Wald Test of Nested Models

Description

waldtest is a generic function for carrying out Wald tests. The default method can be employed for comparing nested (generalized) linear models (see details below).

Usage

 waldtest(object, ...)

 ## Default S3 method:
 waldtest(object, ..., vcov = NULL,
   test = c("F", "Chisq"), name = NULL) 

 ## S3 method for class 'formula':
 waldtest(object, ..., data = list())

 ## S3 method for class 'survreg':
 waldtest(object, ..., test = c("Chisq", "F"))

Arguments

Details

waldtest is intended to be a generic function for comparisons of models via Wald tests. The default method consecutively compares the fitted model object object with the models passed in .... Instead of passing the fitted model objects in ..., several other specifications are possible. For all objects in list(object, ...) the function tries to consecutively compute fitted models using the following updating algorithm:

1. For each two consecutive objects, object1 and object2 say, try to turn object2 into a fitted model that can be compared to (the already fitted model object) object1.
2. If object2 is numeric, the corresponding element of attr(terms(object1), "term.labels") is selected to be omitted.
3. If object2 is a character, the corresponding terms are included into an update formula like . ~ . - term2a - term2b.
4. If object2 is a formula, then compute the fitted model via update(object1, object2).

Consequently, the models in ... can be specified as integers, characters (both for terms that should be eliminated from the previous model), update formulas or fitted model objects. Except for the last case, the existence of an update method is assumed. See also the examples for an illustration.

Subsequently, a Wald test for each two consecutive models is carried out. This is similar to anova (which typically performs likelihood-ratio tests), but with a few differences. If only one fitted model object is specified, it is compared to the trivial model (with only an intercept). The test can be either the finite sample F statistic or the asymptotic Chi-squared statistic (F = Chisq/k if k is the difference in degrees of freedom). The covariance matrix is always estimated on the more general of two subsequent models (and not only in the most general model overall). If vcov is specified, HC and HAC estimators can also be plugged into waldtest.

The default method is already very general and applicable to a broad range of fitted model objects, including lm and glm objects. It can be easily made applicable to other model classes as well by providing suitable methods to the standard generics terms (for deterimining the variables in the model along with their names), update (unless only fitted model objects are passed to waldtest, as mentioned above), residuals (only used for determining the number of observations), df.residual (needed for the F statisic), coef (for extracting the coefficients; needs to be named matching the names in terms), vcov (can be user-supplied; needs to be named matching the names in terms). Furthermore, some means of determining a suitable name for a fitted model object can be specified (by default this is taken to be the result of a call to formula).

The "formula" method fits a lm first and then calls the default method. The "survreg" method just calls the default method, but sets the default test to be Chi-squared.

Value

An object of class "anova" which contains the residual degrees of freedom, the difference in degrees of freedom, Wald statistic (either "F" or "Chisq") and corresponding p value.

See Also

coeftest, anova, linear.hypothesis

Examples

## fit two competing, non-nested models and their encompassing
## model for aggregate consumption, as in Greene (1993),
## Examples 7.11 and 7.12

## load data and compute lags
data(USDistLag)
usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1)))
colnames(usdl) <- c("con", "gnp", "con1", "gnp1")

## C(t) = a0 + a1*Y(t) + a2*C(t-1) + u
fm1 <- lm(con ~ gnp + con1, data = usdl)

## C(t) = b0 + b1*Y(t) + b2*Y(t-1) + v
fm2 <- lm(con ~ gnp + gnp1, data = usdl)

## Encompassing model
fm3 <- lm(con ~ gnp + con1 + gnp1, data = usdl)

## a simple ANOVA for fm3 vs. fm2
waldtest(fm3, fm2)
anova(fm3, fm2)
## as df = 1, the test is equivalent to the corresponding t test in
coeftest(fm3)

## various equivalent specifications of the two models
waldtest(fm3, fm2)
waldtest(fm3, 2)
waldtest(fm3, "con1")
waldtest(fm3, . ~ . - con1)

## comparing more than one model
## (euqivalent to the encompassing test)
waldtest(fm1, fm3, fm2)
encomptest(fm1, fm2)

## using the asymptotic Chisq statistic
waldtest(fm3, fm2, test = "Chisq")
## plugging in a HC estimator
if(require(sandwich)) waldtest(fm3, fm2, vcov = vcovHC)