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2.5. PANEL UNIT-ROOT TESTS 55Table 2.2: Mean and Standard Deviation Adjustments for Levin—Lin τStatistic, reproduced from Levin and Lin [91]τNC ∗ τC ∗ τCT∗˜T K µ ∗˜T σ ∗˜T µ ∗˜T σ ∗˜T µ ∗˜T σ ∗˜T25 9 0.004 1.049 -0.554 0.919 -0.703 1.00330 10 0.003 1.035 -0.546 0.889 -0.674 0.94935 11 0.002 1.027 -0.541 0.867 -0.653 0.90640 11 0.002 1.021 -0.537 0.850 -0.637 0.87145 11 0.001 1.017 -0.533 0.837 -0.624 0.84250 12 0.001 1.014 -0.531 0.826 -0.614 0.81860 13 0.001 1.011 -0.527 0.810 -0.598 0.78070 13 0.000 1.008 -0.524 0.798 -0.587 0.75180 14 0.000 1.007 -0.521 0.789 -0.578 0.72890 14 0.000 1.006 -0.520 0.782 -0.571 0.710100 15 0.000 1.005 -0.518 0.776 -0.566 0.695250 20 0.000 1.001 -0.509 0.742 -0.533 0.603∞ — 0.000 1.000 -0.500 0.707 -0.500 0.500observations NT gets large, but Levin and Lin show that the adjustedstatisticτ ∗ = τ − N ˜TS N τµ ˆσ−2∗˜T ²ˆβ −1D→ N(0, 1), (2.79)σ ∗˜Tas ˜T →∞,N →∞where ˜T = T −¯k−1, and µ ∗˜T and σ ∗˜T are adjustmentfactors reproduced from Levin and Lin’s paper in Table 2.2.Performance of Levin and Lin’s adjustment factors in a controlled environment.Suppose the data generating process (the truth) is, thateach individual is the unit root process∆q it = α i +2Xj=1φ ij ∆q it−j + ² it , (2.80)where ² itiid∼ N(0, σ i ),andeachoftheσ i is drawn from a uniform distributionover the range 0.1 to 1.1. That is, σ i ∼ U[0.1, 1.1]. Also,