Consider the simple regression with no constant: Yi = Xi + ui i = 1, 2, . . . , n where ui IID(0,

Consider the simple regression with no constant: Yi = βXi + ui i = 1, 2, . . . , n where ui ∼ IID(0, σ2) independent of Xi. Theil (1971) showed that among all linear estimators in Yi, the minimum mean square estimator for β, i.e., that which minimizes E(β − β)2 is given by

β = β2n i=1 XiYi/(β2n i=1 X2 i + σ2).

(a) Show that E(β) = β/(1 + c), where c = σ2/β2n i=1 X2 i > 0.

(b) Conclude that the Bias (β) = E(β) − β = −[c/(1 + c)]β. Note that this bias is positive

(negative) when β is negative (positive). This also means that β is biased towards zero.

(c) Show that MSE(β) = E(β−β)2 = σ2/[

n i=1 X2 i +(σ2/β2)]. Conclude that it is smaller than the MSE(βOLS).

Table 3.4 Energy Data for 20 countries RGDP EN Country (in 106 1975 U.S.$’s) 106 Kilograms Coal Equivalents Malta 1251 456 Iceland 1331 1124 Cyprus 2003 1211 Ireland 11788 11053 norway 27914 26086 Finland 28388 26405 Portugal 30642 12080 Denmark 34540 27049 Greece 38039 20119 Switzerland 42238 23234 Austria 45451 30633 Sweden 59350 45132 Belgium 62049 58894 Netherlands 82804 84416 Turkey 91946 32619 Spain 159602 88148 Italy 265863 192453 U.K. 279191 268056 France 358675 233907 W. Germany 428888 352677