Phillips-Perron Test Gauss file(s) unit_ppcv.g, unit_ppsim.g Matlab file(s) unit_ppcv.m, unit_ppsim.m Consider the AR(1) model yt = yt1 + vt , where vt is a stationary

Phillips-Perron Test Gauss file(s) unit_ppcv.g, unit_ppsim.g Matlab file(s) unit_ppcv.m, unit_ppsim.m Consider the AR(1) model yt = φyt−1 + vt ,

where vt is a stationary autocorrelated disturbance with mean zero, variance σ 2 and satisfying the FCLT for autocorrelated processes 1 √ T X [T s] t=1 vt d → ω B(s), where B is standard Brownian motion and ω 2 is the long-run variance ω 2 = P∞ j=−∞ E(vtvt−j ).

(a) Derive the asymptotic distribution of the OLS estimator of φ in the AR(1) model where φ = 1 and show that it depends on the autocorrelation properties of vt .

(b) Suppose that consistent estimators ωb 2 and σb 2 of ω 2 and σ 2 , respectively, are available. Define the transformed estimator φ˜ = φb − 1 2 ωb 2 − σb 2 T −1 PT t=2 y 2 t−1 and derive its asymptotic distribution when φ = 1, hence showing that the asymptotic distribution of φ˜ does not depend on the autocorrelation properties of vt .

(c) Define a unit root test based on φ˜ and simulate approximate asymptotic critical values. This test is the Zα test suggested in Phillips (1986) and extended to allow for constant and time trend in Phillips and Perron (1988).

(d) Carry out a Monte Carlo experiment to investigate the finite sample size properties of the test in part (c). Use T = {100, 200} with vt given by (1 − φL)vt = (1 + θL)εt , εt ∼ iidN(0, 1) where φ = {0, 0.3, 0.6, 0.9} and θ = {−0.8, −0.4, 0, 0.4, 0.8}. Use σb 2 = (T − 1)−1 PT t=2 vb 2 t and let ωb 2 be the Newey-West long-run variance estimator with quadratic spectral lag weights (see Chapter 9) and pre-whitening as suggested by Andrews and Monahan (1992).