Used to fit the Bayesian normal-conjugate linear model with Zellner's g
prior and mean zero coefficient priors. Provides an object similar to the
lm class.
zlm(formula, data = NULL, subset = NULL, g = "UIP")Returns a list of class zlm that contains at least the
following elements (cf. lm):
a named vector of posterior coefficient expected values
the residuals, that is response minus fitted values
the fitted mean values
the numeric rank of the fitted linear model
the residual degrees of freedom
the matched call
the terms object
used
the model frame used
a named vector of coefficient posterior second moments
the log marginal likelihood of the model
a list detailing information on
the g-prior, cf. output value gprior.info in bms
an object of class "formula" (or one that can be coerced to
that class), such as a data.frame - cf. lm
an optional data.frame (or one that can be coerced
to that class): cf. lm
an optional vector specifying a subset of observations to be used in the fitting process.
specifies the hyperparameter on Zellner's g-prior for the
regression coefficients.
g="UIP" corresponds to \(g=N\), the
number of observations (default); g="BRIC" corresponds to the
benchmark prior suggested by Fernandez, Ley and Steel (2001), i.e
\(g=max(N, K^2)\), where K is the total number of covariates;
g="EBL" estimates a local empirical Bayes g-parameter (as in Liang et
al. (2008));
g="hyper" takes the 'hyper-g' prior distribution (as
in Liang et al., 2008) with the default hyper-parameter \(a=3\); This
hyperparameter can be adjusted (between \(2<a<=4\)) by setting
g="hyper=2.9", for instance.
Alternatively, g="hyper=UIP"
sets the prior expected value of the shrinkage factor equal to that of UIP
(above), g="hyper=BRIC" sets it according to BRIC
Stefan Zeugner
zlm estimates the coefficients of the following model \(y = \alpha
+ X \beta + \epsilon\) where \(\epsilon\) ~ \(N(0,\sigma^2)\) and \(X\)
is the design matrix
The priors on the intercept \(\alpha\) and the
variance \(\sigma\) are improper: \(alpha \propto 1\), \(sigma
\propto \sigma^{-1}\)
Zellner's g affects the prior on coefficients:
\(beta\) ~ \(N(0, \sigma^2 g (X'X)^{-1})\).
Note that the prior mean
of coefficients is set to zero by default and cannot be adjusted. Note
moreover that zlm always includes an intercept.
The representation follows Fernandez, C. E. Ley and M. Steel (2001): Benchmark priors for Bayesian model averaging. Journal of Econometrics 100(2), 381--427
See also http://bms.zeugner.eu for additional help.
The methods summary.zlm and predict.lm
provide additional insights into zlm output.
The function
as.zlm extracts a single out model of a bma object (as
e.g. created throughbms).
Moreover, lm for
the standard OLS object, bms for the application of zlm
in Bayesian model averaging.
Check http://bms.zeugner.eu for additional help.
data(datafls)
#simple example
foo = zlm(datafls)
summary(foo)
#example with formula and subset
foo2 = zlm(y~GDP60+LifeExp, data=datafls, subset=2:70) #basic model, omitting three countries
summary(foo2)
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