zlm {BMS}

Bayesian Linear Model with Zellner's g

Description

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.

Usage

zlm(formula, data = NULL, subset = NULL, g = "UIP")

Arguments

formula

an object of class "formula" (or one that can be coerced to that class), such as a data.frame - cf. lm

data

an optional data.frame (or one that can be coerced to that class): cf. lm

subset

an optional vector specifying a subset of observations to be used in the fitting process.

g

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

Details

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.

Value

Returns a list of class zlm that contains at least the following elements (cf. lm):

coefficients

a named vector of posterior coefficient expected values

residuals

the residuals, that is response minus fitted values

fitted.values

the fitted mean values

rank

the numeric rank of the fitted linear model

df.residual

the residual degrees of freedom

call

the matched call

terms

the terms object used

model

the model frame used

coef2moments

a named vector of coefficient posterior second moments

marg.lik

the log marginal likelihood of the model

gprior.info

a list detailing information on the g-prior, cf. output value gprior.info in bms

Author(s)

Stefan Zeugner

References

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.

See Also

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.

Examples

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)