Consider the following subsampling model: y x i mj r ij = + ++ = 0 1 i i ij ( ) 1

Consider the following subsampling model:

y x i mj r ij = + ++ = ββ δε 0 1 i i ij ( ) 1 2,, , ,, , … … and = 1 2 (5.22)

where m is the number of helicopters, r is the number of measured fl ight times for each helicopter, y ij is the fl ight time for the j th fl ight of the i th helicopter, x i is the length of the wings for the i th helicopter, δi is the error term associated with the i th helicopter, and εij is the random error associated with the j th fl ight of the i th helicopter. Assume that the δi s are independent and normally distributed with a mean of 0 and a constant variance σδ

2 , that the εij s are independent and normally distributed with mean 0 and constant variance σ

2

, and that the δi s and the εij s are independent. The total number of observations in this study is n r i m = ∑ =1 i . This model in matrix form is yX Z = + b d + e where Z is a n × m “ incidence ” matrix and δ is a m × 1 vector of random helicopter - to - helicopter errors. The form of Z is

where 1r is a r × 1 vector of ones.

a. Show that Var(y I ) = + σ σδZZ′ 2 2 .

b. Show that the ordinary least squares estimates of β are the same as the generalized least squares estimates.

c. Derive the appropriate error term for testing the regression coeffi cients.

Z= 0 0 0 ... 1, 0 01,