Modelling Changes in the Unconditional Variance of Long Stock Return Series

In this paper we develop a testing and modelling procedure for describing the long-term volatility movements over very long return series. For the purpose, we assume that volatility is multiplicatively decomposed into a conditional and an unconditional component as in Amado and Teräsvirta (2011). The latter component is modelled by incorporating smooth changes so that the unconditional variance is allowed to evolve slowly over time. Statistical inference is used for specifying the parameterization of the time-varying component by applying a sequence of Lagrange multiplier tests. The model building procedure is illustrated with an application to daily returns of the Dow Jones Industrial Average stock index covering a period of more than ninety years. The main conclusions are as follows. First, the LM tests strongly reject the assumption of constancy of the unconditional variance. Second, the results show that the long-memory property in volatility may be explained by ignored changes in the unconditional variance of the long series. Finally, based on a formal statistical test we find evidence of the superiority of volatility forecast accuracy of the new model over the GJR-GARCH model at all horizons for a subset of the long return series.Model specification; Conditional heteroskedasticity; Lagrange multiplier test; Timevarying unconditional variance; Long financial time series; Volatility persistence

Applications of Minkowski Functionals to the Statistical Analysis of Dark Matter Models

A new method for the statistical analysis of 3D point processes, based on the family of Minkowski functionals, is explained and applied to modelled galaxy distributions generated by a toy-model and cosmological simulations of the large-scale structure in the Universe. These measures are sensitive to both, geometrical and topological properties of spatial patterns and appear to be very effective in discriminating different point processes. Moreover by the means of conditional subsampling, different building blocks of large-scale structures like sheets, filaments and clusters can be detected and extracted from a given distribution.Comment: 13 pages, Latex, 2 gzipped tar-files, to appear in: Proc. ``1st SFB workshop on Astro-particle physics'', Ringberg, Tegernsee, 199

Forecasting Equicorrelation

We study the out-of-sample forecasting performance of several time-series models of equicorrelation, which is the average pairwise correlation between a number of assets. Building on the existing Dynamic Conditional Correlation and Linear Dynamic Equicorrelation models, we propose a model that uses proxies for equicorrelation based on high-frequency intraday data, and the level of equicorrelation implied by options prices. Using state-of-the-art statistical evaluation technology, we find that the use of both realized and implied equicorrelations outperform models that use daily data alone. However, the out-of-sample forecasting benefits of implied equicorrelation disappear when used in conjunction with the realized measures.Equicorrelation, Implied Correlation, Multivariate GARCH, DCC

Differentially Private Conditional Independence Testing

Conditional independence (CI) tests are widely used in statistical data analysis, e.g., they are the building block of many algorithms for causal graph discovery. The goal of a CI test is to accept or reject the null hypothesis that $X \perp \!\!\! \perp Y \mid Z$, where $X \in \mathbb{R}, Y \in \mathbb{R}, Z \in \mathbb{R}^d$. In this work, we investigate conditional independence testing under the constraint of differential privacy. We design two private CI testing procedures: one based on the generalized covariance measure of Shah and Peters (2020) and another based on the conditional randomization test of Cand\`es et al. (2016) (under the model-X assumption). We provide theoretical guarantees on the performance of our tests and validate them empirically. These are the first private CI tests with rigorous theoretical guarantees that work for the general case when $Z$ is continuous

Comment on:Pseudo-True SDFs in Conditional Asset Pricing Models. Comparing Fixed-versus Vanishing-Bandwidth Estimators of Pseudo-True SDFs

The paper by Antoine, Proulx, and Renault (2018) (APR) deals with the econometric definition, economic interpretation, and statistical estimation of the pseudo-true stochastic discount factor (SDF) in misspecified conditional asset pricing models. The paper revolves around fundamental issues like the role of conditioning information and omitted risk factors, and has non-trivial interactions with the current debate in the literature on the impact of weak factors (weak identification) for assessing asset pricing models. Building on, and substantially extending, previous contributions in the literature, the approach of the authors to define a pseudo-true SDF relies on the minimizers of econometric criteria based on a conditional version of the Hansen–Jagannathan (HJ) distance, that is, an average across states of squared conditional pricing errors. The authors provide an insightful discussion of the economic interpretation of pseudo-true SDFs. APR advocate the use of a fixed bandwidth (i.e., independent of the sample size) when estimating the conditional pricing errors by kernel regression methods to facilitate statistical analysis. This route leads to bandwidth-dependent pseudo-true SDF parameters and estimators thereof. In our discussion, we investigate the different definitions of pseudo-true SDFs and interpret the fixed-bandwidth proposal as a model calibration which down-weights highfrequency Fourier components of the conditional pricing errors (Section 1). We compare the statistical properties of pseudo-true SDF parameters’ estimators relying on vanishing versus fixed bandwidth, and provide a condition under which the former have a smaller asymptotic variance than the latter (or viceversa). We look at these topics through the lens of misspecified conditional linear SDF models in which priced risk factors are omitted using both simulated and real data (Section 3). We skip regularity conditions and relegate some technical derivations in the Appendix of the paper

On conditional probability and bayesian inference

Measurement theory has dealt with the applicability of the conditional probability formula to the updating of probability assignments when new information is incorporated. In this paper the original probability measure is taken as given, and an assumption on the relation between this probability and a possible conditional probability is imposed. Provided that the original probability is non-atomic, it is proved that there is one and only one transformed probability measure satisfying the assumption. Building on this result, we discuss the hypotheses underlying Bayesian inference. In the Bayesian parametric model, a joint probability distribution on the product of the sample space and the parameter space is assigned. As this probability distribution is shown to be non-atomic, we conclude that, apart from measure-theoretic representability hypotheses, the existence of this joint probability is the only nontechnical hypothesis underlying Bayesian parametric statistical inference

Patterns versus Characters in Subword-aware Neural Language Modeling

Words in some natural languages can have a composite structure. Elements of this structure include the root (that could also be composite), prefixes and suffixes with which various nuances and relations to other words can be expressed. Thus, in order to build a proper word representation one must take into account its internal structure. From a corpus of texts we extract a set of frequent subwords and from the latter set we select patterns, i.e. subwords which encapsulate information on character $n$-gram regularities. The selection is made using the pattern-based Conditional Random Field model with $l_1$ regularization. Further, for every word we construct a new sequence over an alphabet of patterns. The new alphabet's symbols confine a local statistical context stronger than the characters, therefore they allow better representations in ${\mathbb{R}}^n$ and are better building blocks for word representation. In the task of subword-aware language modeling, pattern-based models outperform character-based analogues by 2-20 perplexity points. Also, a recurrent neural network in which a word is represented as a sum of embeddings of its patterns is on par with a competitive and significantly more sophisticated character-based convolutional architecture.Comment: 10 page

deepregression: A Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression

In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library TensorFlow for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as mgcv. The package's modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models