dynRYfit {sae2}

Internal fitting function for Dynamic and Rao-Yu models

Description

Function designed to be called by either eblupDyn or eblupRY to produce EBLUP small area estimates of the dynamic or Rao-Yu time series models through either ML or REML estimation of the variance components. For completeness, the function is documented here, but users are encouraged to base applications on calls to the more convenient eblupDyn or eblupRY.

Usage

dynRYfit(y,  X,  M,  TI,  NV=1,  vcov_e,  maxiter=100, 
         iter.tol=.1e-5,  ncolx=NULL,  sig2_u=1,  sig2_v=1, 
         rho=.8,  rho_u =.4,  delta=NULL,  rho.fixed=NULL, 
         y.include=NULL, ids=NULL,  contrast.matrix=NULL,  
         baby.steps=TRUE,  dampening=NULL,  iter.history=NULL,  
         sig2.min.factor=.0001,  max.rho_u=.98,  max.rho=NULL, 
         tol=.Machine$double.eps, y.rescale=NULL, 
         llike.only=FALSE,  method=c("REML", "ML"), 
         model=c("dyn", "RY"))

Arguments

y

For a univariate model, the dependent variable sorted in ascending order by time within domain. For a multivariate model, the dependent variables sorted in ascending order by time within variable within domain.

X

A matrix of independent variables with the same number of rows as the length of y.

M

The total number of domains, equivalent to D in eblupDyn and eblupRY.

TI

The number of time instances (constant for all domains).

NV

The number of dependent variables.

vcov_e

For the univariate model, the sampling covariance matrix for the direct estimates of the M*TI elements of the dependent variable. The covariance matrix should be in the form of a square matrix with M*TI rows and columns. Non-zero covariances between domains are not allowed, so the matrix must have block diagonal form with M blocks, each of which is a square matrix with TI rows and columns. Note that within domain, non-zero covariances are allowed over time.

For the multivariate model, the square covariance matrix for the M*NV*TI elements of the dependent variables. The matrix should be in the form of a square matrix with M*NV*TI rows and columns. Time should vary within variable, which should vary within domain. Non-zero covariances between domains are not allowed, but non-zero covariances may be present across time and between variables.

maxiter

The maximum number of iterations allowed for the Fisher-scoring algorithm, with a default value of 100.

iter.tol

The convergence tolerance limit for the Fisher-scoring algorithm, with a default value of .000001.

ncolx

For a univariate model, the number of columns of X. For a multivariate model, a vector of length NV must be specified giving the number of columns of X used for each dependent variable.

sig2_u

An initial starting value or values for the variance of the random increments.

sig2_v

An initial starting value or values for a domain level random effect. In the Rao-Yu model, the random effect is constant over time, whereas in the dynamic model it is an initial effect subject to dampening over time.

rho

The correlation across time. This correlation is assumed to be the same for the dependent variables in the multivariate model.

rho_u

For NV > 1 only, the (NV*(NV-1))/2 starting values for the correlations between the random effects of the different dependent variables. If a single value is given, it will be used for the (NV*(NV-1))/2 components. The sort order corresponds to a lower triangle of the covariance matrix.

delta

The random effect components in the preferred internal order. Specification of delta will override any specification of sig2_u, sig2_v, rho, or rho_u. In the univariate case, delta should contain sig2_u, sig2_v, and rho. In the bivariate case, delta should be of length 6 with sig2_u and sig2_v, each of length 2, rho, and rho_u. Similarly, for 3 dependent variables, the length of delta is 10, with the last 3 elements rho_u in lower triangular form.

rho.fixed

If TRUE, the value of rho imbedded in delta, if specified, or else given by rho will remain fixed during the iterations. Among other features, this allows the likelihood function for trial values of rho to be computed at the maximum over the other random effect parameters.

y.include

If specified, vector of length M to indicate which domains to include in the estimation, with 1 signalling inclusion and 0 exclusion. Estimates for the excluded domains will be based on the fixed effects model only.

ids

A data frame with M rows giving ids for each of the domains. The data frame is copied into the returned object.

contrast.matrix

A matrix of coefficients of contrasts. The matrix must have TI*NV rows, but it can contain an arbitrary number of columns. Within each domain, the coefficients are applied to the TI*NV EBLUP estimates.

baby.steps

Unless specified as FALSE, the first five iterations of the Fisher scoring algorithm are dampened by factors of c(.0625, .125, .25, .5, .75). These heuristically derived factors appear to lessen drastic overshooting of the true maximum in the initial iterations.

dampening

A factor used to dampen the changes in the random effect parameters. Unlike baby.steps, its effect persists during all of the iterations until convergence. Note that the "factory setting" of this parameter is 1 for the dynamic model but .9 for the Rao-Yu model.

iter.history

If TRUE, key values are saved during each iteration and included as additional items, described below, in the returned list: delta.hist, llikelihood.hist, adj.hist, s.hist, ix.hist, adj.factor.hist, and warning.hist.

If iter.history is not specified, these items will be returned only if the calculations are begun but not successfully completed.

sig2.min.factor

A factor to multiply the minimum direct variance to use as a minimum value for any of the variance components. The iterations will be constrained not to go below the resulting bounds.

max.rho_u

A maximum allowed value for the estimated rho_u. The default value is .98.

max.rho

A maximum allowed value for rho. By default, eblupRY sets this value to .98.

tol

A tolerance value used by matrix routines to prevent numerical instability. The value may be set to a lower value to encourage covergence, but appropriate caution should be applied.

y.rescale

In the univariate case, a scaler multiplier for all of the y values. If the y values are either too small or too large, the information matrix may become unstable. Setting this value to 1 has no effect; setting it to 10 or 100 rescales very small y values to a more appropriate range. Similarly, positive values less than 1 may be used to rescale large y values. The effect of rescaling is removed before normal return from the function, within the limits of normal precision.

In the multivariate case, y.rescale may be a vector of length NV to rescale each component separately.

llike.only

Compute the log-likelihood (ML) or restricted log-likelihood (REML) without further iteration, typically from values specified by delta.

method

Use restricted maximum likelihood ("REML") or maximum likelihood ("ML").

model

Dynamic ("dyn") or Rao-Yu ("RY").

Details

Many of arguments can be used to control the iterations if the defaults lead to convergence difficulties.

llike.only in combination with delta permits a point-by-point investigation of the likelihood surface.

The primary functions eblupDyn and eblupRY determine X, NV, colx, and model, but the remaining parameters can be passed to dynRYfit through eblupDyn or eblupRY.

Value

eblup

In the univariate case, a vector of length M*TI with the eblup estimates in the same sort order as y. In the multivariate case, a matrix of M*TI rows and NV columns.

fit

A list summarizing the fit of the model with the following:

• model: form of the model: TI - Dynamic or RaoYu; REML or ML.

• covergence: a logical value indicating whether the convergence criterion was met.

parm

A labelled vector with the estimated variance components, correlations, and number of iterations.

coef

A labelled vector of coefficients for the fixed effects of the model or models.

ids

A data frame with D rows and one or more columns of numeric or character domain identifiers.

delta

An ordered vector of the variance components (see above). It may be used as starting values for additional iterations.

eblup.mse

MSE estimates for eblup.

eblup.g1

The g1 term of the MSE estimate.

eblup.g2

The g2 term of the MSE estimate.

eblup.g3

The g3 term of the MSE estimate.

est.fixed

Estimates based on fixed effects only.

est.fixed.var

The variance-covariance matrix for the estimates in coef.

eblup.wt1

Weights given to the direct estimate in forming eblup.

eblup.wt2

Weights given to the direct estimate, including effects through estimating the fixed effect coefficients.

contrast.est

Estimates requested by the specified contrasts.

contrast.mse

MSE estimates for contrast.est.

contrast.g1

The g1 term in the estimation of contrast.mse.

contrast.g2

The g2 term in the estimation of contrast.mse.

contrast.g3

The g3 term in the estimation of contrast.mse.

contrast.fixed.est

Contrast estimates based on the fixed effect model.

contrast.fixed.var

Variance estimates for the fixed effect model.

contrast.wt1

Weight wt1 given to the direct estimate in estimating the contrasts.

contrast.wt2

Weight wt2 in estimating the contrasts.

inf.mat

Information matrix for the components of delta.

var.coef

Variance covariance matrix for coef.

delta.hist

Values of delta at each iteration.

llikelihood.hist

Values of the log-likelihood (ML) or restricted log-likelihood (REML) at each iteration.

adj.hist

Number of cycles in the internal loop to determine adj.factor within each iteration.

inf.mat.hist

Values of inf.mat at each iteration.

s.hist

Vector to be multiplied by the inverse information matrix to determine the change in the parameters.

ix.hist

List of parameters eligible for change at each iteration. Parameters with estimated changes out of bounds will not be eligible.

adj.factor.hist

Adjustment to the vector change in the parameters at each iteration.

warning.hist

A 4-row matrix of warnings at each iteration, where warning 1 is set to 1 for the iteration if the algorithm has not found an increase in the restricted log likelihood or log likelihood, warning 2 is set to 1 if the maximum number of iterations is reached, warning 3 is set to 1 if the estimated variance-covariance matrix becomes singular, and warning 4 is set to 1 if the coefficients of the fixed effects cannot be estimated.

Author(s)

Robert E. Fay, Mamadou Diallo