高次元データのモデリング理論と金融市場の実証分析
慶應義塾大学安道知寛
日本保険・年金リスク学会 研究会（２０１５年６月２４日）

Agenda• Issues to be discussed• Modeling high-dimensional panel data withapplications• SummaryPart 1: Issues

Issue 1• Statistical independence?

Issue 2• Simplicity

Issue 3• Limited access to information

Issue 4• Underlying structurePart 2:Modeling high-dimensional panel data

Details on theoretical results can befound in our papers
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Ando, T. and Bai, J. (2014a). Panel data models with grouped factorstructures under unknown group membership. Journal of Applied
Econometrics, forthcoming.
Ando, T. and Bai, J. (2014b). Selecting the regularization parameters inhigh-dimensional panel data models: consistency and efficiency.
Econometric Reviews, forthcoming.
Ando, T. and Bai, J. (2014c). Asset pricing with general multifactor
structure. Journal of Financial Econometrics, forthcoming.
Ando, T. and Bai, J. (2014d). A simple new test for slope homogeneity inpanel data models with interactive effects. Working Paper.
Ando, T. and Bai, J. (2015). The subprime financial crisis and spillover
effects on global financial markets. Working Paper.
Please see the references thereinSome related econometric literatures• Dynamic exact factor model (Geweke, 1977; Sargent and Sims,1977)• Static approximate factor model (Chamberlain and Rothschild,1983)• Generalized dynamic factor model (Forni et al., 2000; Forni andLippi, 2001; Amengual and Watson, 2007; Hallin and Liska,2007),• Bayesian factor model (Aguilar and West, 2000; Lopes andWest, 2004; Lopes et al., 2008; Ando, 2009; Bhattacharya andDunson, 2011; Tsay and Ando, 2012).
• Approximate factor model (Chamberlain and Rothschild, 1983;Connor and Korajczyk, 1986, 1988; Jones, 2001; Bai and Ng,2002; Bai, 2003, Korajczyk and Sadka, 2008);

Some related econometric literatures• Hierarchical factor model (Moench, Ng, and Potter 2012;
Diebold et al. 2008; Kose et al. 2008; Wang 2014 and Moenchand Ng 2011).
• Panel models with multifactor error structures (Pesaran, 2006;Bai 2009)• Shrinkage estimation (Tibshirani 1996, Zou 2006, Fan and Li
2001, Fan and Peng 2004, Caner 2009; Belloni, Chernozhukovand Hansen 2010; Caner and Zhang 2012; Bai and Liao 2013; Fanand Liao 2013; Liao 2013; Lu and Su 2013).
• Panels with unknown group structures (Bonhomme andManresa 2014).

Asset pricing model
Excess returns

Error
Risk factor 1• Macroeconomic factors• Fundamental factors• Unobservable factors

Risk factor pWorld 20 major stock market

Source: http://www.mapsofworld.com/stock-exchange/

Ando, T. and Bai, J. (2014c).
Asset pricing with general multifactor structure.
Journal of Financial Econometrics, forthcoming.

Model description (1/4)
Excess returns

Common factors
Predictors

Group specific factors

Model description (2/4)• S: the number of groups (known)• Group membership (known)• r: the number of common factors• rj: the number of group-specific factors• N: total number of assets• Nj: the number of assets within the group jModel description (3/4)• Common/group-specific factors– assume orthogonality• Observable risk factors (x)– allow correlations between the common factor and groupspecific factors– but not perfectly correlatedModel description (4/4)• Security-specific returns (ε)– zero mean
– cross-sectional/serial dependence and heteroscedasticity.
– Independent of the observable risk factors, the common
factor and group-specific factors

Shrinkage estimation

subject to the constrains on the factors• SCAD (Fan and Li (2001, JASA)) , Asymptotic principal
components (Connor and Korajczyk (1986, JFE)) are
jointly employedCp criterion• We need to select• the regularization parameter• the number of common factors• the number of group-specific factors.

Cp = Fitness +Penalty on model complexity

China A-shares, B-shares• A-shares– Traded in mainland currencies.
– Generally only available for purchase by mainland
citizens;– Used to be the market for mainlanders.

• B shares– Traded in foreign currencies.
– Was only for foreign investors until February 2001

Empirical questions• What type of observable risk factors explain theindividual excess returns?
• Numbers of common and group-specific factors.

• Meanings of extracted factors.

AB

Data
• Monthly excess returns• March 2002 to December 2012 period• Systematically exclude stocks with missing returns
data.
• 1,039 A-share firms and 102 B-share firms.

Observable risk factors• China macroeconomic variables– (M2, CPI, Macroeconomic climate index)• Exchange rates– (Dollar, EURO, UK, HK, Yen)
• Commodity price index– (Metals, Aluminum, Copper, Oil, Gold, Gas, Nickel,Silver)• Major stock market index– (MSCI world, MSCI EU, MSCI China, SP500, HANG
SENG, FTSE100, TOPIX)

Ando and Bai (2014c).
An impact on observable factors (1/2)

Ando and Bai (2014c).
An impact on observable factors (2/2)

The number of factors• Common factors:–2• A-share group-specific factors:–1• B-share group-specific factors:–3

AB

Interpretation of the factors• Consumer confidence index in China (CCI)• CBOE volatility index (VIX)• Market excess returns of A-shares (ER–A)• Market excess returns of B-shares (ER–B)• Book-to-market ratio (HML)• Market capitalization (SMB)Ando and Bai (2014c)
Common factors

ABAndo and Bai (2014c)
Group-specific factors

Robustness check• Some companies issue “twin” A and B shares.
– Contained 50 “twin” A and B shares.

• The effect of foreign denominated currencies.
– Express the B-share returns in Chinese yuan

Ando, T. and Bai, J. (2014a).
Panel data models with grouped factor structures
under unknown group membership.
Journal of Applied Econometrics, forthcoming.

Issue• Do we know real group membership?

Model description: Ando and Bai (2014a).
Group specific factors

Group specific coefficients• Further unknowns– Number of groups– Group membership indicator• Estimation by the shrinkage methods• Cp type model selection

Theoretical results• Consistency of membership,,• Asymptotic normality of• Variable selection consistency of• Cp criterion for model selection

The number of groups• China stock market• We found 6 groups in our sample

Ando and Bai (2014a) Scatter matrices of the estimatedgroup membership against nominal classification
schemesAndo, T. and Bai, J. (2015).
The subprime financial crisis and spillover effects on
global financial markets, Working Paper

Model and its construction: Ando and Bai (2015)• Further unknowns– Number of groups– Group membership indicator• Estimation by the shrinkage methods• Cp type model selection

Data description (1/2)• Financial industry– Banks– Life Insurance– Nonlife Insurance– Financial Services– Real Estate Investment and Services• 109 international stock exchanges– US (NYSE, NASDAQ, Non-NASDAQ OTC)
– Asia (TSE, Shanghai, Shenzen, Hong Kong,...)– Europe (Frankfurt, London, Luxembourg,...)– Emerging (BSE, Thailand,...)

Data description (2/2)• Periods– May 1 2006 to December 31, 2006
– May 1 2007 to December 31, 2007
– February 1 2008 to August 31, 2008
– September 1 2008 to March 31, 2009
– May 1 2009 to December 31, 2009

• Systematically exclude stocks with missing• More than 6,000 stocks in each of the periods.

Observable risk factors• Averages of lagged returns of each market• More than 50 stocks among approx. 6000stocks.

• 31 predictors, including NYSE, TSE, Shanghai,etc.

Period 1

Period 4

Number of groups• Period 1: S = 11 groups.
• Period 2: S = 13 groups.
• Period 3: S = 15 groups.
• Period 4: S = 17 groups.
• Period 5: S = 14 groups.
• The number of group-specific factors varies
over the groups

Period 2: Market vs groupingPeriod 4: Market vs groupingFactor interpretation• Thomson Routers Global Index• Global, U.S., EU.

• The equity market performance of Financial,Banks, Insurance, Real Estate, and REIT.

Exposure to U.S. factors

Exposure to global factors

Exposure to EU factors

Robustness check• Different sub-period specification• Analysis of daily returns instead of two day
rolling average.

Summary• Assumptions• Reality, and business practice• Recent advancement on econometric methods• Applications