This paper presents the R package HAC, which provides user friendly methods for dealing with hierarchical Archimedean copulae (HAC). Computationally efficient estimation procedures allow to recover the structure and the parameters of HAC from data. In addition, arbitrary HAC can be constructed to sample random vectors and to compute the values of the corresponding cumulative distribution plus density functions. Accurate graphics of the HAC structure can be produced by the plot method implemented for these objects.

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This note complements the inspiring work on dimensional analysis and market microstructure by Kyle and Obizhaeva [18]. Following closely these authors, our main result shows by a similar argument as usually applied in physics the following remarkable fact. If the market impact of a meta-order only depends on four well-defined and financially meaningful variables, then -up to a constant -there is only one possible form of this dependence. In particular, the market impact is proportional to the square-root of the size of the meta-order.This theorem can be regarded as a special case of a more general result of Kyle and Obizhaeva. These authors consider five variables which might have an influence on the size of the market impact. In this case one finds a richer variety of possible functional relations which we precisely characterize. We also discuss the analogies to classical arguments from physics, such as the period of a pendulum.

Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, defined here as the number of trades N , the traded volume V , the asset price P , the squared volatility σ 2 , the bid-ask spread S and the cost of trading C. Different reasonings result in simple proportionality relations ("scaling laws") between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e., N ∼ σ 2 . More sophisticated relations are the so called 3/2-law N 3/2 ∼ σP V /C and the intriguing scaling N ∼ (σP/S) 2 . We prove that these "scaling laws" are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility σ, which turns out to be more subtle than one might naively expect.

We analyze the properties of the Maximum Likelihood (ML) estimator when the underlying log-likelihood function is numerically maximized with the so-called zig-zag algorithm. By splitting the parameter vector into sub-vectors, the algorithm maximizes the log-likelihood function alternatingly with respect to one sub-vector while keeping the others constant. For situations when the algorithm is initialized with a consistent estimator and is iterated sufficiently often, we establish the asymptotic equivalence of the zig-zag estimator and the “infeasible” ML estimator being numerically approximated. This result gives guidance for practical implementations. We illustrate how to employ the algorithm in different estimation problems, such as in a vine copula model and a vector autoregressive moving average model. The accuracy of the estimator is illustrated through simulations. Finally, we demonstrate the usefulness of our results in an application, where the Bitcoin heating 2017 is analyzed by a dynamic conditional correlation model.