Unit Root Tests

Consider the trend-cycle decomposition of a time series yt yt = T Dt + T St + Ct = T Dt + Zt The basic issue in unit root testing is to determine if T St = 0. Two classes of tests, called unit root tests, have been developed to answer this question:

• H0 : T St 6= 0 (yt ∼ I(1)) vs. T St = 0 (yt ∼ I(0))

• H0 : T St = 0 (yt ∼ I(0)) vs. T St 6= 0 (yt ∼ I(1))

1.1

Autoregressive unit root tests

These tests are based on the following set-up. Let yt = φyt−1 + ut, ut ∼ I(0) The null and alternative hypothesis are H0 : φ = 1 (φ(z) = 0 has a unit root) H1 : |φ| < 1 (φ(z) = 0 has root outside unit circle) The most popular of these tests are the Dickey-Fuller (ADF) test and the Phillips-Perron (PP) test. The ADF and PP tests diﬀer mainly in how they treat serial correlation in the test regressions. 1. ADF tests use a parametric autoregressive structure to capture serial correlation φ∗(L)ut = εt φ∗(L) = 1 − φ∗L − · · · − φ∗ Lk 1 k 2. PP tests use non-parametric corrections based on estimates of the long-run variance of ∆yt.

1.2

Stationarity Tests

These tests can be interpreted in two equivalent ways. The ﬁrst is based on the Wold representation ∆yt = ψ ∗(L)εt, εt ∼ iid(0, σ 2) The null and alternative hypotheses are H0 : ψ ∗(1) = 0 (ψ ∗(z) = 0 has a unit root) H1 : ψ ∗(1) > 0 (ψ ∗(z) = 0 has roots outside unit circle) The second is based on the UC-ARIMA model yt μt φ(L)Ct cov(εt, η t) = = = = μt + Ct μt−1 + εt, εt ∼ iid(0, σ 2) ε θ(L)η t, η t ∼ iid(0, σ 2 ) η 0

Here, the null and alternative hypotheses are H0 : σ 2 = 0 (μt = μ0) ε H1 : σ 2 > 0 (μt = μ0 + ε j=1 t X

εj )

Result: Testing for a unit moving average root in ψ ∗(L) is equivalent to testing σ 2 = 0. ε Intuition: Recall the random walk plus noise model. The reduced form is an MA(1) model with moving average root given by θ = −(q + 2) + 2 q

q 2 + 4q

σ2 ε q = 2 ση

If σ 2 = 0 then q = 0, θ = −1 and the reduced form e MA(1) model has a unit moving average root. The most popular stationarity tests are the KitawoskiPhillips-Schmidt-Shin (KPSS) test and the LeyborneMcCabe test. As with the ADF and PP tests the KPSS and Leyborne-McCabe tests diﬀer main in how they treat serial correlation in the test regressions.

1.3

Statistical Issues with Unit Root Tests

Conceptually the unit root tests are straightforward. In practice, however, there are a number of diﬃculties:

• Unit root tests generally have nonstandard and non-normal asymptotic distributions.

• These distributions are functions of standard Brownian motions, and do not have convenient closed form expressions. Consequently, critical values must be calculated using simulation methods.

• The distributions are aﬀected by the inclusion of deterministic terms, e.g. constant, time trend, dummy variables, and so diﬀerent sets of critical values must be used for test regressions with diﬀerent deterministic terms.

1.4

Distribution Theory for Unit Root Tests

Consider the simple AR(1) model yt = φyt−1 + εt, where εt ∼ WN(0, σ 2) The hypotheses of interest are H0 : φ = 1 (unit root in φ(z) = 0) ⇒ yt ∼ I(1)

H1 : |φ| < 1 ⇒ yt ∼ I(0) The test statistic is

ˆ φ−1 tφ=1 = ˆ SE(φ) ˆ φ = least squares estimate If {yt} is stationary (i.e., |φ| < 1) then √ d ˆ T (φ − φ) → N (0, (1 − φ2)) µ ¶ 1 ˆ A φ ∼ N φ, (1 − φ2) T A tφ=φ0 ∼ N (0, 1)

However, under the null hypothesis of nonstationarity the above result gives ˆ A φ ∼ N (1, 0) which clearly does not make any sense. Problem: under the unit root null, {yt} is not stationary and ergodic, and the usual sample moments do not converge to ﬁxed constants. Instead, Phillips (1987) showed that the sample moments of {yt} converge to random functions of Brownian motion: T −3/2 T X

0 t=1 Z T X d 2 1 2 T −2 yt−1 → σ W (r)2dr 0 t=1 Z T X d 2 1 T −1 yt−1εt → σ W (r)dW (r) 0 t=1

yt−1 → σ

d

Z 1

W (r)dr

where W (r) denotes a standard Brownian motion (Wiener process) deﬁned on the unit interval....