AK: AK Estimator (recursive version)
In DanielBonnery/CompositeRegressionEstimation: X

Description Usage Arguments Details References See Also Examples

Consider a sequence of monthly samples (S_m)_{m\in\{1,…,M\}}. In the CPS, a sample S_m is the union of 8 rotation groups: S_m=S_{m,1}\cup S_{m,2}\cup S_{m,3}\cup S_{m,4}\cup S_{m,5}\cup S_{m,6}\cup S_{m,7}\cup S_{m,8}, where two consecutive samples are always such that S_{m,2}=S_{m-1,1}, S_{m,3}=S_{m-1,2}, S_{m,4}=S_{m-1,3}, S_{m,6}=S_{m-1,5}, S_{m,7}=S_{m-1,6}, S_{m,8}=S_{m-1,7}, and one year appart samples are always such that S_{m,5}=S_{m-12,1}, S_{m,6}=S_{m-12,2}, S_{m,7}=S_{m-12,3}, S_{m,8}=S_{m-12,4}.

The subsamples S_{m,g} are called rotation groups, and rotation patterns different than the CPS rotation pattern are possible.

For each individual k of the sample m, one observes the employment status Y_{k,m} (A binary variable) of individual k at time m, and the survey weight w_{k,m}, as well as its "rotation group".

The AK composite estimator is defined in “CPS Technical Paper (2006), [section 10-11]”:

For m=1, \hat{t}_{Y_{.,1}}=∑_{k\in S_1}w_{k,m}Y_{k,m}.

For m≥q 2,

\hat{t}_{Y_{.,m}}= (1-K) \times ≤ft(∑_{k\in S_{m}} w_{k,m} Y_{k,m}\right)~+~K~\times~(\hat{t}_{Y_{.,m-1}} + Δ_m)~+~ A~\times\hat{β}_m

where

Δ_m=η_0\times∑_{k\in S_m\cap S_{m-1}}(w_{k,m} Y_{k,m}-w_{k,m-1} Y_{k,m-1})

and

\hat{β}_m=≤ft(∑_{k\notin S_m\cap S_{m-1}}w_{k,m} Y_{k,m}\right)~-~η_1~\times~≤ft(∑_{k\in S_m\cap S_{m-1}}w_{k,m} Y_{k,m}\right)

For the CPS, η_0 is the ratio between the number of rotation groups in the sample and the number of overlaping rotation groups between two month, which is a constant η_0=4/3; η_1 is the ratio between the number of non overlaping rotation groups the number of overlaping rotation groups between two month, which is a constant of 1/3.

In the case of the CPS, the rotation group one sample unit belongs to in a particular month is a function of the number of times it has been selected before, including this month, and so the rotation group of an individual in a particular month is called the "month in sample" variable.

For the CPS, in month m the overlap S_{m-1}\cap S_{m} correspond to the individuals in the sample S_m with a value of month in sample equal to 2,3,4, 6,7 or 8. The overlap S_{m-1}\cap S_{m} correspond to the individuals in the sample S_m with a value of month in sample equal to 2,3,4, 6,7 or 8. as well as individuals in the sample S_{m-1} with a value of month in sample equal to 1,2,3, 5,6 or 7. When parametrising the function, the choice would be group_1=c(1:3,5:7) and group0=c(2:4,6:8).

Computing the estimators recursively is not very efficient. At the end, we get a linear combinaison of month in sample estimates The functions AK3, and WSrg computes the linear combination directly and more efficiently.

AK(
  list.tables,
  w,
  list.y,
  id = NULL,
  groupvar = NULL,
  groups_1 = NULL,
  groups0 = NULL,
  A = 0,
  K = 0,
  dft0.y = NULL,
  eta0 = 0,
  eta1 = 0
)

`list.tables`	a list of tables
`w`	a character string: name of the weights variable (should be the same in all tables)
`list.y`	a vector of variable names
`id`	a character string: name of the identifier variable (should be the same in all tables)
`groupvar`	a character string: name of the rotation group variable (should be the same in all tables)
`groups_1`	a character string:
`groups0`	if `groupvar` is not null, a vector of possible values for L[[groupvar]]
`eta0`	a numeric value
`eta1`	a numeric value

the function is based on the more general function CompositeRegressionEstimation::composite

“CPS Technical Paper (2006). Design and Methodology of the Current Population Survey. Technical Report 66, U.S. Census Bureau.", “Gurney, M. and Daly, J. F. (1965). A multivariate approach to estimation in periodic sample surveys. In Proceedings of the Social Statistics Section, American Statistical Association, volume 242, page 257."

CompositeRegressionEstimation::composite

library(dataCPS)
data(cps200501,cps200502,cps200503,cps200504,
     cps200505,package="dataCPS") 
list.tables<-list(cps200501,cps200502,cps200503,cps200504,
                  cps200505)
w="pwsswgt";id=c("hrhhid","pulineno");groupvar=NULL;list.y="pemlr";dft0.y=NULL;
groups_1=NULL;
groups0=NULL;
Coef=c(alpha_1=0,alpha0=1,beta_1=0,beta0=0,gamma_1=0)
AK(list.tables,w=w,list.y="pemlr",id=id,groupvar=groupvar)

## With the default choice of parameters for A,K,eta0,eta1
## the composite is  equal to the direct estimator: we check
WS(list.tables = list.tables,weight = w,list.y = list.y)  

## Example of use of a group variable. 
w="pwsswgt";id=NULL;groupvar="hrmis";list.y="pemlr";dft0.y=NULL;
groups_1=c(1:3,5:7);
groups0=c(2:4,6:8);
Coef=c(alpha0=1,alpha_1=0,beta_1=0,beta0=0,gamma_1=0)
AK(list.tables,w=w,list.y="pemlr",id=id,groupvar="hrmis")