arXiv subjects

Mikkel Slot Nielsen

Publications and source records attributed to Mikkel Slot Nielsen.

11 recordsLinked to original sources

arXiv2026

High-dimensional estimation of quadratic variation based on penalized realized variance

In this paper, we develop a penalized realized variance (PRV) estimator of the quadratic variation (QV) of a high-dimensional continuous Itô semimartingale. We adapt the principle idea of regularization from linear regression to covariance estimation in a continuous-time high-frequency setting. We show that under a nuclear norm penalization, the PRV is computed by soft-thresholding the eigenvalues of realized variance (RV). It therefore encourages sparsity of singular values or, equivalently, low rank of the solution. We prove our estimator is minimax optimal up to a logarithmic factor. We derive a concentration inequality, which reveals that the rank of PRV is -- with a high probability -- the number of non-negligible eigenvalues of the QV. Moreover, we also provide the associated non-asymptotic analysis for the spot variance. We suggest an intuitive data-driven subsampling procedure to select the shrinkage parameter. Our theory is supplemented by a simulation study and an empirical application. The PRV detects about three-five factors in the equity market, with a notable rank decrease during times of distress in financial markets. This is consistent with most standard asset pricing models, where a limited amount of systematic factors driving the cross-section of stock returns are perturbed by idiosyncratic errors, rendering the QV -- and also RV -- of full rank.

econ.EM↗

arXiv2021

Tree-based Synthetic Control Methods: Consequences of moving the US Embassy

We recast the synthetic controls for evaluating policies as a counterfactual prediction problem and replace its linear regression with a nonparametric model inspired by machine learning. The proposed method enables us to achieve accurate counterfactual predictions and we provide theoretical guarantees. We apply our method to a highly debated policy: the relocation of the US embassy to Jerusalem. In Israel and Palestine, we find that the average number of weekly conflicts has increased by roughly 103\% over 48 weeks since the relocation was announced on December 6, 2017. By using conformal inference and placebo tests, we justify our model and find the increase to be statistically significant.

econ.EM↗

arXiv2020

Targeting predictors in random forest regression

Random forest regression (RF) is an extremely popular tool for the analysis of high-dimensional data. Nonetheless, its benefits may be lessened in sparse settings due to weak predictors, and a pre-estimation dimension reduction (targeting) step is required. We show that proper targeting controls the probability of placing splits along strong predictors, thus providing an important complement to RF's feature sampling. This is supported by simulations using representative finite samples. Moreover, we quantify the immediate gain from targeting in terms of increased strength of individual trees. Macroeconomic and financial applications show that the bias-variance trade-off implied by targeting, due to increased correlation among trees in the forest, is balanced at a medium degree of targeting, selecting the best 10--30\% of commonly applied predictors. Improvements in predictive accuracy of targeted RF relative to ordinary RF are considerable, up to 12-13\%, occurring both in recessions and expansions, particularly at long horizons.

econ.EM↗

arXiv2020

On non-negative solutions of SDDEs with an application to CARMA processes

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are non-negative. While, to the best of our knowledge, no simple non-negativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\geq 3$. Finally, we extend the result to a multidimensional setting.

math.PR↗

arXiv2020

A surrogate model for estimating extreme tower loads on wind turbines based on random forest proximities

In the present paper we present a surrogate model, which can be used to estimate extreme tower loads on a wind turbine from a number of signals and a suitable simulation tool. Due to the requirements of the International Electrotechnical Commission (IEC) Standard 61400-1, assessing extreme tower loads on wind turbines constitutes a key component of the design phase. The proposed model imputes tower loads by matching observed signals with simulated quantities using proximities induced by random forests. In this way the algorithm's adaptability to high-dimensional and sparse settings is exploited without using regression-based surrogate loads (which may display misleading probabilistic characteristics). Finally, the model is applied to estimate tower loads on an operating wind turbine from data on its operational statistics.

stat.AP↗

arXiv2019

On non-stationary solutions to MSDDEs: representations and the cointegration space

In this paper we study solutions to multivariate stochastic delay differential equations (MSDDEs) which have stationary increments, and we show that this modeling framework is in many ways similar to the discrete-time cointegrated VAR model. In particular, we observe that an MSDDE can always be written in an error correction form and, under suitable conditions, we argue that a process with stationary increments is a solution to the MSDDE if and only if it admits a certain Granger type representation. As a direct implication of these results we obtain a complete characterization of the set of cointegration vectors (the cointegration space). Finally, we exploit the relation between MSDDEs and invertible multivariate CARMA equations to define cointegrated MCARMA processes, and we discuss how this definition is related to earlier literature.

math.PR↗

arXiv2018

Continuous-time models with an autoregressive structure

In this paper we suggest two continuous-time models which exhibit an autoregressive structure. We obtain existence and uniqueness results and study the structure of the solution processes. One of the models, which corresponds to general stochastic delay differential equations, will be given particular attention. We use the obtained results to link the introduced processes to both discrete-time and continuous-time ARMA processes.

math.PR↗

arXiv2018

Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

math.PR↗

arXiv2018

Stochastic differential equations with a fractionally filtered delay: a semimartingale model for long-range dependent processes

In this paper we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.

math.PR↗

arXiv2018

Multivariate stochastic delay differential equations and CAR representations of CARMA processes

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a continuous-time equivalent of the AR($\infty$) representation. Furthermore, we show how this representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

math.PR↗

arXiv2017

Equivalent martingale measures for Lévy-driven moving averages and related processes

In the present paper we obtain sufficient conditions for the existence of equivalent martingale measures for Lévy-driven moving averages and other non-Markovian jump processes. The conditions that we obtain are, under mild assumptions, also necessary. For instance, this is the case for moving averages driven by an $α$-stable Lévy process with $α\in (1,2]$. Our proofs rely on various techniques for showing the martingale property of stochastic exponentials.

math.PR↗