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Financial Markets and Portfolio Management

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15-10-2020

Testing for structural breaks in return-based style regression models

Authors: Yunmi Kim, Douglas Stone, Tae-Hwan Kim

Published in: Financial Markets and Portfolio Management | Issue 1/2021

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Abstract

It is important for investors to know not only the style of a fund manager in which they are interested, but also whether this style is constant or changing through time. The style of a fund manager can be estimated by the so-called style regression, and a great deal of research has been carried out to investigate the statistical properties of style regression methods. However, there has been no formal and statistically valid method to test for a change in manager style when the two typically imposed restrictions (sum-to-one and non-negativity) are jointly present in style analysis. In this study, we apply and extend the results of Andrews (Econometrica 61:821–856, 1993; Estimation when a parameter is on a boundary: theory and application, Yale University, 1997a; A simple counterexample to the bootstrap, Yale University, 1997b; Econometrica 67:1341–1383, 1999; Econometrica 68:399–405, 2000) to develop a valid testing procedure for the possibility wherein the location of any possible change does not need to be specified and the case of multiple shifts is accommodated. When our proposed test is applied to the Fidelity Magellan Fund, it is revealed that the fund’s style changed at least twice between 1988 and 2017.

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In this paper, we attempt to make distinguish between true parameters, estimators, and place holders. On one hand, two notations \( \beta_{0} \) and \( \beta^{*} \) are reserved to indicate the unknown & true parameters; the former \( \beta_{0} \) for the case of style regression without breaks and the latter \( \beta^{*} \) for the case of style regression with structural breaks. On the other hand, \( \tilde{\beta } \) is used to represent an estimator. Finally, \( \beta \) is employed as the place holder in any minimization problem.

The weighting matrix \( \varGamma \) is not necessarily deterministic. We can allow \( \varGamma \) to be a random matrix (denoted by \( \tilde{\varGamma }_{T} \)), which depends on both data and sample size as long as it converges in probability to a non-stochastic, positive-definite matrix. A common example for \( \tilde{\varGamma }_{T} \) is given by \( \tilde{\varGamma }_{T} = R\tilde{D}_{T} R^{\prime}, \) where \( \tilde{D}_{T} = \tilde{M}_{T}^{ - 1} \tilde{V}_{T} \tilde{M}_{T}^{ - 1} \) and \( \tilde{M}_{T} \), \( \tilde{V}_{T} \) are defined later. One can show that \( \tilde{\varGamma }_{T} \) converges to a non-stochastic, positive-definite matrix in probability. This particular weighting matrix is known to be the optimal weighting matrix in the case without non-negativity and sum-to-one restrictions. In the present analysis, this choice is not known to be optimal. Finding an optimal weighting matrix in this case to maximize power would be an interesting issue, which we leave for future research.

For example, suppose \( k = 3 \) and the 1^st and 3^rd elements of \( \beta^{*} \) are known to be zero. Then, the matrix \( Q \) is given by \( Q = \left[ {\begin{array}{*{20}c} { - 1} & 0 & 0 \\ 0 & 0 & { - 1} \\ \end{array} } \right] \).

We note that the bootstrap method may be considered to generate the sampling distribution of relevant test statistics. However, the bootstrap method does not work when some true parameters are on the boundary, which is documented in Andrews (1997b, 2000). In our framework, when some true style coefficients are zero, they are on the boundary so that the bootstrap method does not work in our framework.

The Russell 2000 index is an index measuring the performance of about 2000 smallest-cap American companies included in the Russell 3000 Index, which is made up of 3000 largest U.S. companies. These 3000 companies represent about 98% of all U.S incorporated equity securities. While the Russell 2000 Growth index tracks small-cap companies in the Russell 2000 index that exhibit growth characteristics (i.e., small-cap companies whose earnings are expected grow at an above-average rate relative to the market), the Russell 2000 Value index includes small-cap firms in the Russell 2000 index that are under-priced, measured by some criteria such as book-to-price ratio. The Russell 1000 index includes the top 1000 large-cap stocks included in the Russell 3000 Value index. Both the Russell 1000 Growth index and the Russell 1000 Value index are constructed in an analogous way as in their Russell 2000 counterparts.

All data supporting the findings of this study have been downloaded from Yahoo Finance (https://finance.yahoo.com/quote/FMAGX/history?p=FMAGX) and are also available from the corresponding author (T.-H. Kim) upon request.

All the results in this section are robust to the choice of random seed.

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Title: Testing for structural breaks in return-based style regression models
Authors: Yunmi Kim
Douglas Stone
Tae-Hwan Kim
Publication date: 15-10-2020
Publisher: Springer US
Published in: Financial Markets and Portfolio Management / Issue 1/2021
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-020-00364-2

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