Estimating coefficient distributions in random coefficient regressions
The model is \(Y_ j=a+(b+j)x_ j+e_ j\), \(j=1,2,\dots\) where \(E(b_ j)=E(e_ j)=0\), the \(b_ 1,e_ 1,b_ 2,e_ 2,\dots\) are independent and \(\{b_ 1,b_ 2,\dots\}\) as well as \(\{e_ 1,e_ 2,\dots\}\) are i.i.d. The authors study the estimation of the distribution of \(b_ j\) and \(e_ j\), respectively, for resampling purposes. At first questions of identifiability and consistent estimability of the distributions are discussed. A variety of methods, among them least squares and ``efficient'' estimation are put forward in order to estimate a finite number of moments of \(b_ j\) and \(e_ j\), respectively. From the estimated moments four methods of estimating the distributions are proposed, among them two methods due to Hausdorff (1923). This leads to a computational algorithm. A numerical example is given; proofs are deferred to the appendix. Two critical remarks: The reference Pollard (1984) is not contained in the bibliography. The authors claim that \(a\) and \(b\) can be estimated ``under very general circumstances'' consistently by least squares. But these ``general circumstances'' include necessary and sufficient conditions on the asymptotic behaviour of the regression matrices.
scientific article; zbMATH DE number 166848