Now we need the formula (Henderson and Searle, 1981) which implies Note that to use this block matrix inversion formula we require B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$$_{T$_{l}$}$+ I$_{|}$$_{T}$$_{l}$$_{|}$ to be invertible. However, by the previous result of this theorem only the variance of the intercept term in the form for A − 1 T$_{l}$T$_{l}$ is dependent upon the value of t , so we have that for some V$_{1}$, V$_{2}$ ; dim( V$_{1}$ ) = 1 × 1 , dim( V$_{2}$ ) = | W$_{l}$ | × 1. Now, by Lemma 4.4 det( B − 1 T$_{l}$$_{+1}$$_{T$_{l}$$_{+1}$}$) > 0, which gives det( B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$ ) = { det( B − 1 T$_{l}$$_{+1}$$_{T$_{l}$$_{+1}$}$) } − 1 > 0. By assumption det( A − 1 T$_{l}$$_{T$_{l}$}$) > 0, and thus Therefore M =    E F G H$^{}$  $_{}$, ⇒ M − 1 =    ( E − FH − $^{1}$G ) − 1 − E − $^{1}$F ( H − GE − $^{1}$F ) − 1 − H − $^{1}$G ( E − FH − $^{1}$G ) − 1 ( H − GE − $^{1}$F ) − 1   $_{}$, ( I 2 D − 1 + BA − 1 ) − 1 =    ( B T l +1 T l +1 A − 1 T l T l + I | T l | ) − 1 − ( B T l +1 T l +1 A − 1 T l T l + I | T l | ) − 1 B T l +1 T l +1 A − 1 T l T ′ l 0 I | T ′ l |    . B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ = 1 l    V$_{1}$ 0 V$_{2}$ I$_{|}$$_{T}$$_{l}$$_{|}$$_{−}$$_{1}$$^{}$   = 1 l    V$_{1}$ 0 V$_{2}$ I$_{|}$$_{W}$$_{l}$$_{|}$$^{}$  $_{}$, det( B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$$_{T$_{l}$}$) = det( B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$ )det( A − 1 T$_{l}$$_{T$_{l}$}$) = $^{(}$1 l$^{)}$$^{|}$$^{W$_{l}$}$$^{|}$ V$_{1}$ > 0 , ⇒ V$_{1}$ > 0 . B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ + I$_{|}$$_{T}$$_{l}$$_{|}$ = 1 l    V$_{1}$ + l 0 V$_{2}$ (1 + l ) I$_{|}$$_{W}$$_{l}$$_{|}$$^{}$  $_{}$, ⇒ det( B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ + I$_{|}$$_{T}$$_{l}$$_{|}$ ) > 0 , and B$_{T}$$_{l}$$_{+1}$$_{T}$$_{l}$$_{+1}$A − 1 T$_{l}$T$_{l}$ + I$_{|}$$_{T}$$_{l}$$_{|}$ is therefore invertible as required. Now cov( ˆ β l +1 , ˆ β l +1 ) =     A − 1 T l T l A − 1 T l T ′ l A − 1 T ′ l T l A − 1 T ′ l T ′ l$^{}$    × ×      I 2 D − 1 −    B T l +1 T l +1 A − 1 T l T l B T l +1 T l +1 A − 1 T l T ′ l 0 0       ( B T l +1 T l +1 A − 1 T l T l + I | T l | ) − 1 − ( B T l +1 T l +1 A − 1 T l T l + I | T l | ) − 1 B T l +1 T l +1 A − 1 T l T ′ l 0 I | T ′ l |         . 29