Jef Hendrickx

Street-level Informal Economic Activities: Estimating the Yield of Begging in Brussels

This article develops and applies a method to estimate the revenues of beggars in Brussels. This is relevant for three reasons. First, in the literature on the informal economy, we lack reliable empirical knowledge of informal street-level activities like begging, substantiating the expectation that beggars’ income will be low. Secondly, popular representation of beggars often depicts them as criminal and wealthy. Finally, recent legislation builds on the idea of criminal organisations behind beggars. Building on an analysis of existing attempts to measure beggars’ income, we aim for a triangulation with data from three different sources: observation, self-reports and quasi-experimental observations. This triangulation allows for more reliable and valid conclusions. Hypotheses based upon popular images and the criminalisation of begging are dismissed. The evidence does support the hypothesis based upon the literature on informal activities.

A Kronecker product variant of the FACR method for solving the generalized Poisson equation

We present a fast direct method for the solution of a linear system Mx=y, where M is a block tridiagonal Toeplitzmatrix with A on the diagonal and T on the two subdiagonals (A and T commute). Such matrices are obtained from a finite difference approximation to Poissons equation with nonconstant coefficients in one direction (among others).The new method is called KPCR(l)-method and begins with l steps of cyclic reduction after which the remaining system is solved by a Kronecker product method. For an appropriate choice of l the asymptotic operation count for an n × n grid is O(n2 log2 log2 n), which is faster than either cyclic reduction or the Kronecker product method itself. The algorithm is similar to and has the same complexity as the FACR(l)-algorithm, which is a combination of cyclic reduction and Fourier analysis (or matrix decomposition). However, the FACR(l)-algorithm only reaches this complexity if A (and T) can be diagonalized by a fast transformation, where the new method is fast for every banded A and T. Moreover, the KPCR l-method can be easily generalized to the case where A and T do not commute.

Calculating Value Added of Prostitution with Multiple Data: A New Approach for Belgium

Economic output implies that underground sectors such as prostitution are taken into account. This article presents an innovative methodology to measure turnover and added value in prostitution based on a combination of observational and Internet data. The method is applied to Belgium. Turnover is broken down in transactions and price per segment. The starting point is an observation-based measure of turnover in one locational and visible segment of the market: window prostitution. Fundamental differences between segments make linear generalizations from one segment invalid. Therefore, we estimate the relative size of transactions in other segments (such as brothels or escort) with Internet data. In combination with measures of average price per transaction, a consolidated estimate of turnover in prostitution in Belgium is measured. Estimates of nonresident production are based on data on sex workers’ country of origin. Several bootstrap replications allow for robustness checks of the delta-based standard errors.

Fast direct solution methods for symmetric banded Toeplitz systems, based on the sine transform

We present new fast direct methods for solving a large symmetric banded Toeplitz system of order n with bandwidth p. We make use of structured matrices which can be diagonalized by the discrete sine transform matrix, sometimes called τ -matrices. A first method writes the Toeplitz matrix as the sum of a τ -matrix and a low rank matrix. A second method embeds the Toeplitz matrix in a larger τ -matrix of order m. The methods are similar to Jain [IEEE Trans. Acoust. Speech Signal Process. 26 (1978) 121] and Linzer [Linear Algebra Appl. 170 (1992) 1], who worked with circulant matrices. Both algorithms consist in solving two τ -systems and two smaller systems. A τ -system of order n can be solved in O(n log n) by using a discrete sine transform if n + 1 has small prime factors. Therefore, the second algorithm is preferable, since we can choose m such that m + 1 has small prime factors. On the other hand, in the second method the smaller systems can become large when m differs too much from n ,w hile in the first method the order is always p − 1. In both methods, the small systems have low displacement rank, so we can use fast methods to solve them.