Tài sản của (a) dự toán không đúng trực tiếp, (b) ước tính hiệp phương sai trực tiếp, (c) dự toán không đúng trực tiếp và (d) ước tính hồi quy trực tiếp (Ã, 1000) trên 100 lần nhắc lại.  €  €  € (a) LLTI (1)  78X6, (2)  0X979, yy  79X6 yy yy (1) Syy (2) Syy | SIMULATION STUDIES 337 Table Properties of estimates c direct variance over 100 replications. x T T TT V 1 no Z V 2 a LLTI p1 - p1 a direct variance estimates b direct covariance estimates and d direct regression estimates x 1000 - Pyy - s v S2 yy yy Data Mean SD Mean SD m. - 371 n0 - 18 880 128 m. - 371 n0 - 3776 m. - 371 n0 - 378 m0 - 371 n0 - 38 u T T TT f A T rv P 1 00 2 P 2 1 2 A P 04 7 b LL I CARO 2- yx - 22-8 2- yx - 1-84 yx - 24-7 Data S 1 yx S 2 yx Mean SD Mean SD m. - 371 n - 18 880 115 m. - 371 n0 - 3776 m. - 371 n0 - 378 0 - 371 n0 - 38 c CARO p 1 90 5 P c CARO 2_ xx - 90-5 2 xx - Pxx - S1 xx s xx Data Mean SD Mean SD m0 - 371 n0 - 18 880 399 m0 - 371 n0 - 3776 152 m0 - 371 n0 - 378 m0 - 371 n0 - 38 d LLTI vs. CARO pyx - 252 pyx - 302 Pyx - 255 By B y2x Data Mean SD Mean SD m0 - 371 n0 - 18 880 255 288 m0 - 371 n0 - 3776 254 270 m0 - 371 n0 - 378 256 m0 - 371 n0 - 38 258 338 ANALYSIS OF SURVEY AND GEOGRAPHICALLY AGGREGATED DATA . USING AUXILIARY VARIABLES TO REDUCE AGGREGATION EFFECTS In Section it was shown how individual-level data on y and x can be used in combination with aggregate data. In some cases individual-level data may not be available for y and x but may be available for some auxiliary variables. Steel and Holt 1996a suggested introducing extra variables to account for the correlations within areas. Suppose that there is a set of auxiliary variables z that partially characterize the way in which individuals are clustered within the groups and conditional on z the observations for individuals in area g are influenced by random group-level effects. The auxiliary variables in z may only have a small effect on the individual-level relationships and may not be of any direct interest.
