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(a) [MATH] is continuous and bounded, (b) [MATH] (c) [MATH] then, [EQUATION] Proof. This is the multivariate extension of Theorem 1.3.1 in |
. The proof is almost identical, except that the intervals of size [MATH] become hypercubes with volume [MATH] . A similar result is provided by Theorem 8.3.1 of |
, where condition (a) is replaced by Riemann integrability, condition (b) is replaced by [MATH] , and [MATH] Corollary 3.2 Let [MATH] be a real-valued discrete-time stochastic process, and let [MATH] be an [MATH] -valued discrete-time stochastic process such that every marginal of the joint process [MATH] admits a prob... |
[EQUATION] [EQUATION] [EQUATION] and [EQUATION] In other words, if we can estimate discrete entropy rates, we can estimate differential entropy rates and differential mutual information rates, and consequently incremental diversification. Interestingly, unlike the differential entropy rate, the differential mutual info... |
3.4 Estimating Incremental Diversification Differential mutual information timescale does not always exist. When the process [MATH] is stationary, the differential mutual information timescale [MATH] is guaranteed to exist. Thus, throughout the rest of this paper, we assume that [MATH] is jointly (strongly) stationary ... |
. We note however that these assumptions are not restrictive as they are impossible to invalidate experimentally with a finite sample |
In order to estimate differential mutual information timescale [EQUATION] it is sufficient to be able to estimate the entropy rate of any vector-valued discrete-time stationary ergodic stochastic process [MATH] from a single sample path [MATH] ; this is what we focus our discussion on. |
Considering that differential mutual information rate is invariant by rescaling, we assume coordinate processes of [MATH] all have the same variance [MATH] , and we normalize sample path [MATH] accordingly, if needed. We make this specific choice of variance to ease estimation interpretation and debugging. Indeed, unde... |
In the following, we discuss and compare three estimation approaches, namely model-free estimation in Section 3.4.1 , nonparametric estimation in Section 3.4.2 and maximum-entropy estimation in Section 3.4.3 . The model-free approach places no assumption on the diffusion of [MATH] . The nonparametric approach does not ... |
3.4.1 Model-Free Estimation We recall from Section 3.3 that differential entropy rates can be estimated by first discretizing the input process, then estimating the discrete entropy rate of the discretized process, and finally adjusting for discretization precision. We also note that, when the input process is strongly... |
The notion of complexity of a sequence of characters emitted by a stochastic source is tightly coupled with the discrete entropy rate of the emitting source. Of particular interest is the link between the Lempel-Ziv complexity introduced in |
, and for which we provide a Python implementation in Listing 1, and the discrete entropy rate of a stationary ergodic process , which we recall below. |
Theorem 3.3 Let [MATH] be a discrete-time stationary ergodic stochastic process taking values in a countable set [MATH] , and that has discrete entropy rate [MATH] . If we denote [MATH] the Lempel-Ziv complexity (as per Listing 1) of a sample path of length [MATH] of this process, then |
[EQUATION] Corollary 3.3 Let [MATH] be a discrete-time stationary ergodic stochastic process taking values in a countable set [MATH] , and such that [MATH] . Let us consider a path [MATH] with Lempel-Ziv complexity [MATH] . Let [MATH] with [MATH] and [MATH] be [MATH] independent draws from [MATH] sampled uniformly at r... |
[EQUATION] Proof. Hint: For every [MATH] [MATH] In summary, [MATH] is a consistent estimator of discrete entropy rate. However, in practice, we find the convergence of Theorem 3.3 to be slower than that of Corollary 3.3 . Whence, we choose instead to estimate the discrete entropy rate given a sequence of characters as |
[EQUATION] where [EQUATION] and terms [MATH] represent frequencies of occurrence of distinct symbols in [MATH] . The estimate of Equation ( 36 ) is a consistent estimate of discrete entropy rate for every [MATH] , but larger [MATH] can help reduce estimation variance. Finally, using the results of Section 3.3 , we may ... |
We stress that this approach does not require placing any assumption on the diffusion of [MATH] other than ergodicity and stationarity and, in that sense, is model-free. Algorithm provides a summary. |
Choice of Discretization Precision [MATH] : Corollaries 3.2 and 3.3 guarantee convergence of Algorithm to the true entropy rate as both [MATH] and [MATH] go to infinity. However, for a given sample size [MATH] , as will be the case in practice, estimation error can vary greatly with [MATH] . Too small an [MATH] and the... |
Data Efficiency : Care should be taken before applying this approach when the dimensionality [MATH] of the input process is large. Indeed, if we denote [MATH] the number of distinct characters we expect to commonly see in the discretization of a coordinate process of input [MATH] -valued process [MATH] , then the discr... |
Scalability : Algorithm scales linearly with both the sample size [MATH] and the dimensionality of the input process [MATH] . However, as previously discussed, the number of samples [MATH] needed for accurate estimation itself depends on [MATH] . For a fixed estimation accuracy, the number of samples [MATH] required, a... |
3.4.2 Nonparametric Estimation Analytic formulae to compute entropy rates are not always available. When [MATH] is a stationary Gaussian process however, its entropy rate exists and is available in closed-form. More precisely, if we denote |
[EQUATION] the autocovariance function of [MATH] , and if we assume that [EQUATION] where [MATH] denotes any matrix norm, then the matrix-valued spectral density function |
[EQUATION] is well-defined, forms a Fourier pair with the autocovariance function, [EQUATION] and the entropy rate reads: [EQUATION] |
Thus, entropy rates can be estimated in the stationary Gaussian case by first estimating the spectral density function, and then using Equation ( 39 ), where the integral can be approximated numerically. |
A naïve estimator of the spectral density function is obtained as the piecewise constant extension to [MATH] of the periodogram, defined as |
[EQUATION] with [EQUATION] where [MATH] denotes the transpose of the complex conjugate of [MATH] , and for [MATH] . The periodogram is not a consistent estimator of the spectral density function. It is typically improved and made consistent thanks to smoothing. Reviewing spectral density estimation methods is beyond th... |
and references therein for a more detailed discussion on smoothed periodograms. Remark 3.6 The Gaussian assumption can be relaxed by assuming that, although [MATH] might not be Gaussian, there exists a mapping [MATH] such that the stochastic process [MATH] is Gaussian, stationary and ergodic. Both the periodogram of [M... |
for more details). As for the choice of kernel, the Generalized Spectral Kernels of provide a family that is provably arbitrarily flexible. Equation ( 39 ) can then be used to estimate [MATH] which, although in general differs from [MATH] , can be used as a proxy for incremental diversification. As previously discussed... |
Data Efficiency: As in the model-free case, care should be taken before applying this approach when the dimensionality [MATH] of the input process is too large, but for a different reason. The rationale here is that estimating the spectral density function typically scales poorly with dimensionality [MATH] , and can ha... |
, GP-LVM , autoencoders , manifold learning etc.), and then use the differential mutual information timescale between [MATH] and the compressed version of [MATH] as a proxy for incremental diversification. In Section 3.5 we propose an approximation to incremental diversification that is more data-efficient in that it d... |
Scalability: Time complexity scales cubically with dimensionality [MATH] due to the need to evaluate [MATH] at several frequencies to numerically compute the integral in Equation ( 39 ), and linearly with [MATH] because of the computation of the smoothed periodogram. The integral in Equation ( 39 ) would be very costly... |
can prove more efficient than traditional quadrature techniques, as it typically results in fewer function evaluations. Similarly, memory requirement scales quadratically with [MATH] and linearly with [MATH] . Overall, this approach cannot scale to very large [MATH] as is. In Section 3.5 we propose an approximation of ... |
3.4.3 Maximum-Entropy Estimation Our last estimation approach is based on the principle of maximum-entropy pioneered by E. T. Jaynes in |
. The maximum-entropy principle stipulates that, when faced with an estimation problem, among all models that are consistent with empirical evidence, one should always choose the one that is the most uncertain/ignorant about everything other than what has been observed. |
Given a sample path [MATH] of a vector-valued discrete-time stochastic process [MATH] , the sample autocovariance function defined as |
[EQUATION] with [MATH] provides reliable empirical evidence about the autocovariance of [MATH] , in that it is a consistent and asymptotically unbiased estimator thereof |
. As for measuring ‘uncertainty/ignorance’ about everything else, the entropy rate happens to be the canonical measure for this purpose. Burg’s maximum-entropy theorem |
, which we recall below, provides an answer to the maximum-entropy optimization problem for discrete-time stationary processes under autocovariance constraints. |
Theorem 3.4 Let [MATH] be a stationary [MATH] -valued discrete-time stochastic process. Among all stationary processes whose (matrix-valued) autocovariance functions coincide with that of [MATH] from lag [MATH] to lag [MATH] , the mean-zero Gaussian Vector Autoregressive process of order [MATH] (VAR(p)) has the highest... |
[EQUATION] where [MATH] is the block-matrix such that [EQUATION] with [MATH] The maximum-entropy approach to estimating a differential entropy rate therefore consists of first computing the corresponding sample autocovariance function, and then choosing as [MATH] the largest lag up to which we can reliably estimate aut... |
Remark 3.7 Theorem 3.4 is quite profound. It states that the most principled approach to modeling stochastic processes under an assumption as generic as known autocovariance terms, follows a very simple and well-studied diffusion model whose entropy rate is available in closed-form. A subtle point to note however is th... |
3.5 Scaling Up Incremental Diversification Estimation 3.5.1 The Source of Scalability Issues As previously discussed, all three methods we proposed for estimating the differential entropy rate of an [MATH] -valued stationary ergodic process [MATH] scale poorly with dimensionality [MATH] . This should come as no surpris... |
The model-free approach does not directly suffer from this problem because the discretization step effectively turns the multivariate problem into a univariate one, at the cost of increasing the discrete entropy rate of the resulting discretized process. However, in the best case scenario, the impact of such entropy in... |
The root cause of this lack of scalability is the absence of a structured model expressing how coordinate processes of [MATH] relate to each other. This can for instance be done through a dimensionality reduction technique (e.g. PCA and kernel PCA |
, GP-LVM , autoencoders , manifold learning etc.). We do not follow this idea as it is very sensitive to the dimensionality reduction technique used, and most of them have scalability issues of their own. We choose instead to relax the implicit requirement that we should understand how each asset relates to all the oth... |
3.5.2 Order- [MATH] Incremental Diversification Let [MATH] and [MATH] be a partition of [MATH] into subsets of size [MATH] , where [MATH] for all but at most [MATH] element in the partition. Let [MATH] be a real-valued discrete-time process that is jointly ergodic and stationary with [MATH] -valued process [MATH] . It ... |
[EQUATION] where [MATH] is the vector-valued process whose coordinate processes are the ones of [MATH] whose indices are in [MATH] . Consequently, denoting [MATH] the set of all possible partitions of [MATH] into subsets of size [MATH] , it follows that |
[EQUATION] where [EQUATION] When [MATH] represents the time series of returns of a new asset, and [MATH] those of assets in the reference pool, [MATH] reflects the least amount of incremental diversification the new asset adds to any subset of [MATH] assets in the reference pool. |
Definition 3.3 We denote order- [MATH] incremental diversification a new asset A adds to a reference pool P, the least amount of incremental diversification A adds to a subset of size [MATH] of P, namely |
[EQUATION] where [MATH] is the set of all subsets of P of size [MATH] It follows from Equation ( 42 ) that, as we would expect, if a new asset adds no incremental diversification to any subset of [MATH] assets in the reference pool, then it adds no incremental diversification to the reference pool. Moreover, it is easy... |
[EQUATION] reflects the amount of information about the new asset that can only be obtained from the reference pool by considering more than [MATH] assets at a time. |
As a measure of incremental diversification, [MATH] satisfies Stylized Facts 1 and 2 under the sparsity constraint that the best replicating portfolio does not have more than [MATH] non-zero allocations. [MATH] also satisfies Stylized Fact 3 providing that returns of the new asset do not depend on current and past retu... |
We recall that the guiding principle we used to determine whether an asset adds incremental diversification to a reference pool is that, if it is easy to replicate returns of the new asset using those of the reference pool of assets and factors, then the new asset is not needed. In practice however, if the number of as... |
Scalability : Estimating [EQUATION] using either the model-free approach, or the nonparametric approach, or the maximum-entropy approach scales linearly with the number of assets [MATH] . Rather than taking the [MATH] across all possible partitions of [MATH] into subsets of size [MATH] , which would be intractable, we ... |
3.6 Extension to a Pool of New Assets Our method for quantifying incremental diversification can be extended to quantifying the amount of diversification a universe of new assets [MATH] , for which no asset is fully determined by the others, collectively adds to a reference pool of assets. If we denote |
[EQUATION] the vector-valued time series of returns of assets in the new universe [MATH] , and [MATH] the time series of returns and factor values of the existing reference pool of assets, then the amount of diversification the new universe of assets add to the existing one is |
[EQUATION] and can be computed using previously established results. 3.7 Illustration In this section we empirically illustrate the pertinence of our measure of incremental diversification, as well as estimation methods previously discussed. We first provide a comparative analysis between model free, nonparametric and ... |
3.7.1 Model Comparison We begin by comparing the three approaches we proposed for estimating differential entropy rates on synthetic data, starting with real-valued time series. |
[MATH] Varying [MATH] : In the interest of assessing how our three approaches perform in the presence of memory and leptokurticity, we consider an AR(1) time series with Student-t noise, namely |
[EQUATION] where [MATH] is a Student-t white noise with standard deviation [MATH] , and degree of freedom [MATH] . We generate two sample paths of size [MATH] from our synthetic model, one for which we choose [MATH] so that the innovation term has infinite kurtosis ( [MATH] ), and one for which the innovation term is a... |
with a Hanning window, a window size equals to [MATH] , and a [MATH] % overlap. For the model-free approach, we set [MATH] such that [MATH] is equal to [MATH] -th of the sample standard deviation. As for the ground truth, we recall that the differential entropy rate of any autoregressive process is the differential ent... |
Overall, it can be seen that all three approaches converge. As expected, both the maximum-entropy and the nonparametric approaches, which are the only ones assuming Gaussianity, converge to the entropy rate of the Gaussian AR(1) process that has the same mean and autocovariance function as our Student-t AR(1) process. ... |
The model-free approach on the other hand always converges to the ground truth, even when the excess kurtosis is infinite. In fact, for a fixed sample size [MATH] , the model-free approach converges faster when the excess kurtosis is large, which is understandable as this corresponds to lower entropy rates; in general,... |
To summarize, as much as it appears more flexible than the maximum-entropy approach on the surface, in the univariate case, the nonparametric approach does not add much as far as estimating differential entropy rate is concerned. As for whether one should prefer the model-free or maximum-entropy approach in the univari... |
Varying [MATH] Fixed [MATH] : Next, we consider empirically investigating how the accuracies of our three approaches to estimating differential entropy rates of an [MATH] -valued discrete-time stationary ergodic processes scale with dimensionality [MATH] . To do so, we generate [MATH] samples of an [MATH] -valued proce... |
It can be seen from Figures ( 3a ) and ( 3b ) that the model-free approach is grossly data-inefficient, and should certainly not be used beyond [MATH] for a sample size [MATH] . This is in line with our previous back-of-the-envelop analysis that suggested that the sample size should increase exponentially with [MATH] t... |
As for nonparametric and maximum-entropy approaches, their accuracies decrease (roughly) linearly with dimensionality [MATH] . However, unlike the one-dimensional case, in the multi-dimensional case the maximum-entropy approach is a lot more data-efficient than the nonparametric case, and the difference between the two... |
Another important point worth noting about Figure ( 3c ) is that it gives us a sense of the order of magnitude of the relative error we would make by using the maximum-entropy approach as a function of [MATH] , and consequently, how large an [MATH] we could afford for [MATH] —which corresponds to about [MATH] years of ... |
3.7.2 Stylized Facts Consistency We have previously shown that [MATH] satisfies all 5 Stylized Facts. In this section, we aim to illustrate that the finite-sample maximum-entropy estimator also satisfies all 5 Stylized Facts. To do so, we generate a set of synthetic returns time series that exhibit both cross-sectional... |
[EQUATION] where [MATH] (resp. [MATH] ) is a standard Gaussian matrix with shape [MATH] (resp. [MATH] ) for [MATH] . Columns of [MATH] are thus i.i.d. Gaussian with mean [MATH] and covariance matrix |
[EQUATION] We choose this structure to emulate a low-rank covariance matrix while avoiding numerical instabilities in the OLS estimation of tracking errors due to ill-conditioning. In this spirit, we choose [MATH] so that [MATH] has determinant [MATH] . We introduce temporal dependency from the innovation in an AR(1) f... |
[EQUATION] Each row of [MATH] plays the role of a size [MATH] path of a time series of returns of a different asset. Stylized Facts 1 & 2 : We loop through rows sequentially, and for row [MATH] , we compute both the incremental diversification the corresponding asset adds to the reference pool defined by the first [MAT... |
Stylized Facts 3 : To assess consistency with Stylized Fact 3, we consider a simple momentum strategy on assets defined by rows of [MATH] . For a window size [MATH] , the momentum strategy consists of investing proportionally to the returns of each asset in the pool over the past [MATH] time periods. Returns of this mo... |
Stylized Facts 4 : To illustrate that our approach allows for manager diversification, we consider the universe of assets whose returns are [MATH] (Equation ( 46 )), and we consider [MATH] managers trading these assets long-only, and without leverage. Allocation processes are taken to be independent across managers. At... |
We consider managers one at a time, and for each manager we estimate the incremental diversification his/her fund adds to the funds of all managers previously considered. First we use as [MATH] for the [MATH] -th manager [MATH] . As illustrated in Figure ( 6a ) managers trading the same assets can provide incremental d... |
Stylized Facts 5 : Finally, in order to demonstrate that our finite-sample estimator of incremental diversification is invariant by rescaling, we construct an [MATH] matrix [MATH] such that each of its row is obtained by multiplying the corresponding row of [MATH] by a random scalar drawn from a standard normal. We loo... |
3.7.3 Relation Between Pairwise Correlation and Pairwise Incremental Diversification Considering the widespread, and somewhat excessive, use of correlation by both practitioners and academics as the canonical measure of dependency between assets, the alert reader must be wondering, in the case of two assets, how our me... |
An interesting observation evidencing the limits of using correlation to quantify dependency between assets is that the width of the cloud of red points in Figure ( 8a ) decreases with pairwise correlation. This makes intuitive sense. A strong pairwise correlation between two assets is a strong indication of (linear) d... |
[EQUATION] this is the blue curve illustrated in Figure ( 8a ), which we will refer to as the correlation frontier . Hence, the deviation of the cloud of red points in Figure ( 8a ) away from the correlation frontier is strong evidence that the memoryless Gaussian assumption does not hold for S&P 100 constituents or, e... |
Correlation can easily be adjusted to account for temporal and nonlinear dependencies, by first estimating pairwise incremental diversification, and then inferring the unadjusted correlation value that would be consistent with estimated incremental diversification, under the Gaussian memoryless assumption. We refer to ... |
[EQUATION] as the information-adjusted correlation coefficient It can be seen from Figure ( 8b ) that, more often than not, accounting for temporal and nonlinear dependencies increases correlation. The lower the unadjusted correlation, the higher the difference between unadjusted and information-adjusted correlation co... |
Another interesting difference between pairwise incremental diversification and pairwise correlation is that the former is a lot more sensitive than the latter for smaller pairwise correlations, that is, when it matters, and is less sensitive for larger pairwise correlations, that is, when information redundancy is obv... |
3.7.4 Asset Class Information Clustering A common perception is that diversifying across asset classes provides considerably more benefits than within asset class diversification; the intuition being that assets within the same class share more economic drivers than assets across classes. In our last experiment on incr... |
Specifically, using our measure of incremental diversification, we consider empirically evaluating how much more diversification can be obtained by choosing a new asset to add (to a reference pool made of assets belonging to the same asset class) from a different asset class, compared to choosing a new asset in the sam... |
Figure ( ) however paints a more granular story, one that can hardly be obtained with traditional tools. In effect, we note that U.S. futures on average add nearly as much incremental diversification to our reference pool of currencies as a currency in the pool adds to the rest of the pool on average. In other words, w... |
This could be used as empirical piece of evidence that global currencies already factor-in U.S. futures risk factors, but are also driven by additional risk factors that are unrelated to U.S. futures, and that offer cross asset class diversification opportunities for futures. We stress that we could not arrive to this ... |
Another interesting observation we can make from Figure ( ) is that foreign currencies provide the greatest cross asset class diversification benefits. The largest average incremental diversification ( [MATH] days/bit) is obtained by using global currencies to diversify U.S. futures, or by using U.S. blue chips to dive... |
QUANTIFYING PREDICTABILITY OF RETURNS TIME SERIES Intuitively, a time series of returns [MATH] can be considered sufficiently predictable if there exists a stream of information available at time [MATH] that reduces the uncertainty about future values at time [MATH] . Determining whether future values of a returns time... |
4.1 Auto-Predictability We say of a time series of returns that it is sufficiently auto-predictable when its past values sufficiently reduce the uncertainty about future values (i.e. when it has sufficient memory). |
When [MATH] is (strongly) stationary, the entropy rate [MATH] always exists and it can be shown that [EQUATION] Moreover, it can be shown that [MATH] decreases with [MATH] Thus, |
[EQUATION] which we note does not depend on [MATH] for stationary processes, can be regarded as the maximum reduction in the uncertainty of the return at any point in time [MATH] that one can achieve by knowing all returns prior to [MATH] , which makes it suitable for quantifying predictability of returns. We name [MAT... |
Remark 4.1 The auto-predictability measure is invariant by affine transformations [EQUATION] Proposition 4.1 confirms that memoryless time series are the least auto-predictable. |
Proposition 4.1 Let [MATH] be a stationary discrete-time stochastic process such that [MATH] . Then, [EQUATION] Moreover, [EQUATION] |
if and only if [MATH] are jointly independent for any [MATH] Proof. [EQUATION] [MATH] for every [MATH] and increases with [MATH] . Hence [MATH] if and only if [MATH] for every [MATH] , which holds if and only if [MATH] are jointly independent for every [MATH] |
Another take on the measure of auto-predictability is obtained by noting that [EQUATION] In other words, the measure of auto-predictability is the rate of KL-divergence between the returns time series and its memoryless equivalent. This parallel further confirms Proposition 4.1 and Remark 4.1 . In fact Remark 4.1 can b... |
[EQUATION] for any smooth bijection [MATH] 4.2 Estimation Similarly to incremental diversification, the measure of auto-predictability can be estimated using a model-free approach, a nonparametric approach, or a maximum-entropy approach. |
Model-Free Estimation : In the model-free case, we require the time series to be stationary and ergodic, but not necessarily Gaussian. It follows from Corollaries 3.2 and 3.3 that |
[EQUATION] where [MATH] is the discretized version of [MATH] with precision [MATH] [MATH] the naïve frequency estimator of discrete entropy [MATH] , and [MATH] and [MATH] are as per Corollary 3.3 , is a consistent (in [MATH] ) estimator of the measure of auto-predictability. |
Nonparametric Estimation : The spectral analysis approach requires assuming [MATH] is a stationary and ergodic Gaussian process, in which case |
[EQUATION] and the sample variance provides a consistent estimate of [MATH] thanks to the ergodic assumption. Moreover, we recall that |
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