In Chapter 10 of Your Textbook, Panel Data Estimation Was

Question

Examlex · Accepted Answer

The Answer is (a)Under the extended least least squares assumptions, E(UU' | X)=

\sigma _ { u } ^ { 2 }

I_n.
In the above example of 4 U.S. states and 3 time periods, the identity matrix will be of order 12×12, or (nT)× (nT)in general. Specifically

\left( \begin{array} { c c c c } \sigma _ { u } ^ { 2 } & 0 & \cdots & 0 \0 & \sigma _ { u } ^ { 2 } & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & \sigma _ { u } ^ { 2 }\end{array} ight)

(b)It is reasonable to assume that a shock to an adjacent state would have an effect on its neighboring state, particularly when the shock affects the larger of the two such as the case in Massachusetts. Other examples may be Texas and Arkansas, Michigan and Indiana, California and Arizona, New York and New Jersey, etc. A negative oil price shock, which affects the demand for automobiles produced in Michigan, will have repercussions for suppliers located not only in Michigan, but also elsewhere.
(c)In case of a known variance-covariance matrix of the error terms, the GLS estimator

\hat { \beta }

^GLS = (

X ^ { \prime }

?^-¹^X)^-¹(

X ^ { \prime }

?^-¹^Y)could be used. The variance-covariance matrix would be of the form

\Omega = \left( \begin{array} { c c c c c c c c c c c c } \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } & 0 & 0 \0 & \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } & 0 \0 & 0 & \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } \\sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 \0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } & 0 \0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } \\sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 \0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } & 0 \0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } \\sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 } & 0 & 0 \0 & \sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 } & 0 \0 & 0 & \sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 }\end{array} ight)

(There is a subtle issue here for the case of a feasible GLS estimator, where the variances and covariances have to be estimated. It can be shown, in that case, that the GLS estimator does not exist unless n ? T, which is not the case for most panels. It is easier to see that the variance-covariance matrix is singular for n>T if the data is stacked by time period.)

In Chapter 10 of Your Textbook, Panel Data Estimation Was

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