Chapter2-面板计量习题答案

0 0 0 0 since .`i Ã0T / Q D .`i Ã0T /. I N T P / D .`i Ã0T / .`i Ã0T / D 0. 2.11 Using the Grunfeld (1958) data on investment given as Grunfeld.fil on the Wiley web site this gives the SAS program and output for the estimation of a one-way error component regression. The estimators are OLS, WITHIN, BETWEEN, WALHUS, AMEMIYA, SWAR, NERLOVE and IMLE. The results are reported in Table 2.1.
where there are T ones for the i th individual. This can be rewritten as 2 .` w D ¦¼ ÃT / where `i is the i th column of I N , i.e. `i is a vector that i has one in the i th position and zero elsewhere. ÃT is a vector of ones of dimension T . 0 0 0 (b) .`i Ã0T / P D .`i Ã0T /. I N ÃT Ã0T = T / D .`i Ã0T /. Therefore, in (2.39) " # 2 ¦¼ 1 1 0 1 2 0 0 w D ¦¼ .`i ÃT / P C Q D .`0 Ã0T / 2 2 2 i ¦¹ ¦1 ¦1
VAR_WT=SSQ(WT_RES)/(NT-N-NCOL(X)); /* Equation 2.24 */ V_C_WT=VAR_WT*INV(X‘*Q*X); STD_WT=SQRT(VECDIAG(V_C_WT)); T_WT=WT_BETA/STD_WT; LOOK1=WT_BETA||STD_WT||T_WT; CTITLE={‘PARAMETER’ ‘STANDARD ERROR’ ‘T-STATISTICS’}; RTITLE={‘FIRMV’ ‘CAPITALV’}; PRINT ‘RESULTS OF WITHIN’,, LOOK1(|COLNAME=CTITLE ROWNAME=RTITLE FORMAT=8.5|);
2
Econometric analysis of panel data D .Z 0 Q Z C Z 0 P Z / D . Z 0. Q C P / Z / since Q C P D I N T . (b) GLS on (2.28) yields 2
1 1
. Z 0 Qy C Z 0 P y /
1
X 0 Q Q Qy D . X 0 Q X /
1
e X 0 Qy D þ
(b) For the general linear model y D X þ C u with E .uu 0 / D , a necessary and sufficient condition for OLS to be equivalent to GLS is given by NX D 0; where P NX D I PX and PX D X . X 0 X / 1 X 0 , see Baltagi X0 1 P (1989b). For (2.6), this condition cZ 0
1
Z/
1
Z0
1
y
1 is given by (2.19). where 2 , for i D j and zero otherwise. 2.10 (a) From (2.2) and (2.38), E .u i ;T C S u jt / D ¦¼ The only correlation over time occurs because of the presence of the same individual across the panel. The ¹it ’s are not correlated for different time periods. In vector form 2 .0; : : : ; 0; : : : ; 1; : : : ; 1; : : : ; 0; : : : ; 0/0 w D E .u i ;T C S u / D ¦¼
2 The One-way Error Component Regression Model
NT is idempotent and is therefore its 2.2 (a) Q D I N T P where P D I N J own generalized inverse. The variance–covariance matrix of the disturbance e ¹ D Q ¹ in (2.6) is
0
2P ¦1
3 5 3 5
10
QZ PZ
@
10
0
2Q ¦¹
13 A7 5 1 A
1
0
2P ¦1
Qy Py
@
0
1
2 D4
2 Q =¦¹
0
2 P =¦1
3 5
0
1
2 2 b ŽGLS D [. Z 0 Q Z /=¦¹ C . Z 0 P Z /=¦1 ]
2 2 [. Z 0 Q Z /=¦¹ C . Z 0 P y /=¦1 ]
2 E .e ¹e ¹ 0 / D E . Q ¹¹ 0 Q / D ¦¹ Q 2 . OLS on (2.6) yields with generalized inverse Q =¦¹
b D .X0 Q Q X / þ
1
X 0 Q Qy D . X 0 Q X /
1
X 0 Qy
e given in (2.7). Also, GLS on (2.6) using the generalized inverse which is þ yields b D .X0 Q Q Q X / þ
/* Equation 2.27 */
LOOK1=BW_BETA||STD_BW||T_BW; CTITLE={‘PARAMETER’ ‘STANDARD ERROR’ ‘T-STATISTICS’}; RTITLE={‘INTERCEPT’ ‘FIRMV’ ‘CAPITALV’}; PRINT ‘RESULTS OF BETWEEN’,, LOOK1(|COLNAME=CTITLE ROWNAME=RTITLE FORMAT=8.5|);
*--------- WALLACE & HUSSAIN ESTIMATOR ----------* *--------OF VARIANCE COMPONENTS ---------*; WH_V_V=(OLS_RES‘*Q*OLS_RES)/(NT-N); /* Equation 2.22 with U=OLS residuals */ WH_V_1=(OLS_RES‘*P*OLS_RES)/N; /* Equation 2.21 with U=OLS residuals */
The one-way error component regression model
DATA GRUNFELD; INFILE ‘A:\GRUNFELD.FIL’; INPUT FIRM YEAR INVEST STOCK CAPITAL;
3
PROC IML; USE GRUNFELD; READ ALL INTO TEMP; N=10; T=20; NT=N*T; One=Repeat(1,NT,1); Y=Temp[,3]; X=Temp[,4:5]; Z=One||X; K=NCOL(X); /* I, J, P and Q matrices */ l_t=J(T,1,1); JT=(l_t*l_t‘); Z_U=I(N)@l_t; P=I(N)@JT/T; Q=I(NT)-P; JNT=Repeat(JT,N,N); JNT_BAR=JNT/NT; *--------- OLS ESTIMATORS ------------*; OLS_BETA=INV(Z‘*Z)*Z‘*Y; OLS_RES=Y-Z*OLS_BETA; VAR_REG=SSQ(OLS_RES)/(NT-NCOL(Z)); VAR_COV=VAR_REG*INV(Z‘*Z); STD_OLS=SQRT(VECDIAG(VAR_COV)); T_OLS=OLS_BETA/STD_OLS; LOOK1=OLS_BETA||STD_OLS||T_OLS; CTITLE={‘PARAMETER’ ‘STANDARD ERROR’ ‘T-STATISTICS’}; RTITLE={‘INTERCEPT’ ‘FIRMV’ ‘CAPITALV’}; PRINT ‘RESULTS OF OLS’,, LOOK1(|COLNAME=CTITLE ROWNAME=RTITLE FORMAT=8.5|); *--------- BETWEEN ESTIMATOR ------------------*; BW_BETA=INV(Z‘*P*Z)*Z‘*P*Y; BW_RES=P*Y-P*Z*BW_BETA; VAR_BW=SSQ(BW_RES)/(N-NCOL(Z)); V_C_BW=VAR_BW*INV(Z‘*P*Z); STD_BW=SQRT(VECDIAG(V_C_BW)); T_BW=BW_BETA/STD_BW;
2 N . X 0 Q /. Q =¦¹ / PQ X D 0
using the fact that Q is idempotent, the left-hand side can be written as 2 which is clearly 0, since P NQ X =¦¹ NQ X is the orthogonal projection . X 0 Q/ P of Q X . 2.5 E .u 0 Pu / D E ftr.uu 0 P /g D trf E .uu 0 / P g D tr. P /. From (2.18), P D 2 P since P Q D 0. Hence, from (2.21), E .b 2 / D E .u 0 Pu /=tr. P / D ¦1 ¦1 2 2 0 2 Q / where the tr.¦1 P /=tr. P / D ¦1 . Similarly, E .u Qu / D tr. Q / D tr.¦¹ 2 last equality follows from (2.18) and the fact that Q D ¦¹ Q since P Q D 2 / D E .u 0 Qu /=tr. Q / D tr.¦ 2 Q /=tr. Q / D ¦ 2 . 0. Hence, from (2.22), E .b ¦¹ ¹ ¹ 2.7 (a) OLS on (2.28) yields Ä Â Ã½ 1  à QZ Qy b ŽOLS D . Z 0 Q ; Z 0 P / . Z 0 Q; Z 0 P / PZ Py