.

(5) wecalculate the entire distribution of coefficients across models and do notfocus solely on the bounds of the distribution.

BACE的本质

由于有很多变量，不知道选哪个。那就估计各种可能的形式。变量系数是各个模型系数的加权平均值。

权重是什么？模型的事后概率。事后概率其实是选择模型好坏的一个标准。

什么是好的模型：用最少的变量把整个模型拟合最好（达到最大R^2）的就是最好的。所以，作者在计算模型的概率时，考虑了模型的SSE，并进行了自由度的调整。

再说一遍BACE本质：估计系数是各个模型估计出来的值的加权平均值，权重是模型的事后概率。

其实，计量模型都是那些事，围绕误差做文章。可见我的一个帖子https://bbs.pinggu.org/thread-2854602-1-1.html

模型的事前概率

一句话概括：模型的事前概率等于所有模型平均的变量个数与变量总个数之比。

运算负担

随着变量的增加，需要估计的模型呈指数增长，给运算带来很大的负担。这种情况下怎么设置模型的事前概率呢？

Severalstochastic algorithms have been proposed for dealing with this issue, includingthe Markov-Chain Monte-Carlo Model Composition technique (MC3),stochastic search variable selection (SSVS), and the Gibb’s sampler basedmethod.

We selectmodels by randomly including each variable with independent sampling probabilities.So long as the sampling probabilities are strictly greater than zero andstrictly less than one,

As analternative to model averaging, Leamer (1978) suggests orthogonalizing theexplanatory variables and estimating the posterior means of the effects of theK+1 principle components.

怎么评价模型的估计系数？

方法1

Posteriorinclusion probability is the sum of the posterior probabilities of all of theregressions including that variable.

We can dividethe variables according to whether seeing the data causes us to increase ordecrease our inclusion probability relative to the prior probability. Since ourexpected model size equals 7, the prior inclusion probability is 7/67 = 0.014.There are 18 variables for which the posterior inclusion probability increase.For these variables, our belief that they belong in the regression isstrengthened once we see the data and we call these variables “significant”.

方法2

For eachindividual regression the posterior density is equal to the classical samplingdistribution of the coefficient. In classical terms, a coefficient would be 5percent significant in a two-sided test if 97.5% percent of the probability inthe sampling distribution were on the same side of zero as the coefficientestimate.

评议
1. 模型的多样化，可以用无穷尽来形容。BACE只是将可能的OLS形式做平均，考察的都是线性的函数。
2. 事前概率的选择。只设定模型的事前概率，它是平均的模型变量个数与变量总个数之比。
3. 事后概率的计算。事前概率基础上进行的SSE和自由度的调整。即使用最少变量实现模型最优拟合的模型，其事后概率也大。
4. 估计参数：是各个模型估计值的加权平均。这里的权重是模型的事后概率。
5. BACE and diffuse prior OLS disagree about the relative importance of several important variables.
6. BACE对于理解计量模型估计系数的随机性，有帮助。
7. 稳健性检验：调整事前概率，再估计，得到的估计系数有些波动。

欢迎讨论，指点。