Unit Root Inference

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Definition

Unit root inference concerns the problem of drawing conclusions about the autoregressive parameter ρ\rho in dynamic models when ρ\rho may be at or near 1. In classical statistics, the sampling distribution of the maximum likelihood estimator (MLE) ρ^\hat\rho changes discontinuously at ρ=1\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 ρ=1\rho=1. The two approaches give qualitatively different — and in some cases opposite — probability statements about the same data.

Key Ideas

How It Works

The Model and Setup

Consider the AR(1) model yt=ρ yt−1+εt,εt∼N(0,σ2)y_t = \rho\, y_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2) with the ordinary least squares (OLS) estimator ρ^=∑t=1Tyt−1yt∑t=1Tyt−12\hat\rho = \frac{\sum_{t=1}^T y_{t-1} y_t}{\sum_{t=1}^T y_{t-1}^2} Conditional on y0y_0 and with σ2\sigma^2 known, the likelihood is Gaussian in ρ\rho regardless of whether ρ<1\rho < 1, ρ=1\rho = 1, or ρ>1\rho > 1. This is a classical result that follows from the Gaussianity of the disturbances.

The Helicopter Tour (Sims-Uhlig 1991)

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

Cross-section Direction Shape Interpretation
Fix ρ=1\rho = 1, vary ρ^\hat\rho Along ρ=1\rho=1 plane Asymmetric, skewed left Classical sampling distribution of ρ^\hat\rho under unit root
Fix ρ^=1\hat\rho = 1, vary ρ\rho Along ρ^=1\hat\rho=1 plane Symmetric around ρ=1\rho=1 Posterior p(ρ∣ρ^=1)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 ρ^=0.95\hat\rho = 0.95 with T=100T = 100:

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

The posterior is symmetric: deviations of equal magnitude above and below ρ^\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 π(ρ∣ρ^)\pi(\rho|\hat\rho) can be recovered by comparing the pseudo-posterior to the flat-prior posterior. For each observed ρ^\hat\rho:

The Likelihood Principle Implication

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

Why It Matters

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(pp) model, the F-statistic for the unit-root null H0:δ3=1H_0: \delta_3 = 1 (where δ3\delta_3 is the autoregressive coefficient on the lagged level) has the limiting distribution:

F⇒[∫01W(t) dW(t)]2∫01W(t)2 dtF \Rightarrow \frac{\left[\int_0^1 W(t)\,dW(t)\right]^2}{\int_0^1 W(t)^2\,dt}

where WW is a standard Wiener process. This is the nonstandard Dickey-Fuller distribution. It arises because the restriction δ3=1\delta_3 = 1 constrains the coefficient on the I(1) canonical regressor Zt3Z^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 ρ\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): {ρji}={ρi⋅fj(ρ)/fN(ρ)}\{\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, ≥94%\geq 94\% of the posterior lies below ρ=0.95\rho = 0.95 regardless of prior. Only France shows clear unit-root evidence (median ρ≈0.98\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.

Open Questions

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