THE LOCAL TO UNITY DYNAMIC TOBIT MODEL ANNA BYKHOVSKAYA AND JAMES A. DUFFY Abstract. This paper considers highly persistent time series that are subject to nonlin-

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THE LOCAL TO UNITY DYNAMIC TOBIT MODEL

ANNA BYKHOVSKAYA AND JAMES A. DUFFY

Abstract. This paper considers highly persistent time series that are subject to nonlin-

earities in the form of censoring or an occasionally binding constraint, such as are regularly

encountered in macroeconomics. A tractable candidate model for such series is the dynamic

Tobit with a root local to unity. We show that this model generates a process that converges

weakly to a non-standard limiting process, that is constrained (regulated) to be positive.

Surprisingly, despite the presence of censoring, the OLS estimators of the model parameters

are consistent. We show that this allows OLS-based inferences to be drawn on the overall

persistence of the process (as measured by the sum of the autoregressive coeﬃcients), and for

the null of a unit root to be tested in the presence of censoring. Our simulations illustrate

that the conventional ADF test substantially over-rejects when the data is generated by a

dynamic Tobit with a unit root, whereas our proposed test is correctly sized. We provide an

application of our methods to testing for a unit root in the Swiss franc / euro exchange rate,

during a period when this was subject to an occasionally binding lower bound.

Keywords: non-negative time series, dynamic Tobit, local unit root, unit root test.

1. Introduction

Since the 1950s nonlinear models have played an increasingly prominent role in the analysis

and prediction of time series data. In many cases, as was noted in early work by Moran (1953),

linear models are unable to adequately match the features of observed time series. The eﬀorts

to develop models that enjoy the ﬂexibility aﬀorded by nonlinearities, while retaining the

tractability of linear models, have subsequently engendered an enormous literature (see e.g.

Fan and Yao, 2003; Gao, 2007; Chan, 2009; and Terasvirta et al., 2010).

An important instance of nonlinearity arises when data is bounded by, truncated at, or

censored below some threshold, since such phenomena cannot be adequately captured – even

approximately – by a linear model. Many observed series are bounded below by construction,

and may spend lengthy periods at or near their lower boundary, such as unemployment rates,

prices, gross sectoral trade ﬂows, and nominal interest rates. The non-negativity of interest

rates, and the resulting constraints that this may impose on the eﬃcacy of monetary policy,

has received particular attention in recent years, as central bank policy rates have remained

at or near the zero lower bound for a signiﬁcant portion of the past two decades, across many

economies (see e.g. Mavroeidis, 2021, and the works cited therein).

A tractable model for such series, which generates both their characteristic serial dependence

and censoring, is the dynamic Tobit model. In its static formulation, the model originates

We thank T. Bollerslev, G. Cavaliere, V. Kleptsyn, S. Mavroeidis, and seminar participants at Cambridge,

Cyprus, Duke, NES30 Conference, St Andrews, and Oxford for their helpful comments and advice on earlier

drafts of this work.

arXiv:2210.02599v3 [econ.EM] 8 May 2024

2 ANNA BYKHOVSKAYA AND JAMES A. DUFFY

with Tobin (1958). In its dynamic formulation, the model typically comes in one of two

varieties, which we refer to as the latent and censored models. In the latent dynamic Tobit, an

unobserved process {y∗

t}follows a linear autoregression, with yt= max{y∗

t,0}being observed;

whereas in the censored dynamic Tobit, {yt}is modelled as the positive part of a linear function

of its own lags, and an additive error (see e.g. Maddala 1983, p. 186, or Wei 1999, p. 419).

In both models the right hand side may be augmented with other explanatory variables.

Relative to the latent model, the censored model has the advantage of being Markovian,

which greatly facilitates its use in forecasting. It has also been successfully applied to a range

of censored series, in both purely time series and panel data settings, including: the open

market operations of the Federal Reserve (Demiralp and Jord`a, 2002; de Jong and Herrera,

2011); household commodity purchases (Dong et al., 2012); loan charge-oﬀ rates (Liu et al.,

2019); credit default and overdue loan repayments (Brezigar-Masten et al., 2021); and sectoral

bilateral trade ﬂows (Bykhovskaya, 2023). Recently, Mavroeidis (2021) proposed the censored

and kinked structural VAR model to describe the operation of monetary policy during periods

when the zero lower bound may occasionally bind on the policy rate. If only the actual interest

rate (rather than some ‘shadow rate’) aﬀects agents’ decision making, as assumed in closely

related work by Aruoba et al. (2022), then the univariate counterpart of this model is exactly

the censored dynamic Tobit.

The present work is concerned with the censored, rather than the latent, dynamic Tobit

model. As discussed by de Jong and Herrera (2011, p. 229), the censored model is arguably

more appropriate in settings where yt= 0 results not from limits on the observability of

some underlying y∗

t, but from an economic constraint on the values taken by yt. For example,

Bykhovskaya (2023, Supplementary Material, Appendix A) justiﬁes the application of the

dynamic Tobit with reference to a game-theoretic model of network formation, in which the

zeros correspond to corner solutions of a constrained optimisation problem, i.e. where zeros

are systematically observed because of a non-negativity constraint on agents’ choices. In our

illustrative empirical application, to an exchange rate that is subject to a ﬂoor engineered by

a central bank (Section 5), the most recent values of the exchange rate are taken as suﬃcient

to describe the market equilibrium, and thus the conditional distribution of future rates. Our

focus on the censored model is also motivated by relatively greater need for the development of

relevant econometric theory in this area. In the latent model, the dynamics are simply those

of the latent autoregression, and so are readily understood by standard methods; whereas

in the censored model, the censoring aﬀects the dynamics of {yt}in a non-trivial manner,

making the analysis rather more challenging. Indeed, establishing the stationarity or weak

dependence of the censored dynamic Tobit is far from trivial, as can be seen from Hahn and

Kuersteiner (2010), de Jong and Herrera (2011), Michel and de Jong (2018), and Bykhovskaya

THE LOCAL TO UNITY DYNAMIC TOBIT MODEL 3

(2023). Henceforth, all references to the ‘dynamic Tobit’ are to the censored version of the

model.1

Motivated in part by recent work on modelling nominal interest rates near the zero lower

bound, our concern is with the application of this model to series that are highly persistent,

so that above the censoring point they exhibit the random wandering that is characteristic

of integrated processes. The appropriate conﬁguration of the dynamic Tobit model for such

series, in which the autoregressive polynomial has a root local to unity, has not been considered

in the literature to date – apart from the special case of a ﬁrst-order model with an exact unit

root, as in Cavaliere (2004) and Bykhovskaya (2023). Our results are thus entirely new to the

literature.

Our principal technical contribution, within this setting, is to derive the limiting distri-

butions of both the standardised regressor process, and the ordinary least squares (OLS)

estimates of the parameters of the dynamic Tobit, when that model has an autoregressive

root local to unity. The reader may ﬁnd our focus on OLS surprising, as this method would

usually provide inconsistent estimates in the presence of censoring. However, it turns out that

in our setting consistency (for all model parameters) is restored, and we obtain a usable limit

theory for the estimated sum of the autoregressive coeﬃcients, which conventionally provides

a measure of the overall persistence of a process (cf. Andrews and Chen, 1994; Mikusheva,

2007). While one may contemplate alternatively using maximum likelihood (ML) or least

absolute deviations (LAD) to estimate the model, OLS enjoys the advantages of maintain-

ing only weak distributional assumptions on the innovations (unlike ML), and avoiding the

numerical minimisation of a non-convex criterion function (unlike LAD).

Our asymptotics provide the basis for practical unit root tests for highly persistent, censored

time series. Motivated by our ﬁnding that OLS is consistent, we consider a test based on

the (constant only) augmented Dickey–Fuller (ADF) tstatistic, but which employs critical

values modiﬁed to reﬂect the censoring present in the data generating process. We show,

via Monte Carlo simulations, that as our critical values are larger than the conventional ADF

critical values, their use eliminates the signiﬁcant over-rejection that may result from the naive

application of the ADF test to censored data. (This tendency to over-reject the null of a unit

root appears typical of models that incorporate unit roots and nonlinearities, having been also

found by e.g. Hamori and Tokihisa, 1997; Kim et al., 2002; and Wang and De Jong, 2013.)

Strikingly, the distribution of tstatistic, under censoring, is stochastically dominated by that

obtained from the linear autoregressive model.2

1de Jong and Herrera (2011) alternatively refer to this model as a ‘dynamic censored regression’, but the term

‘dynamic Tobit’ appears more commonly in the literature (see e.g. Hahn and Kuersteiner, 2010; Michel and

de Jong, 2018; Bykhovskaya, 2023).

2This holds only with respect to the distributions: it is not true that if one simulates a linear and a Tobit

model with the same underlying innovations, then the former tstatistic will always be larger than the latter.

4 ANNA BYKHOVSKAYA AND JAMES A. DUFFY

Our work may be construed, more broadly, as extending the analysis of highly persistent

time series, and the associated machinery of unit root testing, from a linear setting to a

nonlinear setting appropriate to time series that are subject to a lower bound. In doing so,

we complement the seminal work of Cavaliere (2005), which similarly sought to extend this

machinery to the setting of bounded time series. Our contribution is to eﬀect this extension

within a class of nonlinear autoregressive models that have been widely applied to censored

time series (as evinced by the works cited above), and which fall outside his framework.

On a technical level, the most closely related works to our own are those of Cavaliere (2004,

2005) and Cavaliere and Xu (2014), who develop the asymptotics of what they term ‘limited

autoregressive processes’ with a near-unit root, which are (one- or two-sided) non-Markovian

censored processes constructed by the addition of regulators to a latent linear autoregression.

While their (one-sided) model has a superﬁcial resemblance to the dynamic Tobit, there are

important, but subtle diﬀerences between the two (see Section 2.4 for a discussion). Perhaps

the most striking similarity is that both models, in the case of an exact unit root, give rise

to processes that converge weakly to regulated Brownian motion; but when roots are merely

local to unity, the limiting processes associated with these two models are distinct (see the

discussion following Theorem 3.1).

Convergence to regulated Brownian motions has also been obtained previously in the set-

ting of ﬁrst-order threshold autoregressive models with an (exact) unit root regime and a

stationary regime, as considered by Liu et al. (2011) and Gao et al. (2013). However, allowing

for both near-unit roots and higher-order autoregressive terms introduces technical challenges

that require us to take a markedly diﬀerent approach from those employed in these earlier

works. With respect to higher-order models, a major diﬃculty relates to the treatment of the

diﬀerences {∆yt}. In a linear autoregressive model with a single unit root, and all other roots

outside the unit circle, these would follow a stationary autoregression; but in our setting they

instead follow a regime-switching autoregression, where the regime depends on the (lagged)

level of yt, and so are inherently non-stationary. Accoridingly, standard arguments for con-

trolling the magnitude of ∆yt, and deriving the limits of functionals thereof, are unavailing.

We provide a striking example in which ∆ytis explosive even though all but one of the autore-

gressive roots (the root at unity) lie outside the unit circle, due to the interactions between

the two autoregressive regimes (see Appendix C). To preclude such cases, we develop a condi-

tion relating to the joint spectral radius of the autoregressive representation for ∆yt, which is

suﬃcient to control the magnitude of ∆ytand plays an essential role in our arguments. (This

concept has been previously employed in the context of stationary autoregressive models, see

e.g. Liebscher (2005); Saikkonen (2008).)

The remainder of this paper is organised as follows. Section 2 discusses the model and

our assumptions. Asymptotic results and corresponding tests are derived in Section 3, while

supporting Monte Carlo simulations are shown in Section 4. Section 5 applies our framework

THE LOCAL TO UNITY DYNAMIC TOBIT MODEL 5

to the exchange rate between the Swiss franc and the euro during a period when this rate was

subject to a lower bound. Finally, Section 6 concludes. All proofs appear in the appendices.

Notation. C,C′,C′′, etc. denote generic constants that may take diﬀerent values in diﬀerent

parts of this paper. All limits are taken as T→ ∞ unless otherwise speciﬁed. p

→and d

→

respectively denote convergence in probability and distribution (weak convergence). We write

‘XT(r)d

→X(r) on D[0,1]’ to denote that {XT}converges weakly to X, where these are

considered as random elements of D[0,1], the space of cadlag functions on [0,1], equipped

with the uniform topology. For p≥1 and Xa random variable, ∥X∥p:= (E|X|p)1/p.

2. The dynamic Tobit model with a near-unit root

2.1. Model and assumptions. Consider a time series {yt}generated by the dynamic Tobit

model of order k≥1, written in augmented Dickey–Fuller (ADF) form,3

(2.1) yt="α+βyt−1+

k−1

i=1

ϕi∆yt−i+ut#+

, t = 1, . . . , T,

where ∆yt:= yt−yt−1, and [x]+:= max{x, 0}denotes the positive part of x∈R. Let

(2.2) B(z) := 1 −βz −(1 −z)

k−1

i=1

ϕizi= (1 −β)z+ (1 −z)ϕ(z),

where ϕ(z) := 1 −Pk−1

i=1 ϕizi. We impose the following on the data generating process (2.1).

Assumption A1.{yt}is initialised by (possibly) random variables {y−k+1, . . . , y0}. Moreover,

T−1/2y0

→b0for some b0≥0.

Assumption A2.{yt}is generated according to (2.1), where:

1. {ut}t∈Zis independently and identically distributed (i.i.d.) with Eut= 0 and Eu2

t=σ2.

2. α=αT:=T−1/2aand β=βT= exp(c/T )for some a, c ∈R.

Assumption A3.There exist δu>0and C < ∞such that:

1. E|ut|2+δu< C.

2. E|T−1/2y0|2+δu< C, and E|∆yi|2+δu< C for i∈ {−k+ 2,...,0}.

Figure 2.1 displays a typical sample path for the dynamic Tobit (2.1), under the preceding

assumptions.

3The autoregressive form is yt= [α+Pk

i=1 βiyt−i+ut]+, where β1=β+ϕ1,βk=−ϕk−1, and βi=ϕi−ϕi−1

for i∈ {2, . . . , k −1}. In particular β=Pk

i=1 βicorresponds to the sum of the autoregressive coeﬃcients.

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THELOCALTOUNITYDYNAMICTOBITMODELANNABYKHOVSKAYAANDJAMESA.DUFFYAbstract.Thispaperconsidershighlypersistenttimeseriesthataresubjecttononlin-earitiesintheformofcensoringoranoccasionallybindingconstraint,suchasareregularlyencounteredinmacroeconomics.AtractablecandidatemodelforsuchseriesisthedynamicTobit...

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