Diffusion Copulas: Identification and Estimation

We propose a new semiparametric approach for modelling nonlinear univariate diffusions, where the observed process is a nonparametric transformation of an underlying parametric diffusion (UPD). This modelling strategy yields a general class of semiparametric Markov diffusion models with parametric dynamic copulas and nonparametric marginal distributions. We provide primitive conditions for the identification of the UPD parameters together with the unknown transformations from discrete samples. Likelihood-based estimators of both parametric and nonparametric components are developed and we analyze the asymptotic properties of these. Kernel-based drift and diffusion estimators are also proposed and shown to be normally distributed in large samples. A simulation study investigates the finite sample performance of our estimators in the context of modelling US short-term interest rates. We also present a simple application of the proposed method for modelling the CBOE volatility index data.

Authors

• 1 publication
• 1 publication
• 4 publications
• Adaptive estimation for small diffusion processes based on sampled data

We consider parametric estimation for multi-dimensional diffusion proces...
08/03/2021 ∙ by Tetsuya Kawai, et al. ∙ 0

• Nonparametric additive factor models using sieve methods

This paper proposes a nonparametric additive factor model where the comm...
03/29/2020 ∙ by Julien Bodelet, et al. ∙ 0

• Nonparametric estimation of marginal distributions for unordered pairs

04/23/2021 ∙ by L. Dumitrescu, et al. ∙ 0

• Identification in a Fully Nonparametric Transformation Model with Heteroscedasticity

The so far most general identification result in the context of nonparam...
03/12/2020 ∙ by Nick Kloodt, et al. ∙ 0

• Estimation of the Transformation Function in Fully Nonparametric Transformation Models with Heteroscedasticity

Completely nonparametric transformation models with heteroscedastic erro...
04/04/2020 ∙ by Nick Kloodt, et al. ∙ 0

• State-dependent jump activity estimation for Markovian semimartingales

The jump behavior of an infinitely active Itô semimartingale can be conv...
11/15/2018 ∙ by Fabian Mies, et al. ∙ 0

• Bayesian semi-parametric inference for diffusion processes using splines

We introduce a semi-parametric method to simultaneously infer both the d...
06/10/2021 ∙ by Paul A. Jenkins, et al. ∙ 0

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Most financial time series have fat tails that standard parametric models are not able to generate. One forceful argument for this in the context of diffusion models was provided by Aït-Sahalia (1996b) who tested a range of parametric models against a nonparametric alternative and found that most standard models were inconsistent with observed features in data.

One popular semiparametric approach that allows for more flexibility in terms of marginal distributions, and so allowing for fat tails, is to use the so-called copula models, where the copula is parametric and the marginal distribution is left unspecified (nonparametric). Joe (1997) showed how bivariate parametric copulas could be used to model discrete-time stationary Markov chains with flexible, nonparametric marginal distributions. The resulting class of semiparametric models are relatively easy to estimate; see, e.g. Chen and Fan (2006). However, most parametric copulas known in the literature have been derived in a cross-sectional setting where they have been used to describe the joint dependence between two random variables with known joint distribution, e.g. a bivariate

-distribution. As such, existing parametric copulas may be difficult to interpret in terms of the dynamics they imply when used to model Markov processes. This in turn means that applied researchers may find it difficult to choose an appropriate copula for a given time series.

One could have hoped that copulas with a clearer dynamic interpretation could be developed by starting with an underlying parametric Markov model and then deriving its implied copula. This approach is unfortunately hindered by the fact that the stationary distributions of general Markov chains are not available on closed-form and so their implied dynamic copulas are not available on closed form either. This complicates both the theoretical analysis (such as establishing identification) and the practical implementation of such models.

An alternative approach to modelling fat tails using Markov diffusions is to specify flexible forms for the so-called drift and diffusion term. Such non-linear features tend to generate fat tails in the marginal distribution of the process. This approach has been widely used to, for example, model short-term interest rates; see, e.g., Aït-Sahalia (1996a,b), Conley et al. (1997), Stanton (1997), Ahn and Gao (1999) and Bandi (2002). These models tend to either be heavily parameterized or involve nonparametric estimators that suffer from low precision in small and moderate samples.

We here propose a novel class of dynamic copulas that resolves the above-mentioned issues: We show how copulas can easily be generated from parametric diffusion processes. The copulas have a clear interpretation in terms of dynamics since they are constructed from an underlying dynamic continuous-time process. At the same time, a given copula-based diffusion can exhibit strong non-linearities in its drift and diffusion term even if the underlying copula is derived from, for example, a linear model. Furthermore, primitive conditions for identification of the parameters are derived; and this despite the fact that the copulas are implicit. Finally, the models can easily be implemented in practice using existing numerical methods for parametric diffusion processes. This in turn implies that estimators are easy to compute and do not involve any smoothing parameters; this is in contrast to existing semi- and nonparametric estimators of diffusion models.

The starting point of our analysis is to show that there is a one-to-one correspondence between any given semiparametric Markov copula model and a model where we observe a nonparametric transformation of an underlying parametric Markov process. We then restrict attention to parametric Markov diffusion processes which we refer to as underlying parametric diffusions (UPD’s). Copulas generated from a given UPD has a clear interpretation in terms of dynamic properties. In particular, standard results from the literature on diffusion models can be employed to establish mixing properties and existence of moments for a given model; see, e.g. Chen et al. (2010). Moreover, we are able to derive primitive conditions for the parameters of the copula to be identified together with the unknown transformation.

Once identification has been established, estimation of our copula diffusion models based on a discretely sampled process proceeds as in the discrete-time case. One can either estimate the model using a one-step or two-step procedure: In the one-step procedure, the marginal distribution and the parameters of the UPD are estimated jointly by sieve-maximum likelihood methods as advocated by Chen, Wu and Yi (2009). In the two-step approach, the marginal distribution is first estimated by the empirical cdf, which in turn is plugged into the likelihood function of the model. This is then maximized with respect to the parameters of the UPD. We provide an asymptotic theory for both cases by importing results from Chen, Wu and Yi (2009) and Chen and Fan (2006), respectively. In particular, we provide primitive conditions for their high-level assumptions to hold in our diffusion setting. The resulting asymptotic theory shows -asymptotic normality of the parametric components. Given the estimates of parametric component, one can obtain semiparametric estimates of the drift and diffusion functions and we also provide an asymptotic theory for these.

Our modelling strategy has parametric ascendants: Bu et al. (2011), Eraker and Wang (2015) and Forman and Sørensen (2014) considered parametric transformations of UPDs for modelling short-term interest rates, variance risk premia and molecular dynamics, respectively. We here provide a more flexible class of models relative to theirs since we leave the transformation unspecified. At the same time, all the attractive properties of their models remain valid: The transition density of the observed process is induced by the UPD and so the estimation of copula-based diffusion models is computationally simple. Moreover, copula diffusion models can furthermore be easily employed in asset pricing applications since (conditional) moments are easily computed using the specification of the UPD. Finally, none of these papers fully addresses the identification issue and so our identification results are also helpful in their setting.

There are also similarities between our approach and the one pursued in Aït-Sahalia (1996a) and Kristensen (2010). They developed two classes of semiparametric diffusion models where either the drift or the diffusion term is specified parametrically and the remaining term is left unspecified. The remaining term is then recovered by using the triangular link between the marginal distribution, the drift and the diffusion terms that exist for stationary diffusions. In this way, the marginal distribution implicitly ties down the dynamics of the observed diffusion process. Unfortunately, it is very difficult to interpret the dynamic properties of the resulting semiparametric diffusion model. In contrast, in our setting, the UPD alone ties down the dynamics of the observed diffusion and so these are much better understood. The estimation of copula diffusions are also less computationally burdensome compared to the Pseudo Maximum Likelihood Estimator (PMLE) proposed in Kristensen (2010).

The remainder of this paper is organized as follows. Section 2 outlines our semiparametric modelling strategy. Section 3 investigates the identification issue of our model. In Section 4, we discuss the estimators of our model while Section 5 investigates their asymptotic properties. Section 6 presents a simulation study to examine the finite sample performance of our estimators. In Section 7, we consider a simple empirical application. Some concluding remarks are given in Section 8. All proofs and lemmas are collected in Appendices.

2 Copula-Based Diffusion Models

2.1 Framework

Consider a continuous-time process with domain , where . We assume that satisfies

 Yt=V(Xt), (2.1)

where is a smooth monotonic univariate function and solves the following parametric SDE:

 dXt=μX(Xt;θ)dt+σX(Xt;θ)dWt. (2.2)

Here, and

are scalar functions that are known up to some unknown parameter vector

, where is the parameter space, while is a standard Brownian motion. We call the underlying parametric diffusion (UPD) and let , , denote its domain.

We call a copula-based diffusion since its dynamics are determined by the implied (dynamic) copula of the UPD , as we will explain below. Given a discrete sample of , , , where denotes the time distance between observations, we are then interested in drawing inference regarding the parameter and the function . Note here that we only observe while remains unobserved since we leave unspecified (unknown to us). For convenience, we collect the unknown component in the structure .

The above class of models allows for added flexibility through the transformation which we treat as a nonparametric object that we wish to estimate together with . By allowing for a broad nonparametric class of transformations , our model is richer and more flexible compared to the fully parametric case with known or parametric specifications of . In particular, as we shall see, any given member of the above class of models is able to completely match the marginal distribution of any given time series.

We will require that the underlying Markov process sampled at , , possesses a transition density ,

 Pr(XΔ∈A|X0=x0)=∫ApX(x|x0;θ)dx, \ \ A⊆X. (2.3)

Moreover, some of our results require to be recurrent, a property which can be stated in terms of the so-called scale density and scale measure. These are defined as

 s(x;θ):=exp{−∫xx∗2μX(z;θ)σ2X(z;θ)dz} and S(x;θ):=∫xx∗s(z;θ)dz (2.4)

for some . We then impose the following:

Assumption 2.1.

(i) and are twice continuously differentiable; (ii) the scale measure satisfies () as (); (iii) .

Assumption 2.2.

The transformation is strictly increasing with inverse , i.e., , and is twice continuously differentiable.

Assumption 2.1(i) provides primitive conditions for a solution to eq. (2.2) to exist and for the transition density to be well-defined, while Assumption 2.1(ii) implies that this solution is positive recurrent; see Bandi and Phillips (2003), Karatzas and Shreve (1991, Section 5.5) and McKean (1969, Section 5) for more details. Assumption 2.1(iii) strengthens the recurrence property to stationarity and ergodicity in which case the stationary marginal density of  takes the form

 fX(x;θ)=ξ(θ)σ2X(x;θ)s(x;θ), (2.5)

where was defined in Assumption 2.1(iii). However, stationarity will not be required for all our results to hold; in particular, some of our identification results and proposed estimators do not rely on stationarity. This is in contrast to the existing literature on dynamic copula models where stationarity is a maintained assumption.

Assumption 2.2 requires to be strictly increasing; this is a testable restriction under the remaining assumptions introduced below which ensures identification: Suppose that indeed is strictly decreasing; we then have , where is increasing and has dynamics . Assuming that the chosen UPD satisfies for , we can test whether indeed is decreasing or increasing.

The smoothness condition on is imposed so that we can employ Ito’s Lemma on the transformation to obtain that the continuous-time dynamics of can be written in terms of as

 dYt=μY(Yt;S)dt+σY(Yt;S)dWt,

with

 μY(y;S) = (2.6) σY(y;S) = σX(U(y);θ)U′(y), (2.7)

where we have used that, with and denoting the first two derivatives of , and . In particular, is a Markov diffusion process. As can be seen from the above expressions, the dynamics of , as characterized by and , may appear quite complex with potentially generating nonlinearities in both the drift and diffusion terms even if and are linear. We demonstrate this feature in the subsequent subsection where we present examples of simple UPD’s are able to generate non-linear shapes of  and

via the non-linear transformation

. At the same time, if we transform by we recover the dynamics of the UPD. As a consequence, the transition density of the discretely sampled process , , can be expressed in terms of the one of as

 pY(y|y0;S)=U′(y)pX(U(y)|U(y0);θ), (2.8)

using standard results for densities of invertible transformations. By similar arguments, the stationary density of satisfies

 fY(y;S)=U′(y)fX(U(y);θ), (2.9)

which shows that any choice for UPD is able to fully adapt to any given marginal density of due to the nonparametric nature of .

The above expressions also highlights the following additional theoretical and practical advantages of our modelling strategy: First, for a given choice of , we can easily compute and since computation of parametric transition densities and stationary densities of diffusion models is in general straightforward, even if they are not available on closed form. Second, inherits all its dynamic properties from ; and in the modelling of , we can rely on a large literature on parametric modelling of diffusion models. Formally, we have the following straightforward results adopted from Forman and Sørensen (2014).

Proposition 2.1

Suppose that Assumptions 2.1(i)–(ii) and 2.2 hold. Then the following results hold for the model (2.1)-(2.2):

1. If Assumption 2.1(iii) hold, then is stationary and ergodic and so is .

2. The mixing coefficients of and coincide.

3. If and for some and , then .

4. If

is an eigenfunction of

with corresponding eigenvalue

in the sense that then is an eigenfunction of with corresponding eigenvalue .

The above theorem shows that, given knowledge (or estimates) of , the properties of in terms of mixing coefficients, moments, and eigenfunctions are well-understood since they are inherited from the specification of . In addition, computations of conditional moments of can be done straightforwardly utilizing knowledge of the UPD. For example, for a given function , the corresponding conditional moment can be computed as

 E[G(Yt+s)|Yt=y]=E[GX(Xt+s)|Xt=U(y)], where GX(x):=G(V(x)).

The right-hand side moment only involves

and so standard methods for computing moments of parametric diffusion models (e.g., Monte Carlo methods, solving partial differential equations, Fourier transforms) can be employed. This facilitates the use of our diffusion models in asset pricing where the price often takes the form of a conditional moment. We refer to Eraker and Wang (2015) for more details on asset pricing applications for our class of models; they take a fully parametric approach but all their arguments carry over to our setting.

The last result of the above theorem will prove useful for our identification arguments since these will rely on the fundamental nonparametric identification results derived in Hansen et al. (1998). Their results involve the spectrum of the observed diffusion process, and the last result of the theorem implies that the spectrum of is fully characterized by the spectrum of together with the transformation. The eigenfunctions and their eigenvalues are also useful for evaluating long-run properties of . In our semiparametric approach, the eigenfunctions and corresponding eigenvalues of are easily computed from and so we circumvent the problem of estimating these nonparametrically as done in, for example, Chen, Hansen and Scheinkman (2009) and Gobet et al. (2004).

2.2 Examples of UPDs

Our framework is quite flexible and in principle allows for any specification of the UPD for . Many parametric models are available for that purpose, and we here present three specific examples from the literature on continuous-time interest rate modelling.

Example 1: Ornstein-Uhlenbeck (OU) model. The OU model (c.f. Vasicek, 1977) is given by

 dXt=κ(α−Xt)dt+σdWt, (2.10)

defined on the domain . The process is stationary if and only if , in which case mean-reverts to its unconditional mean . The scale of is controlled by . Its stationary and transition distributions are both normal, and the corresponding copula of the discretely sampled process is a Gaussian copula with correlation parameter . For this particular model, the resulting drift and diffusion term of the observed process takes the form

 μY(y;S)=κ(α−U(y))U′(y)−12σ2U′′(y)U′(y)3, \ \ σ2Y(y;S)=σ2U′(y)2. (2.11)

In Figure 2 (found in Section 6), we plot these two functions with  and fitted to the 7-day Eurodollar interest rate time series used in Aït-Sahalia (1996b). Observe that  generates non-linear behavior in and despite the UPD being a linear Gaussian process.

Example 2: Cox-Ingersoll-Ross (CIR) model. The CIR process (c.f. Cox et al., 1985) is given by

 dXt=κ(α−Xt)dt+σ√XtdWt. (2.12)

The process has domain and is stationary if and only if , and . Conditional on , admits a non-central

distribution with fractional degrees of freedom while its stationary distribution is a Gamma distribution. To our best knowledge, the corresponding dynamic copula has not been analyzed before or used in empirical work. Figure 4 (in Section

6) displays  and with  and chosen in the same way as in Exampe 1. Compared to this example, the resulting drift and diffusion term of  exhibit even stronger non-linearities.

Example 3: Nonlinear Drift Constant Elasticity Variance (NLDCEV) model. The NLDCEV specification (c.f. Conley et al., 1997) is given by

 dXt=(l∑i=−kαiXit)dt+σXβtdWt (2.13)

with domain . It is easily seen that when and the drift term of the diffusion in (2.13) exhibits mean-reversions for large and small values of . A popular choice for various studies in finance assumes that and or (c.f. Aït-Sahalia, 1996b; Choi, 2009; Kristensen, 2010; Bu, Cheng and Hadri, 2017), in which case the drift has linear or zero mean-reversion in the middle part and much stronger mean-reversion for large and small values of . Meanwhile, the CEV diffusion term is also consistent with most empirical findings of the shape of the diffusion term. It follows that since (2.13) is one of the most flexible parametric diffusions, diffusion processes that are unspecified transformations of (2.13) should represent a very flexible class of diffusion models. Similar to (2.12), the implied copula of the NLDCEV is new to the copula literature.

Examples 1-2 are attractive from a computational standpoint since the corresponding transition densities are available on closed-form thereby facilitating their implementation. But this comes at the cost of the dynamics being somewhat simple. The NLDCEV model implies more complex and richer dynamics but on the other hand its transition density is not available on closed form. However, the marginal pdf of the NLDCEV process, as well as more general specifications, can be evaluated in closed form by (2.5). Moreover, closed-from approximations of the transition density of the NLDCEV model developed by, for example, Aït-Sahalia (2002) and Li (2013) can be employed. Alternatively, simulated versions of the transition density can be computed using the techniques developed in, for example, Kristensen and Shin (2012) and Bladt and Sørensen (2014). In either case, an approximate version of the exact likelihood can be easily computed, thereby allowing for simple estimation of even quite complex underlying UPDs.

2.3 Related Literature

As already noted in the introduction, copula-based diffusions are related to the class of so-called discrete-time copula-based Markov models; see, for example, Chen and Fan (2006) and references therein. To map the notation and ideas of this literature into our continuous-time setting, we set the sampling time distance in the remaining part of this section.

Let us first introduce copula-based Markov models where a given discrete-time, stationary scalar Markov process is modelled through a bivariate parametric copula density111The copula for a given Markov process is defined as

The corresponding copula density is then given by ., say, , together with its stationary marginal cdf , i.e., so that ’s transition density satisfies

 pY(y|y0;θ,FY)=fY(y)cX(FY(y0),FY(y);θ), (2.14)

where . An alternative representation of this model is

 Yi=F−1Y(¯Xi), \ \ ¯Xi+1|¯Xi=x0∼cX(x0,⋅;θ), (2.15)

so that is a transformation of an underlying Markov process ; the latter having a uniform marginal distribution and transition density . Thus, if is induced by an underlying Markov diffusion transition density, the corresponding copula-based Markov model falls within our framework.

Reversely, consider a copula-based diffusion and suppose that the UPD is stationary with marginal cdf . By definition of , its marginal cdf satisfies

 FY(y)=FX(U(y);θ)⇔U(y)=F−1X(FY(y);θ). (2.16)

Substituting the last expression for into (2.8), we see that can be expressed in the form of (2.14) where is the density function of the (dynamic) copula implied by the discretely sampled UPD ,

 cX(u0,u;θ)=pX(F−1X(u;θ)|F−1X(u0;θ);θ)fX(F−1X(u;θ);θ). (2.17)

Thus, any discretely sampled stationary copula-based diffusion satisfies (2.15) with .

However, the literature on copula-based Markov models focus on discrete-time models with standard copula specifications derived from bivariate distributions in an i.i.d. setting. Using copulas that are originally derived in an i.i.d. setting complicates the interpretation of the dynamics of the resulting Markov model, and conditions for the model to be mixing, for example, can be quite complicated to derive; see, e.g., Beare (2010) and Chen, Wu and Yi (2009). This also implies that very few standard copulas can be interpreted as diffusion processes; to our knowledge, the only one is the Gaussian copula which corresponds to the OU process in Example 1.

The reader may now wonder why we do not simply generate dynamic copulas by first deriving the transition density for a given discrete-time Markov model and then obtain the corresponding Markov copula through eq. (2.17)? The reason is that for most discrete-time Markov models the stationary distribution is not known on closed form. Thus, first of all, and thereby also have be approximated numerically. Second, since is now not available on closed form, the analysis of which parameters one can identify from the resulting copula model becomes very challenging. And identification in copula-based Markov models is a non-trivial problem: Generally, for a given parametric Markov model, not all parameters are identified from the corresponding copula as given in (2.17) and some of them have to be normalized.

We here directly generate copulas through an underlying continuous-time diffusion model for . This resolves the aforementioned drawbacks of existing copula-based Markov models: First, we are able to generate highly flexible copulas so far not considered in the literature. Second, given that our copulas are induced by specifying the drift and diffusion functions of , the time series properties are much more easily inferred from our model, c.f. Theorem 2.1. Third, by Ito’s Lemma, eqs. (2.6)-(2.7) provide us with explicit expressions linking the drift and diffusion terms of the observed diffusion process to the UPD through the transformation ; this will allow us to derive necessary and sufficient conditions for identification in the following. Fourth, in terms of estimation, the stationary distribution of a given diffusion model has an explicit form, c.f. eq. (2.5), which allows us to develop computationally simple estimators of copula diffusion models. Finally, some of our identification results will not require stationarity and so expands the scope for using copula-type models in time series analysis.

Our modelling strategy is also related to the ideas of Aït-Sahalia (1996a) and Kristensen (2010, 2011) where is left unspecified while either the drift, , or the diffusion term, , is specified parametrically. As an example, consider the former case where is known up to the parameter . Given knowledge of the marginal density (or a nonparametric estimator of it), the diffusion term can then be recovered as a functional of and as

 μY(y;fY,θ)=12fY(y)∂∂y[σ2Y(y;θ)fY(y)].

So in their setting pins down the resulting dynamics of in a rather opaque manner.

3 Identification

Suppose that a particular specification of the UPD as given in (2.2) has been chosen. Given the discrete sample of , the goal is to obtain consistent estimates of together with . To this end, we first have to show that these are actually identified from data. In order to do so, we need to be precise about which primitives we can identify from data. Given the primitives, we then wish to recover . In the cross-sectional literature, one normally take as given the distribution of data and then establish a mapping between this and the structural parameters. In our setting, we are able to learn about the transition density of our data, , from the population and so it would be natural to use this as primitive from which we wish to recover . However, the mapping from to is not available on closed form in general in our setting and so this identification strategy appears highly complicated. Instead we will take as primitives the drift, , and diffusion term, , of and then show identification of from these. This identification argument relies on us being able to identify and in the first place, which we formally assume here:

Assumption 3.1

The drift, , and the diffusion, , are nonparametrically identified from the discretely sampled process .

The above assumption is not completely innocuous and does impose some additional regularity conditions on the Data Generating Process (DGP). We therefore first provide sufficient conditions under which Assumption 3.1 holds. The first set of conditions are due to Hansen et al. (1998) who showed that Assumption 3.1 is satisfied if is stationary and its infinitesimal operator has a discrete spectrum. Theorem 2.1(4) is helpful in this regard since it informs us that the spectrum of can be recovered from the one of . In particular, if is stationary with a discrete spectrum, then will have the same properties. Since the dynamics of is known to us, the properties of its spectrum are in principle known to us and so this condition can be verified a priori. The second set of primitive conditions come from Bandi and Phillips (2003): They show that as and , the drift and diffusion functions of a recurrent Markov diffusion process are identified. This last result holds without stationarity, but on the other hand requires high-frequency observations.

In order to formally state the above two results, we need some additional notation. Recall that the infinitesimal operator, denoted , of a given UPD is defined as

 LX,θg(x):=μX(x;θ)g′(x)+12σ2X(x;θ)g′′(x),

for any twice differentiable function . We follow Hansen et al. (1998) and restrict the domain of to the following set of functions:

 D(LX,θ)={g∈L2(fX):g′ is a.c., LX,θg∈L2(fX) and limx↓xlg′(x)s(x)=limx↑xug′(x)s(x)=0}.

where a.c. stands for absolutely continuous. The spectrum of is then the set of solution pairs , with and , to the following eigenvalue problem, . We refer to Hansen et al. (1998) and Kessler and Sørensen (1999) for a further discussion and results regarding the spectrum of . The following result then holds:

Proposition 3.1

Suppose that Assumption 2.1(i)-(ii) is satisfied. Then Assumption 3.1 holds under either of the following two sets of conditions:

1. Assumption 2.1(iii) holds and has a discrete spectrum where is the data-generating parameter value.

2. and .

Importantly, the above result shows that Assumption 3.1 can be verified without imposing stationarity. Unfortunately, this requires high-frequency information (). To our knowledge, there exists no results for low-frequency ( fixed) identification of the drift and diffusion terms of scalar diffusion processes under non-stationarity. But by inspection of the arguments of Hansen et al. (1998) one can verify that at least the diffusion component is nonparametrically identified from low-frequency information without stationarity.

We are now ready to analyze the identification problem. Recall that contains the objects of interest and let our model consist of all the structures that satisfy, as a minimum, Assumptions 2.1(i)–(ii) and 2.2. According to (2.6)-(2.7), each structure implies a drift and diffusion term of the observed process. We shall say that two structures and are observationally equivalent, a property which we denote by , if they imply the same drift and diffusion of , i.e.

 ∀y∈Y:μY(y;S)=μY(y;~S) and σY(y;S)=σY(y;~S). (3.1)

The structure is then said to be identified within the model if implies . In our setting, without suitable normalizations on the parameters of the UPD, identification will generally fail. To see this, observe that any given structure is observationally equivalent to the following process: Choose any one-to-one transformation , and rewrite the DGP implied by as

 Yt=~V(~Xt), \ \ ~V(x)=V(T(x)), (3.2)

where solves

 d~Xt=μT−1(X)(~Xt;θ)dt+σT−1(X)(~Xt;θ)dWt, (3.3)

with

 μT−1(X)(x;θ) = (3.4) σT−1(X)(x;θ) = σX(T(x);θ)∂T(x)/(∂x). (3.5)

Suppose now that there exists so that and . Then the alternative representation (3.2)-(3.3) is a member of our model with structure which is observationally equivalent to . The following result provides a complete characterizations of the class of observationally equivalent structures for a given model:

Theorem 3.2

Suppose that Assumptions 3.1 is satisfied. For any two structures and satisfying Assumptions 2.1(i) and 2.2, the following hold: if and only if there exists one-to-one transformation so that

 ~V(x)=V(T(x)) (3.6)

and, with and given in eqs. (3.4)-(3.5),

 (i) μT−1(X)(x;~θ)=μX(x;θ) and (ii) σT−1(X)(x;~θ)=σX(x;θ). (3.7)

In particular, the data-generating structure is identified if and only if there exists no one-to-one transformation such that (3.7) holds for .

Note that the above theorem does not require stationarity since it is only concerned with the mapping which is well-defined irrespectively of whether data is stationary. The first part of the theorem provides a exact characterization of when any two structures are equivalent, namely if there exists a transformation so that (3.6)-(3.7) hold. The second part comes as a natural consequence of the first part: If there exists no such transformation, then the data-generating structure must be identified.

Unfortunately, the above result may not always be useful in practice since it requires us to search over all possible one-to-one transformations and for each of these verify that there exists no for which eq. (3.7) holds. In some cases, it proves useful to first normalize the UPD suitably and then verify eq. (3.7) in the normalized version. First note that for any one-to-one transformation , an equivalent representation of the model is

 Yt=V(¯Xt),

where the ”normalised” UPD solves

 d¯Xt=μ¯X(¯Xt;θ)dt+σ¯X(¯Xt;θ)dWt,

with

 μ¯X(¯x;θ) = μX(¯T(¯x;θ);θ)∂¯T(¯x;θ)/(∂¯x)−12σ2X(¯T(¯x;θ);θ)∂2¯T(¯x;θ)/(∂¯x2)∂¯T(¯x;θ)/(∂¯x)3, (3.8) σ¯X(¯x;θ) = σX(¯T(¯x;θ);θ)∂¯T(¯x;θ)/(∂¯x). (3.9)

Given that the above representation is observationally equivalent to the original model, we can still employ Theorem 3.2 but with and replacing and . Verifying the identification conditions stated in the second part of the theorem for the normalised versions will in some situations be easier by judicious choice of .

Below, we present three particular normalising transformations that we have found useful in this regard. The chosen transformations allow us to provide easy-to-check conditions for a given UPD to be identified. For a given UPD, the researcher is free to apply either of the three identification schemes depending on which is the easier one to implement. The three schemes lead to different normalizations/parametrizations, but they all lead to models that are exactly identified (no over-identifying restrictions are imposed) and so are observationally equivalent: The resulting form of  and  will be identical irrespectively of which scheme is employed.

The three transformations that we consider also highlights three alternative modelling approaches: Instead of starting with a parametric UPD as found in the existing literature, such as Examples 1-3, one can alternatively build a UPD with unit diffusion (), zero drift () or known marginal distribution. As we shall see, either of these three modelling approaches are in principle as flexible as the standard approach where the researcher jointly specifies the drift and diffusion term.

3.1 First Scheme

In our first identification scheme, we choose to normalize by the so-called Lamperti transform,

 ¯Xt=¯T−1(Xt;θ):=γ(Xt;θ), \ \ γ(x;θ)=∫xx∗1σX(z;θ)dz,

for some . The resulting process is a unit diffusion process,

 d¯Xt=μ¯X(¯Xt;θ)dt+dWt,

with domain , where and , and drift function

 μ¯X(¯x;θ)=μX(γ−1(¯x;θ);θ)σX(γ−1(¯x;θ);θ)−12∂σX∂x(γ−1(¯x;θ);θ). (3.10)

For the unit diffusion version of the UPD, the equivalence condition (3.7)(ii) becomes

 1=σ¯X(¯x;θ)=σT−1(¯X)(¯x;~θ)=1∂T(¯x)/(∂x),

which can only hold if for some constant . Thus, we can restrict attention to this class of transformations and (3.7)(i) becomes:

Assumption 3.2.

With given in (3.10): There exists no and such that for all .

Assumption 3.2 imposes a normalization condition on the transformed drift function to ensure identification. When verifying Assumption 3.2 for the transformed unit diffusion defined above, we will generally need to fix some of the parameters that enter and of the original process , see below.

Corollary 3.3

Under Assumptions 2.1(i), 2.2 and 3.1, is identified if and only if Assumption 3.2 is satisfied.

The above transformation result can be applied to standard parametric specifications when is available on closed-form. But it also highlights that in terms of modelling copula diffusions, we can without loss of generality build a model where we from the outset restrict and only model the drift term . For example, we could choose the following flexible polynomial drift model where we have already normalized the diffusion term:

 dXt=(l∑i=1αiXit)dt+dWt, (3.11)

where . Corollary 3.3 shows that this particular copula diffusion specification is identified without further restrictions on . Below we apply Corollary 3.3 to some of the standard parametric diffusions introduced earlier:

Example 1 (continued). The Lamperti transform of the OU process in (2.10) is given by

 d¯Xt=κ(α/σ−¯Xt)dt+dWt.

Since is a location shift of , we need to normalize in order for the identification condition 3.3 to be satisfied; one such is leading to the following identified model,

 d¯Xt=−κ¯Xtdt+dWt. (3.12)

Example 2 (continued). The Lamperti transform of the CIR diffusion in (2.12) is given by

 d¯Xt=[κ(2¯Xtασ2−¯Xt2)−12¯Xt]dt+dWt, (3.13)

which only depends on where . Note that the dimension of the parameter vector reduced from to . Crucially, it also suggests that we can only identify and up to a ratio. Hence, normalization requires fixing either , , or their ratio.

Example 3 (continued). It can be easily verified that the Lamperti transform of the NLDCEV diffusion in (2.13) takes the form

 d¯Xt=[l∑i=−kα∗i¯Xi−β1−βt−β2(1−β)¯X−1t]dt+dWt, (3.14)

where , . Hence, the parameters are identified and the number of parameters is reduced from to . Note that just as (2.10) and (2.12) are special cases of (2.13), both (3.12) and (3.13) are special cases of (3.14).

3.2 Second Scheme

Our second identification strategy transforms by its scale measure defined in eq. (2.4),

 ¯Xt:=S(Xt;θ),

which brings the diffusion process onto its natural scale,

 d¯Xt=σ¯X(¯Xt;θ)dWt,

where the drift is zero (and so known) while

 σ2¯X(¯x;θ)=s2(S−1(¯x;θ);θ)σ2(S−1(¯x;θ);θ). (3.15)

Since the drift term is zero, the identification condition (3.7)(i) becomes

 (3.16)

which can only hold if . We can therefore restrict attention to linear transformations , for some constants , in which case (3.7)(ii) becomes:

Assumption 3.3.

With given in (3.15): There exists no , and such that for all .

In comparison to Assumption 3.2, we here have to impose two normalizations to ensure identification. The intuition for this is that setting the drift to zero does not act as a complete normalization of the process: Any additional scale transformation of still leads to a zero-drift process. Therefore, for the third scheme to work we need both a scale and location normalization.

Theorem 3.4

Under Assumptions 2.1(i)–(ii), 2.2 and 3.1, is identified if and only if Assumption 3.3 is satisfied.

Compared to the first identification scheme, it is noticeably harder to apply this one to existing parametric diffusion models since the inverse of the scale transform is usually not available in closed form. But, similar to the first identification scheme, the result shows that without loss of flexibility, we can focus on UPDs with zero drift and then model the diffusion term in a flexible manner, e.g.,

 (3.17)

Corollary 3.4 shows that this UPD is identified together with without any further parameter restrictions on .

3.3 Third scheme

Our third identification strategy transforms a given stationary UPD by its marginal cdf,

 ¯Xt=FX(Xt;θ). (3.18)

In this case, there is generally no simplification in terms of the drift and diffusion term, which take the form

 μ¯X(¯x;θ) = μX(F−1X(¯x;θ);θ)fX(F−1X(¯x;θ);θ) +12σ2X(F−1X(¯x;θ);θ)f′X(F−1X(¯x;θ);θ)

and

 σ¯X(¯x;θ)=σX(F−1X(¯x;θ);θ)fX(F−1X(¯x;θ);θ). (3.20)

for . But the marginal distribution is now known with and we can directly identify the transformation function by , c.f. eq. (2.16). The identification condition then takes the form:

Assumption 3.4.

With and given in eqs. (3.3)-(3.20), the following hold:

Corollary 3.5

Under Assumptions 2.1-2.2 and 3.1, is identified if and only if Assumption 3.4 is satisfied.

The above result is only useful for showing identification of a given UPD if is available on closed form. But similar to the previous identification schemes, it demonstrates we can restrict attention to diffusions with known marginal distributions in the model building phase. Specifically, one can choose a known density that describes the stationary distribution of together with a parametric specification for, say, the drift function. We can then rearrange eq. (2.5) to back out the diffusion term of the UPD:

 σ2X(x;θ)=2fX(x)∫xxlμX(z;θ)fX(z)dz. (3.21)

If the drift is specified so that for , then Assumption 3.4 will be satisfied for this model. Alternatively, one could choose a parametric specification of the diffusion term and then derive the corresponding drift term of the UPD satisfying

 μX(x;θ)=12fX(x)∂∂x[σ2X(x;θ)fX(x)].

The resulting copula diffusion model is identified as long as the chosen diffusion term satisfies for , then Assumption 3.4 will be satisfied for this model.

Below, we apply the third identification scheme to the OU and CIR model:

Example 1 (continued). The stationary distribution of (2.10) is with and so the marginal density and cdf takes the form and , where and denote the density and cdf of the distribution. Applying the transformation (3.18) yields, after some tedious calculations,

which is independent of and and these therefore have to be fixed, leaving as the only free parameter. This is the same finding as with the first identification strategy.

Example 2 (continued). The stationary distribution of the CIR process is a -distribution with scale parameter and shape parameter . Thus, the marginal density and cdf can be written as

 fX(x;θ) = fX(x;ω,ν)=ωνΓ(ν)xν−1e−ωx FX(x;θ) = FX(x;ω,ν)=1Γ(ν)γ(ν,ωx)

where is the gamma function and is the lower incomplete gamma function. Applying the transformation (3.18) yields

 μ¯X(¯x;θ)=[κ(ν2κ−γ−1(ν,¯xΓ(ν))2κ)+(ν−12−γ−1(ν,¯xΓ(ν))2)]2κΓ(ν)γ−1(ν,¯xΓ(ν))ν−1e−γ−1(ν,¯xΓ(ν))

and

 σ2¯X(¯x;θ)=2κγ−1(ν,¯xΓ(ν))[1Γ(ν)γ−1(ν,¯xΓ(ν))ν−1e−γ−1(ν,¯xΓ(ν))]2.

Note that and