# Approximate Bayes factors for unit root testing

Martin Magris Corresponding author<EMAIL_ADDRESS>Submitted for
consideration at the 2021 annual conference of the International Association
for Applied Econometrics (IAAE). Department of Electrical and Computer
Engineering, Aarhus University, Denmark Alexandros Iosifidis  Department of
Electrical and Computer Engineering, Aarhus University, Denmark

###### Abstract

This paper introduces a feasible and practical Bayesian method for unit root
testing in financial time series. We propose a convenient approximation of the
Bayes factor in terms of the Bayesian Information Criterion as a
straightforward and effective strategy for testing the unit root hypothesis.
Our approximate approach relies on few assumptions, is of general
applicability, and preserves a satisfactory error rate. Among its advantages,
it does not require the prior distribution on model’s parameters to be
specified. Our simulation study and empirical application on real exchange
rates show great accordance between the suggested simple approach and both
Bayesian and non-Bayesian alternatives.

Keywords: Unit root inference; Bayesian analysis; Bayes factor; BIC
JEL classification: C11; C12; C22

##  1 Introduction

In time series analysis the interest is often over some persistence properties
of the process in the medium-long term. In this regard, unit roots are
important since they relate to persistence of shocks and non-mean-reverting
dynamics (e.g. Campbell and Perron, , 1991). In other words, unit roots go in
hand with non-stationarity. This has severe implications in the applicability
of standard econometrics techniques, among which spurious regression (Granger
et al., , 1974; Phillips, , 1986) is the most notable one. Non-stationarity
appears to arise quite commonly in economic time series, and with little
surprise unit root testing has been one of the most proficient research areas
in econometrics. Most theoretical and empirical work in the domain of non-
stationary time series relies on classical frequentist methods, which stand as
the reference approach.

Among others, (Zellner, , 1971; Geweke, , 1988; Poirier, , 1988) showed that
the Bayesian framework is in general well-suited to inferential problems in
econometrics, and it was since (Sims, , 1988) that a number of Bayesian
methods for unit root testing and the corresponding Bayesian unit root
literature developed. The earliest works appear remarkably optimistic and
confident about the impact that a Bayesian approach could have in unit root
testing, even claiming an overall superiority of the Bayesian approach over
classical methods (Sims and Uhlig, , 1991). On the contrary, the adoption and
development of Bayesian methods would shortly appear to be much less
straightforward and very debated. It was in (Phillips, 1991b, ) that a
reconciliation with frequentist methods was first discussed under the light of
impartial and objective Bayesian methods. This work raised several further
issues related to the Bayesian approach in unit root testing that stimulated
active research and strong debates. (Phillips, 1991a, ; Phillips, 1991b, )
advanced the idea that the achievement of impartial Bayesian analysis through
flat priors on the parameters is not well-suited in time series models.
Actually, flat priors over the autoregressive parameter achieve the opposite
effect and have shown to be quite informative (e.g. Kim and Maddala, , 1991;
Schotman and Van Dijk, 1991b, ; Phillips, 1991b, ; Leamer, , 1991, among the
others). A long debate on appropriate uninformative priors for Bayesian unit
root inference followed (see Section 2). The determination of suitable input
information through the prior distribution, which is generally the major
reason for the divergence between the classical and Bayesian approach, is
tightly bonded to the exact hypothesis being tested. For a simple
autoregressive process of order one, hereafter denoted by AR(1), the unit root
inference problem corresponds to testing the null hypothesis of the
autoregressive parameter being equal to one. In general, we shall emphasize
the exact/point-wise/non-interval nature of an hypothesis by referring to it
as point hypothesis. The goal of testing a point null hypothesis cannot be
easily achieved with continuous priors, as the consequent continuous posterior
would assign zero weight to the unit root hypothesis. Feasible tests can
either use discontinuous priors that assign a non-zero mass to the unit root
hypothesis and distribute the reaming one over some interval (e.g. Schotman
and Van Dijk, 1991a, ; DeJong and Whiteman, , 1991), or test closely-related
non-point hypotheses with continuous priors (Koop, , 1994, is instructive as
it considers three possible nulls). The first approach is susceptible to poor
objectivity (Phillips et al., , 1993) while the second one does not properly
match the exact and exclusive purpose of testing the unit root hypothesis
(Schotman and Van Dijk, 1991b, ). Consequently, the Bayesian analysis can be
respectively based on two radically different approaches: on Bayes factors and
odds ratios, or Bayes confidence sets and probability intervals over the
posterior, with the first one being of difficult interpretation in terms of
traditional p-values (Berger and Delampady, , 1987). Furthermore, the
inclusion of an intercept, a trend, or any richer structure in a simple AR(1)
model, are not smooth extensions, as prior beliefs on the autoregressive
parameter generally change according to the particular deterministic component
added to the model (Schotman and Van Dijk, 1991b, ). Also, the particular form
under which a model is expressed, e.g. “structural” or “reduced” from, plays a
role in unveiling feasible directions for the Bayesian analysis. Not less
importantly, (Schotman and Van Dijk, 1991b, ; Uhlig, , 1994; Lubrano, , 1995)
underline the importance of the conditioning set, and in particular the
sensitivity of the whole inferential procedure on the first observation.
Lastly, Bayesian methods are generally known for not being keen on simple
algebra: also in unit root inference numerical methods are often required for
achieving approximate solutions (e.g. Zivot, , 1994).

In this paper, we propose an approximate Bayesian unit root testing procedure
that mitigates the above-mentioned criticalities, especially the choice of the
prior. In particular, we focus on the simple AR(1) dynamics, which is a
fundamental process and a major ingredient in the theory of non-stationary
time series and unit root testing, and on testing the point null unit root
hypothesis. We apply standard approximation results to obtain a generic
Bayesian testing procedure based on approximate Bayes factors and thus
approximate posterior odds. With an asymptotic error rate sufficient to
guarantee the applicability of the proposed method also for samples of
moderate size, the approximate form of the Bayes factor is independent on the
choice of the priors, it is remarkably simple to compute, and scales to more
complex models as long as their maximum likelihood estimates are attainable.
Indeed, our approximate Bayes factor is formulated as a simple function of the
well-known Bayesian Information Criteria (BIC), being thus easy to implement
and attractive for empirical research. Although the BIC approximation of Bayes
factors has proved to be a valid tool in different fields, its use in unit
root testing has not been investigated, though it appears to be very well-
suited for this class of problems. In the empirical sections, we propose a
Monte Carlo experiment and an empirical application on real exchange rates,
and analyze the performance of our BIC approach with respect to other
competing Bayesian methods and the most widespread frequentist Dickey-Fuller
test (Dickey and Fuller, , 1981). Our experiment validates the proposed
approach, as it stands out as a feasible and viable simple alternative for
Bayesian unit root inference.

This paper is organized as follows. Section 2 reviews the literature on
approaches, issues and advances in Bayesian unit root testing, underlying the
numerous problems that arise in this context. Section 3 introduces Bayes
factors and generally discusses the Bayesian testing framework based on
evidence. Section 4 introduces our suggested testing approach based on the BIC
approximation of Bayes factors. Section 5 reports the results of the Monte
Carlo study, while an empirical application on real exchange rates is
presented in Section 6. Section 7 concludes and suggests directions for future
research. The Appendix collects details on some benchmark Bayesian unit root
tests and further results from the simulation study.

##  2 Bayesian unit root testing

On the premise that the asymptotic distribution theory changes discontinuously
between the stationary and unit root case, with the classical hypothesis
testing appearing as a not reasonable inferential procedure opposed to a
Bayesian flat prior, the Bayesian analysis of unit root models was first
suggested in (Sims, , 1988). Since then, a wide and rich literature on the
field pinpointed the advantages and flaws of the Bayesian approach making it
complex and long-debated. First and most notably, the identification of a
suitable prior is in this context remarkably difficult and widely discussed.
Besides this, there are several issues associated with model specification and
formulation, the role played by the initial observation, issues related to the
invariance of the prior under different sampling frequencies, and
computational arguments. In the following, we review the most relevant aspects
of these issues. For a more comprehensive overview on the topic see e.g.
(Maddala and Kim, , 1998, Ch. 8), and the articles in following dedicated
special issues: Journal of Applied Econometrics (1991, vol. 6, n.4),
Econometric Theory (1994, vol. 10) and Journal of Econometrics (1995, vol. 69,
n.1).

###  2.1 The choice of the prior

Here we shortly outline the setting based on flat priors adopted, among the
others, by (Sims and Uhlig, , 1991; Sims, , 1988; Geweke, , 1988; Thornber, ,
1967; Zellner, , 1971; Schotman and Van Dijk, 1991b, ). Consider the simple
AR(1) model:

$x_{t}=\rho x_{t-1}+u_{t}\text{,}$

and assume a flat prior for $\left(\rho,\sigma\right)$,
$\pi\left(\rho,\sigma\right)\propto 1/\sigma$ with $-1<\rho<1$, and $\sigma>0$
being the standard deviation of the normal i.i.d. error $u$. $\rho$ is the
autoregressive parameter and $\rho=1$ corresponds to the unit root hypothesis
of interest, under which the AR(1) model reduces to a Brownian motion. Be
$x_{0}$ the initial starting value of $T$ consecutive observations. With
Gaussian likelihood

$L\left(x|\rho,\sigma,x_{0}\right)=\left(2\pi\right)^{-T/2}\sigma^{-T}\exp\left(-\frac{\sum_{t=1}^{T}\left(x_{t}-\rho
x_{t-1}\right)^{2}}{2\sigma^{2}}\right)$

the joint posterior for $\left(\rho,\sigma\right)$ is given by

$\pi\left(\rho,\sigma|x,x_{0}\right)\propto\sigma^{-T-1}\exp\left(-\frac{\sum_{t=1}^{T}\left(x_{t}-\rho
x_{t-1}\right)^{2}}{2\sigma^{2}}\right)=\sigma^{-T-1}\exp\left[-\frac{R+\left(\rho-\hat{\rho}\right)^{2}Q}{2\sigma^{2}}\right]\,\text{,}$

with $\hat{\rho}=\sum x_{t}x_{t-1}/\sum x_{t-1}^{2}$ being the OLS estimate
for $\rho$,
$R=\sum\hat{\epsilon}^{2}_{t}=\sum\left(x_{t}-\rho\hat{x_{t}}\right)^{2}$ the
residual sum of squares, and $Q=\sum x_{t-1}^{2}$. For the above joint
posterior, one obtains the following margins:

$\displaystyle\pi\left(\rho|x,x_{0}\right)$
$\displaystyle\propto{R+\left(\rho-\hat{\rho}\right)^{2}Q}^{-T/2}$
$\displaystyle\pi\left(\sigma|x,x_{0}\right)$
$\displaystyle\propto\sigma^{-T}\exp\left(-\frac{R}{2\sigma^{2}}\right)\text{.}$

Our notation distinguishes between priors and posteriors depending on the
conditioning set: $\pi\left(\cdot\right)$ opposed to
$\pi\left(\cdot|\text{data}\right)$, respectively.

The marginal posterior for $\rho$ has the form of a symmetric univariate
t-distribution, centered around the OLS estimate $\hat{\rho}$, while the
marginal posterior for $\sigma$ is an inverted gamma-2 distribution (Zellner,
, 1971). Sims and Uhlig,  (1991) conclude that classical methods relying on
asymmetric distributions of the OLS estimator of $\rho$, such as the Dickey-
Fuller statistics, attribute too much weight to large values of $\rho$, while
the above Bayesian framework based on flat prior is a more logical and sounder
basis for inference than classical testing. This argument is advanced by
comparing the distributions $\rho|\hat{\rho}=1$ and $\hat{\rho}|\rho=1$, with
the first being the posterior distribution of the true parameter with the
estimated parameter taken as given (Bayesian approach), and the second being
the sampling distribution of the estimated parameter under the value of the
true parameter (classical approach). While classical methods are generally
based on an asymmetric and nonstandard distribution for the autoregressive
parameter, Bayesian methods lead to a symmetric and standard posterior. The
asymmetry in $\hat{\rho}|\rho=1$ drives the argument that classical procedures
based on p-values are misleading.

A similar approach is that developed in (Schotman and Van Dijk, 1991b, ), see
Appendix A.1 for a detailed description. This appears to be the most
widespread setting for Bayesian unit root testing, commonly referenced also in
the recent literature and here used as a benchmark. It takes
$\rho\in\\{S,1\\}$, $S=\\{\rho|-1<a\leq\rho<1\\}$, and specifies the priors
for $\rho$ and $\sigma$ as

$\Pr\left(\rho=1\right)=\pi_{0}\,\text{,}\qquad\text{Pr}\left(\rho|\rho\in
S\right)=\frac{1}{1-a}\,\text{,}\qquad\text{Pr}\left(\sigma\right)\propto\frac{1}{\sigma}\,\text{.}$

That is, $\rho$ is taken uniform over $S$ and with probability mass $\pi_{0}$
on $\rho=1$, $\sigma$ and $\rho$ are independent. The mass at $\rho=1$ is
intended to allow for a feasible testing of the null hypothesis $H_{0}:\rho=1$
(see Section 2.2). For clarity, we shall refer to such an exact/point-
wise/non-interval null hypothesis as point null. Restrictions over the domain
of $\rho$ are also adopted in (Geweke, , 1988) and (DeJong and Whiteman, ,
1991), with the latter also providing an empirical analysis and a comparative
study over the classical approach on the Nelson-Plosser data. The arbitrarity
in selecting the restricted domain for the autoregressive parameter and the
values of the statistics supporting a unit root decision are pointed out and
criticized in (Sowell, , 1991) and (Phillips, 1991b, ).

Opposed to the use of non-flat priors like Normal-Wishart conjugates, which
are known to be informative about the properties of the model (Zellner, ,
1971; Phillips, 1991b, ) and that correspond to a prior belief that explosive
roots are unlikely when centered around the unit root (Uhlig, , 1994), the
choice of flat priors is generally attractive. This is due to that flat priors
often appear as a suitable approach for attributing a degree of “neutrality”
or “objectivity” to Bayesian analyses, while being convenient in terms of
computations, often leading to algebraic solutions (e.g. Zellner, , 1971). The
early Bayesian inference as of (Sims, , 1988) strongly favors the above
approach, but the use of flat priors is not innocuous as it may appear.
Phillips, 1991b raised concerns on uniform priors since the inference on
$\rho$ is conditional on the observed sample moments and sufficient
statistics, which depend on the value taken by $\rho$ and are radically
different for $\rho=1$ and $|\rho|<1$. The use of flat priors does not
correspond to uninformativeness and indeed is shown to downweight large values
of $\rho$, i.e. of unit roots and explosive processes. This is because when
$|\rho|$ is large, the data is more informative about $\rho$ and treating all
the values of $\rho$ as equally-likely implicitly corresponds to downweighting
large values of $\rho$. Consequently, the testing strategy based on regression
with or without unit roots taken with the same likelihood irrespectively of
the value of $\rho$ is inadequate. Furthermore, Phillips, 1991b finds that
discrepancy in the results between standard and Bayesian methods in unit root
testing for macroeconomic US time series is largely due to the use of the
misleading flat prior.

As an objective setting (Phillips, 1991b, ) proposed a Jeffreys’ prior
(Jeffreys, , 1946; Perks, , 1947). Jeffreys’ priors (often called “ignorance
priors”) are defined up to a proportionality factor as $\propto\sqrt{|i|}$,
where $|i|$ is the determinant of the expected Fisher information matrix $i$.
Jeffreys’ priors render the posterior invariant under one-to-one
reparametrizations and enjoy a number of desirable properties (Ly et al., ,
2016). Such priors are often interpreted as reflecting the properties of the
sampling process and emphasizing data evidence, being interpretable as
equivalent to the information that a typical single observation, on average,
provides (Xia and Griffiths, , 2012). Yet, the general use of the Jeffrey’s
prior with respect to the flat prior and its ability to convey a state of
“ignorance” about the existence of a unit root has been readily argued in
(Leamer, , 1991). Furthermore, in an extensive Monte Carlo study Kim and
Maddala,  (1991) find that the above approach favors big values of $\rho$
distorting the sample evidence. (Uhlig, , 1994) finds however that for the
univariate AR(1) model the major differences between flat and informative
priors are limited to the explosive regions. Further details on the approach
of (Phillips, 1991b, ) are provided in Appendix A.2.

###  2.2 The null hypothesis

As argued in (Schotman and Van Dijk, 1991b, ), a genuine and exclusive
interest in the unit root hypothesis formally corresponds to testing the null
$\rho=1$. Therefore, this hypothesis should be preferred for a Bayesian
treatment of unit root inference. If a continuous prior density for $\rho$ is
to be adopted, the probability associated with $\rho=1$ is zero and so its
posterior probability (e.g. Zivot, , 1994). In other words, testing a point
null $\rho=1$ – which corresponds to the only exact hypothesis on the
existence of a unit root – is not trivial. Indeed, a continuous prior
$\pi\left(\rho\right)$ requires a full Bayesian analysis for retrieving the
corresponding posterior $\pi\left(\rho|x,x_{0}\right)\propto
L\left(x|\rho,x_{0}\right)\pi\left(\rho\right)$, for some data $x$ and initial
value $x_{0}$.

A posterior confidence interval at the probability level $\alpha$ is a subset
$C_{\alpha}$ such that

$\text{Pr}\left(C_{\alpha}|x,x_{0}\right)=\int_{C_{\alpha}}\pi\left(\rho|x,x_{0}\right)\,d\rho\geq
1-\alpha\,\text{,}$

where typically the subset $C_{\alpha}$ is chosen is such a way that its size
is minimal (highest posterior density criterion). The unit root hypothesis can
be rejected by checking whether $\rho=1$ does not belong to the posterior
confidence interval $C_{\alpha}$. However, it is not unlikely for the
posterior to be bimodal (e.g. Kim and Maddala, , 1991) and $C_{\alpha}$ may
result in a disconnected set. Since the interest is around the unit root, an
alternative is to redefine $C_{\alpha}$ so that

$\text{Pr}\left(C_{\alpha}|x,x_{0}\right)=\int_{\rho_{\inf}}^{\rho_{\sup}}\pi\left(\rho|x,x_{0}\right)\,d\rho\geq
1-\alpha\,\text{,}$

with $\rho_{\inf}$ ($\rho_{\sup}$) being any convenient truncation point of
the likelihood or $-\infty$ ($+\infty$). Alternatively, one can consider the
probability

$\Pr\left(\rho\geq
1|x,x_{0}\right)=\int_{1}^{+\infty}\pi\left(\rho|x,x_{0}\right)\,d\rho$ (1)

and decide to reject the unit root when its value is below a certain
threshold, say 5%. This however is not anymore a test on a point null
hypothesis as now $H_{0}$ jointly covers the unit and explosive roots cases:

$H_{0}:\rho\geq 1\qquad\qquad\qquad H_{1}:\rho<1\,\text{.}$

This is the setting adopted by (e.g. Koop, , 1994; DeJong and Whiteman, ,
1991; Phillips, 1991b, ).

The continuity problem and the impossibility for testing a point null could be
technically resolved by using a discontinuous prior. In fact, a natural
solution is to give $\rho=1$ a positive probability $\pi_{0}$ and assign to
the values of $\rho$ over some interval $S$ the density
$\left(1-\pi_{0}\right)\pi\left(\rho\right)$, where $\pi\left(\rho\right)$ is
a proper prior on $S$ (as e.g. in Schotman and Van Dijk, 1991b, ; DeJong and
Whiteman, , 1991). As in (Schotman and Van Dijk, 1991b, ; Zivot, , 1994) a
common choice for $S$ is the stationary region $|\rho|<1$. With this
procedure, a discontinuous prior can be easily adopted and the following
hypotheses can be tested

$H_{0}:\rho=1\qquad\qquad\qquad H_{1}:\rho\in S\,\text{.}$

In general, $H_{1}$ is not a generic non- unit root alternative since
explosive processes are ruled out, and neither a properly stationary
alternative, since the lower bound of $S$ could potentially be either within
or outside the stationary region. Furthermore, is it possible to draw a data-
driven criterion for selecting $S$ in the testing procedure (see Appendix
A.1). In this case, the specific formulation of the alternative depends on the
OLS estimate of $\rho$ and on the sample size. It is therefore the test itself
that defines the possible values of $\rho$ under $H_{1}$, and such implicitly
imposed restrictions on $S$ are de facto analogous to adopting a strong prior.

###  2.3 Other issues

In addition to the above difficulties, the whole inferential procedure is
remarkably sensitive on model formulation, and the effect of nuisance
parameters is not negligible.

Schotman and Van Dijk, 1991b warn on the effects of extending the AR(1) model
with trends and intercepts since the inclusion of such elements changes with
(or implicitly corresponds to) prior beliefs. Their study shows that the use
of Jeffreys’ priors as in (Phillips, 1991b, ) downweights the unit root
hypothesis relative to a flat prior in models with trend and intercept. By
using the alternative reduced-form parametrization of the AR model with trend
Schotman and Van Dijk, 1991b are able of explaining the observed bias towards
stationarity under flat priors.

Lately, Phillips, 1991b explains that such behavior is dependent on the
initial value and that conditioning the likelihood on the $x_{0}$ resolves the
issue. The apparently secondary role played by the initial value is thus shown
to have an enormous impact inference. Analyses in this regard can be found in
(Zivot, , 1994) and (Lubrano, , 1995). The latter shows that the treatment of
the first observation produces results that are more or less in accordance
with the classical results and that e.g. the fixed or random treatment of
$x_{0}$ does make a difference when an intercept is included or not (opposed
to the simple AR(1) as outlined in (Zellner, , 1971)). Following (Thornber, ,
1967), Lubrano,  (1995) extends the discussion suggesting the use of
uninformative Beta densities.

Further issues related to the choice of priors are interactions/correlations
between the different elements in multivariate parameters (i.e. appropriate
priors specifications with non-diagonal covariance matrices), their often
improper nature, and computational issues (e.g. Zivot, , 1994). Also, the
conceptual problem of adopting priors that are irrespective and insensible to
the sampling frequency of the observations has been pointed out e.g. in
(Leamer, , 1991) and (Sims, , 1991).

###  2.4 Some recent developments

The literature review would be certainly incomplete without references to more
recent works. Though the major advances in the theory of Bayesian unit root
testing come from the 20th century, Bayesian unit root testing keeps being an
area of active research.

Closely related to the above literature is the Full Bayesian Significance Test
(FBST) of de Bragança Pereira and Stern,  (1999). This procedure allows to
test for the point null unit root hypothesis, has no limiting requirements on
the prior, allows for flexible error distributions, applies to small sample
sizes, and is invariant with respect to models’ parametrization. Though
attractive, the FBST test has not gained popularity in Bayesian unit root
testing as the only application is that of (Diniz et al., , 2011) where the
FBST performance is compared against the Augmented Dickey-Fuller but not
against existing alternative Bayesian methods, in both their simulation study
and application.

Recently also unit root testing in Stochastic Volatility (SV) models where the
underlying volatility process is unobservable has attracted several research
contributions. So and Ll,  (1999) develop a Monte Carlo Markov Chain
approximation of the odds for certain SV models. Their method is improved in
(Li and Yu, , 2010) with a more robust algorithm and increased test power.
Extensions with leverage effects within an SV model on an AR(1) process are
considered in (Li et al., , 2012). Kalaylıoğlu and Ghosh,  (2009) focus on the
role played by priors in unit root Bayesian inference in SV models. They
introduce a class of non-informative priors to develop a testing procedure
that is feasible based on Gibbs-sampled approximations of the posterior but
unpractical for adopting Bayes factors. Simulation methods for approximating
posterior credibility intervals are also adopted in (Kalaylıoğlu et al., ,
2013), where correlations between returns’ series errors and the latent SV
with potential unit roots are allowed to be non-zero. Extensions for
heteroskedasticity are considered in (Chen et al., , 2013). Severe distortions
in the size of the Dickey-Fuller test statistics under unit roots in an AR(1)
with SV are reported in an extensive simulation study in (Zhang et al., ,
2013), where a Bayesian testing approach is introduced as a remedy.
Developments over more complex dynamics over the simple AR(1) model include
non-normal innovations (Hasegawa et al., , 2000), polynomial trends
(Chaturvedi and Kumar, , 2005), nonlinear smooth transitions (Chen and Lee, ,
2015) and structural breaks (Park and Shintani, , 2016; Vosseler, , 2016),
also in panel data (Kumar and Agiwal, , 2018).

Not strictly relevant for our analysis, yet closely related to Bayesian unit
root literature, are the dozens of research works on Bayesian cointegration
testing, appeared since (Koop, , 1991).

##  3 Evidence

Consider a random variable $X$ with density parametrized over
$\theta\in\Theta$. The hypothesis testing problem we are concerned about
consists of deciding among the null hypothesis $H_{0}:\theta=\theta_{0}$ and
the alternative $H_{1}:\theta\neq\theta_{0}$. This is achieved by considering
suitable measures of evidence of a hypothesis against the other, such as the
widespread p-value, or Bayes factors and Bayesian posterior probabilities.

###  3.1 P-value

Let us denote by $T\left(\cdot\right)$ a test statistics, and by
$t=T\left(x\right)$ its value when data $X=x$ is observed. The null hypothesis
$H_{0}$ is rejected in favor of the alternative $H_{1}$ if $T\left(x\right)$
is more extreme than one would expect if $H_{0}$ was true. By choosing a
significance level $\alpha$, $H_{0}$ is rejected when the probability of
$T\left(X\right)$ being greater than $T\left(x\right)$ is small (i.e. lower or
equal to $\alpha$), given that $H_{0}$ is true. Formally, the hypothesis
$H_{1}$ is accepted if

$\Pr\left(\lvert T\left(X\right)\rvert\geq T\left(x\right)\lvert
H_{0}\right)\leq\alpha\,\text{,}$

that is, extreme values of the statistics are deemed to provide evidence
against $H_{0}$.

While it is straightforward to identify in higher values of $|t|$ stronger
evidence against the null hypothesis, the problem of evaluating the strength
of evidence for $H_{0}$ against $H_{1}$ is left open. Frequentists use a scale
of evidence set from Fisher’s work in the 1920s: a common interpretation is
that $\alpha=0.99$ corresponds to very strong evidence, $\alpha=0.95$ to
strong evidence and so on, with neutral evidence at around $\alpha=0.9$. In
other words, the higher the evidence the lower the Type I error of rejecting
the true hypothesis. Bayesian literature provides different answers to the
problem, based on quantities known as Bayes factor and posterior odds.

###  3.2 Posterior odds for a point null

Considering the unknown model probabilities as random, the Bayes’ rule yields
posterior probabilities given the observed data. For the hypothesis $H_{1}$
(and analogously for $H_{0}$):

$\text{Pr}\left(H_{1}|x\right)=\frac{\Pr\left(x|H_{1}\right)\text{Pr}\left(H_{1}\right)}{\text{Pr}\left(x\right)}\,\text{,}$

where $\text{Pr}\left(H_{1}\right)$ is the prior probability of hypothesis
$H_{1}$ being true, and
$\text{Pr}\left(x\right)=\Pr\left(x|H_{0}\right)\text{Pr}\left(H_{0}\right)+\Pr\left(x|H_{1}\right)\text{Pr}\left(H_{1}\right)$
is the marginal density of $X=x$. The quantity $\Pr\left(x|H_{1}\right)$ is
referred to as marginal likelihood or marginal probability (of the data) under
$H_{1}$. In this context, the language and notation encountered in the
literature can be heterogeneous and slightly abused as it is common to
indistinctly use the terms (joint) “density”, “likelihood”, (joint)
“probability” and their corresponding notations, e.g. $f$, $L$, $\Pr$. Here we
adopt the notation and terminology of (Kass and Raftery, , 1995).

By the law of the total probability the marginal probability
$\Pr\left(x|H_{1}\right)$ is obtained by marginalizing out the parameter
$\theta_{1}$ under $H_{1}$:

$\Pr\left(x|H_{1}\right)=\int\Pr\left(x|\theta_{1},H_{1}\right)\pi\left(\theta_{1}|H_{1}\right)d\theta_{1}\,\text{,}$
(2)

where $\pi\left(\theta_{1}|H_{1}\right)$ is a continuous density. From a
frequentist perspective, $\pi\left(\theta_{1}|H_{1}\right)$ is a mere weight
function to allow the computation of the average likelihood. For a Bayesian,
$\pi\left(\theta_{1}|H_{1}\right)$ would be the prior density for $\theta_{1}$
conditional on $H_{1}$ being true (Berger and Delampady, , 1987). Note the
difference between the prior probability of a hypothesis or model being true
opposed to the prior density referring to its corresponding parameter.

The ratio of the posterior probabilities for the two hypotheses is referred to
as posterior odds ratio, or posterior odds:

$\frac{\text{Pr}\left(H_{0}|x\right)}{\text{Pr}\left(H_{1}|x\right)}=\frac{\Pr\left(x|H_{0}\right)}{\Pr\left(x|H_{1}\right)}\,\frac{\text{Pr}\left(H_{0}\right)}{\text{Pr}\left(H_{1}\right)}\,\text{.}$

Analogously, one may define the prior odds as
$\text{Pr}\left(H_{0}\right)/\text{Pr}\left(H_{1}\right)$. Posterior odds
quantify the evidence of $H_{0}$ over $H_{1}$ after data $x$ has been
observed. On the other hand, prior odds do not convey any evidence as they
solely quantify the prior plausibility of $H_{0}$ over $H_{1}$ before any data
is observed. The interpretation of odds ratios is straightforward as they
correspond to simple probability ratios. Odds ratios $K_{1}$ greater than one
(or $\log K_{1}>0$) stand for evidence in favour of the null hypothesis, with
a corresponding probability $K_{1}/\left(1+K_{1}\right)$. This is aligned with
the general Bayesian rationale. After observing $x$, the prior probability
$\text{Pr}\left(H_{0}\right)=\pi_{0}$ and corresponding prior odds
$K_{0}=\pi_{0}/\left(1-\pi_{0}\right)$ reflecting the prior belief on $H_{0}$
being the true model are updated into the posterior odds $K_{1}$ and the
corresponding new posterior probability for $H_{0}$ being true.

For a point hypothesis $H_{0}:\theta=\theta_{0}$, the assignment of a positive
probability will be rarely thought possible for $\theta=\theta_{0}$ to hold
exactly: this is to be understood as a realistic approximation of the
hypothesis $H_{0}:|\theta-\theta_{0}|\leq b$, for some small $b$, so that
$\pi\left(\theta_{0}|H_{0}\right)$ in fact represents the prior probability
assigned to $\\{|\theta-\theta_{0}|\leq b\\}$ (see Berger and Sellke, , 1987).
The way to depict such prior is through a smooth density with a sharp peak
around $\theta_{0}$.

###  3.3 Bayes factors

The focus of this paper is on the ratio

$B_{01}=\frac{\Pr\left(x|H_{0}\right)}{\Pr\left(x|H_{1}\right)}\,\text{,}$

referred to as Bayes factor. With this definition, the posterior odds for a
null hypothesis $H_{0}$ and the alternative $H_{1}$, as discussed so far, can
be written as:

$\frac{\text{Pr}\left(H_{0}|x\right)}{\text{Pr}\left(H_{1}|x\right)}=B_{01}\frac{\Pr\left(H_{0}\right)}{\Pr\left(H_{1}\right)}\text{.}$

Bayes factors can be interpreted as either the ratio quantifying the
plausibility of observing the data $x$ under $H_{0}$ over $H_{1}$, or as the
degree by which the observed data updates the prior odds
$\text{Pr}\left(H_{0}\right)/\text{Pr}\left(H_{1}\right)$. With respect to the
posterior odds which involve prior probabilities on the hypotheses, the
interest in Bayes factors arises from the fact that they appear as actual odds
implied by the observed data only. Moreover, Bayes factors are of attractive
interpretation since they can be viewed as likelihood ratios obtained by
averaging likelihoods $\Pr\left(x|\theta_{k},H_{k}\right)$ across
$\theta_{k}|H_{k}$, with weights $\pi\left(\theta_{k}|H_{k}\right)$,
$k=\\{0,1\\}.$

Not less importantly, the calculation of the Bayes factor requires the
prescription only of the prior distributions
$\pi\left(\theta_{k}|H_{k}\right)$, while the full Bayesian analysis leading
to posterior odds requires the additional specification of the prior
probabilities $\text{Pr}\left(H_{k}\right)$, for $k=\\{0,1\\}$. The
interpretation from above still applies: a Bayes factor of e.g. $1/10$ means
that $H_{1}$ is supported by an evidence 10 times as high as $H_{0}$ is.
Furthermore, under the appealing “neutral” choice
$\text{Pr}\left(H_{0}\right)=\text{Pr}\left(H_{1}\right)=1/2$, Bayes factors
coincide with posterior odds, further enforcing $B_{01}$ as a suitable
alternative to p-values. Similar to the frequentist approach where decisions
are based on the critical level $\alpha$, the higher the Bayes factor the
stronger the evidence in favor of $H_{0}$ over $H_{1}$. (Jeffreys, , 1961)
provides a scale for interpreting $B_{01}$ as the degree to which $H_{0}$ is
supported by the data over $H_{1}$, with ratios greater than 100 being
decisive. A comparison between the Bayesian and frequentist decision scales
can be found in (Efron et al., , 2001).

####  3.3.1 Bayes factors for unit root testing

We show how the above description on Bayes factor and posterior odds
practically applies to test the set of hypotheses

$H_{0}:\rho=1\qquad\qquad\qquad H_{1}:\rho\in S\text{.}$

Here we extend the discussion to any generic AR-like model, holding the usual
interpretation of $\rho$ as the autoregressive parameter on interest. First,
to compare the average likelihood of a model over the complementary region
through posterior odds as outlined above, we assign a certain positive weight
$\pi_{0}$ to the point null hypothesis $H_{0}$ and share the complement
$\left(1-\pi_{0}\right)$ over the interval $S$ relevant under $H_{1}$. Second,
as a general case the likelihood function is parametrized over a rich
$K$-dimensional parameter $\\{\rho,\theta\\}$ defined over some appropriate
set $\\{S,\Theta\\}\in\mathbb{R}^{K}$. Therefore, the computation of the Bayes
factor in this setting involves a multidimensional marginalization over the
elements of $\theta$, thus posteriors odds read:

$K_{1}=\frac{\pi_{0}}{1-\pi_{0}}\frac{\int_{\Theta}L\left(x|\rho=1,\theta,x_{0}\right)\pi\left(\theta|\rho=1\right)d\theta}{\int_{S}\int_{\Theta}L\left(x|\rho,\theta,x_{0}\right)\pi\left(\theta|\rho\right)\pi\left(\rho\right)\,d\theta
d\rho}=\frac{\Pr\left(\rho=1|x\right)}{\Pr\left(\rho\in S|x\right)}\,\text{.}$
(3)

Eq.(3) embeds a feasible and largely adopted specification for the prior over
$\left(\rho,\theta\right)$ that imposes conditional independence on the
parameters, allowing for a convenient factorization of the joint prior as a
product of conditionally independent factors, that is
$\pi\left(\rho,\theta\right)=\pi\left(\theta|\rho\right)\pi\left(\rho\right)$.
Rather than the above probability notation ($\Pr$) typical in introductory
discussions on Bayes factors, here we use the likelihood notation ($L$) as it
is more common in the related econometric literature.

For inference on the simple AR(1) process

$x_{t}=\rho x_{t-1}+u_{t}\,\text{,}$

$K=2$, as $\theta$ generally includes the unknown variance $\sigma$ of the
innovations. In this case, posterior odds have the simple form

$K_{1}=\frac{\pi_{0}}{1-\pi_{0}}\frac{\int_{0}^{\infty}L\left(x|\rho=1,\sigma,x_{0}\right)\pi\left(\sigma\right)d\sigma}{\int_{S}\int_{0}^{\infty}L\left(x|\rho,\sigma,x_{0}\right)\pi\left(\sigma\right)\pi\left(\rho\right)\,d\sigma
d\rho}\,\text{.}$

The conditional independence assumption among the parameters is generally not
very restrictive and commonly extended to unconditional independence, thus
$\sigma$ is provided with its own prior
$\pi\left(\sigma|\rho\right)=\pi\left(\sigma\right)$. Further information can
be found e.g. in (Schotman and Van Dijk, 1991b, ; Zivot, , 1994; Zellner and
Siow, , 1980, among the others).

In the following section we show how to approximate the general Bayes factor
involved in Eq.(3) with a friendly form of moderate error that neither
requires integration nor the priors to be specified.

##  4 Hypothesis testing with BIC

###  4.1 Laplace approximation

For $k=\\{0,1\\}$ consider the densities
$\Pr\left(x|H_{k}\right)=\int\Pr\left(x|\theta_{k},H_{k}\right)\pi\left(\theta_{k}|H_{k}\right)d\theta_{k}$
involved in the definition of Bayes factors. Be $\theta_{k}$ the parameter
under $H_{k}$, $\pi\left(\theta_{k}|H_{k}\right)$ its prior, and
$\Pr\left(x|\theta_{k},H_{k}\right)$ the prior density of $x$ given the values
of $\theta_{k}$. $\theta_{k}$ in general represents a vector parameter with
dimension $d_{k}$. In the following, we shall refer to the marginal
probability of the data (or marginal likelihood) $\Pr\left(x|H_{k}\right)$ as
$I$ and adopt a simplified notation where we drop $k$ and rewrite the marginal
likelihood as

$I=\int\Pr(x|\theta,H)\pi\left(\theta|H\right)d\theta\,\text{.}$

Except for some elementary cases where the above integral can be evaluated
analytically, the computation of the marginal likelihood is intractable and
requires numerical methods. In fact, analytic solutions for $I$ are limited to
exponential family distributions and conjugate priors, including normal linear
models (e.g. DeGroot, , 2005; Zellner, , 1971). A general description of the
different approaches for evaluating $I$ with numerical methods is provided in
(Evans et al., , 1995).

To recover a first useful approximation for $I$, assume that the posterior
density, proportional to $\Pr(x|\theta,H)\pi\left(\theta|H\right)$, is peaked
around its maximum $\tilde{\theta}$, which is the posterior mode. This is
generally the case for large samples if the likelihood function of the data
$\Pr(x|\theta,H)$ is peaked around its maximum $\hat{\theta}$ (Kass and
Raftery, , 1995). Let
$g\left(\theta\right)=\log\left(\Pr(x|\theta,H)\pi\left(\theta|H\right)\right)$
and consider its Taylor expansion around $\tilde{\theta}$:
$g(\theta)=g(\tilde{\theta})+(\theta-\tilde{\theta})^{T}g^{\prime}(\tilde{\theta})+\nicefrac{{1}}{{2}}(\theta-\tilde{\theta})^{T}g^{\prime\prime}(\tilde{\theta})(\theta-\tilde{\theta})+o(||\theta-\tilde{\theta}||^{2})$.
Since $g^{\prime}(\tilde{\theta})=0$ as $g$ reaches its maximum at
$\tilde{\theta}$, it follows that:

$\displaystyle I$
$\displaystyle=\int\text{exp}[g(\theta)]d\theta\,\,\approx\text{exp}[g(\tilde{\theta})]\int\text{exp}\left[\nicefrac{{1}}{{2}}(\theta-\tilde{\theta})^{T}g^{\prime\prime}(\tilde{\theta})(\theta-\tilde{\theta})\right]d\theta\,\text{,}$
(4)

where we recognize in the integrand the kernel of a generic $d$-dimensional
multivariate normal distribution with mean $\tilde{\theta}$ and covariance
matrix $\tilde{\Sigma}$. $\tilde{\Sigma}$ also corresponds to minus the
inverse Hessian matrix of the second order derivatives of $g(\theta)$
evaluated at $\theta=\tilde{\theta}$, i.e.
$\tilde{\Sigma}^{-1}=-g^{\prime\prime}(\tilde{\theta})$. The integrand in
Eq.(4) therefore equals $\left(2\pi\right)^{d/2}|\tilde{\Sigma}|^{1/2}$, from
which the following approximation is known as Laplace approximation (e.g.
Konishi and Kitagawa, , 2008):

$\tilde{I}=\left(2\pi\right)^{d/2}|\tilde{\Sigma}|^{1/2}\Pr(x|\tilde{\theta},H)\pi(\tilde{\theta}|H)\,\text{.}$
(5)

In particular, as $n$ diverges $I=\tilde{I}(1+O\left(n^{-1}\right))$, see e.g.
(Tierney et al., , 1989; Kass et al., , 1991). Eq.(5) can be applied to any
regular statistical model and stands as a viable general approach for
evaluating the marginal likelihoods involved in the definitions of Bayes
factors with an approximation error of order $O\left(n^{-1}\right)$. Slate,
(1994) discusses requirements on the sample size for reaching posterior
normality, and the accuracy of Laplace’s method has been more generally
investigated in (e.g. Efron and Hinkley, , 1978; Kass and Vaidyanathan, ,
1992). An empirical rule is provided in (Kass and Raftery, , 1995): sample
sizes of at least $5d$ provide a satisfactory accuracy in well-behaved
problems, with $20d$ applicable in most situations.

The use of Eq.(5) is impractical since $\tilde{\theta}$ refers to the
posterior mode and $\tilde{\Sigma}$ to the negative inverse Hessian of
$g\left(\theta\right)$, while maximum likelihood estimates and information
matrices are of common use and generally readily available as standard outputs
in any statistical software. Indeed, a variation over Eq.(5) that has
attracted much attention uses the maximum likelihood estimator $\hat{\theta}$,
applies to large samples where $\tilde{\theta}\approx\hat{\theta}$, and relies
on the covariance matrix $\hat{\Sigma}$ so that $\hat{\Sigma}^{-1}$
corresponds to the observed information matrix, i.e. the negative Hessian of
the log-likelihood evaluated at $\hat{\theta}$ (Tierney et al., , 1989; Kass
and Vaidyanathan, , 1992):

$\hat{I}=\left(2\pi\right)^{d/2}|\hat{\Sigma}|^{1/2}\Pr(x|\hat{\theta},H)\pi(\hat{\theta}|H)\,\text{.}$
(6)

The relative error in this case is still the best rate $O\left(n^{-1}\right)$.
If one replaces the observed information matrix with the expected information
matrix $i$, the asymptotic error rate moves to the larger order
$O\left(n^{-1/2}\right)$. The expected information matrix $i$ is a $d\times d$
matrix whose $\left(h,k\right)$ element is

$-\mathbb{E}\left[\left.\frac{\partial\log\Pr\left(x_{1}|\theta,H\right)}{\partial\theta_{h}\partial\theta_{k}}\right|_{\theta=\hat{\theta}}\right]\,\text{,}$

and the expectation is taken over $x_{i}$ with $\theta$ held constant.
Therefore, in large samples the observed information matrix
$\hat{\Sigma}^{-1}$ can be approximated based on the expected information
matrix $\hat{\Sigma}^{-1}\approx ni$, and
$n^{d}i=|\hat{\Sigma}^{-1}|=|\hat{\Sigma}|^{-1}$. With this substitution
Eq.(6) rewrites as:

$\hat{I}=\left(2\pi\right)^{d/2}n^{-d/2}|i|^{-1/2}\Pr(x|\hat{\theta},H)\pi(\hat{\theta}|H)\,\text{,}$

from which

$\displaystyle\log
I=\log\Pr(x|\hat{\theta},H)+\log\pi(\hat{\theta}|H)+\frac{d}{2}\log\left(2\pi\right)-\frac{d}{2}\log
n-\frac{1}{2}\log|i|+O\left(n^{-1/2}\right)\,\text{.}$ (7)

Note that the prior density $\pi\left(\theta|H\right)$ needs to be fully
specified, as it is involved throughout the approximation procedure.

This leads to the conclusive approximation form that does not involve prior
densities:

$\displaystyle\log I=\log\Pr(x|\hat{\theta},H)-\frac{d}{2}\log
n+O\left(1\right)\,\text{.}$ (8)

This last approximation is in virtue of the fact that in Eq.(7) besides
$\Pr(x|\hat{\theta},H)$ and $\log n$ which are respectively of order
$O\left(n\right)$ and $O\left(\log n\right)$, all the remaining terms are of
order $O\left(1\right)$ or lower. From Eq.(8) we have that the marginal
likelihood is thus equal to the maximized likelihood $\Pr(x|\hat{\theta},H)$
minus a correction term where the approximation is $O\left(1\right)$. Even
though the $O\left(1\right)$ term does not vanish, because all the other terms
tend to infinity as $n$ increases, the error is dominated and vanishing as a
proportion of $I$. Raftery,  (1995) shows that in reality the error term is
not as high as one might think, although an $O\left(1\right)$ error suggests
that the approximation is in general quite crude. In fact, the error can be of
a smaller order of magnitudes given a reasonable choice of the prior.

As a remark, the definition of the sample size $n$ should reflect the rate at
which the Hessian matrix of the log-likelihood grows, i.e. satisfactory for
the approximation $\hat{\Sigma}^{-1}\approx ni$. This $n$ turns to be the
number contributions to the summation appearing in the definition of the
Hessian (Raftery, , 1995; Kass and Raftery, , 1995) – e.g. in survival
analysis, $n$ would match the number of non-censored observations rather than
the total number of observations.

###  4.2 BIC Approximation of the Bayes factor

The above discussion provides the basis for the following approximation of the
Bayes factor. Hereafter we, focus on the case where the null hypothesis is
nested. That is, we assume some parametrization under $H_{1}$ of the form
$\theta_{1}=\left(\rho,\beta\right)$ such that $H_{0}$ is obtained from
$H_{1}$ by imposing the restriction $\rho=\rho_{0}$ for some $\rho_{0}$. Both
$\rho$ and $\beta$ can be vectors. Let $\theta_{1}$ denote the parameter under
$H_{1}$ with prior
$\pi\left(\theta_{1}|H_{1}\right)=\pi\left(\rho,\beta|H_{1}\right)$, and for
$H_{0}:\rho=\rho_{0}$ let its prior be
$\pi\left(\theta_{0}|H_{0}\right)=\pi\left(\beta|H_{0}\right)$.

Based on Eq.(6), by applying the definition of the Bayes factor to the log-
ratio of the marginal likelihoods, one obtains:

$2\log
B_{10}\approx\Lambda+\log|\hat{\Sigma}_{1}|-\log|\hat{\Sigma}_{0}|+2\log\pi(\hat{\theta}_{1}|H_{1})-2\log\pi(\hat{\theta}_{0}|H_{0})+\left(d_{1}-d_{0}\right)\log\left(2\pi\right)$

where
$\Lambda_{10}=2(\log\Pr(x|\hat{\theta}_{1},H_{1})-\log\Pr(x|\hat{\theta}_{0},H_{0}))$
corresponds to the log-likelihood ratio statistics with $d_{1}-d_{0}$ degrees
of freedom. Refer to (Raftery, , 1996) for an additional discussion, and for
the approximation of the Bayes factor under to Eq.(5). On the other hand,
based on the approximation in Eq.(8), one obtains:

$\displaystyle 2\log B_{10}$
$\displaystyle\approx\Lambda_{10}-\left(d_{1}-d_{0}\right)\log\left(n\right)=2S_{10}\,\text{,}$
(9)

where

$\displaystyle 2S_{10}$
$\displaystyle=2\log\Pr(x|\hat{\theta}_{1},H_{1})-2\log\Pr(x|\hat{\theta}_{0},H_{0})-\left(d_{1}-d_{0}\right)\log\left(n\right)$
$\displaystyle=\left[d_{0}\log\left(n\right)-2\log\Pr(x|\hat{\theta}_{0},H_{0})\right]-\left[d_{1}\log\left(n\right)-2\log\Pr(x|\hat{\theta}_{1},H_{1})\right]\,\text{.}$

The following consistency result for $n\to\infty$ known as Schwarz criterion

$\left(S_{10}-\log B_{10}\right)/\log B_{10}\to 0\,\text{,}$ (10)

is attractive as it establishes $S_{10}$ as a standardized quantity to be used
even when the priors are hard to set and a useful reference quantity in
scientific reporting (Kass and Raftery, , 1995). The $O\left(1\right)$ error
implies that even in large samples $S_{10}$ does not lead to the correct value
in absolute terms, the error does go to zero in terms of proportion with
respect to the actual log of the Bayes factor (e.g. Kass and Raftery, , 1995;
Raftery, , 1995, 1996). Importantly, for certain classes for priors the error
of approximation reduces to $O\left(n^{-1/2}\right)$. One class is that of
Jeffrey’s priors with a specific choice of the constant preceding them,
another class is that of unit information priors (Raftery, , 1995; Wasserman,
, 2000; Wagenmakers, , 2007). With respect to subjectively determined priors,
a surprisingly good agreement between the Schwarz criterion and actual Bayes
factors is observed in Kass and Wasserman,  (1995). In general when the sample
size $n$ is sufficiently large, the approximation is very satisfactory for
most of purposes and is of widespread use, including applications in
psychology (e.g. Wasserman, , 2000), ecology (e.g. Aho et al., , 2014), and
computer vision (e.g. Stanford and Raftery, , 2002). Kass and Wasserman,
(1995) further show that for the intuitive and reasonable choice of the unit
information prior $\text{exp}\left(S_{10}\right)/B_{10}\to 1$ with an error of
order $O\left(n^{-1/2}\right)$. This provides a direct interpretation of the
Schwartz criterion in terms of Bayes factors and evidence.

For a given model $k$, recall the definition of the Bayesian Information
Criterion (BIC):

$\text{BIC}_{k}=d_{k}\log n-2\log\Pr(x|\hat{\theta}_{k},H_{k})\,\text{.}$ (11)

It is easy to recognize that the right side of Eq.(8) is closely related to
BIC as $-2\log I=\text{BIC}+O\left(1\right)$, and that
$2S_{10}=\text{BIC}_{0}-\text{BIC}_{1}=\Delta\text{BIC}_{01}$. Eq.(9) then is
equivalent to

$\log B_{10}\approx\frac{1}{2}\Delta\text{BIC}_{01}\,\text{.}$ (12)

Eq.(12) establishes $\Delta\text{BIC}_{01}$ as an approximate measure of the
log-evidence in support of the hypothesis $H_{1}$ over $H_{0}$. We shall refer
to either Eq.(12) or its exp-version
$B_{10}\approx\exp\left(\nicefrac{{1}}{{2}}\text{BIC}_{01}\right)$ as the BIC
approximation of the Bayes factor.

For the above BIC approximation the approximation error is generally
$O\left(1\right)$, but in virtue of the Schwartz criterion
$\left(\Delta\text{BIC}_{10}-\log B_{10}\right)/\log B_{10}\to 0$ the error
approaches zero as a proportion of the Bayes factor. As discussed above, it
can further reduce to $O\left(n^{-1/2}\right)$ for certain priors. Eq.(12)
formally justifies the extensive practice of model selection based on the
smallest BIC value. Indeed, the higher the evidence in support of model 1,
i.e. the higher the log-Bayes factor, the more positive
$\Delta\text{BIC}_{01}$ and the smaller $\text{BIC}_{1}$ with respect to
$\text{BIC}_{0}$.

In the context of linear models with normal errors, BIC rewrites in the
alternative convenient form

$\text{BIC}_{k}=n\log\left(1-R^{2}_{k}\right)+d_{k}\log n\,\text{,}$

with $R^{2}$ being the usual R-squared. The proportion $1-R^{2}_{k}$ of the
variance that model $k$ fails to explain relates to the sum of squares table
through the equivalence $1-R^{2}_{k}=SSE_{k}-SS_{\text{total}}$, where
$SSE_{k}$ is the sum of squared errors for model $k$ and $SS_{\text{total}}$
the total sum of squared errors. This leads to the following expression:

$\Delta\text{BIC}_{01}=n\log{\frac{SSE_{0}}{SSE_{1}}}+\left(d_{0}-d_{1}\right)\log\left(n\right)\,\text{.}$
(13)

Applied examples on the use of Eq.(13) are provided e.g. in (Wagenmakers, ,
2007; Masson, , 2011).

Furthermore, for nested models such that $d_{1}-d_{0}=1$, $\Lambda_{10}\approx
t^{2}$ with $t$ being the t-statistics for testing the significance of the
parameter in model $1$ that is set to zero in model $0$, and $\Lambda_{10}$
the corresponding likelihood ratio statistics. From Eq.(12):

$2\log B_{10}\approx\Delta\text{BIC}_{01}=\Lambda_{10}-\log n\approx
t^{2}-\log n\,\text{.}$ (14)

This underlines a proportionality between $t$, $\Delta\text{BIC}$ and $B$,
which means that the $t$ statistics can be directly translated into BIC and
into grades of evidence through Bayes factors (Johnson, , 2005). High values
of $t$ support the statistical significance of the additional parameter in the
full model. In turn, $\text{BIC}_{1}$ is smaller than $\text{BIC}_{0}$, so
$\Delta\text{BIC}_{01}>0$ and $\log B_{10}>0$, which is indicative of evidence
against the reduced model. By reparametrization, the above extends to
hypotheses where an element $\theta^{\prime}$ of $\theta$ is set to a fixed
value (rather than zero), e.g. $H_{0}:\theta^{\prime}=1$ is analogous to
$H_{0}:\theta^{\prime\prime}=0$ by taking
$\theta^{\prime\prime}=\theta^{\prime}-1$.

###  4.3 Unit root testing based on BIC approximation

The major contribution of this paper is to test for unit roots in financial
time series by the BIC approximation of the Bayes factor, Eq.(12). We shall
list the major points in favor of this approach.

  * i.

As reviewed above, the choice of the prior is the principal problem in
Bayesian testing of unit roots. On the contrary, our proposed BIC
approximation does not require a full specification of the priors. Neither
that of the autoregressive parameter, nor of any other parameter. The
independence of the BIC approximation on prior specifications is also
attractive from the point of objectivity in Bayesian analysis.

  * ii.

Bayes factors and posterior odds allow test point nulls on the autoregressive
parameter, which can be problematic as shown earlier. This setting is however
natural for the BIC approximation.

  * iii.

The BIC approximation to Bayes factor is a general procedure that does not
depend on the model form or parametrization. Regardless of whether the AR
model under investigation has an intercept, a trend component, exogenous
regressors, or any richer structure, the BIC procedure applies. In general,
the error is $O\left(1\right)$ but reduces to zero as a proportion of the
Bayes factor (cf. Eq.(10)).

  * iv.

Testing based on BIC approximation does not require any integration and does
not present major computational issues: it only requires the maximum
likelihood estimates, as of definition in Eq.(11).

  * v.

The applicability of the method depends on the feasibility of the
approximation in Eq.(6), i.e. on the feasible hypotheses that the data $x$
consist of i.i.d. observations, that the posterior is peaked around its
maximum, and that the sample size is sufficient for
$\tilde{\theta}\approx\hat{\theta}$ and $\hat{\Sigma}^{-1}\approx ni$ to be
satisfactory approximations. A sample size of about 20 times the number of
parameters appears to be generally fair for well-behaved problems where the
likelihood is not grossly non-normal. For the simple AR(1) model with unknown
innovations’ variance, this corresponds to about 40 points, e.g. about two
months of daily market data.

Lastly, consider that even if the applicability of the above procedure is
quite broad, time series models involved in classical unit root testing
applications are broadly of linear form, so the simplified alternative form in
Eq.(13) commonly applies. Also, the point form of the null hypothesis
naturally suggests a nested hypotheses structure where the only restriction is
on the autoregressive parameter. That is, applications will generally
encounter linear model specifications where $d_{1}-d_{0}=1$.

##  5 Simulation Study

To validate our proposed testing methodology and compare it with some existing
alternatives, we develop a Monte Carlo simulation study. In particular, we
simulate 20,000 simple AR(1) processes $x_{t}=\rho x_{t-1}+u_{t}$, with
$t=\\{1,\dots,T\\}$ and independent standard normal innovations, by
considering different sample lengths $T=\\{50,100,200,500,1000,5000\\}$ and
different values of the autoregressive parameter
$\rho=\\{0.2,0.5,0.8,0.9,0.99,0.999,1\\}$. We test the point null unit root
hypothesis $H_{0}:\rho=1$ and summarize in Table 1 the corresponding results.
All the tables related to the Monte Carlo study report averages across the
simulated samples.

Table 1 includes probabilities associated with different alternative testing
methods. For the approach of Schotman and Van Dijk, 1991b we both adopt a
constant lower integration bound $a$ for the autoregressive parameter fixed to
$-1$ (SVD) and the data-driven one (SVD*), see Appendix A.1. As a reference
measure, Table 1 includes the average p-value for the Dickey-Fuller test (DF)
and the posterior non-stationary probabilities $\Pr_{\rho\geq
1}=\text{Pr}\left(\rho\geq 1|x\right)$ from (Phillips, 1991b, ), computed
through Eq.(1). These posterior probabilities are reported only for $T\leq
200$ as larger $T$ quickly drive Eq.(17) below machine precision and
integration turns problematic: recovering such probabilities for large $T$ is
here beyond our scope. For the SVD, SVD* and BIC entries in Table 1, we report
as our main result the probabilities corresponding to the Bayes factors in
Table 3, since of easier interpretation. Furthermore, assuming prior odds
equal to one, these probabilities also interpret as posterior odds.

Our simulation results show a smooth and coherent behavior of the proposed BIC
approximation leading to posterior probabilities behaving in accordance with
how one might expect from this controlled setting. (i) The acceptance
probabilities for the null are progressively higher as $\rho$ approaches the
unity. (ii) For a fixed $\rho$, larger samples reduce the posterior unit root
probability for small values of $\rho$ while increase it for $\rho$ around 1.
Indeed larger samples embed higher evidence in support of the true hypothesis,
for which we observe increasing posterior probabilities (and log- Bayes
factors). With $\rho=0.8$ and $T=50$, for the BIC approximation we compute an
average posterior probability for $H_{0}$ of .240, however as $T$ increases
the evidence towards the true hypothesis $\rho=0.8$ increases as well, and the
posterior probability of the null moves rapidly towards zero (e.g. from
$T=200$ ahead). Similarly, for $\rho=1$ and $T=50$, the small size of the
sample advocates for a stationary dynamics with a considerable 0.211
probability, while at larger $T$ the probability associated with far-from-
unity values of $\rho$ sharply decreases to zero.

Furthermore, our simulation study leads to posterior probabilities that are
also well-aligned across the BIC, SVD and SVD* testing approaches, following
the same trends across different values of $\rho$ for fixed $T$. The BIC
approximation is however leading to probabilities that are not uniformly
greater or smaller than those from SVD*. In fact, for small to moderate sample
sizes BIC returns higher posteriors probabilities for the unit root
hypothesis, while smaller for large $T$ with respect to SVD*. This behavior
could be partially explained by the different specifications of the
alternative hypothesis. While for SVD the reported probabilities are those of
a unit root against a stationary alternative, for BIC the alternative is a
generic $H_{1}:\rho\neq 1$. It is thus reasonable that the rejection
probabilities are larger for BIC than SVD, as by construction for BIC the
feasible parameter space under the alternative $H_{1}$ is broader. Also with
respect to posterior probabilities of explosive roots/non-stationarity
dynamics $\Pr_{\rho\geq 1}$ under the ignorance prior, the BIC approximation
appears coherent and well-aligned. Though BIC and $\Pr_{\rho\geq 1}$
probabilities refer to different hypotheses, we observe that indeed the higher
the posterior probability of a unit root, the higher the posterior probability
supporting the non-stationary option.

As expected, p-values associated with the most classical frequentist Dickey-
Fuller test cannot exclude the non-stationary hypothesis with increasing
confidence as $\rho$ moves towards one. Accordingly, the evidence in support
of a unit root is reflected in increasing posterior BIC probabilities. All the
above discussion applies as well to the Bayes factors reported in Appendix
A.3, where negative signs stand for evidence against the unit root null.

This study confirms an overall very satisfactory behavior of the proposed
method with respect to some Bayesian and non-Bayesian alternatives for unit
root hypothesis testing. Despite the general $O(1)$ error associated with BIC
approximation and its complete independence on the prior specification, our
simulated posterior probabilities are well-behaving (i.e. showing desirable
smooth monotonicity over $\rho$ for $T$ fixed, and the other way around),
coherent with their expected behavior, and aligned with the decisions over the
null the other approaches suggest.

$T=50$ |  | $T=500$
---|---|---
$\rho$ | SVD | SVD* | BIC | DF | $\Pr_{\rho\geq 1}$ |  | $\rho$ | SVD | SVD* | BIC | DF
0.200 | .000 | .000 | .000 | .001 | .015 |  | 0.200 | .000 | .000 | .000 | .001
0.500 | .003 | .002 | .004 | .001 | .107 |  | 0.500 | .000 | .000 | .000 | .001
0.800 | .292 | .124 | .240 | .032 | .244 |  | 0.800 | .000 | .000 | .000 | .001
0.900 | .686 | .341 | .545 | .111 | .306 |  | 0.900 | .000 | .000 | .000 | .001
0.990 | .955 | .663 | .787 | .390 | .445 |  | 0.990 | .957 | .336 | .801 | .115
0.999 | .973 | .719 | .797 | .485 | .514 |  | 0.999 | .994 | .652 | .924 | .406
1.000 | .975 | .729 | .798 | .501 | .529 |  | 1.000 | .996 | .701 | .926 | .491
$T=100$ |  | $T=1000$
0.200 | .000 | .000 | .000 | .001 | .002 |  | 0.200 | .000 | .000 | .000 | .001
0.500 | .000 | .000 | .000 | .001 | .078 |  | 0.500 | .000 | .000 | .000 | .001
0.800 | .041 | .011 | .031 | .003 | .184 |  | 0.800 | .000 | .000 | .000 | .001
0.900 | .458 | .125 | .321 | .032 | .243 |  | 0.900 | .000 | .000 | .000 | .001
0.990 | .965 | .608 | .827 | .332 | .425 |  | 0.990 | .900 | .131 | .623 | .034
0.999 | .983 | .701 | .849 | .474 | .528 |  | 0.999 | .996 | .618 | .942 | .355
1.000 | .985 | .714 | .850 | .495 | .546 |  | 1.000 | .998 | .697 | .947 | .488
$T=200$ |  | $T=5000$
0.200 | .000 | .000 | .000 | .001 | .000 |  | 0.200 | .000 | .000 | .000 | .001
0.500 | .000 | .000 | .000 | .001 | .057 |  | 0.500 | .000 | .000 | .000 | .001
0.800 | .000 | .000 | .000 | .001 | .138 |  | 0.800 | .000 | .000 | .000 | .001
0.900 | .084 | .012 | .049 | .004 | .182 |  | 0.900 | .000 | .000 | .000 | .001
0.990 | .968 | .529 | .844 | .249 | .389 |  | 0.990 | .001 | .000 | .000 | .001
0.999 | .990 | .686 | .888 | .457 | .534 |  | 0.999 | .996 | .351 | .932 | .123
1.000 | .992 | .706 | .889 | .492 | .562 |  | 1.000 | 1.000 | .697 | .976 | .487

Table 1: Simulation results. SVD, SVD* and BIC: unit root posterior
probabilities (prior odds equal to one). DF: p-values of the Dickey-Fuller
test. $\Pr_{\rho\geq 1}$: posterior probabilities $\text{Pr}\left(\rho\geq
1|x\right)$.

##  6 Empirical Application

In our empirical application, we analyze Real Exchange Rates (RERs) time
series for nine major currencies. RERs are obtained by deflating nominal
exchange rates by the relative price of domestic vs. foreign goods and
services, thus reflecting the competitiveness of a country with respect to a
reference basket. Common choices for deflation are the consumer price index
(CPI), the producer price indices, or GDP-based deflators. An increase in RER
implies that exports become more expensive and imports turn cheaper,
indicating a loss in trade competitiveness, for instance in response to an
appreciation of the domestic currency, or in response to increased domestic
inflation. We extract official monthly RERs (CPI deflated) time series
distributed by the European Commission and available from the Statistical data
warehouse of the European Central Bank111Data and its documentation are
available at https://sdw.ecb.europa.eu/browse.do?node=9691113., for the period
between January 2010 and November 2020. This corresponds to 131 records for
each of the nine222Australian dollar (AUD), Canadian dollar (CAD), Swiss franc
(CHF), Chinese yuan (CNY), euro (EUR), British pound (GBP), Hong Kong dollar
(HKD), Japanese yen (JPY), US dollar (USD). major currencies we consider in
the analysis.

Real exchange rates relate to the long-run equilibrium condition – known as
Purchasing Power Parity (PPP) – which implies a steady long-term level and a
constant unconditional mean for RERs series. The existence of a unit root
would contradict this theory. There have been considerable efforts in
empirically verifying the PPP theory and several papers discussed mid- and
long- term departures from the expected RER stationarity, leading to a
controversial debate. The essence is that conclusions based on empirical
research strongly depend on the exact definition of equilibrium that one
adopts, on the methods used to test it, on the underlying hypotheses on the
time series, and on its length. This applies to unit root analyses as well,
which often lead to opposite results on PPP’s validity (see e.g. MacDonald, ,
1995, and the references therein). In this regard, the unit root analysis of
Parikh and Wakerly,  (2000) suggests that such equilibrium is expected to be
generally observed over a time-span of at least 50 years. Non-short-lived
disequilibrium periods in RER dynamics are thus common. At first sight, Figure
1 seems to confirm such a tendency, suggesting a recent disequilibrium period
for some of the major currencies. For instance, we observe an upward trend for
the US dollar series and an apparent non-mean-reversing behavior for the
Chinese yuan and Japanese yen, which gradually adjust towards new RER levels,
suggesting the presence of unit roots.

Figure 1: Real exchange rates for some selected currencies.

Posterior probabilities of the unit root hypothesis in RERs are reported in
Table 2. Our results indicate ubiquitous evidence in favor of the unit root
hypothesis for all the currencies. Bayesian results are aligned with the
p-values from the Dickey-Fuller (DF) test and posterior probabilities from
(Phillips, 1991b, ). Indeed, higher posterior probabilities are both
associated with higher p-values of the Dickey-Fuller statistics and higher
posterior probabilities $\Pr_{\rho\geq 1}=\Pr\left(\rho\geq 1|x\right)$
associated with the non-stationary alternative. Explanatory are the results
for the Euro-zone where the .047 p-value of the DF test indicates a mid
rejection of the unit root hypothesis. This is aligned with Figure 1 where the
EUR series indeed displays a much stationary behavior than any other series.
Accordingly, BIC probabilities take their smallest value across all the
countries analyzed, with the stationary hypothesis having .455 probability,
much smaller than the average .880 observed for all the other currencies,
where the DF p-value is on average .625. This is also aligned with the
conclusion one would draw from the SVD* approach, where the highest evidence
in favor of stationarity is again associated with the euro series. This is
reasonable, as euro-zone countries have a significant presence in the basket
for determining RER’s CPI correction (19 out of 37 countries in the basket
adopt euro). The observed greater plausibility of the stationary hypothesis is
here not surprising but expected and coherent. Lastly, note that BIC
probabilities appear to uniformly dominate SVD* ones, this could perhaps be
explained in terms of the typical bias towards stationarity implied by the use
of uniform priors.

| $\log BF_{01}$ | Prob. |  |
---|---|---|---|---
Currency | SVD* | BIC | SVD* | BIC | DF | $\Pr_{\rho\geq 1}$
AUD | -0.333 | 1.076 | .418 | .746 | .090 | .288
CAD | 1.455 | 2.431 | .811 | .919 | .676 | .705
CHF | 0.996 | 2.282 | .730 | .907 | .442 | .569
CNY | 1.848 | 2.044 | .864 | .885 | .904 | .875
EUR | -1.194 | 0.180 | .232 | .545 | .047 | .161
GBP | 0.403 | 1.802 | .599 | .858 | .240 | .374
HKD | 1.812 | 2.120 | .860 | .893 | .883 | .857
JPY | 1.154 | 2.369 | .760 | .914 | .533 | .553
USD | 1.641 | 2.349 | .838 | .913 | .802 | .778

Table 2: SVD* and BIC: log- Bayes factors and their corresponding posterior
probabilities for the unit root hypothesis (prior odds equal to one). DF:
p-values of the Dickey-Fuller test. $\Pr_{\rho\geq 1}$: posterior
probabilities $\Pr\left(\rho\geq 1|x\right)$.

##  7 Conclusion

Unit root testing has historically been among the most active areas in
econometric research. With classical frequentist methods following Dicker
Fuller (DF) statistics, being broadly adopted and established, Bayesian
methods did not gain much attention in empirical research and applications.
However, research and debates on Bayesian unit root testing have been very
active and sound. Indeed, the econometric research in the field pointed out a
series of unique criticalities that turn the Bayesian approach for unit root
inference particularly challenging.

On the other hand, the testing procedure based on the BIC approximation of the
Bayes factor addressed in this paper appears to provide a simple and
satisfactory method for Bayesian unit root testing. With such an approach, the
integration problem involved in Bayes factors turns to a standard maximum
likelihood estimation problem, and the Bayes factors have the very simple form
of Eq. (12). Notably, (i) priors are not involved in the approximated form
(though they determine the asymptotic error rate), (ii) the procedure smoothly
scales to more complex time series models, and (iii) allows to target the
exact point null unit root hypothesis. The ratio of the Bayes factor
discriminates between the hypotheses according to the common interpretation
scale of (Jeffreys, , 1961).

The simulation study confirms the validity of our proposed approach showing
that, in this controlled setting, the BIC approximation, leads to decisions
that are entirely coherent with the expectations, under a wide range of values
for the autoregressive parameter, and sample sizes. The posterior
probabilities associated with the null are furthermore aligned with those of
other Bayesian procedures and in accordance with the p-values from the DF
test. The same coherence between BIC, other Bayesian methods, and the
frequentist DF test also arises from the the analysis of real exchange rates
series. In particular, our BIC-based conclusion of non-stationarity matches
the decisions one would draw based on the DF and SVD tests. The results are
furthermore aligned with the posterior probabilities from (Phillips, 1991b, ),
in support of an apparent violation of the real exchange rate and purchasing
power parity equilibrium in the last decade.

Recognizing that BIC is just a model Information Criterion (IC) among many
others that perhaps reduce to BIC as special cases, it would be interesting to
explore the use of such alternatives. Among these, generalized variants of BIC
(see e.g. Konishi and Kitagawa, , 2008, Ch. 9), the FIC (Wei, , 1992), and the
ICs based on predictive distributions (Phillips and Ploberger, , 1994, 1996)
serve a broader model selection scope (than e.g. dampening the drawbacks
associated to priors’ selection in unit root testing) and rely on different
theoretical and motivation bases. Generalized ICs are potentially superior for
the generic purpose of model selection, however, they do not necessarily have
a direct connection with well-identified Bayesian inference problems, though
the use of some of them in unit root testing could now be explained by our
results on BIC. Perhaps a direction for related future works could be that of
investigating to what extent ICs and model selection techniques have a clear
link with certain problems in Bayesian inference. This would shed light on
whether it is possible to rely on ICs and model selection methods as general
tools for approximated Bayesian inference in situations where e.g. priors are
of difficult specification.

## Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 890690.

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##  Appendix A Appendices

###  A.1 The model of Schotman and Van Dijk (1991b)

Consider the simplest autoregressive process of order one with zero-mean

$x_{t}=\rho x_{t-1}+u_{t}\,\text{.}$ (15)

Assume that (i) $x_{0}$ is a known constant, implying that we work
conditionally on the initial observation, (ii) $u_{t}$ are independent and
identically distributed (i.i.d.) normal random variables with mean zero and
unknown variance $\sigma^{2}$, (iii) $\rho\in\\{S,1\\}$;
$S=\\{\rho|-1<a\leq\rho<1\\}$. We assume (iv) to observe a sample of $T$
observations on a time series $\\{x_{t}\\}$. The Bayesian analysis is carried
out via posterior odds. For the simple model here under consideration we have

$K_{1}=K_{0}\frac{\int_{0}^{\infty}L\left(x|\rho=1,\sigma,x_{0}\right)\pi\left(\sigma\right)\,d\sigma}{\int_{S}\int_{0}^{\infty}L\left(x|\rho,\sigma,x_{0}\right)\pi\left(\sigma\right)\pi\left(\rho\right)\,d\sigma
d\rho}=\frac{\text{Pr}\left(\rho=1|x,x_{0}\right)}{\text{Pr}\left(\rho\in
S|x,x_{0}\right)}\,\text{.}$ (16)

where $K_{0}$ represents the prior odds ratio in favour of the hypothesis
$\rho=1$, and $K_{1}$ the corresponding posterior odds ratio. The ratio
between the integrals corresponds to the Bayes factor,
$\pi\left(\sigma\right)$ and $\pi\left(\rho\right)$ represent the prior
densities for $\sigma$ and $\rho\in S$, $L\left(x|\cdot\right)$ the likelihood
function for the observed data $x$.

The prior odd $K_{0}$ expresses the relative weight of the null hypothesis
against its stationary alternative such that the point $\rho=1$ is given the
probability mass $\pi_{0}=K_{0}/\left(1+K_{0}\right)$, and analogously
$K_{1}/\left(1+K_{1}\right)$ provides the posterior probability of the null
hypothesis $\rho=1$.

Schotman and Van Dijk, 1991b specify the marginal distribution of $\rho$ and
$\sigma$ as

$\Pr\left(\rho=1\right)=\pi_{0}\,\text{,}\qquad\text{Pr}\left(\rho|\rho\in
S\right)=\frac{1}{1-a}\,\text{,}\qquad\text{Pr}\left(\sigma\right)\propto\frac{1}{\sigma}\,\text{,}$

that is, $\rho$ is taken uniform over $S$ and with probability mass $\pi_{0}$
on $\rho=1$, with $\sigma$ and $\rho$ independent. Besides the fact that the
density for $\rho$ depends only on the parameter $\alpha$ with great
simplification of the integration problem in the denominator of Eq.(16), the
overall solution even in this simple setting is not obvious:

$K_{1}=\frac{\pi_{0}}{1-\pi_{0}}\frac{C_{T}^{-1}}{\left(T-1\right)^{1/2}}\left(\frac{\sigma^{2}_{0}}{\hat{\sigma}^{2}}\right)^{-\frac{T}{2}}\frac{1-a}{s_{\hat{\rho}}}\left[F\left(\frac{1-\hat{\rho}}{s_{\hat{\rho}}};T-1\right)-F\left(\frac{a-\hat{\rho}}{s_{\hat{\rho}}};T-1\right)\right]^{-1}\,\text{.}$

$\hat{\rho}$ is the OLS estimator of $\rho$, $s^{2}_{\hat{\rho}}$ the squared
OLS standard error of $\hat{\rho}$, $\hat{\sigma}^{2}$ the estimated variance
of the residuals, $\sigma^{2}_{0}$ the variance of the first differences in
$x$, $F\left(\cdot,\nu\right)$ the cumulative density of the t-distribution
with $\nu$ degrees of freedom,
$C_{T}=\Gamma\left(\left(T-1\right)/2\right)\Gamma\left(1/2\right)/\Gamma\left(T/2\right)$
a constant and $\pi_{0}/\left(1-\pi_{0}\right)$ the prior odds ratio $K_{0}$.

By choosing a prior equal balance between the stationary and random walk
hypothesis $\pi_{0}=1/2$ Schotman and Van Dijk, 1991b recover a feasible
operational test procedure based on an empirical determination of the lower
bound $a$,

$a^{*}=\hat{\rho}+s_{\hat{\rho}}F^{-1}\left(\alpha
F\left(-\hat{\tau}\right)\right)\,\text{,}$

where $\tau=\frac{1-\rho}{s_{\hat{\rho}}}$ is the Dickey-Fuller statistics,
and $0<\alpha<1$ a constant (typically between 0.001 and 0.1) such that the
posterior contains $1-\alpha$ of its probability mass in
$\left[a^{*},1\right)$. Therefore Bayes factors turn to be entirely functions
of the data, and are computed as:

$K_{1}=\frac{C_{T}^{-1}}{\left(T-1\right)^{1/2}}\left(\frac{\sigma_{0}^{2}}{\hat{\sigma}_{0}^{2}}\right)^{-\frac{T}{2}}\left(\frac{-\hat{\tau}-F^{-1}\left(\alpha
F\left(-\hat{\tau}\right)\right)}{F\left(-\hat{\tau}\right)}\right)\,\text{.}$

###  A.2 The model of Phillips (1991b)

Phillips, 1991b adopts the information matrix prior from (Jeffreys, , 1946).
In particular, for a generic family of densities with parameter
$\theta=\left(\rho,\sigma\right)$ and information matrix $i$, the
uninformative Jeffreys’ prior he considers it is defined as
$\pi\left(\theta\right)\propto|i|^{\frac{1}{2}}$.

For the AR(1) model $x_{t}=\rho x_{t-1}+u_{t}$ with $u_{t}$ i.i.d. zero-mean
normal with variance $\sigma^{2}$, and initial value $x_{0}$, the above prior
becomes

$\pi\left(\rho,\sigma\right)\propto\frac{1}{\sigma}I_{\rho}^{\frac{1}{2}}\,\text{,}$

where the continuous function $I_{\rho}$, for $-\infty<\rho<+\infty$ and
sample size $T$, is defined as:

$I_{\rho}=\begin{dcases}\frac{T}{1-\rho^{2}}-\frac{1}{1-\rho^{2}}\frac{1-\rho^{2T}}{1-\rho^{2}}+\left(\frac{x_{0}}{\sigma}\right)^{2}\frac{1-\rho^{2T}}{1-\rho^{2}}&\text{if
}\rho\neq 1\text{,}\\\
\frac{T\left(T-1\right)}{2}+T\left(\frac{x_{0}}{\sigma}\right)^{2}&\text{if
}\rho=1\text{.}\end{dcases}$

This choice of the prior achieves tighter confidence sets for large values of
$|\rho|$, is invariant to transformations of the parameter and enjoys other
desirable properties. The prior depends on $x_{0}$ and its information grows
with the sample size $T$ at a geometric rate when $\rho>1$. Under the Gaussian
likelihood for the observed sample $x$ and the above prior, the posterior
distribution of $\rho$ reads

$\pi\left(\rho|x\right)\propto\alpha_{0}^{\frac{1}{2}}\left[R+\left(\rho-\hat{\rho}\right)^{2}Q\right]^{-\frac{T}{2}}\text{.}$
(17)

The practical contribution to the posterior from the choice of the prior is
embedded in the factor $\alpha_{0}$:

$\alpha_{0}=\begin{dcases}\frac{T}{1-\rho^{2}}-\frac{1}{1-\rho^{2}}\frac{1-\rho^{2T}}{1-\rho^{2}}&\text{if
}\rho\neq 1\text{,}\\\ \frac{T\left(T-1\right)}{2}&\text{if
}\rho=1\text{.}\end{dcases}$

On the other hand, the sum of squared residuals $R$, the quantity $Q=\sum
x_{t-1}^{2}$ and the OLS estimate of the autoregressive coefficient
$\hat{\rho}$ are entirely depend on the data and the model choice. The
posterior in Eq.(17) is generally less susceptible of the downward bias than
the posterior based on a flat prior, is not symmetric around $\hat{\rho}$, and
depending on the values of the data-dependent quantities described above may
have a second mode for $\rho>1$.

###  A.3 Simulation study: complement

| $T=50$ | $T=100$ | $T=200$
---|---|---|---
$\rho$ | SVD | SVD* | BIC | SVD | SVD* | BIC | SVD | SVD* | BIC
0.200 | -11.35 | -11.60 | -11.13 | -23.81 | -24.25 | -23.60 | -48.97 | -49.54 | -48.77
0.500 | -5.72 | -6.25 | -5.62 | -12.61 | -13.36 | -12.52 | -26.59 | -27.51 | -26.51
0.800 | -0.88 | -1.95 | -1.15 | -3.16 | -4.52 | -3.43 | -8.08 | -9.66 | -8.36
0.900 | 0.78 | -0.66 | 0.18 | -0.17 | -1.95 | -0.75 | -2.39 | -4.43 | -2.97
0.990 | 3.06 | 0.68 | 1.31 | 3.31 | 0.44 | 1.56 | 3.42 | 0.12 | 1.68
0.999 | 3.60 | 0.94 | 1.37 | 4.08 | 0.85 | 1.73 | 4.58 | 0.78 | 2.07
1.000 | 3.68 | 0.99 | 1.38 | 4.19 | 0.91 | 1.73 | 4.76 | 0.88 | 2.08
| $T=500$ | $T=2000$ | $T=5000$
$\rho$ | SVD | SVD* | BIC | SVD | SVD* | BIC | SVD | SVD* | BIC
0.200 | -125.05 | -125.74 | -124.85 | -252.40 | -253.15 | -252.20 | -1273.42 | -1274.25 | -1273.21
0.500 | -69.28 | -70.35 | -69.20 | -140.86 | -142.01 | -140.78 | -715.38 | -716.66 | -715.30
0.800 | -23.43 | -25.23 | -23.71 | -49.41 | -51.34 | -49.69 | -259.36 | -261.48 | -259.65
0.900 | -9.61 | -11.93 | -10.20 | -22.09 | -24.57 | -22.69 | -123.80 | -126.54 | -124.41
0.990 | 3.10 | -0.68 | 1.39 | 2.20 | -1.89 | 0.50 | -7.01 | -11.63 | -8.74
0.999 | 5.17 | 0.63 | 2.50 | 5.56 | 0.48 | 2.78 | 5.45 | -0.62 | 2.62
1.000 | 5.57 | 0.85 | 2.53 | 6.21 | 0.84 | 2.88 | 7.78 | 0.83 | 3.69

Table 3: Simulation results, log- Bayes factors. Refer to Section 5 for a
discussion.