This paper treats the problem of estimating the Mahalanobis distance for the purpose of detecting outliersin high-dimensional data. Three ridge-type estimators are proposed and risk functions for deciding anappropriate value of the ridge coefficient are developed. It is argued that one of the ridge estimator hasparticularly tractable properties, which is demonstrated through outlier analysis of real and simulated data.

This article treats the problem of linking the relation between excess return and risk of financial assets when the returns follow a factor structure. The authors propose three different estimators and their consistencies are established in cases when the number of assets in the cross-section (n) and the number of observations over time (T) are of comparable size. An empirical investigation is conducted on the Stockholm stock exchange market where the mean-standard deviation ratio is calculated for small- mid- and large cap segments, respectively.

In some multivariate contexts there is a close relation between the number of parameters (p) and the number of observations (n). In a situation where p grows with n it frequently happens that the statistic does not converge to its true parameter. An additional issue is if the data set also contain missing observations and , for example, as p grows with n so does the number of missing observations. This situation arises in the context of empirical applications of the Arbitrage Pricing Theory model, where the data is incomplete due to the nature of stocks leaving or entering the stock exchange. In this thesis two situations of increasing dimension are considered: firstly, the case of complete data sets where the statistic of interest is the inverse covariance matrix where three types of shrinkage estimators of the inverse covariance matrix are investigated, particularly as an ingredient of a composite estimator, specifically Zellners seemingly unrelated regression models and the Mahalanobis distance. Secondly, the case arising in empirical application of the APT model where the data set is incomplete and the interest is to model the underlying covariance structure among the variables by a few factors. Two possible solutions to the problem are considered and a case study using the Swedish OMX data is conducted for demonstration.

5.

Karlsson, Peter

Högskolan i Jönköping, Internationella Handelshögskolan, IHH, Economics, Finance and Statistics.

The Incompleteness Problem of the APT model.2011Ingår i: Computational Economics, ISSN 0927-7099, E-ISSN 1572-9974, Vol. 38, nr 2, s. 129-151Artikel i tidskrift (Refereegranskat)

Abstract [en]

The Arbitrage Pricing Theory provides a theory to quantify risk and the reward for taking it. While the theory itself is sound from most perspectives, its empirical version is connected with several shortcomings. One extremely delicate problem arises because the set of observable asset returns rarely has a history of complete observations. Traditionally, this problem has been solved by simply excluding assets without a complete set of observations from the analysis. Unfortunately, such a methodology may be shown to (i) lead for any fixed time period to selection bias in that only the largest companies will remain and (ii) lead to an asymptotically empty set containing no observations at all. This paper discusses some possible solutions to this problem and also provides a case study containing Swedish OMX data for demonstration.

This thesis consists of four individual essays and an introduction chapter. The essays are in the field of multivariate statistical analysis of High dimensional data. The first essay presents the issue of estimating the inverse covariance matrix alone and when it is used within the Mahalanobis distance in High-dimensional data. Three types of ridge-shrinkage estimators of the inverse covariance matrix are suggested and evaluated through Monte Carlo simulations. The second essay deals with incomplete observations in empirical applications of the Arbitrage Pricing Theory model and the interest is to model the underlying covariance structure among the variables by a few common factors. Two possible solutions to the problem are considered and a

case study using the Swedish OMX data is conducted for demonstration. In the third essay the issue of outlier detection in High-dimensional data is treated. A number of point estimators of the Mahalanobis distance are suggested and their properties are evaluated. In the fourth and last essay the relation between the second central moment of a distribution to its first raw moment is considered in an financial context. Three possible estimators are considered and it is shown that they are consistent even when the dimension increases proportionally to the number of observations.

This paper concerns the problem of estimating the Mahalanobis distance when the dimension of the data matrix is comparable to the sample size. Two different ridge-shrinkage estimators are considered and estimators of related risk functions are derived. The properties of these point estimators are investigated in terms of excess risk and bias relative to the traditional estimator.

8.

Kim Karlsson, Hyunjoo

et al.

The Linnaeus University, Växjö, Sweden.

Karlsson, Peter

Högskolan i Jönköping, Internationella Handelshögskolan, IHH, Statistik. The Linnaeus University, Växjö, Sweden .

Månsson, Kristofer

Högskolan i Jönköping, Internationella Handelshögskolan, IHH, Statistik.

Sjölander, Pär

Högskolan i Jönköping, Internationella Handelshögskolan, IHH, Statistik.

In this article we investigate if the Swedish consumer prices for electricity are adjusted equally fast regardless of whether the NordPool power market prices are decreased or increased. Due to relatively moderate variations in the variables, we have applied quantile regression, since it is mainly the large changes (above the median) that essentially tend to have a considerable effect on the consumer prices. Moreover, in order to adjust for stochastic- and deterministic trends, autocorrelation, structural breaks as well as to measure APT effects in the short- and in the medium-run, we apply a wavelet decomposition approach. Our results show evidence that significantly positive asymmetric price transmission (APT) effects exist in this market. More specifically, in the short-run (based on the wavelet decomposition D1 for 1–2 months cycles), we find that that there is a higher propensity to rapidly and systematically increase the consumer prices subsequently to an increase in the NordPool market price, compared with the propensity to decrease their customers prices subsequently to a corresponding drop in the NordPool market prices. However, no significant APT effects were detected in the medium- or in the long-run (i.e. the asymmetric price transmission effects are observed only in the short-run). In summary, we could isolate significant APT effects in the short-run (1–2 months decomposition cycles), and for large changes in the dependent variable (percentiles = 0.9). Therefore, only large changes in the NordPool prices lead to feedback effects in the form of asymmetric price transmission effects. Our evidence supports the notion of firms’ downward stickiness of retail prices for maximizing profit, which are not expected to be found on a fully efficient market. Although our finding shows that the price inefficiency is short-lived, these large temporal inefficiencies are still costly for the consumers. It should be noted that blunt traditional powerless methods do not detect these APT effects, while our wavelet quantile methods are powerful and make a significant contribution in the literature by providing new empirical evidence.