bayesian nonparametric adaptive spectral density ... · bayesian framework where we use a mcmc...

Bayesian Nonparametric Adaptive Spectral Density Estimationfor Financial Time Series

Nick James 1 2 Roman Marchant 1 Richard Gerlach 3 Sally Cripps 1 4

AbstractDiscrimination between non-stationarity and long-range dependency is a difficult and long-standingissue in modelling financial time series. This pa-per uses an adaptive spectral technique whichjointly models the non-stationarity and depen-dency of financial time series in a non-parametricfashion assuming that the time series consists of afinite, but unknown number, of locally station-ary processes, the locations of which are alsounknown. The model allows a non-parametricestimate of the dependency structure by mod-elling the auto-covariance function in the spec-tral domain. All our estimates are made within aBayesian framework where we use a ReversibleJump Markov Chain Monte Carlo (RJMCMC)algorithm for inference. We study the frequen-tist properties of our estimates via a simulationstudy, and present a novel way of generating timeseries data from a nonparametric spectrum. Re-sults indicate that our techniques perform wellacross a range of data generating processes. Weapply our method to a number of real examplesand our results indicate that several financial timeseries exhibit both long-range dependency andnon-stationarity.

1. IntroductionModelling the volatility of financial time series has beenthe subject of much interest since the deregulation of worldfinancial markets, which began in the late 1970’s. It is a diffi-cult task. First financial time series are often non-stationary,by which we mean that the statistical properties change overtime, making the development of statistical models problem-atic. Second, even stationary financial time series exhibitnon-standard features such as volatility clustering (Mandel-brot, 1963) and related kurtosis. Third, the signal to noiseratio is high making it difficult to detect any underlyingtrends. The primary contribution of this paper is to workin the spectral domain to capture and distinguish betweenfeatures of time series data, such as non-stationarity andlong-range dependency and compare estimates of these fea-

tures with estimates obtained using parametric time domainmodels.

Models for the time-varying nature of volatility in financialmarkets began with Engle (1982), who introduced the Au-toRegressive Conditionally Heteroskedastic (ARCH) model.Bollerslev (1986) extended this model to the more parsimo-nious Generalized ARCH (GARCH), while Taylor (1982;1986a;b) developed the Stochastic Volatility (SV) frameworkover the same period. These first generation volatility mod-els are conditionally Gaussian, with the dynamic volatilitycomponent meant to account for the leptokurtosis present inmost financial return series. Bollerslev (1987) allowed forconditionally Student-t returns in a GARCH process, specif-ically increasing the level of this aspect able to be captured.Various salient features of observed financial returns, e.g.the leverage effect, whereby volatility is higher in falling,compared to rising, markets, and the non-stationary aspects,including the time-varying nature of the conditional distri-bution of returns and possible structural break components,are allowed for in subsequent extended GARCH models inthe literature. These include the EGARCH (Nelson, 1991),GJR-GARCH (Glosten et al., 1993), T-GARCH (Zakoian,1994) and T-SV (So et al., 2002) models, all attemptingto capture the leverage effect; and via Markov SwitchingGARCH (MS-GARCH) (Cai, 1994; Hamilton & Susmel,1994; Gray, 1996; Haas et al., 2004) and MS-SV models(So et al., 1998).

Bayesian estimation has been prominent in this area, espe-cially for SV models where the likelihood, without condi-tioning on the latent stochastic process, does not exist inclosed form and simulation-based and/or data augmentationmethods, including Markov Chain Monte Carlo (MCMC),are standard. As high frequency data became more andmore available, volatility modelling moved first to directlymodel realized measures, such as realized variance (Ander-sen et al., 2003), and then to extensions of GARCH and SVprocesses, allowing realized measures as inputs that drivevolatility changes in the model, e.g. the GARCH-X modelof Hwang & Satchell (2007). More recently, Hansen et al.(2011) developed the realized GARCH framework, allowingan extra measurement equation capturing the contemporane-ous relationship between the latent volatility and the realizedmeasure. All these models make assumptions about distri-

arX

iv:1

902.

0335

0v1

[q-

fin.

ST]

9 F

eb 2

019

Bayesian Nonparametric Adaptive Spectral Density Estimation for Financial Time Series

bution of the noise and have parametric representations ofthe evolution of the volatility and although some methodsmay explicitly model regime shifts and stochastic behaviour,if the parametric form of the model does not resemble theunderlying phenomenology of the data generation processit will perform poorly.

Flexibly estimating the time-dependency of a phenomenonvia the spectral density goes back to the 1950’s (Whittle,1957). However, it is not often applied to financial timeseries, despite several appealing reasons for doing so aspointed out by Chaudhuri & Lo (2015). First, studyingfrequency components of security’s return processes canprovide insight into previously unseen economic structuredriving price movements. Secondly, as investment timehorizons can range from microseconds to many years, time-specific risks can be accounted for in portfolio constructiondecisions. Thirdly, frequency domain analysis can alsocompare strategies that operate on different timescales, andmay provide diversification across investment strategies op-erating on varying timescales. Finally, frequency-domainmeasurements offer more understandable representationsof the complex periodic dynamics financial markets mayexhibit.

In addition to these advantages, latest developments in spec-tral analysis, in particular the concept of local stationaritydeveloped by Dahlhaus (1997) and built on by Rosen et al.(2009; 2012), have led to the development of flexible non-parametric methods for estimating time varying spectra.

We use the technique of Rosen et al. (2012) to jointly esti-mate a time-varying non-parametric spectrum for financialtime series data, and to distinguish between non-stationarityand long-range dependency, as evidenced by volatility clus-tering. We use this flexible time-varying spectrum to simu-late ”ground truth” data in the spectral domain and convertback into time domain. This enables us to compare time do-main models of volatility with each other and with spectraltechniques. All our parameter estimates are made within aBayesian framework where we use a MCMC algorithm forinference. Rather than modelling volatility with a condition-ally stationary process as proposed by ARCH/GARCH/SV-style models, we assume that the data generating processis non-stationary, and consists of an unknown number andlocation of locally stationary processes.

The remainder of this paper is organised as follows. Section2 describes the model, priors and estimation procedure. Sec-tion 3 shows validation over simulated and real-world data.Finally, Section 4 draws conclusions from our experiments.

2. Model, Priors and Estimation2.1. Model for Non-stationary processes

Suppose ytt=1,...,T is a time series with observations froma Dahlhaus locally stationary process with evolutionaryspectrum f(ν, t), which we wish to estimate. To do this, weassume that the time series consists of K piecewise station-ary processes, each of length ns for s = 1, . . . ,K. Given apartition of K segments, we define the partition points to beξK = (ξ0,K , ξ1,K . . . ξK,K), with ξ0,K = 0 and ξK,K = Tso that the set As is given by As = t; ξs−1 + 1 < t < ξsas in (Rosen et al., 2012). Therefore, we can rewrite

yt =

K∑s=1

y(s)t δ(t, As,K) (1)

where, δ(t, As,K) = 1, if t ∈ As and δ(t, As,K) = 0

otherwise, and where the y(s)t ’s are independent stationaryprocesses, for s = 1, . . . ,K, each with spectral densityfs,K(ν).

The joint probability density function of a realizationy = (y1, . . . , yT ) given the individual spectra FK =(f1,K(ν), . . . , fK,K(ν)), the number of segments K, andthe partition points ξK is

p(y|FK ,K, ξK) =

K∏s=1

p(yξ(s−1,K)+1, . . . , yξ(s,K)

|fs,K(ν))(2)

2.2. Priors

2.2.1. PRIOR FOR SPECTRA

Given a partition defined by K segments and their respec-tive parition points ξK , and a realization ys), our goal is toestimate the unknown spectra fs,K(ν), for ν ∈ (0.5, 0.5).To motivate a prior for fs,K(ν), we frame the problem of es-timating the autocovariance structure of a time series, givenby the spectrum, as a nonparametric regression estimationproblem. In effect turning a covariance estimation probleminto a mean estimation problem, which is more parsimo-nious and tractable.

To elaborate, define the Discrete Fourier Transform (DFT)for segment s of length ns, at frequency νk to be

xs(νk) =1√ns

ns∑t=1

yt+ξs−1+1× (3)

(cos(2πνkt)− i sin(2πνkt)) ,

where νk = k/ns ∀k ∈ 0, 1, . . . , (ns − 1). Let the peri-odogram at frequency νk, I(νk), be the squared modulus ofthe DFT

Is(νk) =∣∣xs(νk)xs(νk)

∣∣ . (4)


Then Whittle (Whittle, 1957) showed that the distributionof xs = (xs(ν1) . . . , xs(νns)), under certain regularity con-ditions, is complex normal so that

xs ∼ns∏k=1

1

πfs(νk)exp

(− Is(νk)

fs(νk)

). (5)

This representation suggests that the Is(νk) are i.i.d. withIs(νk) ∼ exp(fs(νk)) and therefore

log(Is(νk)) = log(fs(νk)) + εk; εk ∼ log(exp(1)) (6)

Letting ws(νk) = log (Is(νk)) and gs(νk) = log (fs(νk))we have

ws(νk) = gs(νk) + εk, (7)

To place a prior on the unknown function gs(νk) we de-compose it into its linear and non-linear components so thatgs(νk) = αs0 + hs(νk) and place a Gaussian Process priorover the unknown function hs(νk), see for example (Wahba,1990). Specifically we assume

hs(νk) = τsW (νk) (8)

or equivalently,

hs = (hs(ν1), . . . , hs(νns)) ∼ N

(0, τ2sΩ

)(9)

where W (.) is a Wiener process, τ2s is a smooth-ing parameter and the ith, jth element of Ω, ωij =cov(hs(νi), hs(νj)) = min(νi, νj).

For computational convenience we write hs as a linearcombination of basis functions by performing an eigen-value decomposition on Ω = QDQ′. Specifically we letX = QD1/2 be the design matrix and βs ∼ (0, τ2s Ins

) bethe vector of regression coefficients, so that hs = Xβs hasthe required distribution. We follow Wood et al. (Woodet al., 2002) and Rosen et al. (Rosen et al., 2009) and keeponly those basis functions corresponding to the 30 largesteigenvalues, for computational speed.

2.2.2. PRIOR FOR PARTITION

The partition is defined by the number of of locally station-ary segments K and the partition points, ξK , given K. Theprior on the partition Pr(K, ξS) = Pr(ξs|K) Pr(K) ξs,Kis as follows;

Pr(K) =1

S(10)

where S is the the upper limit for the number of segments,in the experiments which follow this is typically set to be 30.Given K we decompose the prior on ξK into a sequence ofdiscrete uniform priors,so that

Pr(ξK | K) =

K−1∏s=1

Pr(ξs,K |ξs−1,K) , (11)

where Pr(ξj,m = t | m) = 1/ps,K , for s = 1, . . . ,K − 1,ps,K is the number of available locations for partition pointξs,K and is equal to T − ξs−1,K − (K − s + 1)tmin + 1.The quantity tmin is a user chosen number. It represents theminimum number of observations that are deemed sufficientfor the Whittle likelihood approximation to hold. In thispaper we set this to be 50, however we note that this isarbitrary, and indeed there is a substantial literature whichdiscusses the quality of the Whittle approximation.

The prior in Equation 11 states that the first partition point isequally likely to occur at any point in the time series subjectto the constraint that there are at least tmin observations ineach of the K segments. The prior on subsequent partitionpoints is similar and states that, conditional on the previouspartition point, the next partition point is equally likely tooccur in any available location, again subject to the sameconstraint see (Rosen et al., 2012) for details.

2.3. Generation of Temporal Data

A contribution of this paper is to use the time-varying spec-tra estimated as in (Rosen et al., 2012) to generate a timeseries, without assuming the time domain data generatingprocess. This is achieved using the result that the DFT’sof the realization of a process, are approximately normallydistributed if the joint cumulants of that process, of ordersgreater than 2, are absolutely summable (Brillinger, 1975).

Let

xr = (x(0,r), ..., x(ns−1,r)) ,

xi = (x(0,i), ..., x(ns−1,i)) ,

be the real and imaginary components of the DFT for a setof realizations from a locally stationary process s of lengthns. The distribution of these quantities for a zero-meanprocess are;

x(0,r) ∼ N (0, fs(ν0))

x(0,i) ∼ δ(0)

x(1:ns2 −1,r)

∼ N(

0, fs(ν1:ns2 −1)/2

)x(1:ns

2 −1,i)∼ N

(0, fs(ν1:ns

2 −1)/2)

where δ(.) is the Dirac delta function. If n is even then

x(ns2 ,r)

∼ N(0, f(νns/2)

)x(ns

2 ,i)∼ δ(0) .

To ensure symmetry we set

x(n2 +1:n−1,r) = x(n

2−1:1,r)x(n

2 +1:n−1,i) = −x(n2−1:1,i)

,


So that given a time-varying spectrum f(ν, t), for ξs−1,S <t ≤ ξs,S we generate xr and xi and form x = xr + ixiand apply the Inverse-DFT to generate the time series corre-sponding to each locally stationary process and so obtain atime domain realization from a non-stationary process.

2.4. Estimation

In this paper we take a Bayesian approach and estimate theunknown time-varying spectrum by its posterior mean

E[f(ν, t)|y] =

S∑K=1

p(K,T )∑j=1

f(ν, t)|y,K, ξK)

×Pr(ξS |K,y) Pr(K|y)

where the sum is over all possible partitions and

E [f(ν, t)|y,K, ξK ] = (12)∫E[f(ν, t)|y,K, ξS ,FK ]p(FK |y,K, ξK)dFK .

We use Reversible Jump MCMC (RJMCMC) to performthe required transdimensional integration, see (Rosen et al.,2012) for details.

3. ExperimentsThis section validates the use of more flexible, adaptivenon-parametric models for estimating spectrum of financialtime series and its volatility. The experiment setup is asfollows, we evaluate the goodness of fit for different tech-niques over data with a known generative process and overreal-world data from the daily returns and squared returnsof the NASDAQ Index from 2002-2018 and the GBP:USDfrom 2010-2018. Section 3.1 presents details on the datageneration processes and evaluation of results for a knowntime-varying spectral density. Section 3.3 shows the re-sults of fitting different models over the returns and squaredreturns of the NASDAQ and GBP:USD.

3.1. Simulated Data

To compare the performance of various models for financialreturns and volatility, in terms of the ability of the modelto recover the true data generating process, we simulatedata using three models for time series. The first model isa stationary process, while the second and third models arenon-stationary processes. The first model is a GARCH (1,1)process. The second model is a regime-switching GARCH(1,1) process (Haas et al., 2004; Ardia, 2016) and the thirdis the AdaptSpec model of Rosen et al. (2012). The datagenerating process of a GARCH(1,1) model 1d is given by

yt ∼ N (µ, σ2t ) (13)

σ2t = α0 + α1η

2t−i + β1σ

2t−1, (14)

where ηt−1 = yt−1 − µ. We set µ = 0, α0 = 1, α1 =0.1, β1 = 0.1., so that the process is stationary with anunconditional variance, σ2

uc = α0

(1−α1−β1). Figure 1a shows

a sample spectrum and Figure 1d the associated realisationin time.

The second model we generate data from is a Regime-Switching GARCH model 1e as in (Ardia, 2016; Haaset al., 2004). Specifying a model which allows for regime-switching is one way of accounting for non-stationarity.For each point in time t, a latent state variable st fort ∈ 1, 2, .., T, determines the regime from which theobservation is generated. Let Pr(st = j|y) be the probabil-ity that an observation at time t was generated by regimej, for j = 1, . . . , NR, where NR is the number ofpossibleregimes. Our Regime-Switching model is the following(Bauwens et al., 2014; Haas et al., 2004)

yt|st=j ∼ N(µj , σ

2jt

)(15)

σ2jt|st=j = α0,j + β1,jσ

2t−1 + α1,jη

2t−1 (16)

For our simulation we set NR = 2. Define KR to be thenumber of segments generated by the NR regimes, so thatKR ≥ NR. The location of the regime switches are definedby the cutpoints c = (c1, . . . , cKR

). Let r = (r1, . . . , rKR)

be an indicator vector denoting the regime which generatesthe data in segment k, so that rk = j, if segment k wasgenerated by regime j. For our simulation we set KR = 3,c = (1000, 3000, 5000) and r = (1, 2, 1) . Our set ofparameters in our regime switching model are, α0,1 = 1,α1,1 = 0.1, β1,1 = 0.1, α0,2 = 1, α1,2 = 0.3 and β1,2 =0.2.

The third model for generating data is now described. Us-ing the model in (Rosen et al., 2012) we obtained an es-timate of the posterior mode of the number of locally sta-tionary processes for the NASDAQ daily returns from 2002to March 2018, denoted by KNAD and an estimate of theposterior mean of the spectra1c corresponding to those lo-cally stationary processes. We generated 50 realizations 1fof the real and imaginary components of the DFT’s, xs,rand xs,i respectively each of length ns,KNAD

, then theinverse-DFT was applied to obtain 50 time series, ys eachof length ns,KNAD

, for s = 1, . . . , KNAD as described inSection 2.3. These KNAD time series were concatenated,so that 50 realizations of a non-stationary process, of length∑s=1 nsKNAD

, were obtained.

In what follows we shall refer to these three data generatingprocesses as GARCH, Regime and AdaptSpec.

3.2. Metrics to measure performance

To assess the relative performances of the GARCH, Regime,and the AdaptSpec models we use Mean Squared Error(MSE) and Symmetric Kullback Liebler (SKL) divergence.


(a) GARCH Log Spectrum (b) Regime Log Spectrum(c) AdaptSpec Log Spectrum

(d) GARCH Time Series (e) Regime Time Series(f) AdaptSpec Time Series

Figure 1. Spectra and example realisations.

We define the quantities as follows

SKL =

T∑t=1

T−1∑k=0

f(νk, t) logf(νk, t)

f(νk, t)

+ f(νk, t) logf(νk, t)

f(νk, t)

MSE =

T∑t=1

n∑k=1

(f(νk, t)− f(νk, t))2

where f(ν, t) is the true time-varying spectrum and f(ν, t)is an estimate of this true spectrum. In what follows weuse the subscripts G, R, and AD, to refer to the GARCH,Regime and AdaptSpec models respectively. Plots of thetrue log spectra fG(ν, t), fR(ν, t) and fAD(νt, t), used togenerate the data along with an example of a realizationappear in Figure 1.

Boxplots of the log(SKL) and log(MSE) for all threeestimators and all three data generating models appear inFigure 3. We chose to plot the log of these validation metrics,rather than the metrics itself, because the difference betweenthe values of the SKL and MSE for three estimators isvery large.

As expected, when data are generated from a particularmodel, the estimates obtained from the method which as-sumes that particular model provide the best fit, (except

in certain circumstance with the Regime model which willbe discussed later). However, the plots also show that theestimates obtained from the AdaptSpec model when thedata are generated from the GARCH or Regime models arealways the next best. For example 3, where the true modelis a single GARCH model, which is the same as a Regimemodel where the number of regimes is equal to one, Adapt-Spec outperforms the estimate obtained using the REGIMEmodel. In other words, the improvement gained by using aflexible model, when flexibility is required, exceeds the lossof using a flexible model when flexibility is not required.

The performance of the Regime model when data are gen-erated from a GARCH model warrants further explanation.Our experience of using the model by (Ardia, 2016), showsthat unless the true number of regimes is equal to the user-setnumber of regimes, results are highly variable. Part of theissue is an over-identification problem. If a single GARCHmodel is the truth but one estimates the spectrum using aregime switching model, where the number of regimes isgreater one, then there are infinitely many different combi-nations which could recover the truth. While this shouldnot necessarily present a problem with the estimated fit orprediction (as opposed to parameter inference), it does. Thisappears to be due to the fact that the probability of beingin a particular regime can change abruptly on a daily basis.These estimated probabilities in turn, are very sensitive tothe specification of the particular type of GARCH model


(a) GARCH Generated Data (b) Regime Generated Data (c) AdaptSpec Generated Data

Figure 2. Boxplot of the log(SKL) divergence for three estimators from 50 realisations generated from each respective process.

(a) GARCH Generated Data(b) Regime Generated Data (c) AdaptSpec Generated Data

Figure 3. Boxplot of the log(MSE) for three estimators from 50 realisations generated from each respective process.

assumed to generate data in the different regimes.

For example, we reproduced the ”smoothed” probabilitiesobtained for the time series of the daily returns for the SwissMarket Index, which was analyzed by Ardia (2016) Theseprobabilities are the blue line in Figure 6, and are estimatedusing a Regime model assuming two GJR-GARCH pro-cesses. However, if we assume that the underlying datagenerating process for the regimes is a GARCH(1,1) ratherthan a GARCH(1,1) with a GJR variance specification (Haaset al., 2004), then we obtained the estimated smoothed prob-abilities given by the red line in Figure 6. The difference isstriking.

These results also explain why AdaptSpec performs wellacross a range of data generating process; Adaptspec is anon-parametric model, so that by estimating the dependencyin the frequency domain we avoid making any assumptionsabout the data generating process in the time domain.

3.3. Real Examples: NASDAQ, GBP:USD

It is well known that the distribution of many financial assetsare non-normally distributed, and exhibit volatility cluster-ing. Whether this volatility clustering is evidence of long-range dependence in a stationary process, or attributable tonon-stationarity is less clear. In this section we attempt to an-swer this question by estimating the potentially time-varyingspectrum of a financial time series’ actual and squared re-

turns. The time-varying spectrum of the actual return seriesis a non-parametric estimate of the evolution of the sec-ond moment of the return series’ distribution, while thetime-varying spectrum of the squared return series is a non-parametric estimate of the evolution of the fourth-moment.We choose the NASDAQ daily returns and the GBP:USDexchange rate daily returns from 2002-2018 and 2010-2018respectively to demonstrate the technique.

Figure 5a 5b show the actual return series for the NASDAQindex and its estimated time-varying spectrum, while panels5c 5d show the squared returns for the NASDAQ index andits corresponding estimated time-varying spectrum. Figure4 is an analogous plot for the GBP:USD exchange rate.

Figure 5 provides several insights into the stationarity anddependency of the NASDAQ returns. First, the series isdefinitely non-stationary. The posterior mode of the numberof locally stationary segments for the return series is 12.Second, it would appear that the market for the NASDAQindex is weak-form inefficient at several points in time. Aweak-form efficient market is characterised by having zeroautocorrelation in the first moment of the return distribution,and hence a flat spectrum. 5b of Figure 5 shows several peri-ods of time where the assumption of weak-form efficiency isviolated, of particular note is the spectrum during the GlobalFinancial Crisis (GFC) in 2008-2009, which shows a clearpeak. 5d of Figure 5 shows that the volatility clustering isnot removed even after accounting for non-stationarity. If


(a) GBP:USD Returns (b) GBP:USD Returns Log Spec-trum (c) GBP:USD Squared Returns (d) GBP:USD Squared Returns

Log Spectrum

Figure 4. GBP:USD Returns and Log Spectra

(a) NASDAQ Returns(b) NASDAQ Returns Log Spec-trum (c) NASDAQ Squared Returns

(d) NASDAQ Squared ReturnsLog Spectrum

Figure 5. NASDAQ Returns and Log Spectra

Figure 6. Estimated smoothed probabilities of the second regime

non-stationarity accounted for volatility clustering then wewould expect the locally stationary spectra of the squaredreturn to be flat, however 5d shows that there is still strongpositive correlation of the squared returns, as evidenced bythe peak in power at low frequency for most of the timeperiods.

Figure 4 paints a similar picture for the GBP:USD exchangerate; the time series is clearly non-stationary, showing anoverall increase in variability at the time of the Brexit votewith an accompanying dependency in the first moment of

the series at that time, indicating violations of weak-formefficiency. However, the time varying spectral density ofthe GBP:USD squared returns as seen in 4d provides someinteresting insights - distinguishing the behaviour of theGBP:USD’s volatility with that of the NASDAQ Index. Inparticular, it indicates that non-stationarity drives the volatil-ity clustering behaviour of the returns. This is clear be-cause unlike the NASDAQ squared returns spectrum 5d, theGBP:USD squared returns spectrum 5d is predominantlyflat within any candidate segment - suggesting that the largernon-stationary process is in fact piecewise stationary.

4. ConclusionsOur experiments indicate that given a non-stationary datagenerating process, nonparametric models outperform para-metric models, where the latter assumes a constant structureover time. Our simulations demonstrate that there is lessestimation error in applying a flexible method such as Adapt-SPEC to a parametric data generating process, than applyinga parametric model to a non-stationary data generating pro-cess. For validation, we generate ”ground truth” data inthe spectral domain, and compare the resulting estimationfrom time domain models with spectral analysis techniques.The time series we generate after converting our groundtruth spectrum into a time series strongly resembles manyfinancial time series (such as the NASDAQ), and illustratesthe need for flexible nonparametric models to capture thecomplex, non-stationary structure of the underlying time


series.

ReferencesAndersen, T., Bollerslev, T., Diebold, F., and Labys, P. Mod-

eling and forecasting realized volatility. Econometrica,71(2):579–625, 2003.

Ardia, D. Markov-Switching GARCH Models in R: TheMSGARCH Package. Journal of Statistical Software,2016.

Bauwens, L., Backer, B., and Dufays, A. A BayesianMethod of Change-Point Estimation with RecurrentRegimes: Application to GARCH Models. Journal ofEmpirical Finance, 29:207–229, 2014.

Bollerslev, T. Generalized Autoregressive Conditional Het-eroskedasticity. Journal of Econometrics, 31:307–327,1986.

Bollerslev, T. A Conditionally Heteroskedastic Time SeriesModel for Speculative Prices and Rates of Return. TheReview of Economics and Statistics, 69(3), 1987.

Brillinger, D. Time Series: Data Analysis and Theory. Holt,Rinehart, and Winston, 1975.

Cai, J. A markov model of switching-regime arch. Journalof Business & Economics Statistics, 12:309–316, 1994.

Chaudhuri, A. and Lo, A. Spectral Analysis of stock-returnvolatility, correlation and beta. In IEEE Signal Processingand Signal Processing Education Workshop, 2015.

Dahlhaus, R. Fitting time series models to nonstationaryprocesses. Annals of Statistics, 1997.

Engle, R. Autoregressive Conditional Heteroskedasticitywith Estimates of the Variance of United Kingdom Infla-tion. Econometrica, 50(4):987–1007, 1982.

Glosten, L., Jagannathan, R., and Runkle, D. On the rela-tion between the expected value and the volatility of thenominal excess returns on stocks. The Journal of Finance,1993.

Gray, S. Modelling the conditional distribution of interestrates as a regime-switching process. Journal of FinancialEconometrics, 2:211–250, 1996.

Haas, M., Mittnik, S., and Paolella, M. Mixed normal con-ditional heteroskedasticity. Journal of Financial Econo-metrics, 2:211–250, 2004.

Hamilton, J. and Susmel, R. Autoregressive conditionalheteroskedasticity and changes in regime. Journal ofEconometrics, 64:307–333, 1994.

Hansen, P., Huang, Z., and Shek, H. Realized GARCH:A Joint Model for Returns and Realized Measures ofVolatility. Journal of Applied Econometrics, 2011.

Hwang, S. and Satchell, S. GARCH Model with Cross-sectional Volatility: GARCHX Models. Applied Finan-cial Economics, 15:203–216, 2007.

Mandelbrot, B. The variation of certain speculative prices.Journal of Business, (36):394–419, 1963.

Nelson, D. Conditional heteroskedasticity in asset returns:A new approach. Econometrica, 59:347–370, 1991.

Rosen, O., Stoffer, D., and Wood, S. Local spectral analysisvia a bayesian mixture of smoothing splines. Journal ofThe American Statistics Association, 104:249–262, 2009.

Rosen, O., Wood, S., and Stoffer, D. AdaptSPEC: Adap-tive Spectral Estimation for Non-stationary time series.Journal of The American Statistics Association, 107:1575–1589, 2012.

So, M., Lam, K., and Li, W. A stochastic volatility modelwith markov switching. Journal of Business & EconomicStatistics, 16:244–253, 1998.

So, M., Li, W., and Lam, K. On a Threshold StochasticVolatility Model. Journal of Forecasting, 22:473–500,2002.

Taylor, S. Financial Returns Modelled by the product oftwo stochastic processes, a study of daily sugar prices1961-1979. Time Series Analysis: Theory and Practice 1,pp. 203–226, 1982.

Taylor, S. Modelling Financial Time Series. New York,Wiley, 1986a.

Taylor, S. Modelling stochastic volatility. MethematicalFinance, 4(183-204), 1986b.

Wahba, G. Spline Models for Observational Data. Societyfor industrial and applied mathematics, 1990.

Whittle, P. Curve and periodogram smoothing. Journal ofthe Royal Statistical Society B, 19:38–47, 1957.

Wood, S., Jiang, W., and Tanner, M. Bayesian mixture ofsplines for spatially adaptive nonparametric regression.Biometrika, 89(513-528), 2002.

Zakoian, J. Threshold heteroskedastic models. Journal ofEconomic Dynamics and Control, 18:931–955, 1994.

bayesian nonparametric adaptive spectral density ... · bayesian framework where we use a mcmc...

Documents