Nonstationary Portfolios: Diversification in the Spectral Domain
Bruno Scalzo, Alvaro Arroyo, Ljubisa Stankovic, Danilo P. Mandic
NNONSTATIONARY PORTFOLIOS: DIVERSIFICATION IN THE SPECTRAL DOMAIN
Bruno Scalzo , Alvaro Arroyo , Ljubi ˇ sa Stankovi´c , Danilo P. Mandic Department of EEE, Imperial College London, London, SW7 2BT, UK Faculty of Electrical Engineering, University of Montenegro, Podgorica, 81000, MontenegroEmails: { bruno.scalzo-dees12, alvaro.arroyo17, d.mandic } @imperial.ac.uk, [email protected] ABSTRACT
Classical portfolio optimization methods typically determine an op-timal capital allocation through the implicit, yet critical, assumptionof statistical time-invariance. Such models are inadequate for real-world markets as they employ standard time-averaging based estima-tors which suffer significant information loss if the market observ-ables are non-stationary. To this end, we reformulate the portfoliooptimization problem in the spectral domain to cater for the nonsta-tionarity inherent to asset price movements and, in this way, allowfor optimal capital allocations to be time-varying. Unlike existingspectral portfolio techniques, the proposed framework employs aug-mented complex statistics in order to exploit the interactions betweenthe real and imaginary parts of the complex spectral variables, whichin turn allows for the modelling of both harmonics and cyclostation-arity in the time domain. The advantages of the proposed frameworkover traditional methods are demonstrated through numerical simu-lations using real-world price data.
Index Terms — Financial signal processing, portfolio optimiza-tion, spectral analysis, augmented complex statistics, nonstationary
1. INTRODUCTION
The principle of diversification has become the cornerstone ofdecision-making in finance and economics ever since the introduc-tion of modern portfolio theory (MPT) by Harry Markowitz in 1952[1]. The MPT suggests an optimal strategy for the investment, basedon the first- and second-order moments of the asset price returns,which can be formulated as a quadratic optimization task commonlyreferred to as the mean-variance optimization (MVO).Consider the vector, x ( t ) ∈ R N , which contains the returns of N assets at a time t , the i -th entry of which is given by x i ( t ) = p i ( t ) − p i ( t − p i ( t − (1)where p i ( t ) denotes the value of the i -th asset at a time t . The MVOasserts that the optimal vector of asset holdings, w ∈ R N , is ob-tained through the following optimization problem max w { w T m − λ w T Rw } (2)where m = E { x } ∈ R N is a vector of expected future returns, R = cov { x } ∈ R N × N is the covariance matrix of returns, and λ isa Lagrange multiplier, also referred to as the risk aversion parameter.In practice, it is usually necessary to impose additional constraintson the values of w , for instance, to constrain the portfolio leverage.The increasing availability of computational power has naturallymade MVO a ubiquitous tool for financial practitioners, however, the validity of its underlying theory remains perhaps the most debatedtopic in the field to date. Among issues that make MVO unreliable inpractice, a major caveat is the notorious challenge of estimating themoments, m and R , of nonstationary asset price movements. It hasbeen shown that standard time-averaging based estimators of m and R typically yield portfolios that are far from truly optimal, and henceexhibit poor out-of-sample performance [2, 3, 4, 5, 6]. Moreover,this issue is further amplified by the well-established sensitivity ofMVO to perturbations of the estimates, m and R , whereby smallchanges in the inputs may generate portfolio holdings with vastlydifferent compositions [7, 8, 9, 10, 11].The information loss incurred by sample estimators in nonsta-tionary environments can be demonstrated using von Neumann’s mean ergodic theorem [12] and Koopman’s operator theory [13].Consider an idealised case whereby the asset price returns evolve intime according to x ( t ) = S x ( t − , with S : C N (cid:55)→ C N denot-ing the unitary shift operator in a Hilbert space. The mean ergodictheorem asserts that the sample mean approaches the orthogonal sub-space of x ( t ) , that is lim T →∞ T T − (cid:88) t =0 x ( t ) = lim T →∞ T T − (cid:88) t =0 S t ( x (0)) = lim T →∞ T T − (cid:88) t =0 P x (0) (3)with the boundedness property governed by (cid:107)P x ( t ) (cid:107) ≤ (cid:107) x ( t ) (cid:107) ,which arises from the Cauchy-Schwarz inequality.To overcome the limitations of MVO in the presence of non-stationarity, there has been an increasing interest in the use of spec-tral analysis techniques. While spectral analysis has a long historyin econometrics [14, 15, 16], with applications ranging from busi-ness cycle analysis [17], option valuation [18], empirical analysis[19, 20, 21, 22, 23, 24], through to causality analysis [25], its appli-cation in portfolio optimization has been rather sparse. To this end, spectral portfolio theory [26, 27, 28] was recently introduced withthe aim to enhance portfolio performance by allowing the investorsto benefit from diversifying not only across assets but also acrossfrequencies, whereby the cyclical components of the variance andcovariance of asset returns are accounted for respectively by usingthe periodogram and cross-spectra [15].Despite mathematical elegance and physical intuition, there re-main issues that need to be addressed prior to a more widespreadapplication of spectral analysis to portfolio optimization. For exam-ple, spectral estimation is an inherently complex-valued task, how-ever, spectral measures such as the power spectral density (PSD)employed in [27] are magnitude-only based and hence cannot ac-count for the information within the phase spectrum. From the max-imum entropy viewpoint [29, 30, 31, 32], such models make animplicit, yet fundamental, assumption that the phase information,which is intrinsic to complex-valued spectral data, is uniform and a r X i v : . [ q -f i n . S T ] J a n hus not informative. Mathematically, this is equivalent to assert-ing that the variable is wide-sense stationary in the time domain[33]. Furthermore, spectral measures such as the PSD are absolute(or non-centred) spectral moments and hence cannot distinguish be-tween the information attributed by the spectral mean from that bythe spectral covariance, yet these designate respectively the harmon-ics and cyclostationarity in the time domain.To this end, we formulate a spectral portfolio theory using aclass of spectral estimators for nonstationary signals, whereby theharmonic and cyclostationary time-domain signal properties are des-ignated respectively by the mean and covariance of the associatedspectral representation. Unlike existing methods, the proposed spec-tral portfolio framework is intrinsically complex-valued and thusbenefits from augmented complex statistics [34, 35, 33] in order toallow for a precise description of the interaction between the realand imaginary parts of complex spectral variables, and thus of thetime-phase alignment. In this way, the proposed approach is shownto enable creation of time-varying capital allocation schemes. Theadvantages of the proposed framework over traditional methods aredemonstrated through simulations based on real-world price data.
2. A CLASS OF NONSTATIONARY SIGNALS
We begin by consider a real-valued signal, x ( t ) ∈ R N , which admitsthe following time-frequency expansion [36, 35] x ( t ) = (cid:90) ∞−∞ e ωt (cid:120) ( t, ω ) dω (4)where (cid:120) ( t, ω ) ∈ C N is the realisation of a random spectral processat an angular frequency, ω , and time instant, t . The Hermitian sym-metry, (cid:120) ∗ ( t, ω ) = (cid:120) ( t, − ω ) , holds so that x ( t ) is real-valued.To cater for a broad variety of deterministic and stochastic time-domain signals, the spectral process is assumed to be multivariategeneral complex Gaussian distributed [37], i.e. (cid:120) ( t, ω ) follows thelinear model (cid:120) ( t, ω ) = (cid:109) ( ω ) + (cid:115) ( t, ω ) (5)where (cid:115) ( t, ω ) ∈ C N is a zero-mean stochastic process, while the spectral mean , (cid:109) ( ω ) ∈ C N , defined as (cid:109) ( ω ) = E { (cid:120) ( t, ω ) } (6)is time-invariant. The spectral covariance and spectral pseudo-covariance are also time-invariant and defined respectively as (cid:82) ( ω ) = cov { (cid:120) ( t, ω ) } = E ¶ (cid:115) ( t, ω ) (cid:115) H ( t, ω ) © (7) (cid:80) ( ω ) = pcov { (cid:120) ( t, ω ) } = E ¶ (cid:115) ( t, ω ) (cid:115) T ( t, ω ) © (8)where the bound (cid:107) (cid:80) ( ω ) (cid:107) ≤ (cid:107) (cid:82) ( ω ) (cid:107) holds, by virtue of theCauchy-Schwarz inequality.As with multivariate complex variables in general, the spectralprocess admits a compact augmented representation of the form (cid:120) ( t, ω ) = ï (cid:120) ( t, ω ) (cid:120) ∗ ( t, ω ) ò ∈ C N (9)which compactly parametrizes the pdf of (cid:120) ( t, ω ) as follows [37] p ( (cid:120) , t, ω ) = exp î − ( (cid:120) ( t, ω ) − (cid:109) ( ω )) H (cid:82) − ( ω )( (cid:120) ( t, ω ) − (cid:109) ( ω )) ó π N det ( (cid:82) ( ω )) (10) with (cid:109) ( ω ) = E { (cid:120) ( t, ω ) } = ï (cid:109) ( ω ) (cid:109) ∗ ( ω ) ò (11) (cid:82) ( ω ) = cov { (cid:120) ( t, ω ) } = ï (cid:82) ( ω ) (cid:80) ( ω ) (cid:80) ∗ ( ω ) (cid:82) ∗ ( ω ) ò (12)being respectively the augmented spectral mean and covariance.Therefore, (cid:120) ( t, ω ) is said to be distributed according to (cid:120) ( t, ω ) ∼ CN ( (cid:109) ( ω ) , (cid:82) ( ω )) (13)Furthermore, if the time-frequency representations exhibit non-orthogonal bin-to-bin increments , then it is necessary to also con-sider the following dual-frequency statistics (for ω (cid:54) = ν ) (cid:82) ( ω, ν ) = cov { (cid:120) ( t, ω ) , (cid:120) ( t, ν ) } = E ¶ (cid:115) ( t, ω ) (cid:115) H ( t, ν ) © (14) (cid:80) ( ω, ν ) = pcov { (cid:120) ( t, ω ) , (cid:120) ( t, ν ) } = E ¶ (cid:115) ( t, ω ) (cid:115) T ( t, ν ) © (15)which are referred to respectively as the dual-frequency spectral co-variance and dual-frequency spectral pseudo-covariance . These ex-hibit the following properties (cid:82) ( ω, ν ) = (cid:82) ∗ ( ν, ω ) (16) (cid:80) ( ω, ν ) = (cid:80) ( ν, ω ) (17) (cid:107) (cid:80) ( ω, ν ) (cid:107) ≤ (cid:107) (cid:82) ( ω, ν ) (cid:107) ≤ (cid:107) (cid:82) ( ω ) (cid:107) (cid:107) (cid:82) ( ν ) (cid:107) (18)owing to the Cauchy-Schwarz inequality [35]. Remark 1.
Notice that the spectral moments in (6)-(8) are centred ,which contrasts the usual spectral statistics based on the absolute or non-centred moments as in [27]. It is therefore possible to expressthe standard PSD, denoted by ˜ (cid:82) ( ω ) , in terms of the spectral meanand covariances as follows ˜ (cid:82) ( ω ) = E ¶ (cid:120) ( t, ω ) (cid:120) H ( t, ω ) © = (cid:109) ( ω ) (cid:109) H ( ω ) + (cid:82) ( ω ) (19)This shows that the mean and covariance information become entan-gled when employing the absolute (non-centred) spectral statistics.This result also highlights that the power spectrum is inadequate fordetecting harmonics in low signal-to-noise ratio environments, since (cid:107) (cid:82) ( ω ) (cid:107) (cid:29) (cid:107) (cid:109) ( ω ) (cid:107) . The PSD of the harmonics would therefore bedominated by the power associated with the noise, thereby renderingthe harmonic indistinguishable from the noise.The linearity property of the Fourier transform in (4) dictatesthat if the spectral processes are multivariate complex Gaussiandistributed, that is, (cid:120) ( t, ω ) ∼ CN ( (cid:109) ( ω ) , (cid:82) ( ω )) , then their time-domain counterpart, x ( t ) , is also multivariate Gaussian distributed,since a linear function of Gaussian random variables is also Gaussiandistributed. The signal, x ( t ) , is thus distributed according to x ( t ) ∼ N ( m ( t ) , R ( t )) (20)where m ( t ) ∈ R N and R ( t ) ∈ R N × N are the time-varying meanvector and covariance matrix, defined respectively as m ( t ) = E { x ( t ) } (21) R ( t ) = cov { x ( t ) } (22)which are a function of the introduced spectral statistics, as is shownnext. .1. Mean From (21), consider a statistical expectation of the spectral expan-sion of x ( t ) , as in (4), to yield m ( t ) = E { x ( t ) } = (cid:90) ∞−∞ e ωt E { (cid:120) ( t, ω ) } dω = (cid:90) ∞−∞ e ωt (cid:109) ( ω ) dω (23)Therefore, the time-varying mean of x ( t ) is a multivariate real-valued harmonic signal. Notice that for ω = 0 , the signal reduces toa multivariate DC component. Following from the relation in (22), and upon introducing the centredsignal, s ( t ) = x ( t ) − m ( t ) , consider the covariance of the spectralexpansion of x ( t ) , as in (4), to obtain R ( t ) = cov { x ( t ) } = E ¶ s ( t ) s T ( t ) © = (cid:90) ∞−∞ (cid:90) ∞−∞ e ( ω − ν ) t (cid:82) ( ω, ν ) + e ( ω + ν ) t (cid:80) ( ω, ν ) dωdν (24)Therefore, the time-varying covariance of x ( t ) consists of a sumof cyclostationary components, each modulated at an angular fre-quency, ω . Example 1.
With reference to Remark 1, we next demonstrate thebenefits of employing the proposed centred spectral moments overthe legacy absolute second-order spectral moments (power spectrumand complementary spectrum). Consider a single realisation of aunivariate general nonstationary signal in Figure 1(a). The signalconsists of two harmonics at different angular frequencies embed-ded in general cyclostationary noise (shown in Figure 1(b)). Observethat the signal constituents are completely identifiable when employ-ing the centred spectral moments in Figure 1(c), whereby: (i) M ( ω ) designates the harmonics; (ii) R ( ω ) designates the WSS component;and (iii) P ( ω ) designates the degree of cyclostationarity. In contrast,the absolute second-order moment (PSD), ˜ R ( ω ) , cannot distinguishbetween the harmonic and stochastic components, as shown in Fig-ure 1(d). Time A m p li t ud e (a) Nonstationary signal.
Time A m p li t ud e (b) Constituent signals in (a).
Angular frequency ( ω ) M ag n i t ud e M ( ω ) R ( ω ) P ( ω ) (c) First- and second-ordermoments of the spectrum.
Angular frequency ( ω ) M ag n i t ud e ˜ R ( ω ) (d) Absolute second-ordermoments of the spectrum.
Fig. 1 : Spectral analysis of a real-valued nonstationary signal. (a) Asingle realisation. (b) The constituents of the signal in (a). (c) Thecentred spectral moments. (d) The absolute spectral moments.
3. COMPACT SPECTRAL REPRESENTATION
Consider a nonstationary signal which exhibits a discrete frequencyspectrum, consisting of M frequency bins, ω = [ ω , ..., ω M ] T . Thediscrete spectral expansion of x ( t ) in (4) therefore becomes x ( t ) = 1 √ M M (cid:88) m =1 Ä e ω m t (cid:120) ( t, ω m ) + e − ω m t (cid:120) ∗ ( t, ω m ) ä (25) Remark 2.
Unlike the conventional DFT, the normalization by theconstant, √ M in (25), provides a rigorous mapping of coordinatesfrom the time-domain to the time-frequency domain through a purerotation in the complex plane, thus preserving both the desired or-thogonality and norm properties [38].To facilitate the analysis in this work, we express (25) in thefollowing compact form x ( t ) = Φ ( t, ω ) (cid:120) ( t, ω ) (26)where Φ ( t, ω ) ∈ C N × MN is the augmented spectral basis , definedas Φ ( t, ω ) = (cid:2) Φ ( t, ω ) Φ ∗ ( t, ω ) (cid:3) (27) Φ ( t, ω ) = 1 √ M (cid:2) e ω t I N · · · e ω M t I N (cid:3) (28)with I N ∈ R N × N being the identity matrix, and (cid:120) ( t, ω ) ∈ C MN the augmented spectrum representation , given by (cid:120) ( t, ω ) = ï (cid:120) ( t, ω ) (cid:120) ∗ ( t, ω ) ò , (cid:120) ( t, ω ) = (cid:120) ( t, ω ) ... (cid:120) ( t, ω M ) (29)With the augmented spectrum representation, (cid:120) ( t, ω ) in (29), it isnow possible to jointly consider all of the dual-frequency spectralcovariances in ω through the proposed compact formulation. To seethis, consider the following probabilistic model (cid:120) ( t, ω ) ∼ CN ( (cid:109) ( ω ) , (cid:82) ( ω )) (30) (cid:109) ( ω ) = E { (cid:120) ( t, ω ) } , (cid:82) ( ω ) = cov { (cid:120) ( t, ω ) } (31)where (cid:109) ( ω ) ∈ C MN denotes the augmented spectral mean , de-fined as (cid:109) ( ω ) = ï (cid:109) ( ω ) (cid:109) ∗ ( ω ) ò , (cid:109) ( ω ) = (cid:109) ( ω ) ... (cid:109) ( ω N ) (32)and (cid:82) ( ω ) ∈ C MN × MN denotes the augmented spectral covari-ance , given by (cid:82) ( ω ) = ï (cid:82) ( ω ) (cid:80) ( ω ) (cid:80) ∗ ( ω ) (cid:82) ∗ ( ω ) ò (33) (cid:82) ( ω ) = (cid:82) ( ω ) · · · (cid:82) ( ω , ω M ) ... . . . ... (cid:82) ( ω M , ω ) · · · (cid:82) ( ω M ) (34) (cid:80) ( ω ) = (cid:80) ( ω ) · · · (cid:80) ( ω , ω M ) ... . . . ... (cid:80) ( ω M , ω ) · · · (cid:80) ( ω M ) (35)o derive estimators of (cid:109) ( ω ) and (cid:82) ( ω ) , we starting from theleast squares (LS) estimate of (cid:120) ( t, ω ) based on (26), that is ˆ (cid:120) ( t, ω ) = Φ + ( t, ω ) x ( t ) ≡ Φ H ( t, ω ) x ( t ) (36)with the symbol ( · ) + as the pseudo-inverse operator. Next, since ˆ (cid:120) ( t, ω ) is stationary in time and hence ergodicity applies, we cansimply approximate the expectation operators in (31) with the time-averages, to obtain the following method of moment estimators ˆ (cid:109) ( ω ) = 1 T T − (cid:88) t =0 Φ H ( t, ω ) x ( t ) (37) ˆ (cid:82) ( ω ) = 1 T T − (cid:88) t =0 Φ H ( t, ω )ˆ s ( t )ˆ s T ( t ) Φ ( t, ω ) (38)with ˆ s ( t ) = x ( t ) − ˆ m ( t ) = x ( t ) − Φ ( t, ω ) ˆ (cid:109) ( ω ) . Remark 3.
The estimator of (cid:109) in (37) is, in essence, the discreteFourier transform (DFT) of x ( t ) . Similarly, the estimator of (cid:82) in(38) is the power spectrum matrix of the centred variable, s ( t ) .
4. SPECTRAL PORTFOLIO OPTIMIZATION
We next derive a spectral portfolio optimization framework basedon the considered class of nonstationary signals. While the standardMVO in (2) operates in the time-domain and with a constant capitalallocation, w , we instead consider optimizing the spectral content ofthe time-varying capital allocation, w ( t ) . This is made possible byconsidering the following spectral decomposition, as in (26) w ( t ) = Φ ( t, ω ) (cid:119) ( ω ) (39)In this way, the spectral MVO is formulated as max (cid:119) ( ω ) { (cid:119) H ( ω ) (cid:109) ( ω ) − λ (cid:119) H ( ω ) (cid:82) ( ω ) (cid:119) ( ω ) } (40) s.t. (cid:119) H ( ω ) (cid:82) ( ω ) (cid:119) ( ω ) = σ whereby we maximise the mean portfolio return based on the spec-tral mean, while constrain the variance of the portfolio to a targetlevel, σ , based on the spectral covariance. Upon inspecting thestationary points of the objective function in (40), we obtain the La-grangian multiplier λ = 12 σ » (cid:109) H ( ω ) (cid:82) − ( ω ) (cid:109) ( ω ) (41)which, in turn, yields the optimal augmented spectral weights (cid:119) opt ( ω ) = 12 λ (cid:82) − ( ω ) (cid:109) ( ω ) = σ (cid:82) − ( ω ) (cid:109) ( ω ) » (cid:109) H ( ω ) (cid:82) − ( ω ) (cid:109) ( ω ) (42)The optimal time-varying capital allocation can finally be retrievedthrough the augmented spectral basis, as in (26), to yield w opt ( t ) = Φ ( t, ω ) (cid:119) opt ( ω ) (43) Remark 4.
In contrast to the standard MVO in (2), the proposedspectral portfolio framework allows for optimal time-varying capi-tal allocation schemes. In this way, the investor is better positionedto exploit seasonal trends of asset prices, designated by the spec-tral mean, (cid:109) ( ω ) , and seasonal variations of the correlation betweenasset price movements, designated by the spectral covariance, (cid:82) ( ω ) .
5. SIMULATIONS
The performance of the proposed spectral MVO was investigatedusing monthly historical price data comprising of the commod-ity futures contracts constituting the Bloomberg Commodity Index,in the period 2010-01-01 to 2020-05-01. The data was split into: (i)the in-sample dataset (2010-01-01 to 2014-12-31) which was used toestimate the spectral moments, (cid:109) ( ω ) and (cid:82) ( ω ) , and to compute theoptimal spectral weights, (cid:119) ( ω ) ; and (ii) the out-sample data (2015-01-01 to 2020-05-01), used to objectively quantify profitability ofthe asset allocation strategies. For simplicity, the frequencies chosenfor ω corresponded to periodicities of year (A), months (S) and months (Q) ( business quarter). The standard equally-weighted(EW) and MVO (MVO) portfolios were also simulated for compari-son purposes, with the results displayed in Fig. 2.Observe from Fig. 2 (a)–(b) that the proposed spectral MVOconsistently delivered greater returns than the standard EW andMVO portfolios in the out-of-sample dataset, thereby attaining ahigher Sharpe ratio , i.e. the ratio of the mean to the standard devi-ation of portfolio returns. Fig. 2 (c) illustrates that by accountingfor the augmented spectral information, the portfolio was betterpositioned to exploit time-dependent dynamics in the market, whichcontrasts classical approaches that assume a constant optimal allo-cation. W e a l t h Spectral MVO (A)Spectral MVO (A, S)Spectral MVO (A, S, Q)MVOEqual Weights (a) Out-of-sample performance.
Spectral MVO(A) Spectral MVO(A, S) Spectral MVO(A, S, Q) MVO EW .
91 1 .
31 1 . − . − . (b) Annualised out-of-sample Sharpe ratios. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecDate6420246 C a p i t a l A ll o c a t i o n ( % ) MVOSpectral MVO (A)Spectral MVO (A, S)Spectral MVO (A, S, Q) (c)
Allocation to gold futures by month of the year.
Fig. 2 : Investment performance for the standard MVO and the pro-posed spectral MVO, with varying frequency spectra, ω . The targetportfolio volatility, σ , was set to per annum.
6. CONCLUSIONS
A spectral portfolio theory has been introduced which employs aug-mented complex statistics in order to account for the full interactionbetween the real and imaginary parts of the complex spectra of assetprice movements. This has been shown to enable the optimal capi-tal allocation to be time-varying, which allows for the modelling ofboth harmonics and cyclostationarity in asset returns. Simulationshave demonstrated the advantages of the proposed framework overconventional portfolio techniques, including a full utilization of thevariation of the mean and covariance of asset returns in time. . REFERENCES [1] H. Markowitz, “Portfolio selection,”
Journal of Finance , vol. 7,no. 1, pp. 77–91, 1952.[2] R. W. Klein and V. S. Bawa, “The effect of estimation riskon optimal portfolio choice,”
Journal of Financial Economics ,vol. 3, no. 3, pp. 215–231, 1976.[3] J. D. Jobson and B. Korkie, “Estimation for Markowitz effi-cient portfolios,”
Journal of the American Statistical Associa-tion , vol. 75, no. 371, pp. 544–554, 1980.[4] R. C. Merton, “On estimating the expected return on the mar-ket: An exploratory investigation,”
Journal of Financial Eco-nomics , vol. 8, no. 4, pp. 323–361, 1980.[5] P. Jorion, “Bayesian and capm estimators of the means: Impli-cations for portfolio selection,”
Journal of Banking & Finance ,vol. 15, no. 3, pp. 717–727, 1991.[6] M. Britten-Jones, “The sampling error in estimates of mean-variance efficient portfolio weights,”
The Journal of Finance ,vol. 54, no. 2, pp. 655–671, 1999.[7] R. Michaud, “The Markowitz optimization enigma: Is opti-mized optimal?”
Financial Analysts Journal , vol. 45, pp. 31–42, 1989.[8] ——,
Efficient Asset Allocation: A Practical Guide to StockPortfolio Optimization and Asset Allocation . Harvard Busi-ness School Press, 1998.[9] M. J. Best and R. R. Grauer, “On the sensitivity of mean-variance-efficient portfolios to changes in asset means: someanalytical and computational results,”
Review of FinancialStudies , vol. 4, no. 2, pp. 315–342, 1991.[10] V. K. Chopra and W. T. Ziemba, “The effect of errors in means,variances, and covariances on optimal portfolio choice,”
TheJournal of Portfolio Management , vol. 19, no. 2, pp. 6–11,1993.[11] I. Kondor, S. Pafka, and G. Nagy, “Noise sensitivity of portfo-lio selection under various risk measures,”
Journal of Banking& Finance , vol. 31, no. 5, pp. 1545–1573, 2007.[12] J. von Neumann, “Proof of the quasi-ergodic hypothesis,”
Pro-ceedings of the National Academy of Sciences of the UnitedStates of America , vol. 18, pp. 70–82, 1932.[13] B. O. Koopman, “Hamiltonian systems and transformation inHilbert space,”
Proceedings of the National Academy of Sci-ences of the United States of America (PNAS) , vol. 17, no. 5,pp. 315–318, 1931.[14] C. Granger and M. Hatanaka,
Spectral Analysis of EconomicTime Series . Princeton University Press, 1964.[15] E. R., “Band spectrum regressions,”
International EconomicReview , vol. 15, no. 1, pp. 1–11, 1974.[16] C. W. J. Granger and R. F. Engle, “Applications of spectralanalysis in econometrics,”
Handbook of Statistics , vol. 3, pp.93–109, 1983.[17] M. Baxter and R. G. King, “Measuring business cycles: Ap-proximate band-pass filters for economic time series,”
Reviewof Economics and Statistics , vol. 81, pp. 575–593, 1990.[18] P. P. Carr and D. B. Madan, “Option valuation using the fastFourier transform,”
Journal of Computational Finance , vol. 2,pp. 61–73, 1999. [19] C. Croux, M. Fourni, and L. Reichlin, “A measure of comove-ment for economic variables: Theory and empirics,”
The Re-view of Economics and Statistics , vol. 83, no. 2, pp. 232–241,2001.[20] J. Ramsey, “Wavelets in economics and finance: Past and fu-ture,”
Studies in Nonlinear Dynamics & Econometrics , vol. 6,no. 3, pp. 1–27, 2002.[21] N. Huang, M. Wu, W. Qu, S. Long, S. Shen, and J. Zhang, “Ap-plications of Hilbert-Huang transform to non-stationary finan-cial time series analysis,”
Applied Stochastic Models in Busi-ness and Industry , vol. 19, no. 3, pp. 245–268, 2003.[22] P. Crowley, “A guide to wavelets for economists,”
Journal ofEconomic Surveys , vol. 21, no. 2, pp. 207–267, 2007.[23] A. Rua, “Measuring comovement in the time-frequencyspace,”
Journal of Macroeconomics , vol. 32, no. 2, pp. 685–691, 2010.[24] ——, “Wavelets in economics,”
Economic Bulletin and Finan-cial Stability Report Articles , pp. 71–79, 2012.[25] J. Breitung and B. Candelon, “Testing for short- and long-runcausality: A frequency domain approach,”
Journal of Econo-metrics , vol. 132, no. 2, pp. 363–378, 2006.[26] S. E. Chaudhuri and A. W. Lo, “Spectral analysis of stock-return volatility, correlation, and beta,”
In Proceedings ofthe IEEE Signal Processing and Signal Processing EducationWorkshop , pp. 232–236, 2015.[27] ——, “Spectral portfolio theory,”
SSRN Electronic Journal.10.2139/ssrn.2788999 , 2016.[28] F. M. Bandi, S. E. Chaudhuri, A. W. Lo, and A. Tamoni, “Spec-tral factor models,”
Johns Hopkins Carey Business School Re-search Paper , no. 18–17, 2019.[29] E. T. Jaynes, “Information theory and statistical mechanics,”
Physical Review , vol. 106, no. 4, pp. 620–630, 1957.[30] J. P. Burg, “Maximum entropy spectral analysis,”
In Proceed-ings of 37th Meeting, Society of Exploration Geophysics , 1967.[31] A. van den Bos, “Alternative interpretation of maximum en-tropy spectral analysis,”
IEEE Transactions on InformationTheory , vol. 17, no. 4, pp. 493–494, 1971.[32] T. M. Cover, “An information-theoretic proof of Burg’s max-imum entropy spectrum,”
Proceedings of the IEEE , vol. 72,no. 8, pp. 1094–1096, 1984.[33] D. P. Mandic and V. S. L. Goh,
Complex Valued NonlinearAdaptive Filters: Noncircularity, Widely Linear and NeuralModels . Wiley, 2009.[34] F. D. Naseer and J. L. Massey, “Proper complex random pro-cesses with applications to information theory,”
IEEE Trans-actions on Information Theory , vol. 39, no. 4, pp. 1293–1302,1993.[35] P. J. Schreier and L. L. Scharf, “Stochastic time-frequencyanalysis using the analytic signal: Why the complementarydistribution matters,”
IEEE Transactions on Signal Processing ,vol. 51, no. 12, pp. 3071–3079, 2003.[36] M. Lo`eve,
Probability Theory . Springer-Verlag, 1977.[37] A. van den Bos, “The multivariate complex normal distribution– a generalization,”
IEEE Transactions on Information Theory ,vol. 41, no. 2, pp. 537–539, 1995.[38] J. O. Smith,