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

  • Piotr Kokoszka EMAIL logo , Hong Miao and Ben Zheng
Published/Copyright: March 25, 2017
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Statistics & Risk Modeling
From the journal Statistics & Risk Modeling Volume 34 Issue 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⁒πšͺ^⁒𝐅^-1+N+MM⁒(𝐅^*)-1⁒πšͺ^*⁒(𝐅^*)-1.

It was shown, in [23, Lemma 4], that 𝐅^β†’a.s.𝐅. By assumption, πšͺ^→𝑃πšͺ. With analogous relations for the second sample, we obtain

(A.2)𝚫^N,M→𝑃1θ⁒𝐅-1⁒πšͺ⁒𝐅-1+11-θ⁒(𝐅*)-1⁒πšͺ*⁒(𝐅*)-1.

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

[𝐖𝐖*]∼𝐍⁒(,[πšͺπšͺ*]).

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⁒πšͺ⁒𝐅-1+11-θ⁒(𝐅*)-1⁒πšͺ*⁒(𝐅*)-1),

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

Now Let

𝐘p:=1θ⁒𝐅-1⁒𝐖+11-θ⁒(𝐅*)-1⁒𝐖*,
𝚺p:=1θ⁒𝐅-1⁒πšͺ⁒𝐅-1+11-θ⁒(𝐅*)-1⁒πšͺ*⁒(𝐅*)-1,

where 𝚺p is nonsingular. Then (A.3) becomes 𝐘p∼N⁒(,𝚺p). This follows from the identity 𝐘pT⁒𝚺p-1⁒𝐘pβˆΌΟ‡2⁒(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⁒πšͺ⁒𝐅-1+11-θ⁒(𝐅*)-1⁒πšͺ*⁒(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
β†’π‘‘πœΌT⁒𝚫-1⁒𝜼,

where 𝜼T⁒𝚫-1⁒𝐁∼N⁒(0,𝜼T⁒𝚫-1⁒𝜼). This implies TN,Mβ†’π‘ƒβˆž. ∎

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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

Β© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A double clustering algorithm for financial time series based on extreme events
  3. Improved algorithms for computing worst Value-at-Risk
  4. Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price
  5. Company rating with support vector machines
  6. Loan pricing under estimation risk
https://doi.org/10.1515/strm-2016-0010
Keywords for this article
Asymmetric beta; cumulative intraday return curves; functional data analysis; two sample test

Articles in the same Issue

  1. Frontmatter
  2. A double clustering algorithm for financial time series based on extreme events
  3. Improved algorithms for computing worst Value-at-Risk
  4. Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price
  5. Company rating with support vector machines
  6. Loan pricing under estimation risk
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