Unit Root Inference

Definition

Unit root inference concerns the problem of drawing conclusions about the autoregressive parameter $\rho$ in dynamic models when $\rho$ may be at or near 1. In classical statistics, the sampling distribution of the maximum likelihood estimator (MLE) $\hat\rho$ changes discontinuously at $\rho=1$ (Dickey-Fuller 1979), producing non-standard asymptotic distributions that require separate tables. In Bayesian statistics with a flat prior, the posterior is proportional to the likelihood, which retains the same Gaussian shape whether or not $\rho=1$ . The two approaches give qualitatively different — and in some cases opposite — probability statements about the same data.

Key Ideas

The likelihood function in an autoregressive (AR) model is Gaussian-symmetric around $\hat\rho$ , exactly as in a static regression, regardless of whether $\rho=1$ .
The asymmetric Dickey-Fuller distribution is a property of the MLE's sampling distribution, not of the likelihood or the Bayesian posterior.
Flat-prior Bayesian analysis leads to standard symmetric posteriors; using Dickey-Fuller (DF) p-values as posterior probabilities implicitly imposes a prior that irrationally favors explosive $\rho > 1$ values.
The likelihood principle (Berger-Wolpert 1984) states that inference should be based on the likelihood function, not on the sampling behavior of estimators; AR models satisfy the principle even at the unit root boundary.

How It Works

The Model and Setup

Consider the AR(1) model $y_t = \rho\, y_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2)$ with the ordinary least squares (OLS) estimator $\hat\rho = \frac{\sum_{t=1}^T y_{t-1} y_t}{\sum_{t=1}^T y_{t-1}^2}$ Conditional on $y_0$ and with $\sigma^2$ known, the likelihood is Gaussian in $\rho$ regardless of whether $\rho < 1$ , $\rho = 1$ , or $\rho > 1$ . This is a classical result that follows from the Gaussianity of the disturbances.

The Helicopter Tour (Sims-Uhlig 1991)

Treat $\rho$ as having a flat (uniform) prior over $(0.80, 1.10)$ and construct the joint probability density function (pdf) of $(\rho, \hat\rho)$ by Monte Carlo (31 values of $\rho$ ; 10,000 simulations each; $T = 100$ ). The joint pdf forms a 3D surface. Two cross-sections reveal the classical/Bayesian contrast:

Cross-section	Direction	Shape	Interpretation
Fix $\rho = 1$ , vary $\hat\rho$	Along $\rho=1$ plane	Asymmetric, skewed left	Classical sampling distribution of $\hat\rho$ under unit root
Fix $\hat\rho = 1$ , vary $\rho$	Along $\hat\rho=1$ plane	Symmetric around $\rho=1$	Posterior $p(\rho\mid\hat\rho=1)$ under flat prior

Both cross-sections are views of the same surface; the asymmetry is a property of how you slice it, not of the underlying likelihood.

DF P-values vs. Posterior Probabilities

For an observed $\hat\rho = 0.95$ with $T = 100$ :

Hypothesis	DF one-sided p-value	Flat-prior posterior probability
$\rho = 0.9$ (lower tail)	0.04 (reject at 5%)	$P(\rho < 0.9\mid\hat\rho=0.95) \approx 0.07$
$\rho = 1.0$ (upper tail)	0.12 (fail to reject)	$P(\rho > 1.0\mid\hat\rho=0.95) \approx 0.07$

The posterior is symmetric: deviations of equal magnitude above and below $\hat\rho$ are equally (un)likely. The DF test procedure treats them very differently.

The Implicit Prior

If a researcher systematically interprets DF p-values as posterior probabilities, the implied prior $\pi(\rho|\hat\rho)$ can be recovered by comparing the pseudo-posterior to the flat-prior posterior. For each observed $\hat\rho$ :

The implied prior puts 2–3 $\times$ more weight on $\rho \approx 1$ than on $\rho \approx 0.9$ .
The implied prior is sample-dependent: it changes with $\hat\rho$ , so no single prior rationalizes the procedure.
The implied prior keeps increasing into $\rho > 1$ without bound — it assigns greater prior probability to explosive dynamics than to moderate stationarity.

The Likelihood Principle Implication

The likelihood principle asserts that all information about $\rho$ from the data is contained in the likelihood function $L(\rho \mid y)$ . Since $L(\rho \mid y)$ is symmetric around $\hat\rho$ in AR models, any inference satisfying the likelihood principle must treat $\rho = \hat\rho + \delta$ and $\rho = \hat\rho - \delta$ as equally supported by the data for any $\delta$ . The Dickey-Fuller framework violates this because it incorporates information about the behavior of $\hat\rho$ in samples other than the one observed.

Why It Matters

Tail sensitivity (Geweke 1993): Unit root conclusions in the Nelson-Plosser (1982) macroeconomic series are not robust to the assumption of Gaussian disturbances. Replacing normality with a Student-t likelihood (estimated degrees of freedom typically 3–7) systematically reduces posterior odds in favor of difference stationarity; the trend-coefficient posterior standard deviation (SD) shrinks as well. The same data, under a different tail assumption, can support appreciably different conclusions. See Geweke (1993).
Reporting practice: Conventional $t$ and $F$ statistics describe the likelihood shape correctly in autoregressions; they should be reported even when DF corrected p-values are also provided.
Model simplification decisions: Whether to difference, allow for cointegration, etc., should be based on the posterior probability of near-unit-root behavior, not on DF p-values that carry implicit explosive priors.
Bayesian priors for macro models: Flat or near-flat priors on $\rho$ are well-behaved near the unit root boundary; the apparent "non-standard asymptotics" are an artifact of the frequentist framing.
Connects to Sims (1988): The graphical and quantitative results here extend the analytical argument in Sims (1988) "Bayesian Skepticism on Unit Root Econometrics."

Dickey-Fuller as a Special Case (Sims-Stock-Watson 1990)

Sims, Stock, and Watson (1990) show that the Dickey-Fuller unit-root test is a special case of their general canonical-regressor framework (Section 5). In the univariate AR( $p$ ) model, the F-statistic for the unit-root null $H_0: \delta_3 = 1$ (where $\delta_3$ is the autoregressive coefficient on the lagged level) has the limiting distribution:

$F \Rightarrow \frac{\left[\int_0^1 W(t)\,dW(t)\right]^2}{\int_0^1 W(t)^2\,dt}$

where $W$ is a standard Wiener process. This is the nonstandard Dickey-Fuller distribution. It arises because the restriction $\delta_3 = 1$ constrains the coefficient on the I(1) canonical regressor $Z^3_t$ , not on a mean-zero stationary canonical regressor — exactly the condition identified in Theorem 2 as producing a nonstandard limit. The classical Dickey-Fuller asymptotic theory thus emerges as a consequence of the general principle rather than a sui generis result.

Prior Robustness in Real Data: OECD Unemployment (Summers 2002)

Summers (2002) applies Bayesian unit root inference to unemployment rates in 16 Organisation for Economic Co-operation and Development (OECD) countries, using five priors on $\rho$ : N(1,1000), N(1,0.5), Lubrano's (1995) extended Beta, Beta(10,2), and Berger-Yang (1994). Posterior draws from the base N(1,1000) prior are reweighted to alternative priors via importance sampling (Geweke 1999): $\{\rho_j^i\} = \{\rho^i \cdot f_j(\rho)/f_N(\rho)\}$ — no additional MCMC required.

The key empirical finding is that unlike the stylized AR(1) example in Sims-Uhlig (1991), OECD unemployment posteriors are nearly identical across all five priors. For 12 of 16 countries, $\geq 94\%$ of the posterior lies below $\rho = 0.95$ regardless of prior. Only France shows clear unit-root evidence (median $\rho \approx 0.98$ , 24–32% mass above 1.0); Italy and Spain are borderline.

This illustrates an important distinction: prior sensitivity in unit root inference is high in data-poor or misspecified settings (stylized AR(1) with no structural breaks) but low when the data are informative — as with real macroeconomic series modelled with structural breaks. The theoretical case (unemployment is bounded) and the empirical posterior both point toward stationarity.

'Objective' Posterior-Odds Tests (Koop 1992a)

Koop (1992a) turns the Bayesian critique into a usable testing recipe. Rather than reporting a Dickey-Fuller p-value, he computes posterior odds comparing a difference-stationary (unit-root) model against stationary and trend-stationary alternatives — a Bayes factor weighted by model priors — so the output is a direct probability statement about the competing hypotheses. To avoid the subjectivity and computational burden of the informative priors in Koop (1991), he uses two classes of reference priors that are informative enough to yield well-defined odds but need minimal subjective input (hence "objective"). Applied to the Nelson-Plosser and Shiller series (and a Monte Carlo study), the odds show that the classical failure to reject a unit root — a consequence of low power against trend-stationary alternatives — is weak evidence: the data frequently do not strongly favour difference over trend stationarity. The reframing from "reject the unit root?" to "how do the odds compare?" is the same move Sims (1988) argued for analytically, now delivered as a reproducible procedure.

Open Questions

The model analyzed (AR(1), known $\sigma^2$ , condition on $y_0$ ) is deliberately simplified. The extension to multivariate systems (vector autoregression, VAR), unknown $\sigma^2$ , and multiple lags involves a higher-dimensional joint pdf that is harder to visualize but the qualitative conclusion extends.
The practical significance of the explosive-prior implication depends on whether applied researchers actually interpret DF p-values as posteriors. If they are used purely for size-correct hypothesis testing, the critique has less force.
Uhlig (1994) extended the Bayesian perspective to the multivariate case, arguing for agnosticism about exact unit roots in macro models.