wals {WALS}

Weighted-Average Least Squares for linear regression models

Description

Performs model averaging for linear regression models using the Weighted-Average Least Squares method by Magnus et al. (2010). See also De Luca and Magnus (2011), Kumar and Magnus (2013) and Magnus and De Luca (2016).

Usage

wals(x, ...)

## S3 method for class 'formula'
wals(
  formula,
  data,
  subset = NULL,
  na.action = NULL,
  weights = NULL,
  offset = NULL,
  prior = weibull(),
  model = TRUE,
  keepY = TRUE,
  keepX = FALSE,
  sigma = NULL,
  ...
)

## S3 method for class 'matrix'
wals(
  x,
  x2,
  y,
  subset = NULL,
  na.action = NULL,
  weights = NULL,
  offset = NULL,
  prior = weibull(),
  keepY = TRUE,
  keepX = FALSE,
  sigma = NULL,
  ...
)

## Default S3 method:
wals(x, ...)

Arguments

x

Design matrix of focus regressors. Usually includes a constant (column full of 1s) and can be generated using model.matrix.

...

Arguments for workhorse walsFit.

formula

an object of class "Formula" (or one that can be coerced to that class, e.g. "formula"): a symbolic description of the model to be fitted. The details of model specification are given under ‘Details’.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment which the function is called from.

subset

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

na.action

not implemented yet.

weights

not implemented yet.

offset

not implemented yet.

prior

Object of class "familyPrior". For example weibull or laplace.

model

if TRUE (default), then the model.frame is stored in the return.

keepY

if TRUE (default), then the response is stored in the return.

keepX

if TRUE, then the model matrices are stored in the return. the return.

sigma

if NULL (default), then the variance of the error term is estimated. See walsFit for more details.

x2

Design matrix of auxiliary regressors. Usually does not include a constant column and can also be generated using model.matrix.

y

Response as vector.

Details

R port of MATLAB code wals.m (version 2.0, revision 18 December 2013) by J.R. Magnus and G. De Luca, available from https://www.janmagnus.nl/items/WALS.pdf. Calculates WALS estimates when focus regressors (X1) are present in all submodels and model averaging takes place over the auxiliary regressors (X2).

Formulas typically contain two parts, i.e. they are of the form "y ~ X11 + X12 | X21 + X22", where the variables before "|" are the focus regressors (includes a constant by default) and the ones after "|" are the auxiliary regressors. If only a one-part formula is specified, then all regressors are considered as auxiliary regressors and only a constant is employed as focus regressor, i.e. "y ~ X1 + X2" is equivalent to "y ~ 1 | X1 + X2".

WARNING: Interactions in formula do not work properly yet. It is recommended to manually create the interactions beforehand and then to insert them as 'linear terms' in the formula.

wals.default() raises an error if x is not an object of class "matrix" or a class that extends "matrix". Otherwise it calls wals.matrix(). It is a modified version of glmboost.default from the mboost package version 2.9-8 (2023-09-06) (Hofner et al. 2014).

Value

wals.formula() returns an object of class "wals". This is a list that contains all elements returned from walsFit and additionally

y

If keepY = TRUE, contains the response vector.

x

list. If keepX = TRUE, then it is a list with elements x1 and x2 containing the design matrices of the focus and auxiliary regressors, respectively.

weights

returns the argument weights.

offset

returns the argument offset.

cl

Call of the function.

formula

formula used.

terms

List containing the model terms of the focus and auxiliary regressors separately, as well as for the full model.

levels

List containing the levels of the focus and auxiliary regressors separately, as well as for the full model.

contrasts

List containing the contrasts of the design matrices of focus and auxiliary regressors.

model

If model = TRUE, contains the model frame.

See returns of walsFit for more details.

wals.matrix() returns an object of class "walsMatrix", which inherits from "wals". This is a list that contains all elements returned from walsFit and additionally the response y, the list x with model matrices x1 and x2, the call cl, offset and weights.

wals.default() raises an error if x is not an object of class "matrix" or a class that extends "matrix". Otherwise returns an object of class "walsMatrix". See above for more details.

References

De Luca G, Magnus JR (2011). “Bayesian model averaging and weighted-average least squares: Equivariance, stability, and numerical issues.” The Stata Journal, 11(4), 518–544. doi:10.1177/1536867X1201100402.

Hofner B, Mayr A, Robinzonov N, Schmid M (2014). “Model-based Boosting in R: A Hands-on Tutorial Using the R Package mboost.” Computational Statistics, 29, 3–35.

Kumar K, Magnus JR (2013). “A characterization of Bayesian robustness for a normal location parameter.” Sankhya B, 75(2), 216–237. doi:10.1007/s13571-013-0060-9.

Magnus JR, De Luca G (2016). “Weighted-average least squares (WALS): A survey.” Journal of Economic Surveys, 30(1), 117-148. doi:10.1111/joes.12094.

Magnus JR, Powell O, Prüfer P (2010). “A comparison of two model averaging techniques with an application to growth empirics.” Journal of Econometrics, 154(2), 139-153. doi:10.1016/j.jeconom.2009.07.004.

Examples

## Replicate table on p. 534 of De Luca & Magnus (2011)
fitDM <- wals(gdpgrowth ~ lgdp60 + equipinv + school60 + life60 + popgrowth |
                  law + tropics + avelf + confucian, data = GrowthMPP,
                prior = laplace())
tableDM <- cbind("coef" = coef(fitDM), "se" = sqrt(diag(vcov(fitDM))))
print(round(tableDM, 7))


## Replicate first panel of Table I in Amini & Parmeter (2012)
data("datafls", package = "BMS")

# NOTE: Authors manually scale data, then rescale the resulting coefs and se.
X <- model.matrix(y ~ ., data = datafls)
Xscaled <- apply(X, MARGIN = 2, function(x) x/max(x))
Xscaled <- Xscaled[,-1]
scaleVector <- apply(X, MARGIN = 2, function(x) max(x))
flsScaled <- as.data.frame(cbind(y = datafls$y, Xscaled))

# NOTE: prescale = FALSE, still used old version of WALS in Magnus et al. (2010).
# Not recommended anymore!
fitFLS <- wals(y ~ 1 | ., data = flsScaled, prescale = FALSE, eigenSVD = FALSE,
               prior = laplace())
tableFLS <- cbind('coef' = coef(fitFLS)/scaleVector,
                  'se' = sqrt(diag(vcov(fitFLS)))/scaleVector)
printVars <- c("(Intercept)", "GDP60", "Confucian", "LifeExp", "EquipInv",
               "SubSahara", "Muslim", "RuleofLaw")
print(round(tableFLS[printVars,], 4))


## Replicate third panel of Table I in Amini & Parmeter (2012)
data("SDM", package = "BayesVarSel")

# rescale response
SDM$y <- SDM$y / 100

# NOTE: Authors manually scale data, then rescale the resulting coefs and se.
X <- model.matrix(y ~ ., data = SDM)
Xscaled <- apply(X, MARGIN = 2, function(x) x/max(x))
Xscaled <- Xscaled[,-1]
scaleVector <- apply(X, MARGIN = 2, function(x) max(x))
SDMscaled <- as.data.frame(cbind(y = SDM$y, Xscaled))

# NOTE: prescale = FALSE, still used old version of WALS in Magnus et al. (2010).
# Not recommended anymore!
fitDW <- wals(y ~ 1 | ., data = SDMscaled, prescale = FALSE, eigenSVD = FALSE,
              prior = laplace())
tableDW <- cbind(coef(fitDW)/scaleVector, sqrt(diag(vcov(fitDW)))/scaleVector)
printVars <- c("(Intercept)", "EAST", "P60", "IPRICE1", "GDPCH60L", "TROPICAR")
print(round(tableDW[printVars,], 5))

## Example for wals.matrix()
X <- model.matrix(mpg ~ disp + hp + wt + vs + am + carb, data = mtcars)
X1 <- X[,c("(Intercept)", "disp", "hp", "wt")] # focus
X2 <- X[,c("vs", "am", "carb")] # auxiliary
y <- mtcars$mpg

wals(X1, X2, y, prior = weibull())