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 coefficent 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 coefficents of the following model y = α + X β + ε where ε ~ N(0,σ^2) and X is the design matrix
The priors on the intercept α and the variance σ are improper: alpha \propto 1, sigma \propto σ^{-1}
Zellner's g affects the prior on coefficients: beta ~ N(0, σ^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 through`bms`).
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 0.3.1 Index]