Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price

Piotr Kokoszka; Hong Miao; Ben Zheng

doi:10.1515/strm-2016-0010

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Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price

Piotr Kokoszka , Hong Miao und Ben Zheng

Veröffentlicht/Copyright: 25. März 2017

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Aus der Zeitschrift Statistics & Risk Modeling Band 34 Heft 1-2

Abstract

We introduce a functional factor model to investigate the dependence of cumulative return curves of individual assets on the market and other factors. We propose a new statistical test to determine whether the dependence in two sample periods are equal. The statistical properties of the test are established by asymptotic theory and simulations. We apply this test to study the impact of the recent financial crisis and trends in oil price on individual stock and sector ETFs. Our analysis reveals the significance of the daily oil futures curves and their different impact on individual stocks and sector ETFs. It also shows that the functional approach has an information content different from that obtained from scalar factor models for point-to-point returns.

Keywords: Asymmetric beta; cumulative intraday return curves; functional data analysis; two sample test

MSC 2010: 62G10

A Proof of Theorem 4.6

Proof.

Since we assume that the two samples are independent, the limits in (4.8) and (4.9) are independent. Combining (4.8), (4.9) and (4.6), we obtain

(A.1)N+M⁢(𝜷^-𝜷^*)=N+MN⁢N⁢(𝜷^-𝜷)-N+MM⁢M⁢(𝜷^*-𝜷)→𝑑1θ⁢𝐅-1⁢𝐖+11-θ⁢(𝐅*)-1⁢𝐖*.

Define

𝚫^N,M=N+MN⁢𝐅^-1⁢𝚪^⁢𝐅^-1+N+MM⁢(𝐅^*)-1⁢𝚪^*⁢(𝐅^*)-1.

It was shown, in [23, Lemma 4], that 𝐅^→a.s.𝐅. By assumption, 𝚪^→𝑃𝚪. With analogous relations for the second sample, we obtain

(A.2)𝚫^N,M→𝑃1θ⁢𝐅-1⁢𝚪⁢𝐅-1+11-θ⁢(𝐅*)-1⁢𝚪*⁢(𝐅*)-1.

Observe that 𝐖 and 𝐖* are two p-dimensional random vectors satisfying

[𝐖𝐖*]∼𝐍⁢(,[𝚪𝚪*]).

Since the matrices 𝐅 and 𝐅*, defined by (4.7), are nonsingular, we can conclude that

(A.3)1θ⁢𝐅-1⁢𝐖+11-θ⁢(𝐅*)-1⁢𝐖*∼N⁢(,1θ⁢𝐅-1⁢𝚪⁢𝐅-1+11-θ⁢(𝐅*)-1⁢𝚪*⁢(𝐅*)-1),

where the variance matrix of 1θ⁢𝐅-1⁢𝐖+11-θ⁢(𝐅*)-1⁢𝐖* has rank p, and also 𝚪 and 𝚪* are assumed to be full rank.

Now Let

𝐘p:=1θ⁢𝐅-1⁢𝐖+11-θ⁢(𝐅*)-1⁢𝐖*,

𝚺p:=1θ⁢𝐅-1⁢𝚪⁢𝐅-1+11-θ⁢(𝐅*)-1⁢𝚪*⁢(𝐅*)-1,

where 𝚺p is nonsingular. Then (A.3) becomes 𝐘p∼N⁢(,𝚺p). This follows from the identity 𝐘pT⁢𝚺p-1⁢𝐘p∼χ2⁢(p). By Slutsky’s theorem, the claim of the theorem follows by combining (A.1) and (A.2). ∎

B Proof of Theorem 4.7

Proof.

Let

𝐁N,M:=N+MN⁢N⁢(𝜷^-𝜷)-N+MM⁢M⁢(𝜷^*-𝜷*),

𝐁:=1θ⁢𝐅-1⁢𝐖+11-θ⁢(𝐅*)-1⁢𝐖*,

𝚫:=1θ⁢𝐅-1⁢𝚪⁢𝐅-1+11-θ⁢(𝐅*)-1⁢𝚪*⁢(𝐅*)-1.

Then, by (A.1) and (A.3), 𝐁N,M→𝑑𝐁, where 𝐁∼N⁢(,𝚫). Since 𝜷*=𝜷+𝜼, we have

N+M⁢(𝜷^-𝜷^*)=N+MN⁢N⁢(𝜷^-𝜷)-N+MM⁢M⁢(𝜷^*-𝜷*+𝜼)

=N+MN⁢N⁢(𝜷^-𝜷)-N+MM⁢M⁢(𝜷^*-𝜷*)-N+M⁢𝜼

=𝐁N,M-N+M⁢𝜼.

Let T~N,M:=𝐁N,MT⁢𝚫^N,M-1⁢𝐁N,M. Then, by Theorem 4.6, T~N,M→𝑑χ2⁢(p). Now we have

TN,M=(N+M)⁢(𝜷^-𝜷^*)T⁢(𝚫^M,N)-1⁢(𝜷^-𝜷^*)

=(𝐁N,M-N+M⁢𝜼)T⁢(𝚫^M,N)-1⁢(𝐁N,M-N+M⁢𝜼)

=𝐁N,MT⁢(𝚫^M,N)-1⁢𝐁N,M-2⁢N+M⁢𝜼T⁢(𝚫^M,N)-1⁢𝐁N,M+(N+M)⁢𝜼T⁢(𝚫^M,N)-1⁢𝜼

=T~N,M-2⁢N+M⁢𝜼T⁢(𝚫^M,N)-1⁢𝐁N,M+(N+M)⁢𝜼T⁢(𝚫^M,N)-1⁢𝜼.

It follows that

(N+M)-1⁢TN,M=(N+M)-1⁢T~N,M-2⁢(N+M)-1⁢𝜼T⁢(𝚫^M,N)-1⁢𝐁N,M+𝜼T⁢(𝚫^M,N)-1⁢𝜼

→𝑑(N+M)-1⁢χ2⁢(p)-2⁢(N+M)-1⁢𝜼T⁢𝚫-1⁢𝐁+𝜼T⁢𝚫-1⁢𝜼

→𝑑𝜼T⁢𝚫-1⁢𝜼,

where 𝜼T⁢𝚫-1⁢𝐁∼N⁢(0,𝜼T⁢𝚫-1⁢𝜼). This implies TN,M→𝑃∞. ∎

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Received: 2016-6-7

Revised: 2016-11-18

Accepted: 2017-2-17

Published Online: 2017-3-25

Published in Print: 2017-6-1

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Artikel in diesem Heft

https://doi.org/10.1515/strm-2016-0010

Schlagwörter für diesen Artikel

Asymmetric beta; cumulative intraday return curves; functional data analysis; two sample test