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12.15 Bootstrapping DD The Distance Difference test statistic for testing the composite null Ho : h(0) = 0 is defined as DD - n min On(q) - min On(q) . q:h(q)-0 4 where On(g) is the GMM objective function On(q) where Omm consistently estimates Omm - E [m (2, 0) m (2,0)'] . It is known that, as the sample size n tends to infinity, DD - Xaim(). Write out a detailed formula for the bootstrap statistic DD".13.7 Durbin-Watson statistic and panel data Consider the standard one-way error component model with random effects: Wit = D48 + hit vi, i=1,..., n, t=1, ....T, (13.1) where S is k x 1, p; are random individual effects, , ~ /ID(0, o2), we are idiosyncratic shocks, un ~ IID(0, 59), and at; and ve are independent of zy for all i and t and mutually. The equations are arranged so that the index / is faster than the index i. Consider running OLS on the original regression (13.1) and running OLS on the GLS-transformed regression vit - Agi - (24 - #2;)'8+ (1- #); ton - Adj, i=1, ....n, t=1, . ...T, (13.2) where # is a consistent (as n - co and T stays fixed) estimate of a = 1-0.//} + To2. When each OLS estimate is obtained using a typical regression package, the Durbin-Watson (DW) statistic is provided among the regression output. Recall that if 21, 62, ..., eN-1, ex is a series of regression residuals, then the DW statistic is "(8; - ej-1)2 DW = N 1. Derive the probability limits of the two DW statistics, as n - co and T stays fixed. 2. Using the obtained result, propose an asymptotic test for individual effects based on the DW statistic [Hint: That the errors are estimated does not affect the asymptotic distribution of the DW statistic. Take this for granted.]