1

THE X FACTOR: GROUPING SECURITIES, DEFINING “SIMILAR”

AND FORMING ESTIMATES

Nick WadeNorthfield Information Services2011

2

WHAT’S THE MAIN POINT?

Most of the models we use feature1: a fixed factor structure that does not allow for change or evolution in the way that

markets convert information into price change and are estimated using one of many techniques that assume all the data in the

sample follow some nice, clean, simple rules that bear no resemblance to real life We contend that risk models:

Need some kind of adaptive structure that allows them to change as new or transient effects appear

Need adjustment for “regimes” in the data – this impacts asset allocation and “insurance”

Should harness current or forward-looking information as a conditioning input Should be estimated using a flexible technique that allows for evolution in the

underlying data We also offer some thoughts about emergent techniques, new

candidates for factors, and directions for future research

1 There are a lot more problems, but the focus for today is JUST the idea that things change over time

3

ROADMAP

Introduction to Factor Models Choices of Factors Pros and Cons A Hybrid Approach Better Factors Choices of Estimation Methodology Data Regimes Better Estimation Methodology Conclusions

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WHAT IS A FACTOR MODEL?

The main purpose of a factor model is to find a set of common themes that explain the variability in security prices that is shared across securities.

Having defined a set of factors, we then look to estimate the return associated with those factors and the individual security exposures/sensitivities/betas to those factors

The end result is a model of how the portfolio will behave – how much the securities will move together and how much they will behave uniquely

That can lead us to various useful risk characteristics

5

THE LINEAR MODEL

Relationship between R and F is linear ∀F There are N common factor sources of return Relationship between R and H is linear ∀H There is no correlation between F and H ∀ F,H The distribution of F is stationary, Normal, i.i.d. ∀F There are M stock-specific sources of return There is no correlation between H across stocks The distribution of H is stationary, Normal, i.i.d. ∀H (Implicitly also the volatility of R and F is stationary)

N

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ESTIMATING A RISK MODEL

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i

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The Variance of a portfolio is given by the double sum over the factors contributing systematic or common factor risk, plus a weighted sum of the stock-specific or residual risks.

7

COMMON FACTOR CHOICES

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HOW MANY FACTORS…?

The academic consensus seems to be that there is not much difference going from 5 to 10 to 15 factors. In other words, 5 do the job. Lehmann & Modest (1988) Connor & Korajczyk (1988) Roll & Ross (1980)

If somebody is suggesting a 90 factor model to you, tell them to try harder

9

ENDOGENOUS MODEL

The Endogenous or Fundamental Model seeks to estimate Fit assuming Eit by regression.

Typical factors include E/P, D/E, Industry membership, Country membership… King (1966) Rosenberg and Guy (1975) etc.

Model is pre-specified These models are hugely popular, partly

because we’ve been building them for so long we’ve become habituated

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EXOGENOUS MODEL

The Exogenous or Macro model seeks to estimate Ei from Fit.

Typical factors include Market, Sector, Oil, Interest-Rates… Ross (1976) Chen (1986)

Model is pre-specified

11

STATISTICAL MODEL

Assume N Use Factor Analysis or Principle

Components to estimate Ei

Use Regression to estimate Fit

Errors in Variables

PROS AND CONS OF EACH APPROACHApproach Pros Cons

Fundamental (micro) model

• Suitable for concentrated portfolios • Number of factors is fixed thus unchanging

• Dependent on accounting statement accuracy

• Dependent on accounting standards comparability

•Membership factors for industry/country/sector

• Errors will be in factor returns, hence in covariance matrix, and hence not diversifiable

Macro-economic model

• No dependence on accounting data

• The response of each security to changes in market/sector/industry/ whatever to be different across securities

• Errors will be in loadings (exposures), thus diversifiable

• Factor number fixed and unchanging

• Exposure to factors is stationary over time

Statistical model • All correlation is information

• Captures new, or transient effects

• Adaptive to the market

• Great for a short-term model

• Attribution of risk is difficult

• Issues with noise in data

• Errors in variables

• Number of factors is either pre-specified or sample-dependent

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OTHER APPROACHES

Combined Models: Northfield Hybrid Model Stroyny (2001)

Simultaneous Estimation Black et al (1972) Heston and Rouwenhorst (1994, 1995) Satchell and Scowcroft (2001) GMM Hansen (1982) McElroy and Burmeister (1988) using NLSUR (which is

assymptotically equivalent to ML) Bayesian Approach:

Pohlson and Tew (2000) Ericsson and Karlsson (2002)

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SIMULTANEOUS ESTIMATION

Removing the limitation of binary or membership variables (such as industry, country, sector, region etc). Marsh and Pfleiderer (1997) Scowcroft and Satchell (2001)

Start with an estimate of the exposures (e.g. 1.00 for all companies) use that estimate to solve for the factor return, then use that factor return in turn to re-solve for a revised set of exposures, thus converging iteratively on a better solution for both Eit and Fit. Black et al (1972) Heston and Rouwenhorst (1994, 1995) Scowcroft and Satchell (2001)

Given various limiting restrictions we can ensure that the model converges and that it is unique.

This is the idea behind the UBS range of models and it’s good. But it’s not adaptive.

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HYBRID MODEL (NORTHFIELD 1998)

Combine macro, micro, and statistical factors An observable factor core, and statistical

factors trawling the residual return to find new factors

Gain the advantages of each, whilst mitigating the limitations of each Intuitive, explainable, justifiable observable

factors Minimal dependence on accounting information Rapid inclusion of new or transient factors

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THOUGHTS

Notice we are already trying to allow for an adaptive set of factors – we will pick this theme up later

In a minute we’ll be allowing for time-dependent volatility and correlations as well

But are all price movements based on fundamentals? Apple?

What about news? What about liquidity impacts? What about index membership or common ownership?

Exposed to “have to” sales as index weight changes for example

Some securities are held in many funds, high “centrality”

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BETTER FACTOR CHOICES

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A FEW SUGGESTIONS FOR BETTER FACTORS

A factor can be any shared behavior HISTORICAL: Semantic clustering (text mining)

Dig into everything published on a universe of companies and look for similarities by phrase comparison etc

PREDICTIVE: News flows Look at instances of occurrence in news, sentiment

Inference from other asset classes What does a bond spread change tell us about equity vol? What about a change in option implied volatility/implied

correlation? Social network analysis

Apply emergent techniques to look at influence within groups, measures of asset centrality, flow of information, diversification?

Influence: types of network shape

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NEWS AS A FACTOR

Mitra, Mitra, diBartolomeo (2008)

Since Dan’s going into depth later I will restrain myself – just note that you can use news incidence/sentiment to condition risk forecasts

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OWNERSHIP DATA

Not a well-explored source of information Starmine: [see Dirk Renick “Research on

sentiment based smart money”] Use ownership data to reverse out factor

preferences Funds are attracted to companies that are “like” ones

they already own Funds exhibit biases toward companies with certain

fundamental characteristics, and these biases change over time

1999 2001 2003 2004 2005 2007

StarMine PriceMo EPS_CAGR3 ROE ROE ROE ROELTG ROE Profit Margin Interest Coverage F12m E/P F12m E/PG5 EPS Profit Margin Interest Coverage F12m E/P Interest Coverage Interest CoverageDebt/Assets Debt/Assets LTG Profit Margin Profit Margin Profit MarginInterest Coverage LTG F12m E/P LTG StarMine EQ StarMine PriceMo

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COUNTRY, INDUSTRY, SECTOR, REGION…

A useful (I hope) digression into the world of factor selection.

It is pretty much standard practice to take note of membership in, or exposure to, one or more countries or regions, and one or more industries or sectors

Problems: multinational firms, globalization, index domination Heston and Rouwenhorst (1994, 1995) Scowcroft and Sefton (2001) Diermeier and Solnik (2000) MacQueen and Satchell (2001)

Suggestions: Estimate a different kind of index FTSE (1999), Bacon and Woodrow (1999) Split into “global” market and “domestic” market either by some cut off on a variable like foreign sales (Diermeier

and Solnik 2000) or by some statistical process (MacQueen and Satchell 2001) Solve Model iteratively using Heston and Rouwenhorst (1994, 1995) approach Or extensions to that: Scowcroft and Sefton (2001).

The real problem is that something as bland as “industry” is one-dimensional and does not pick up enough nuance about company relationships

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SEMANTIC CLUSTERING

Let’s get rid of Industry Classifications Why? Are all banks the same? Nope.

Semantic clustering, or text mining can be used to: Help predict ratings changes [Starmine] Help update risk/return estimates

[Ravenpack etc] Measure distance between companies

based on published information [Quid.com]

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SOCIAL NETWORK ANALYSIS

A million followers, and still not winning… [thanks Charlie] Recent developments in SNA look at the different kinds of

groups and influence between/within groups Solis (2009) stocks form a “small world” network “Six degrees of separation” etc. Measures of “centrality”

Degree centrality Reach centrality Flow centrality Betweeness centrality Kritzman: Asset Centrality Google: page rank algorithm

Intuitive groups: index membership, ownership in mutual funds

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IN CONTEXT

For our purposes, how is a group connected? Think of localized version of CAPM – which

asset best represents “the market” Hubs? What degree of connectedness? Ownership commonality across mutual funds

Why do we care? Investment strategy, asset allocation Co-movement (risk), diversification, crowding Information flow

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ASSET CENTRALITY

We can take this idea from Social Network Analysis and apply it to a variety of contexts: Was Lehman too big to fail? Can we

quantify it’s centrality? Is BHP more influential than Rio, or less

with the Resources sector? Which sectors are the most/least

“democratic”? Note that this links again with James’ work

on diversification

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ABSORPTION RATIO (KRITZMAN 2011) On a related note – how tightly connected is the market, or

a particular sector? Lo (2008) Yenilmez and Saltoglu (2011)

Absorption ratio quantifies this by looking at the proportion of variance explained by common themes. As this number rises, the level of “systemic” risk rises, since

assets are more tightly connected. This is one requirement for a crash – just add panic A signal for when to apply costly insurance – e.g. zero-cost collar You could use Dispersion and get a similar result (but without

allowing for idiosyncratic vol to move indep of syst vol) You could use Implied Correlation and get a similar forward-

looking result

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ARTIFICIAL IMMUNE SYSTEMS

At a high level, our immune system consists of two pieces: Innate immunity Learned immunity

In our context The factors we believe to be useful at t=0 Plus the factors the model learns along the way

Tune the model Criteria for accepting a new factor Criteria for archiving / forgetting factors Memory length for previously useful factors

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SINGLE-PASS CLUSTERING AND RELATED METHODS

Concept: high frequency data in high volumes presents a storage problem, so need techniques that can analyze data as it arrives rather than data-mine.

Issues: it’s a goldfish. No memory.

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ISSUES WITH ESTIMATION

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SOME PROBLEMS

We are dealing with an evolving data set, not a static one Explore how this impacts our common techniques Look at more advanced / better techniques to fit evolving data

sets We are (potentially) dealing with different regimes in the

data, not one uniform set Look at models that explicitly allow for regime change (not in

a George Bush sense) We are dealing with complex behavior within groups

For example, some groups play follow the leader Some groups herd. There is no leader THOUGHTS: sefton: beta compression, social network analysis,

influence

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THE WORLD IS VERY OBVIOUSLY TIME-VARYING

Non-stationary volatility (ARCH, GARCH, etc) We spend an heroic amount of time trying to forecast non-

stationary volatility But we often just ignore it when we calculate correlation,

or perform regression analysis, or run factor analysis (or PCA)

Non-stationary mean (Trend) We often build models to capture the alpha in momentum,

reversals, and other manifestations of a non-stationary mean

But we often ignore those when we calculate correlation, or perform regression analysis, or run factor analysis

Read the fine print…

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SIMPLE ADJUSTMENTS (1)

Non-stationary factor return series will lead to the model underestimating portfolio risk

Adjust by changing variance calculation to include trend component of return

2

1nn

xV i

2

1nn

xxV i

Adjust Model for the influence of non-stationary factor returns

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SIMPLE ADJUSTMENTS (2)

What about security volatility? We observe:

Serial correlation (not i.i.d.) Bid-ask bounce Non-Normal distributions

Parkinson volatility

Adjust Model for the influence of non-stationary security returns

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CONTEMPORANEOUS OR FORWARD-LOOKING SIGNALS

You could make the argument that these are “factors” rather than an estimation approach, but we use them to condition existing factors.

Take a model that has been estimated on purely historical data Find true forward-looking signals

E.g. option-implied volatility Find other contemporaneous signals

E.g. dispersion measures, range measures, volume Adjust the parameters of the “historical” model so that the forecasts

of the model match the signals from “now” and the “future” Update it daily so that it stays “current”

The Advantage: we have kept the same factor structure but removed the sole dependency on the past.

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GENERALIZE THE IDEA: NORTHFIELD “ADAPTIVE NEAR-HORIZON RISK MODELS”(ANISH SHAH, 2008)

The richest source of information about the future is not the past – increasingly it is consensus estimates about the future from e.g. option markets, prediction markets…

Take any risk model. e.g. one of our models estimated monthly

Add “flexibility points” and fit to information about current conditions

Adjust for statistical differences between short and long term returns

Many benefits of this approach Avoid statistical complexities of high frequency data Keep familiar factor structure Common factor structure for long and short horizons permits

interpolating any horizon in between Works with any factor model

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Northfield Asia ex. Japan Risk ModelsTracking Error LH Tracking Error SH Linear (Tracking Error SH)

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DATA WITH REGIMES

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KEY QUESTIONS

Before we get carried away… What evidence of detectable regimes? How many regimes? What kind of model can fit multiple

regimes? Can any of these fit multiple regimes on

evolving data? i.e. learn new regimes as they appear

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EVIDENCE FOR REGIMES: TIME-VARYING CORRELATIONS

Increasing attention is being paid to the issue of correlations varying over time: (stocks) De Santis, G. and B. Gerard (1997), International asset pricing and portfolio

diversification with time-varying risk, Journal of Finance, 52, 1881-1912. (stocks) Longin, F. and B. Solnik (2001), Extreme correlation of international equity

markets, Journal of Finance, LVI(2), 646-676. (bonds) Hunter, D.M. and D.P. Simon (2005), A conditional assessment of the

relationships between the major world bond markets, European Financial Management, 11(4), 463-482.

(bonds) Solnik, B., C. Boucrelle and Y.L. Fur (1996), International market correlation and volatility, Financial Analysts Journal, 52(5), 17-34.

Markov switching model: Chesnay, F and Jondeau, E “Does Correlation Between Stock Returns really increase during turbulent periods?” Bank of France research paper.

To date little explored – however, Implied Correlation also seems useful, and more powerful than historical correlation in forecasting (we saw the same result with volatility):

Campa, J.M. and P.H.K. Chang (1998), The forecasting ability of correlations implied in foreign exchange options, Journal of International Money and Finance, 17, 855-880.

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CORRELATION STABILITY

One of the first… Kaplanis (1988): STABLE Tang (1995), Ratner (1992), Sheedy (1997): STABLE –

although crash of 1987 regarded as an “anomaly” Bertero and Mayer (1989), King and Wadwhani (1990)

and Lee and Kim (1993): correlation has increased, but STABLE

Not quite so stable? Erb et al (1994) – increases in bear markets

Longin and Solnick (1995) – increases in periods of high volatility

Longin and Solnick (2001) – increases in bear markets

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DETECTING REGIMES

Use Viterbi’s algorithm (Viterbi 1967) to detect states.

Use Jennrich tests (Jennrich 1970) to decide whether correlation differences between states are significant

Mahalanobis Distance (Mahalanobis 1939, Kritzman 2009)

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MAHALANOBIS DISTANCE

MD is one example of a “Bregman divergence” , a group of distance measures.

Clustering: classifies the test point as belonging to that class for which the Mahalanobis distance is minimal. This is equivalent to selecting the class with the maximum likelihood.

Regression: Mahalanobis distance and leverage are often used to detect outliers, especially in the development of linear regression models. A point that has a greater Mahalanobis distance from the rest of the sample population of points is said to have higher leverage since it has a greater influence on the slope or coefficients of the regression equation. Specifically, Mahalanobis distance is also used to determine multivariate outliers. A point can be an multivariate outlier even if it is not a univariate outlier on any variable.

Factor Analysis: recent research couples Mahalanobis distance with Factor Analysis and use MD to determine whether a new observation is an outlier or a member of the existing factor set. [Zhang 2003]

MD depends on covariance (S^-1 is the inverse of the covariance matrix), so is exposed to the same stationarity issues that affect correlation, however as described above it can help us reduce correlation’s outlier dependence.

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MAHALANOBIS IN ACTION

Borrowed from Kritzman: Skulls, financial turbulence, and theimplications for risk management. July 2009

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TURBULENCE IN THE MARKET

Kritzman (2009): Correlation of US and foreign stocks when

both markets’ returns are one standard deviation above their mean: -17%

Correlation of US and foreign stocks when both markets’ returns are one standard deviation below their mean: +76%

“Conditional correlations are essential for constructing properly diversified portfolios”

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CORRELATION REGIMES

If it’s not stable, how about Markov switching models? Ramchand and Susmel (1998), Chesnay and

Jondreau (2001) – correlation, conditioned on market regime, increases in periods with high volatility

Ang and Bekaert (1999) – evidence for two regimes; a high vol/high corr, and a low vol/low corr.

Don’t forget this needs to adapt as our world changes: Hidden Markov Experts on evolving data

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FLEXIBLE ESTIMATION TECHNIQUES

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TECHNIQUES FOR EVOLVING DATA

Most of our favorite tools are designed to fit static data sets where behaviors are mostly unchanged Neural network, Kalman filter, OLS/GLS regression, PCA,

ICA, factor analysis, variance, correlation… just about all of them

Recent developments in cluster analysis are encouraging Artificial Immune Systems Single-pass clustering Regime-switching models e.g. HME etc [recent] EPCIA [very recent] HME on evolving data

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REGRESSION WITH NONSTATIONARY DATA Techniques have been developed

specifically to allow time-varying sensitivities FLS (flexible least-squares) FLS is primarily a descriptive tool that

allows us to gauge the potential for time-evolution of exposures

T

ttt

T

tttttt xy

11

1

11

2

Minimze both sum of squared errors and sum of squared dynamic errors (coefficient estimates)

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FLS EXAMPLE

An example from Clayton and MacKinnon (2001) The coefficient apparently exhibits structural shift in 1992

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CLUSTER ANALYSIS WITH NONSTATIONARY DATA

Guedalia, London, Werman; “An on-line agglomerative clustering method for nonstationary data” Neural Computation, February 15, 1999, Vol. 11, No. 2, Pages 521-54

C. Aggarwal, J. Han, J. Wang, and P. S. Yu, On Demand Classification of Data Streams, Proc. 2004 Int. Conf. on Knowledge Discovery and Data Mining (KDD'04), Seattle, WA, Aug. 2004.

G. Widmer and M. Kubat, “Learning in the Presence of Concept Drift and Hidden Contexts”, Machine Learning, Vol. 23, No. 1, pp. 69-101, 1996.

Again, there are techniques available to conquer the problem

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FACTOR ANALYSIS WITH NONSTATIONARY DATA

Dahlhaus, R. (1997). Fitting Time Series Models to Nonstationary Processes. Annals of Statistics, Vol. 25, 1-37.

Del Negro and Otrok (2008): Dynamic Factor Models with Time-Varying Parameters: Measuring Changes in International Business Cycles (Federal Reserve Bank New York)

Eichler, M., Motta, G., and von Sachs, R. (2008). Fitting dynamic factor models to non-stationary time series. METEOR research memoranda RM/09/002, Maastricht University.

Stock and Watson (2007): Forecasting in dynamic factor models subject to structural instability (Harvard).

There are techniques available, and they are being applied to financial series.

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EVOLVING PRINCIPAL COMPONENT INNOVATION ANALYSIS (EPCIA)

You want PCA but your factor structure is changing EFA (evolving factor analysis): keep adding

factors EFWFA (evolving fixed-window factor

analysis): keep adding new factors, but forget old ones to make room!

EPCIA allows for new factors to emerge

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LAYER-EMBEDDED NETWORKS

Within our networks, information and interactions may flow at multiple levels E.g. Pasquel and de Weck (2011 working paper) E.g. Luttrell (2010 working paper)

Enter multi-layer modeling At a high level, a layer-embedded network captures

the effects of multiple interacting processes at different frequencies or across different groups.

E.g. combining short and long-term alpha signals Combining global and local factors Combining pervasive and short-term factors

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SUMMARY THOUGHTS

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CONCLUSIONS

Our world changes This requires an adaptive risk model factor structure This requires the ability to accommodate regimes in our risk

models, our portfolio construction, hedging, and asset allocation by harnessing contemporaneous and forward-looking signals

The market is not driven solely by fundamentals We need to leverage news/perception We need to explore nuanced relationships beyond bland

membership Techniques exist to address all of these issues, and are

being applied today.

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TAKE HOME

Northfield: risk models that utilize implied volatility since 1997 adaptive hybrid risk models since 1998 risk models utilizing cross-sectional dispersion since

2003 using implied volatility and dispersion in our entire

range of short-horizon adaptive models since 2009 If you’re doing some kind of time-series analysis

on financial data you need to keep time-dependence, regimes, and evolving data in mind

There are techniques to conquer all of these challenges, but they’re not the easy ones that come as part of Excel!

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Bulsing M., Scowcroft A., and Sefton J., “Understanding Forecasting: A Unified framework for combining both analyst and strategy forecasts” UBS Working Paper, 2003.

Chen N.F. Roll R. Ross S.A. “Economic Forces and the Stock Market” Journal of Business 59, 1986. Connor G and Korajczyck R.A. “Risk and Return in an equilibrium APT: application of a new test

methodology” Journal of Financial Economics 21, 1988. diBartolomeo D. “Why Factor Risk Models Often Fail Active Quantitative Managers. The

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REFERENCES II Lintzenberger R. and Ramaswamy K. “The effects of dividends on common

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MacQueen J. “Alpha: the most abused term in Finance” Northfield Conference, Montebello, 2005

MacQueen J. and Satchell S. “An Enquiry into Globalisation and Size in World Equity Markets”, Quantec, Thomson Financial, 2001.

Mandelbrot B. “The variation of certain speculative prices” Journal of Business, 36. 1963.

Markowitz, H.M. “Portfolio Selection” 1st edition, John Wiley, NY, 1959. Marsh T. and Pfleiderer P. “The Role of Country and Industry Effects in

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McElroy M.B., Burmeister E. “Arbitrage Pricing Theory as a restricted non-linear multivariate regression model” Journal of Business and Economic Statistics 6, 1988.

Northfield Short Term Equity Risk Model Northfield Single-Market Risk Model (Hybrid Risk Model) Pfleiderer, Paul “Alternative Equity Risk Models: The Impact on Portfolio

Decisions” The 15th Annual Investment Seminar UBS/Quantal, Cambridge UK 2002.

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REFERENCES III

Pohlson N.G. and Tew B.V. “Bayesian Portfolio Selection: An empirical analysis of the S&P 500 index 1970-1996” Journal of Business and Economic Statistics 18, 2000.

Pope Y and Yadav P.K. “Discovering Errors in Tracking Error”. Journal of Portfolio Management, Winter 1994.

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Ross S.A. “The Arbitrage Theory of Capital Asset Pricing” Journal of Economic Theory, 13, 1976.

Satchell and Scowcroft “A demystification of the Black-Litterman model: managing quantitative and traditional portfolio construction” Journal of Asset Management 1, 2000.

Scowcroft A. and Sefton J. “Risk Attribution in a global country-sector model” in Knight and Satchell 2005 (“Linear Factor Models in Finance”)

Scowcroft A. and Sefton J. “Do tracking errors reliably estimate portfolio risk?”. Journal of Asset Management Vol 2, 2001.

Shanken J. “The Arbitrage Pricing Theory: Is it testable?” Journal of Finance, 37, 1982. Sharpe W. “Capital Asset Prices: a theory of market equilibrium under conditions of

risk” Journal of Finance, 19, 1964. Stroyny A.L. “Estimating a combined linear model” in Knight and Satchell 2005

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60

REFERENCES IV

Ang, A. and Bekaert, G. (1999) ‘International Asset Allocation with time-varying Correlations’, working paper, Graduate School of Business, Stanford University and NBER.

Banerjee, Arindam; Merugu, Srujana; Dhillon, Inderjit S.; Ghosh, Joydeep (2005). "Clustering with Bregman divergences". Journal of Machine Learning Research 6: 1705–1749. http://jmlr.csail.mit.edu/papers/v6/banerjee05b.html.

Bertero, E. and Mayer, C. (1989) ‘Structure and Performance:Global Interdependence of Stock Markets around the Crash of October 1987’, London, Centre for Economic Policy Research.

Chesnay, F. and Jondeau, E. (2001) ‘Does Correlation between Stock Returns really increase during turbulent Periods?’, Economic Notes by Banca Monte dei Paschi di Siena SpA, Vol. 30,No. 1, pp.53–80.

Jim Clayton and Greg MacKinnon (2001), "The Time-Varying Nature of the Link Between REIT, Real Estate and Financial Asset Returns" (pdf,6.3M), Journal of Real Estate Portfolio Management, January-March Issue

Erb, C.B., Harvey, C.R. and Viskanta, T.E. (1994) ‘Forecasting international Equity Correlations’, Financial Analysts Journal,pp.32–45.

Jakulin A & Bratko I (2003a). Analyzing Attribute Dependencies, in N Lavra\quad{c}, D Gamberger, L Todorovski & H Blockeel, eds, Proceedings of the 7th European Conference on Principles and Practice of Knowledge Discovery in Databases, Springer, Cavtat-Dubrovnik, Croatia, pp. 229-240

Jennrich R. (1970) ‘An Asymptotic χ2 Test for the Equality of Two Correlation Matrices’, Journal of the American Statistical Association,Vol. 65, No. 330.

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REFERENCES V

R. Kalaba, L. Tesfatsion. Time-varying linear regression via flexible least squares. International Journal on Computers and Mathematics with Applications, 1989, Vol. 17, pp. 1215-1245.

Kaplanis, E. (1988) ‘Stability and Forecasting of the Comovement Measures of International Stock Market Returns’, Journal of International Money and Finance, Vol. 7, pp.63–75.

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