zlm {BMS} R Documentation

## 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

### 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

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

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.

### 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)

[Package BMS version 0.3.5 Index]