Unit Roots, Structural Breaks, and Persistence

1. Integrated processes

A process is integrated of order \(d\), \(I(d)\), if it must be differenced \(d\) times to become stationary. The prototype \(I(1)\) process is the random walk with drift:

Compare the trend-stationary model \(y_t=\alpha+\mu t+u_t\), with \(u_t\) stationary.

Beveridge–Nelson decomposition. Any \(I(1)\) process with \(\Delta y_t=\mu+\psi(L)\varepsilon_t\) can be written as a random-walk permanent component \(\tau_t=\tau_{t-1}+\mu+\psi(1)\varepsilon_t\) plus a stationary transitory component. The long-run impact of a shock is \(\psi(1)=\sum_j\psi_j\). A process that is "nearly" trend-stationary has \(\psi(1)\) close to zero.

2. Why unit roots break standard inference

In \(y_t=\phi y_{t-1}+\varepsilon_t\) with \(\phi=1\), the OLS estimator is super-consistent: \(T(\widehat\phi-1)\) converges to a non-degenerate distribution, so \(\widehat\phi\) converges at rate \(T\), not \(\sqrt T\). The limit is non-normal and skewed to the left. It is expressed through a standard Brownian motion \(W(r)\) on \([0,1]\):

These results follow from the functional CLT (\(T^{-1/2}\sum_{s\le rT}\varepsilon_s/\sigma\Rightarrow W(r)\)) and the continuous mapping theorem. The second limit is the Dickey–Fuller distribution. Its 5% critical value is about \(-1.95\) with no deterministics, \(-2.86\) with a constant, and \(-3.41\) with a constant and trend, far from the normal \(-1.645\).

Spurious regression. Regressing one independent random walk on another yields "significant" \(t\) statistics with probability approaching one, and \(R^2\) does not go to zero (Granger and Newbold, 1974; Phillips, 1986). The \(t\) statistic diverges at rate \(\sqrt T\). A telltale sign is a high \(R^2\) together with a Durbin–Watson statistic near zero. Lecture 12 explains when a regression in levels of \(I(1)\) variables is meaningful (cointegration).

3. Dickey–Fuller tests

Rewrite the AR(1) as \(\Delta y_t=\pi y_{t-1}+\varepsilon_t\), with \(\pi=\phi-1\), and test \(H_0:\pi=0\) (unit root) against \(H_1:\pi<0\) (stationarity). Three versions differ in their deterministic terms:

Choosing deterministics. The alternative must be plausible: for a trending series (log GDP), include a trend, so the alternative is trend-stationarity. For a non-trending series (interest rates, unemployment), include only a constant. Including unnecessary trends costs power; omitting necessary ones can make the test inconsistent.

Augmented DF (ADF). Serial correlation in \(\Delta y_t\) is absorbed by lagged differences:

The \(t\) statistic on \(\pi\) has the same DF limit distribution. Choose \(p\) by the modified AIC of Ng and Perron (2001), or by a general-to-specific sequence starting from \(p_{\max}=\lfloor12(T/100)^{1/4}\rfloor\). Too few lags distort size; too many lose power.

Phillips–Perron corrects the DF statistic non-parametrically with a long-run variance estimate instead of adding lags. It has poor size when there is a large negative MA root, as in many inflation series.

Low power and DF-GLS. Against near-unit-root alternatives such as \(\phi=0.95\), ADF tests have very low power in samples of typical macro length. DF-GLS (Elliott, Rothenberg and Stock, 1996) first detrends by GLS under a local alternative, then runs the ADF regression without deterministics. It is nearly efficient and should be the default.

KPSS (stationarity as the null). Kwiatkowski, Phillips, Schmidt and Shin (1992) test \(H_0\): stationary (around a level or trend), using the scaled sum of squared partial sums of residuals with a long-run variance estimate. Using ADF/DF-GLS and KPSS together gives four possible outcomes:

4. Structural breaks

Known break date (Chow test). For a break at \(T_b\), interact all regressors with a post-break dummy and test the interactions jointly with a robust Wald test. The break date must be chosen without looking at the data. Choosing the date where the series visibly changes invalidates the critical values.

Unknown break date (Andrews, 1993; Quandt). Compute the Chow statistic \(F(\tau)\) for every candidate break fraction \(\tau\in[\tau_0,1-\tau_0]\), typically with 15% trimming, and use

which has a non-standard limit (a supremum of a squared Brownian-bridge process) with tabulated critical values. Andrews and Ploberger (1994) propose average and exponential versions with optimal power properties. The break date estimator \(\widehat T_b=\arg\min SSR(T_b)\) is consistent at rate \(T\).

Multiple breaks (Bai and Perron, 1998, 2003). These methods sequentially test \(\ell\) versus \(\ell+1\) breaks, estimate break dates by global minimization of the SSR using dynamic programming, and provide confidence intervals for the break dates.

Parameter instability without discrete breaks. CUSUM tests of recursive residuals (Brown, Durbin and Evans, 1975) and the Nyblom (1989) test detect gradual drift. Time-varying-parameter models estimate it directly.

5. Breaks and unit roots

Perron (1989) showed that a stationary series with a one-time break in level or trend looks highly persistent to an ADF test that ignores the break. The test is then biased toward non-rejection of the unit root. Allowing for the 1929 crash and the 1973 oil-price slowdown, he rejected unit roots in many of the Nelson–Plosser (1982) series. When the break date is unknown, treating it as known biases the test the other way. Zivot and Andrews (1992) and Lumsdaine–Papell search over break dates under the unit-root null, with appropriately larger critical values.

Practical conclusion. Unit-root and break tests are complements. Persistence, breaks, nonlinearity (for example, threshold adjustment) and long memory (fractional integration) can all produce similar-looking data. Choose the model that the economics supports, and show robustness to the alternatives.

import numpy as np
from statsmodels.tsa.stattools import adfuller, kpss

rng = np.random.default_rng(10)
T, R = 200, 2000
# 1. DF distribution under the null (constant only)
tstats = [adfuller(np.cumsum(rng.normal(size=T)), maxlag=0, regression="c", autolag=None)[0] for _ in range(R)]
print("simulated 5% critical value:", np.quantile(tstats, 0.05).round(2), "(tabulated -2.88 for T=200)")

# 2. power against phi = 0.95
def ar1(phi):
    y = np.zeros(T)
    for t in range(1, T):
        y[t] = phi * y[t - 1] + rng.normal()
    return y
power = np.mean([adfuller(ar1(0.95), regression="c", autolag="AIC")[1] < 0.05 for _ in range(500)])
print("ADF power vs phi=0.95, T=200:", power)

# 3. stationary series with a level break
y = ar1(0.5) + 3 * (np.arange(T) > 100)
print("ADF p-value ignoring break:", round(adfuller(y, regression="c")[1], 3),
      "| KPSS p-value:", kpss(y, regression="c", nlags="auto")[1])

import numpy as np, statsmodels.api as sm
from statsmodels.tsa.stattools import adfuller, kpss, zivot_andrews
y = np.log(sm.datasets.macrodata.load_pandas().data["realgdp"].to_numpy())
# TODO: adfuller(y, regression="ct", autolag="AIC"); kpss(y, regression="ct", nlags="auto")

import numpy as np, statsmodels.api as sm
from statsmodels.tsa.stattools import adfuller, kpss, zivot_andrews
m = sm.datasets.macrodata.load_pandas().data
y = np.log(m["realgdp"].to_numpy()); dy = np.diff(y)
a = adfuller(y, regression="ct", autolag="AIC"); k = kpss(y, regression="ct", nlags="auto")
print(f"log GDP:   ADF(ct) t = {a[0]:.2f} (5% cv {a[4]['5%']:.2f}, p = {a[1]:.3f}, lags {a[2]});  KPSS(ct) = {k[0]:.3f} (5% cv {k[3]['5%']})")
a2 = adfuller(dy, regression="c", autolag="AIC"); k2 = kpss(dy, regression="c", nlags="auto")
print(f"growth:    ADF(c)  t = {a2[0]:.2f} (p = {a2[1]:.4f});  KPSS(c) = {k2[0]:.3f} (5% cv {k2[3]['5%']})")
za = zivot_andrews(y, regression="ct", autolag="AIC")
bp = int(za[4]); yr = m['year'].iloc[bp]; q = m['quarter'].iloc[bp]
print(f"Zivot-Andrews (break in level and trend): stat = {za[0]:.2f}, p = {za[1]:.3f}, break at {int(yr)}Q{int(q)}")

log GDP:   ADF(ct) t = -2.38 (5% cv -3.43, p = 0.389, lags 2);  KPSS(ct) = 0.207 (5% cv 0.146)
growth:    ADF(c)  t = -6.97 (p = 0.0000);  KPSS(c) = 0.324 (5% cv 0.463)
Zivot-Andrews (break in level and trend): stat = -3.70, p = 0.697, break at 1968Q2

import numpy as np, statsmodels.api as sm
from statsmodels.tsa.stattools import adfuller
d = sm.datasets.nile.load_pandas().data
y = d["volume"].to_numpy(); years = d["year"].astype(int).to_numpy()
# TODO: for each break b, F = (SSR_r - SSR_u) / (SSR_u / (n - 2)) with a one-mean vs two-means model

import numpy as np, statsmodels.api as sm
from statsmodels.tsa.stattools import adfuller
d = sm.datasets.nile.load_pandas().data
y = d["volume"].to_numpy(); years = d["year"].astype(int).to_numpy(); n = len(y)
ssr_r = ((y - y.mean())**2).sum()
F = {}
for b in range(int(0.15 * n), int(0.85 * n)):
    ssr_u = ((y[:b] - y[:b].mean())**2).sum() + ((y[b:] - y[b:].mean())**2).sum()
    F[b] = (ssr_r - ssr_u) / (ssr_u / (n - 2))
b = max(F, key=F.get)
print(f"sup-F = {F[b]:.1f} at a break starting in {years[b]} (Andrews 5% cv ~ 8.7)")
print(f"mean flow before: {y[:b].mean():.0f}, after: {y[b:].mean():.0f}")
a1 = adfuller(y, regression="c", autolag="AIC")
z = y.copy().astype(float); z[:b] -= y[:b].mean(); z[b:] -= y[b:].mean()
a2 = adfuller(z, regression="n", autolag="AIC")
print(f"ADF raw series:            t = {a1[0]:.2f}, p = {a1[1]:.3f}")
print(f"ADF break-adjusted series: t = {a2[0]:.2f}, p = {a2[1]:.4f}")

sup-F = 75.9 at a break starting in 1899 (Andrews 5% cv ~ 8.7)
mean flow before: 1098, after: 850
ADF raw series:            t = -4.05, p = 0.001
ADF break-adjusted series: t = -8.38, p = 0.0000