To find the asymptotic distribution of(Br-B), the approach in last section was to multiply(4)by VT, resulting in T(ax-B)=(1/) (1/V⑦∑xet (5) t=1 The usual assumption was that(1/T)> xtx converge in probability to non- singular matrix M while(1/VT)2L x,Et converges in distribution to a N(, v) random val riables, implying that VT(Br-B)N(O, (M-IVM-1)) For xt given in (5), we note that implying that T T(T+1)/ O(T1)O(T2) ∑t∑t2 T(T+1)/2T(T+1)(2T+1)/6 O(TO(TS) In contrast to the usual results as(2), the matrix(1/T)2tixixt in(5) diverges. To obtain converge and nondegenerate limiting distribution, we can T1/20 0T3/2 and obtains /2 3/2 0T-3/2 t=1 T-1∑1T-2∑t according to(6 Turning next to the second term in(4)and premultiplying it by rT Et (1/VT∑et XtE 0 T t=1 (1/T∑(t/T)etTo find the asymptotic distribution of (βˆ T − β), the approach in last section was to multiply (4) by √ T, resulting in √ T(βˆ T − β) = " (1/T) X T t=1 xtx 0 t #−1 " (1/ √ T) X T t=1 xtεt # . (5) The usual assumption was that (1/T) PT t=1 xtx 0 t converge in probability to non￾singular matrix M while (1/ √ T) PT t=1 xtεt converges in distribution to a N(0, V) random variables, implying that √ T(βˆ T − β) L−→ N(0,(M−1VM−1 )). For xt given in (5), we note that 1 T v+1 X T t=1 t v → 1 v + 1 , (6) implying that X T t=1 xtx 0 t =  P1 P P t t Pt 2  =  T T(T + 1)/2 T(T + 1)/2 T(T + 1)(2T + 1)/6  ≡  O(T 1 ) O(T 2 ) O(T 2 ) O(T 3 )  . (7) In contrast to the usual results as (2), the matrix (1/T) PT t=1 xtx 0 t in (5) diverges. To obtain converge and nondegenerates limiting distribution, we can think of premultiplying and postmultiplying hPT t=1 xtx 0 t i by the matrix Υ−1 T =  T 1/2 0 0 T 3/2 −1 , and obtains ( Υ−1 T "X T t=1 xtx 0 t # Υ−1 T ) =  [T −1/2 0 0 T −3/2   P1 P P t t Pt 2   [T −1/2 0 0 T −3/2  =  T −1 P1 T −2 Pt T −2 Pt T −3 Pt 2  → Q, where Q =  1 1 2 1 2 1 3  (8) according to (6). Turning next to the second term in (4) and premultiplying it by Υ−1 T , Υ−1 T "X T t=1 xtεt # =  T −1/2 0 0 T −3/2   P P εt tεt  =  (1/ √ T) Pεt (1/ √ T) P(t/T)εt  . (9) 9