Lecture 25 – Unit Root Tests IIIThe Ng-Perron Unit Root Test(Improved Finite Sample Performance)The ADF and PP unit root tests are known (fromMC simulations) to suffer potentially severe finite sample power and size problems.1. Power – The ADF and PP tests are known to have low power against the alternative hypothesis that the series is stationary (or TS) with a large autoregressive root. (See, e.g., DeJong, et al, J. of Econometrics, 1992.)2. Size – The ADF and PP tests are known to have severe size distortion (in the direction of over-rejecting the null) when the series has a large negative moving average root. (See, e.g., Schwert. JBES, 1989: MA = -0.8, size = 100%!) A variety of alternative procedures have been proposed that try to resolve these problems, particulary, the power problem, but the ADF andPP tests continue to be the most widely used unitroot tests. That may be changing.Digression –Recall that there are two ways of thinking about the roots of the polynomial associated with the MA or AR component of a time series. Suppose,yt = a1yt-1 + a2yt-2 + … + apyt-p + εt , εt ~ iid Sometimes we define the roots of the AR as the solutions to 1-a1z -…-apzp = 0(and, for the AR process, the stationarity condition is these roots are greater than one in modulus)Sometimes we define the roots of the AR as the solutions tozp – a1zp-1 - … - ap = 0(and, for the AR process, the stationarity condition is these roots are less than one in modulus)(How are the roots in these two approaches related to one another?)When we talk about about yt having a large AR root we are thinking in terms of the second approach and referring to the largest root being close to one.Similarly, we can calculate the roots of an MA polynomial, b(L), in either of these two ways. When we talk about yt having a large MA root we are thinking in terms of the second approach and referring to the largest root being close to one. Suppose, for example, yt = εt – 0.9εt-1 , εt ~ iid (0,σ2)The MA root defined by the first approach is 1/0.9 = 1.1. The MA root defined by the second approach is 0.9, a large positive root. This process can be approximated by an AR(p) for sufficiently large p with AR coefficients: .9, .92,… Note that the larger the MA root, the larger the value of p that will be needed for a good approximation.Suppose,yt = εt + 0.9εt-1The MA root (defined by the second approach) is -0.9, a large negative root.This process can be approximated by an AR(p) for sufficiently large p with AR coefficients: -0.9, (-0.9)2,… Note, again, that the larger the MA root, the larger the value of p that will be needed for a good approximation.Among the macroeconomic time series whose differences are thought to be characterized by a large negative MA root: inflation rate, unemployment rateInsert Table 7 from Ng and PerronInsert Table 1 from Phengpis and CrowderEnd DigressionNg and Perron (Econometrica, 2001), building on some of their own work (Perron and Ng, Rev. of Econ. Studies, 1996) and work by Elliott, Rothenberg, and Stock (Ecta, 1996), new tests to deal with both of these problems. Their tests, in constrast to many of the other “new” unit root tests that have been developed over the years, seems to have caught on as a preferred alternative to the traditional ADF and PP tests.(Note that for a test to become widely used, it must work well and be relatively easy to apply.)The family of NP tests (which includes among others, modified DF and PP test statistics) share the following features – First, the time series is de-meaned or detrended by applying a GLS estimator. Thisstep turns out to improve the power of the tests when there is a large AR root and reduces size distortions when there is a largenegative MA root in the differenced series.The second feature of the NP tests is a modified lag selection (or truncation selection) criteria It turns out that the standard lag selection procedures used in specifying the ADF regression (or for calculating the long run variance for the PP statistic) tend to underfit, i.e., choose too small a lag length, when there is a large negative MA root. This creates additional size distortion in unit root tests. The NP modified lag selection criteria accounts for this tendency.Example – NP’s DF-GLS test using a modified AIC (MAIC) to select the lag length for the ADF regression.H0: Yt ~ I(1), no driftHA: Yt ~ I(0), no restriction on the mean(Yt = inflation rate, unemployment rate, interest rate,…)Step 1 – De-mean the data by applying GLS using a “local-to-unity” transformation:Y1* = Y1 , Yt* = Yt – (1-7/T)Yt-1 for t = 2,…,T11* = 11t* = 1- (1-7/T) for t = 2,…,TRegress Yt* on 1t*, bhat; yt = Yt –bhat.Step 2 – Find the optimal lag length, p, for the ADF regression of yt on yt-1,Δyt-1,…,Δyt-p+1. (Note that there will not be an intercept in the ADFregression. Why not?)-Select pmax-Fit the ADF regression for p = 1,…,pmax (holding the sample size fixed: ptp ,ˆ,ˆ-Choose p to minimize the MAIC: MAIC(p) = max2)(*2ˆlogpTprppwhere TpptppT12,max2maxˆ1ˆ21212ˆ/*)1ˆ(pTptppMAXyr- Fit the ADF regression (no intercept or trend) to yt using the optimal p:ρ-hat, τμADF-GLSwhere τμADF-GLS is the t-statistic associated with H0: ρ = 1 in the ADF regression (or H0: ρ = 0 in the regression of Δyt on yt-1, Δyt-1,…, Δyt-p+1).- The asymptotic null distribution of τμADF-GLS is the DF distribution for τμ.Note – If the original series Yt is assumed to have a linear deterministic trend component under H0 and HA, the procedure above is modified as follows:1.Revise Step 1 – De-mean the data by applying GLS using a “local-to-unity” transformation:Y1* = Y1 , Yt* = Yt – (1-13.5/T)Yt-1 for t = 2,…,T11* = 11t* = 1- (1-13.5/T) for t = 2,…,Tt1* = 1tt* = t-(1-13.5/T)*(t-1) for t = 2,…,T Regress Yt* on 1t*, tt*: bhat; yt = Yt –bhat.2. Revise Step 2- Fit the ADF regression (no intercept or trend) to yt using the optimal p:ρ-hat, τtADF-GLSwhere τtADF-GLS is the t-statistic associated with H0: ρ = 1 in the ADF regression (or H0: ρ = 0 in the regression of Δyt on yt-1, Δyt-1,…, Δyt-p+1).- The asymptotic null distribution of τtADF-GLS is not the DF distribution forτt. Percentiles of the asymptotic null distribution of τtADF-GLS are provided in NP, Table 1 (See the p = 1