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54 CHAPTER 2. SOME USEFUL TIME-SERIES METHODStrend) and k i lags of ∆˜q it . 23∆˜q it = a i + b i t +˜q it−1 = a 0 i + b 0 it +k iXj=1k iXj=1c ij ∆˜q it−j +ê it , (2.73)c 0 ij∆˜q it−j +ˆv it , (2.74)where ê it and ˆv it are OLS residuals. Now run the regressionê it = δ iˆv it−1 +û it , (2.75)set ˆσ ei 2 = 1 P Tt=kiT −k i −1 +2 û2 it, and form the normalized observationsẽ it = êitˆσ ei,ṽ it = ˆv itˆσ ei. (2.76)Denote the long run variance of ∆q it by σ 2 qi = γ i 0 +2 P ∞j=0 γ i j,whereγ i 0 = E(∆q 2 it) andγ i j = E(∆q it ∆q it−j ). Let ¯k = 1 NP Ni=1k i and estimateσ 2 qi by Newey and West [114]ˆσ 2 qi =ˆγi 0 +2 kXj=1µ1 − j ˆγ j i k +1, (2.77)where ˆγ j i = 1 P Tt=2+j∆˜qT −1it ∆˜q it−j . Let s i = ˆσ qiˆσ ei, S N = 1 Nrun the pooled cross-section time-series regressionP Ni=1s i andẽ it = βṽ it−1 +˜² it . (2.78)The studentized coefficient is τ = ˆβ P N P Tt=1i=1 ṽ it−1 /ˆσ˜² where ˆσ˜² =1 P Ni=1 P Tt=1˜²NTit . As in the univariate case, τ is not asymptoticallystandard normally distributed. In fact, τ diverges as the number of23 To choose k i , one option is to use AIC or BIC. Another option is to use Hall’s [69]general-to-speciÞc method recommended by Campbell and Perron [19]. Start withsome maximal lag order ` and estimate the regression. If the absolute value of thet-ratio for ĉ i` is less than some appropriate critical value, c ∗ , reset k i to ` − 1 andrepeattheprocessuntilthet-ratiooftheestimatedcoefficient with the longest lagexceeds the critical value c ∗ .