Commentary: Friendships and Emotions

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Biases in Macroeconomic Forecasts: Irrationality or Asymmetric Loss?

Survey data on expectations frequently find evidence that forecasts are biased, rejecting the joint hypothesis of rational expectations and symmetric loss. While the literature has attempted to explain this bias through forecasters' strategic behavior, we propose a simpler explanation based on asymmetric loss. We establish that existing rationality tests are not robust to even small deviations from symmetry and hence have little ability to tell whether the forecaster is irrational or the loss function is asymmetric. We propose new and more general methods for testing forecast rationality jointly with flexible families of loss functions that embed quadratic loss as a special case. An empirical application to survey data on forecasts of nominal output growth shows strong evidence against rationality and symmetric loss. There is considerably weaker evidence against rationality once asymmetric loss is permittedrationality testing, forecasting, asymmetric loss

Friends, Neighbours and Distant Partners: Extending or Decentring Family Relationships?

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Using Computer Algebra Packages to Complement the Spreadsheet Construction of Binomial Option Trees: The Example of Mathcad

In this paper we show how the mathematical programming package Mathcad can be used to complement the construction of a binomial option tree in Excel. A binomial option tree is first constructed in Excel using standard spreadsheet 'cut and paste' operations. The same binomial tree is then constructed in Mathcad. We conclude that spreadsheet construction of the tree provides students with a sound concept of the underlying mechanics of the option tree. Additionally, the Mathcad construction reinforces the mathematical notation found in many option pricing texts (e.g. summation signs and indices) and allows for the construction of a more flexible lattice that may be easily altered (e.g. the number of steps). In the process students are provided with an understanding of how to construct option trees in the increasingly important world of computer algebra packages.

The Importance of Revenue Sharing for the Local Economic Impacts of a Renewable Energy Project: A Social Accounting Matrix Approach

As demand for electricity from renewable energy sources grows, there is increasing interest, and public and financial support, for local communities to become involved in the development of renewable energy projects. In the UK, “Community Benefit” payments are the most common financial link between renewable energy projects and local communities. These are “goodwill” payments from the project developer for the community to spend as it wishes. However, if an ownership stake in the renewable energy project were possible, receipts to the local community would potentially be considerably higher. The local economic impacts of these receipts are difficult to quantify using traditional Input-Output techniques, but can be more appropriately handled within a Social Accounting Matrix (SAM) framework where income flows between agents can be traced in detail. We use a SAM for the Shetland Islands to evaluate the potential local economic and employment impact of a large onshore wind energy project proposed for the Islands. Sensitivity analysis is used to show how the local impact varies with: the level of Community Benefit payments; the portion of intermediate inputs being sourced from within the local economy; and the level of any local community ownership of the project. By a substantial margin, local ownership confers the greatest economic impacts for the local community.renewable energy; rural economic impacts; revenue sharing; community ownership

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space \sM_{\dd}^n of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices \bW with \bW^T \bQ_{D,n} \bW = \bQ_{W,n} where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and \bQ_{D,n} and \bQ_{W,n} are their corresponding symmetric matrices. There are natural actions on the parameter space \sM_{\dd}^n. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions one can find rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings beginning with math.MG/0010298. Revised and extended. Added: Apollonian groups and Apollonian Cluster Ensembles (Section 4),and Presentation for n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system \sM_\DD consisting of those $4 \times 4$ real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where \bQ_D is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and \bQ_W of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space \sM_\DD. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 200