Do trees read the Wall Street Journal?

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Universiteit van Amsterdam

Master Thesis

Do Trees read the Wall Street Journal?

Financial Econometrics

Author:

Maarten Bakkerode (10851267)

Supervisor:

dr. N.P.A. van Giersbergen

Second reader:

dr. S.A. Broda

April 7, 2016

Abstract

In this study, a family of two dimensional copula models is used to characterize the cross-sectional dependence between the daily equity returns of public companies investing in timberlands and three major stock market indexes. Innovative GARCH filters extract the marginal distributions that compose the main ingredient for copula modeling. The copulas reveal a positive relation between market volatility and the probability of joint extreme events between stock and market index. Evidence of these events is found only during the 2008 financial crisis. The analysis presents evidence that timberland investments provide a certain stability against financial turmoil.

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1 Introduction

1.1 Background

Approximately a third of the total land area of the United States is covered by forested areas. This abundance of timber resources has given rise to many investment opportunities. Timberland investments are used primarily by large institutional investors. The two main underlying assets are tree farms and managed natural forests. Timberland assets possess several unique characteristics that are attractive to investors, including the ability to hedge against inflation (Wan et al., 2013) and the opportunity for forest owners to delay the harvest of the trees, when the prices of timber are below a reservation price (Conroy and Miles, 1989). This produces higher returns. Additional returns are generated by the timber price appreciation and land price appreciation. However, the main driver for the total returns on timberland assets is the biological growth (Caulfield, 1998) of trees, which is independent of bare land prices, timber prices and other business cycles. Mei et al. (2013) simulate the returns under dynamic timber price assumptions and also identifiy biological growth as the dominant contributor. Trees continue to grow, independent of financial turmoil. One may therefore expect a low correlation between returns on timberland investments and the major stock markets, creating a significant diversification potential for investment managers.

Investing in US timberland assets has become increasingly popular over the last decades and is done through either private equity timberland assets or through public equity mar-kets. There are both advantages and disadvantages associated with investing in securitized timberlands. On the one hand are tax advantages, greater liquidity than private ownership and the ease of valuation. On the other hand there are the risk of natural disasters and the systematic, market-related risk that comes with being a publicly traded stock, which cannot be diversified away (Ruppert, 2011). Unsystematic risk is specific to individual firms and can be diversified away in a portfolio. Publicly traded companies are always under pressure from shareholders, demanding growth. Stagnation of earnings or dividend growth is reflected in a declining share price.

This study examines the comovements between public companies specialized in managing timberlands and major stock market indexes and reﬂects on their dependence structure over

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haviour. The characteristics of these series (i.e. fat tails, asymmetry) cannot be reproduced by linear correlation approaches based on properties of the multivariate normal distribution (Giacomini et al., 2012). A copula-based measure speciﬁes the dependence structure and accounts for non-linearity without the constraint of normality. It is investigated how the dependence structures behave in times of increased market volatility.

1.2 Overview of timberland investment companies

There are three major vehicles of timberland securitization: letter stocks (LS), master limited partnerships (MLP) and real estate investment trusts (REIT). LSs are created by detaching a business unit from the ﬁrm’s assets, providing returns to investors independently. It makes analyses of the ﬁrms prospects more transparant. Drawbacks are that LSs operate within a corporate structure and are therefore taxed twice. LSs are not widely used for timberland securitization. MLPs, introduced in the 1980s, are of high liquidity and tradable on major exchanges. Limited partners provide capital and purchase units of the securitized MLP. The general partner retains control over MLP assets. The partnership is largely limited to retail investors, as institutional investors such as pension funds are prohibited by their charters from investing in MLPs.

Timber REITs are companies that have specialized in owning and managing timberlands and have become the most common channel. They provide great liquidity and allow in-stitutional investors to participate in a manner MLPs can not (Caulﬁeld and Flick, 2000). Furthermore, REITs allow for earning income taxed at capital gains rates, rather than at ordinary income tax rates. A drawback is that REITs often track the market direction more than most other vehicles do.

Historically, six public companies have converted to a timber REIT. This study focusses on Plum Creek Timber (PCL), Potlatch (PCH), Rayonier (RYN) and Weyerhaeuser (WY) only, as Longview Fibre (LFB) was taken private in 2007 and CatchMark (CTT) did not exist prior to 2013. The table below gives a brief summary of the four companies under investigation.

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Table 1: Summary of public timber REITs

Plum Creek Potlatch Rayonier Weyerhaeuser

Ticker PCL PCH RYN WY

Exchange NYSE NASDAQ NYSE NYSE

Market index listed on S&P500 S&P400 S&P400 S&P500

Market cap. 2015 ($ billion) 8.42 1.27 2.82 15.68

Revenues 2014 ($ billion) 1.48 0.61 0.60 7.40

Net income 2014 ($ million) 214 90 98 1,782

Number of states present 2014 19 5 9 9

Land size (million acres) 6.6 1.6 2.3 6.7

REIT conversion date 01.01.1999 01.01.2006 01.01.2004 01.01.2010

1.3 Literature review

Sun (2013b) evaluates the change in market risk of five timber stocks after converting to REIT using a (time-varying) symmetrized Joe-Clayton copula, showing strong evidence of upper-and lower tail dependence before upper-and after conversion to REIT upper-and a significantly increased dependence after the conversion. Tail dependence captures the behaviour of random variables during extreme events. It measures the probability of a large positive or negative shock in the stock price of a timber REIT, given a large positive of negative shock in the market index. Sun (2013a) also shows that there is a larger correlation coefficient with the S&P500 index after conversio and adopts four approaches to compare the risk of three timber REITs during the 2008 financial crisis.

A cointegration analysis by La and Mei (2015), using both Engle-Granger and Johansen tests, shows no general trends among the S&P 500 index and timber REIT equities, indicat-ing a long run diversiﬁcation potential. Their analyses use daily stock prices from December 2009 to December 2013.

The diversification potential of timberland investments from the risk perspective is quan-tified by Scholtens and Spierdijk (2010) in a mean-variance framework to the quarterly re-turns of U.S. private equity timberland, which are indexed by the NCREIF1 Timberland Index. Initially, private equity timberlands seem to improve the mean-variance frontier, al-though the result breaks down when the appraisal smoothing bias is removed from the model. This efficient frontier of the mixed portfolio is significantly improved under the

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conditional-framework, according to Wan et al. (2015).

Wang et al. (2011) investigate the time-varying dependence structures between the Chi-nese market and other major world markets using copula models. A major result is that time-varying copulas outperform constant copulas.

Textbooks on copulas include the work of Joe (1997) and Nelsen (1999). Patton (2001) was a pioneer with the introduction of time-varying dependence structures for copula models. More recently, Patton et al. (2012) reviewed the growing literature on copula-based models for ﬁnancial time series data.

1.4 Research question

The main research question is: How does the dependence between publicly traded timber REITs and major stock market indexes behave in times of ﬁnancial turmoil? The following research questions will form the foundation of the investigation:

• What models peform best for modeling the marginal distribution of the ﬁnancial time

series?

• Which copulas perform best for describing the dependence structure between the stocks

and market indexes?

• Do time-varying copulas outperform constant copulas?

• Does allowing for a structural break in the dependence structure improve the model? • Does the dependence of publicly traded timber REITs with the market increase in

times of ﬁnancial turmoil?

• Are the results robust to large-, mid- and small-cap indexes?

The rest of this study is organized as follows. Section 2 brieﬂy explains the means of data collection and transformations to prepare for copula modeling, choosing from various stochastic volatility models. Next, theory on copulas is provided, including the family of copulas used. It is examined if allowing for a structural break in a time-varying dependence structure improves the models. A method is provided to answer the main research question, comparing the estimated conditional market volatility with the time-varying copula models. Section 3 provides the results through a selection of tables and plots for visualization. Section 4 and 5 summarize the main ﬁndings and make recommendations for further investigation.

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2 Methods

All computations for this study are done in Matlab (MathWorks, Inc., 2015). The scripts and functions are a combination of original work, the documentation from the MathWorks website and the code available from Andrew Patton’s website.

2.1 Data collection

The data that are used and transformed are the daily closings of the four public timber REITs and the daily closings of three major stock market indexes. Ghysels et al. (2006) support the use of daily data. They claim that daily data leads to the best volatility forecasts and that higher frequency data yields no further improvement. The data are collected from DataStream and go back as far as February 17, 1994, the earliest date of which the daily closings of all four public companies is available. The end date is November 6, 2015, one day before Weyerhaeuser conﬁrmed the acquisition of Potlatch2, forming ”the world’s premiere timberland and forest-products company”, boosting Potlatch’s stock with 17 percent that day. The raw data consist of 5,470 observations of daily closings and span the trading dates February 17, 1994 to November 6, 2015.

For the market indexes, the S&P 500, S&P 400 and Russell 2000 are selected, representing large-, mid- and small-cap, respectively. The S&P 500 (SP500) is based on the market capitalizations of 500 large companies that are listed on either the New York Stock Exchange (NYSE) or the NASDAQ. To be included in the index, the company must have a market capitalization greater than $5.3 billion. The S&P400 (SP400) serves as a barometer for the US mid-cap equities. A market capitalization between $1.4 billion and $5.9 billion is required. The Russell 2000 Index (R2000) is the most widely quoted measure for the performance of small-cap company stock.

Daily closings pt are transformed to daily returns rt to prepare the data for subsequent

modeling. The returns are equal to the ﬁrst diﬀerences of the prices: rt = pt+1− pt. The

return series therefore contain one observation less than the daily closings. The motivation for the ﬁrst diﬀerences is that the daily closings appear integrated of order 1, whereas the returns do not. They include the number of observations, mean, minimum- and maximum

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gate additional properties of the return series. The Jarque-Bera test evaluates normality of the series, the Augmented Dickey-Fuller tests for a unit root, the ARCH-LM test evalu-ates conditional heteroskedasticity and the Ljung-Box Q test evaluevalu-ates independence of the residuals.

The presence of fat tails, or excess kurtosis, in the distribution of a random variable in-creases the likelihood of extreme events. Quantile-quantile plots display the sample quantiles of the return series against the theoretical quantiles of a normal distribution. Alternatively, one can observe the value of the kurtosis, which indicates the presence of fat tails when it exceeds 3. In that case the Student’s t−distribution fits the return series more closely, as it provides more flexibility than the normal distribution. It is well known that financial returns series often exhibit excess kurtosis.

2.2 Copulas

The objective is to model the dependence between the daily returns of the timber stocks and the market indexes. Correlation only works with normal distributions, wheras financial time series often exhibit non-normal behaviour such as skewness, a measure for asymmetry, and fat tails. An approach introduced by Sklar (1959) in the form of Sklar’s Theorem is cop-ula modeling. Copcop-ulas are nonparametric measures of the dependence of random variables that capture dependence between non-normal series. The advantage of copula modeling is that the univariate marginal distributions can be separated from the dependence structure, bringing great flexibility in evaluating joint distributions. The dependence structure is then represented by a copula. Nowadays, the literature on copulas is comprehensive (Joe, 2014) and in practice copulas are extensively used to solve many financial problems. Some copulas have the ability to capture tail dependence, the comovement of the random variables during extreme events. Copulas come in all shapes and sizes. Below is the definition of a bivariate copula, with the property that the copula function is defined overU(0, 1) variables u and v.

Deﬁnition 1. _{A two-dimensional copula is a function C : [0, 1]}2 → [0, 1] with the following

three properties

1. C(u, v) is increasing in u and v

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3. For all u, u, v, v ∈ [0, 1] such that u < u and v < v, it follows that C(u, v)−

C(u, v) − C(u, v) + C(u, v) ≥ 0

Sklar (1959) states that a n−dimensional joint distribution can be decomposed into n univariatie marginal distributions and an n−dimensional copula. This study assumes n = 2 and two-dimensional copula models. The ﬁrst dimension represents a stock, the second dimension represents a market index.

Theorem 1. _{Let F (r) be a joint distribution function with marginal distribution functions}

F1(r1) = P (R1≤ r1) and F2(r2) = P (R2≤ r2). Then there exists a copula C : [0, 1]2→ [0, 1]

such that for all r = (r1, r2)∈ R2 it holds that

F (r1, r2) = C (F1(r1), F2(r2))

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A copula function C of random vector R maps the univariate marginal distributions F1 and

F2 to the joint distribution F with R ∼ F = C(F1, F2). If marginal distributions F1 and

F2 are continuous, then copula C is unique. Conversely, if C is a copula and F1 and F2

are univariate marginal distribution functions, then the multivariate function F (r), deﬁned above, is a joint function with marginal distributions F1 and F2.

Let u = F1(r1) and v = F2(r2), both standard uniformly distributed. Then copula

func-tion C(u, v) describes the joint distribufunc-tion of random variables R1 and R2. The last prop-erty, the converse of Sklar’s theorem, is most used in empirical studies. For a 2-dimensional distribution function with continuous margins F1 and F2 a unique copula can be obtained

for all u, v ∈ U(0, 1):

C(u, v) = F (F₁−1(u), F₂−1(v)) (2)

The joint distribution function F (r1, r2) can be diﬀerentiated with respect to r1 and r2,

resulting in joint density f :

f (r1, r2, θr1, θr2, θc) = ∂ 2_[C(F₁_(r₁_{), F}₂_(r₂_))] ∂F1(r1)∂F2(r2) × n i=1 fi(ri) for i = 1, 2 (3)

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with θr1,θr2 and θc the parameter vectors of distribution functions F1, F2 and F respectively.

The log-likelihood function L of f is then given by

L(θr1, θr2, θc) = T i=1 log f1(r1; θr1) + T i=1 log f2(r2; θr2) + T i=1 log c(F1(r1), F2(r2); θr1, θr2, θc) (4) = Lr1(θr1) + Lr2(θr2) + Lc(F1(r1), F2(r2); θC)

with T = 5, 469 equal to the length of the return series. The maximum likelihood estimator (MLE) is found by maximizing log-likelihood with respect to all parameters. The properties that the MLE should satisfy are consistency, asymptotic eﬃciency and asymptotic normality.

2.3 Models for the marginal distributions

Most financial return series exhibit some degree of autocorrelation, and more importantly, conditional heteroskedasticity. The autocorrelation and conditional heteroskedasticity in the return series therefore need to be filtered out, in order to prepare the series for subse-quent modeling. The return series should be filtered in such a way that i.i.d. residuals are generated.

To extract a series of i.i.d. observations using model objects for stationary linear time series models, autoregressive models are ﬁtted to the conditional mean of the returns to compensate for autocorrelation. Generalized autoregressive conditional heteroskedasticity models (GARCH, Bollerslev (1986)) are ﬁtted to the conditional variance to compensate for heteroskedasticity. An autoregressive GARCH model should be able to control for both autocorrelation and heteroskedasticity.

Assume the return series are given by rt = μ + t with t = σtzt with ν degrees of

freedom and ztan i.i.d. t−distributed process. To investigate the autocorrelation, two types

of autoregressive models are investigated for the return series.

• AR(p) model speciﬁes autocorrelation as follows

rt= μ + p i=1 φirt−i+ t (5)

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• ARMA(p, q) model includes lagged error terms and speciﬁes autocorrelation as follows rt= μ + p i=1 φirt−i+ t+ q j=1 θjt−j (6)

To ﬁlter out the heteroskedasticity, various GARCH models are investigated.

• GARCH(P, Q) model speciﬁes the volatility as follows

σt2= κ + P i=1 γiσt−i2 + Q j=1 αj2t−j (7)

Glosten et al. (1993) introduce a flexible GARCH model for volatility clustering including a leverage effect that captures the greater effect that negative shocks generally have on volatility than positive shocks have. Indicator function I[t−j<0] equals 1 if t−j < 0 for

j = 1, ..., t − 1 and 0 otherwise.

• GJR-GARCH(P,Q) model speciﬁes the volatility as follows

σ2t = κ + P i=1 γiσ2t−i+ Q j=1 αj2t−j+ Q j=1 ξj2t−jI[t−j<0] (8)

Combining the ﬁlters leads to the following seven models, containing various modiﬁca-tions to the returns and conditional variances.

1. ARMA(p, q) equals (6)

2. GARCH(P, Q) equals (7)

3. AR(p)-GARCH(P, Q) combines (5) and (7)

4. ARMA(p, q)-GARCH(P, Q) combines (6) and (7)

5. GJR-GARCH(P, Q) equals (8)

6. AR(p)-GJR-GARCH(P, Q) combines (5) and (8)

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These models return the optimized log-likelihood objective function. Based on the Akaike information criterion (AIC), the optimal values of p, q, P and Q are determined. AIC is a model ﬁt statistic that considers goodness-of-ﬁt and parsimony:

AIC = −2L(ˆθ; r) + 2q

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with L(ˆθ; r) the maximum likelihood value of the copula and q a vector containing the

dimensions of the parameter spaces. Unlike the optimized log-likelihood value, AIC penalizes more complex models, i.e., models with additional parameters. Models with low AIC values have higher levels of empirical support. The model with the lowest AIC value is assumed to be the best supported model.

The optimal number of autoregressive lags p is determined by ﬁtting the model over a grid of up to 10 lags and the volatility models are examined up to order (3,3). This results in seven optimal models to describe the dynamics of the four stocks and seven models to describe the dynamics of the three market indexes. The decision is based on the lowest sum of the AICs of the four stocks and the lowest sum of the AICs of the three market indexes. The seven models for the stocks and seven models for the market indexes undergo Ljung-Box Q-tests for remaining autocorrelation and ARCH LM-tests for heteroskedasticity. The two ﬁlters that perform best, decided based on the p−values of the tests, are selected to model the marginal distributions for the stocks and market indexes.

Once the optimal ﬁlters are derived, the ﬁltered residuals ˆ_t and conditional variances ˆ

σ2t for each return series are extracted. Subsequently, the standardized residuals are derived

by dividing the estimated residuals by the square root of the estimated variances. Math-ematically, this is equivalent to ˆ_z_t = ˆ_t_/_σˆ_t2 with standardized residuals ˆ_z_t representing the underlying zero mean, unit variance i.i.d. t−distributed series. Similar to the return series, summary statistics of standardized residuals ˆ_z_t of each series are provided, includ-ing the mean, standard deviation, skewness, kurtosis and the percentage of observations that lie outside 95% conﬁdence interval [−1.96, 1.96] of a standard normal random variable. Furthermore, Jarque-Bera tests, ARCH-LM tests and Ljung-Box Q tests provide additional information.

Comparing the ACFs of the return series with the ACF of the corresponding standardized residuals should reveal if the standadized residuals are approximately i.i.d., and thus much

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more manageable in subsequent modeling.

Finally, the standardized residuals are transformed to standard uniform distributions using the empirical CDF. A Kolmogorov-Smirnov (KS) test evaluates the null hypothesis that the transformed data, stored in vectors u and v, are indeed U(0, 1) distributed. An ordinary KS test examines if data areN (0, 1)−distributed. However, a subtle modiﬁcation tests for the desired speciﬁcation.

2.4 Constant copulas

This section provides the set of copulas that are used (Patton (2006) and Joe (2014)): the Normal copula, the (rotated) Clayton copula, the Plackett copula, the Frank copula , the (rotated) Gumbel copula, Student’s t copula and Symmetrized Joe-Clayton copula. In this study, the two dimensional versions of these copulas will be used. For each copula the dependence parameter, the log-likelihood and possible tail dependences are computed.

In ﬁnancial literature, often the Student’s t copula is used for its ability to capture tail dependence, whereas the Normal copula can not. The Gumbel copula also captures tail dependence, but only upper tail dependence and thus fails to capture lower tail dependence. As potential joint losses are often more interesting than potential joint proﬁts, the Gumbel copula can be rotated, such that it captures dependence of negative extreme events. The Clayton copula also has this property, whereas the rotated Clayton copula captures upper tail dependence. Below is the family of copulas, including the restrictions on the parameters and values of the tail dependence.

1. Normal or Gaussian copula with tail dependence (0, 0)

C1(u, v; R) = Φ2(Φ−1_{(u), Φ}−1_{(v); R), 0 < u, v < 1} (10)

with Φ is the cdf of a standard normal distribution and Φ₂ is a standard bivariate normal distribution with 2× 2 correlation matrix R ∈ (−1, 1)2. The Normal copula has τU = τL = 0 for correlation less that one (Embrechts et al., 2002), which implies that the variables are independent in the tails of the distribution. The dependence

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function associated with bivariate normality is given by C1(u, v|ρ1) = _Φ−1_(u) −∞ _Φ−1_(v) −∞ 1 2π(1− ρ2₁)exp −(x2 1− 2ρ1x1x2+ x22) 2(1− ρ2) dx1 dx2, −1 < ρ1 < 1 (11)

2. Clayton copula with tail dependence (2−1/δ2_{, 0)}

C2(u, v; δ2) = (u−δ2+ v−δ2− 1)−1/δ2, 0 ≤ u, v ≤ 1

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with dependence parameter δ2 ∈ [0, ∞). Note that as δ2 goes to zero, the marginals

become independent. The density is given by

c2(u, v; δ2) = (1 + δ2)(uv)−δ2−1(u−δ2 + v−δ2 − 1)−2−1/δ2, 0 < u, v < 1

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3. Rotated Clayton copula with tail dependence instead (0, 2−1/δ2₎

C3(u, v; δ3) = C2(1− u, 1 − v; δ₃_{), 0 ≤ u, v ≤ 1} (14)

with dependence parameter δ3∈ [0, ∞). The density is given by

c3(u, v; δ3) = c2(1− u, 1 − v; δ3), 0 < u, v < 1

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4. Plackett copula with tail dependence (0, 0)

C4(u, v; δ4) = 1₂η−1

1 + η(u + v) − [(1 + η(u + v))2− 4δ4ηuv]1/2

, 0 ≤ u, v ≤ 1

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for 0≤ δ₄_{< ∞ with η = δ}₄− 1. The density is given by

c4(u, v; δ4) = δ4[1 + η(u + v − 2uv)]

[(1 + η(u + v))2− 4δ4ηuv]3/2, 0 < u, v < 1

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5. Frank copula with tail dependence (0, 0) C5(u, v; δ5) =−δ5−1log 1−(1− e −δ5u₎₍₁_{− e}−δ5v₎ 1− e−δ5 , 0 ≤ u, v ≤ 1 (18)

with δ5 ∈ (−∞, ∞). The density is given by

c5(u, v; δ5) = δ5(1− e

−δ5_)e−δ5(u+v)

[1− e−δ5 − (1 − e−δ5u₎₍₁_{− e}−δ5v_)]2, 0 < u, v < 1

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6. Gumbel copula with tail dependence (0, 2 − 21/δ6₎

C6(u, v; δ6) = exp

−([−log u]δ6_{+ [}_{−log v]}δ6₎1/δ6 _{, 0 ≤ u, v ≤ 1} (20)

with δ6 ∈ (1, ∞) and density

c6(u, v; δ6) = exp −(˜uδ6 _{+ ˜}_vδ6₎1/δ6 _[˜_uδ6_{+ ˜}_vδ6_]1/δ6_{+ δ} 6− 1 ... (21) ˜ uδ6 _{+ ˜}_vδ6 1/δ6−2_(˜_u˜_v)δ6−1_(uv)−1_{, 0 < u, v < 1}

with ˜_{u = −log u, ˜}_{v = −log v and δ ∈ (1, ∞)}

7. Rotated Gumbel copula with tail dependence (2− 21/δ6_{, 0)}

C7 = C6(1− u, 1 − v; δ7), 0 ≤ u, v ≤ 1

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with δ7 ∈ (1, ∞) and density

c7 = c6(1− u, 1 − v; δ7), 0 < u, v < 1

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8. Student’s tν copula with degrees of freedom parameter ν, where tν+1(.) is the cdf of the

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is 2× t_ν+1 −(ν+1)(1−ρ)_1+ρ , 2 × tν+1 −(ν+1)(1−ρ)_1+ρ C8(u, v; ρ8, ν) = tν(t−1ν (u), t−1ν (v)) (24) = _t−1_(v) 0 _t−1_(u) 0 (2π) −1₍₁_{− ρ}2 8)−1/2[1 + v−1(x21− ρ8x1x2+ x22)]−(ν+2)/2dx1dx2

with −1 < ρ₈_{< 1, x}₁ _{= t}−1_ν _{(u), x}₂ _{= t}−1_ν _{(v) and density}

c8(u, v; ρ8, ν) = t2,ν(T −1 ν (u), Tν−1(v); ρ8) tν(Tν−1(u))tν(Tν−1(v)) (25) = 1 1− ρ2₈ Γ ((ν + 2)/2) Γ(ν/2) Γ2_{((ν + 1)/2)} 1 +x21+x22−2ρ8x1x2 ν(1−ρ2₈) _−1−ν/2 1 +x21 ν _{−1/2−ν/2} 1 +r22 ν _{−1/2−ν/2} with t2,ν(y; ρ8) = (1− ρ28)−1/2Γ ((ν + 2)/2)_{Γ(ν/2)[πν]} 1 +x21+ x22− 2ρ8x1x2 ν(1 − ρ2₈) _−1−ν/2 (26)

The Normal copula has no tail dependence, while the Students-t copula allows diﬀerent degrees of symmetric tail dependence through the degrees of freedom ν. A smaller ν implies greater tail dependence, As ν grows large (≈ 30), the students-t copula con-verges to the Normal copula. The Student’s t copula is capable of describing ﬁnancial data much more accurately than the Normal copula does, due to its ability to monitor tail dependence.

9. Symmetrized Joe-Clayton copula with coeﬃcients of lower and upper tail dependence

τL∈ (0, 1) and τU ∈ (0, 1). First consider the ordinary Joe-Clayton copula

CJC(u, v|τL, τU) = 1−

1− (1− (1 − u)ω2₎−ω1_{+ (1}_{− (1 − v)}ω2₎−ω1_{− 1}−1/ω1 1/ω2 (27)

with ω1 and ω2 are two parameters with ω1 ≥ 0 and ω2 ≥ 1. Furthermore ω1 =

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dependence.

The Joe-Clayton copulas allows upper and lower tail dependence between 0 and 1. A disadvantage of this copula is that there still exists asymmetry in the copula, caused by its functional form, even when τU = τL. An improvement is the symmetrized Joe-Clayton copula (SJC), with tail dependences measures completely determining the presence or absence of asymmetry. The SJC copula modiﬁes the JC copula as follows

CSJC(u, v|τL, τU) = 0.5 ×

CJC(u, v|τL, τU) + CJC(1− u, 1 − v|τL, τU) + u + v − 1

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The SJC density can be derived as follows

cSJC(u, v|τL, τU) = ∂ 2_C SJC(u, v|τL, τU) ∂u∂v (29) = 0.5 × ∂2CJC(u, v|τL, τU) ∂u∂v + ∂2CJC(1− u, 1 − v|τL, τU) ∂(1 − u)∂(1 − v)

which allows the tail dependence measures to be estimated, following the two-step maximum likelihood method by Joe and Xu (1996). The copula is symmetric when

τU = τL. In the ﬁgure below are contour plots of some of the constant copulas.

Figure 1: Contour plots of two dimensional copulas

2.5 Copula performance

The various copulas are non-nested. Hence, to compare their performance, a method diﬀerent than comparing AIC values must be applied. Vuong (1989) developed a likelihood-ratio test for model selection of nested, non-nested or overlapping models with exogenous i.i.d.

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variables. Let

t,i,j = logfi(Yt|rt,i, ˆθi

)

fj(Yt|rt,j, ˆθj)

= log fi(Yt|rt,i, ˆθi)− log fj(Yt|rt,j, ˆθj)

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for t = 1, ..., T and 0 < i < j ≤ 9 to cover all combinations of copulas. Deﬁne E(t,i,j) = μ and the null- and alternative hypothesis by

H0 : μ = 0 vs Ha: μ = 0

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Vuong showed that 1/T times the likelihood ratio statistic consistently estimates μ:

1

TLRT(ˆθi, ˆθj)→p E(t,i,j) = μ with ˆθi and ˆθj the maximum likelihood estimates and

LRT(ˆθi, ˆθj) = T

t=1

log fi(Yt|rt,i, ˆθi)− log fj(Yt|rt,j, ˆθj)

+ q(i) − q(j) 2 log T (32) = T t=1

t,i,j+ q(i) − q(j)₂ log T

where the last part of the equation penalizes a higher number of parameters q in the com-plexer copulas. The test statistic is then given by

Z = LRT(ˆθi, ˆθj) T ω_T2 →d_{N (0, 1)} (33) with ω2_T = _T1 T_t=12_t,i,j − 1 T _T t=1t,i,j ₂

. _{Under the null hypothesis, test statistic Z} converges to a standard normal distribution. Under the alternative, either μ < 0, when

Z < −Φ−1(1− α/2), indicating that copula i signiﬁcantly underperforms compared to cop-ula j, or μ > 0, when Z > Φ−1(1− α/2), indicating that copula i signiﬁcantly outperforms copula j.

2.6 Time-varying copulas

The dependence structure of constant copulas is captured by a single number. This may lead to unreliable results when the dependence is measured over a longer period of time, as macroeconomic changes may signiﬁcantly aﬀect this dependence. Time-varying copulas allow the dependence structure to vary over time. The time-varying copulas in this study are

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the Normal copula, the Rotated Gumbel copula and the Symmetrized Joe-Clayton copula. Notice that the Normal copula does not capture tail dependece, the rotated Gumbel copula captures lower tail dependence only and the SJC copula captures both upper tail and lower tail dependence separately.

For the dependence parameter ρ of the Normal copula, ρt models the evolution of the

tail dependence as follows

ρt= Λ ⎛ ⎝θ1+ θ2ρt−1+ θ3σˆt+ θ4 1 m m j=1 Φ(ut−j)Φ−1(vt−j) ⎞ ⎠ (34)

with Λ(x) ≡ (1 − e−x)(1 + e−x)−1 a logistic transformation to ensure −1 < ρ_t _{< 1 for all t} with the assumption that dependence parameter ρtfollows a non-linear ARMA(1, m) process.

Lagged parameter ρt−1captures any persistence. To answer the main research question, the

relationship between the time-varying dependence structure and the market volatility is also investigated. The market volatility ˆ_σ_t_{inﬁltrates the model through parameter θ}₃and reveals the relation between the dependence structures of the timber REITs with the market indexes and the market volatility for non-zero values of θ3. The mean of the last m observations

captures any variation in the dependence. For m = 10, 10 trading days are considered, equivalent to a period of 2 weeks over which the average variation in the dependence is computed. Other choices for m are 42, 63, 84 and 126 tradings days, representing a period of two, three, four and six months, respectively. This results in a loss of m observations at the beginning of the study period starting February 17, 1994. Furthermore, the standard errors are derived, which are equal to the square root of the diagonal elements of the inverse Hessian matrix of the optimization:

SE_¯θ=

diag(H_i,j−1) with Hi,j = ∂

2_f

∂ri∂rj

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The model with the lowest standard errors delivers the highest precision, and therefore the most accurate estimators.

The dependence parameter for the Rotated Gumbel copula, ζt, is given by

⎛

m

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with Φ(x) ≡ 1 + x2 a quadratic transformation, such that ζt> 1 for all t. The dependence

parameters for the SJC copula, τtU and τtL, are given by

τtU = Ψ ⎛ ⎝θ1+ θ2τt−1U + θ3σˆt+ θ4 1 m m j=1 |ut−j− vt−j| ⎞ ⎠ (37) τtL= Ψ ⎛ ⎝θ1+ θ2τt−1L + θ3σˆt+ θ4 1 m m j=1 |ut−j− vt−j| ⎞ ⎠ (38)

with Ψ(x) ≡ (1 + e−x)−1 _{a logistic transformation, such that 0 < τ}U

t , τtL < 1 for all t.

The speciﬁcation is again analogous to a non-linear ARMA(1, m) process, where the same values of m are examined. Similar to the constant copulas, the relative performance of the time-varying copulas is determined by Vuong’s likelihood-ratio test. Additionally, the performance of the time varying copulas is compared to the best performing constant copula. This will indicate whether time-varying copulas outperform constant copulas.

2.7 Structural break

The study investigates whether the model ﬁt improves when parameters are added to the time-varying dependence to allow for a structural break in the observations. Sun (2013b) argues that converting to a public REIT structure increases the market dependence. The conversion date can be considered as a change point in the dependence structure. Following the literature, this study investigates change point tbreak = [1230, 2990, 2486, 3997],

corre-sponding to the four diﬀerent conversion dates given in Table 1. As the model allows to conveniently investigate alternative change points, one may consider the burst of the dot-com bubble in 2000 or the ﬁnancial crisis in 2008.

The evolution of the dependence parameter of the Normal copula is now given by

ρt= Λ ⎛ ⎝(θ1+ Δθ5) + (θ2+ Δθ6)ρt−1+ (θ3+ Δθ7)ˆσt+ (θ4+ Δθ8) 1 m m j=1 Φ−1_(u_t−j)× Φ−1_(v_t−j) ⎞ ⎠ (39) with Δ = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 _{for t < t}_break 1 _{for t ≥ t}_break

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with Λ as deﬁned above. Hence, parameter θ1 before conversion (θ1,b) equals θ1, whereas

parameter θ1 after conversion (θ1,a) equals θ1 + θ5. Dummy variable Δ allows the extra

parameters to inﬁltrate the model at tbreak. The time-varying dependence structure of the

rotated Gumbel copula is given by

ζt= Φ ⎛ ⎝(θ1+ Δθ5) + (θ2+ Δθ6)ζt−1+ (θ3+ Δθ7)ˆσt+ (θ4+ Δθ8) 1 m m j=1 |ut−j− vt−j| ⎞ ⎠ (40)

with Φ and Δ deﬁned above. Similarly, the time-varying upper and lower tail dependence parameters of the symmetrized Joe-Clayton, with Ψ and Δ deﬁned above, are given below.

τtU = Ψ ⎛ ⎝(θ1+ Δθ5) + (θ2+ Δθ6)τt−1U + (θ3+ Δθ7)ˆσt+ (θ4+ Δθ8) 1 m m j=1 |ut−j− vt−j| ⎞ ⎠ (41) τtL= Ψ ⎛ ⎝(θ1+ Δθ5) + (θ2+ Δθ6)τt−1L + (θ3+ Δθ7)ˆσt+ (θ4+ Δθ8) 1 m m j=1 |ut−j− vt−j| ⎞ ⎠ (42)

2.8 Relation between dependence structures and market volatility

The main research question is answered by investigating the relation between the time-varying dependence structures and the market volatility. Heteroskedasticity-and-autocorrelation consistent estimators (HAC) are derived that describe the eﬀect of market volatility on the dependence structures of the time-varying copulas. The motivation for HAC estimates is that the error terms ηtare not i.i.d. and the results would otherwise be unreliable. For these

regressions, the inverses of transformations Λ, Φ and Ψ, with the dependence structures ˆ_ρ_t, ˆ

ζt, ˆτtU and ˆτtL as inputs, are the dependent variables. However, to avoid multicollinearity,

ˆ

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below. Λ−1( ˆ_ρ_t_{) = α + βˆ}_σ_t_{+ η}_t with Λ−1_{(x) = log} 1 + x 1− x (43) Φ−1(ˆ_ζ_t_{) = α + βˆ}_σ_t_{+ η}_t with Φ−1_{(x) =}√_{x − 1} (44) Ψ−1(ˆ_τ_tU_{) = α + βˆ}_σ_t_{+ η}_t with Ψ−1_{(x) = log} 1− x x (45) Ψ−1(ˆ_τ_tL_{) = α + βˆ}_σ_t_{+ η}_t with Ψ−1_{(x) = log} 1− x x (46)

The estimates ˆ_{β measure the impact of the independent variables of market volatility on} market dependence of the stocks.

The 2008 financial crisis hit the stock markets on September 15, 2008, after one of the most tumultuous weekends in Wall Street history. Lehman Brothers had filed for bankruptcy protection that day and Merrill Lynch had sold itself to Bank of America3. It is that day that market volatilities started to rise at a very fast pace. As markets collapsed, inevitably the four companies in this study suffered, too. With both market volatilities and the dependence structures approaching their all time highs, this unique period may sketch an unrepresen-tative image of market volatilities’ impact on the dependence parameters. Therefore, the period from September 15, 2008 to May 5, 2009 will be excluded from an extra set of regres-sions. This is a period of almost 8 months, 161 trading days, during which volatility levels peaked. At the end of this period, the volatilities of the market indexes have returned to approximately their pre-crisis levels.

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3 Results

This section provides the results that lead to the answers to the research questions.

3.1 Stocks and market indexes

The table below displays summary statistics of the daily closings of the four stocks and three market indexes, consisting of 5,470 trading days over a period from February 17, 1994 to November 6, 2015.

Table 2: Summary statistics of the daily closings

PCL ($) PCH ($) RYN ($) WY ($) SP500 SP400 R2000

Number of observations 5,470 5,470 5,470 5,470 5,470 5,470 5,470

Start value 29.25 39.84 6.94 19.70 470.34 180.65 264.88

End value 40.29 32.69 24.15 30.40 2099.20 1463.32 1199.75

Average change (×100) 0.20 -0.13 0.31 0.20 29.78 23.45 17.09

Minimum daily change (%) -14.84 -26.97 -19.43 -17.16 -9.03 -10.89 -11.85

Maximum daily change (%) 24.49 19.80 16.90 14.07 11.58 10.48 9.27

Note: Start date is February 17, 1994 and the end date is November 6, 2015

The ﬁgure below shows the relative price movements of the stocks and market indexes. Their initial levels have been normalized to unity to illustrate their relative performance.

Figure 2: Relative price movements of the stocks (left) and market indexes (right)

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high levels of excess kurtosis, both in line with what is expected of financial time series. For the hypothesis tests the results are as follows. The Jarque-Bera test statistics are all highly significant, indicating rejection of the null hypotheses of normality in the return series. The Augmented Dickey-Fuller test statistics are also highly significant, indicating rejection of the null hypothesis of a unit-root in the return series. The ARCH LM test statistics are all highly significant, indicating rejection of the null hypothesis of no conditional heteroskedas-ticity, revealing significant ARCH effects in the return series. The critical value is 3.842 for rejection at the 5% significance level. The Ljung-Box Q-test statistics are all highly signifi-cant, indicating rejection of the no residual autocorrelation null hypothesis. Enemies for an i.i.d. series, required for copula modeling, are autocorrelation and heteroskedasticity in the residuals.

Table 3: Summary statistics of the daily returns

PCL PCH RYN WY SP500 SP400 R20000 Number of observations 5,469 5,469 5,469 5,469 5,469 5,469 5,469 Mean (×1000) 0.06 -0.03 0.23 0.09 0.27 0.38 0.28 Minimum (×100) -16.06 -31.43 -21.61 -18.82 -9.47 -11.53 -12.61 Maximum (×100) 21.90 18.07 15.61 13.16 10.96 9.96 8.86 Standard dev (×100) 1.84 2.13 1.92 2.06 1.19 1.29 1.42 Skewness 0.34 -0.57 -0.29 -0.13 -0.25 -0.39 -0.34 Kurtosis 15.63 22.75 15.04 7.74 11.30 9.94 8.69 Jarque-Bera test 36437.8 89007.9 33091.3 5155.6 15743.4 11099.1 7488.3 [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

Aug. Dick-Fuller test -87.8 -86.3 -84.3 -75.6 -78.7 -74.7 -76.7

[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

ARCH LM test 425.5 228.9 242.4 137.9 243.5 336.2 483.5

[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

Ljung-Box Q-test 194.6 194.7 149.4 31.2 106.9 77.7 79.3

[0.000] [0.000] [0.000] [0.053] [0.000] [0.000] [0.000]

Note: The minimum and maximum returns are approximations of the minimum and maximum percent-age of change in the daily closings in Table 3. The numbers in brackets are p−values

Figure 8 in the Appendix shows scatter plots of all combinations of the returns of the stocks with corresponding returns of the market indexes. The majority of the observations lies in the lower left or upper right quadrant of the plots, indicating a positive correlation.

The figure below shows that the sample ACF and PACF of the returns of Plum Creek Timber exhibit significant autocorrelation. The sample ACF has significant autocorrelation at lags 1, 9, 10 and 16, whereas the PACF has significant autocorrelation at lags 1, 2, 9, 10

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and 16. The sample ACF function of the squared returns implies that GARCH modeling may signiﬁcantly condition the data used in the subsequent tail estimation process.

Figure 3: Plum Creek Timber (PCL) returns

In Figure 9 in the Appendix are the quantile-quantile plots for all return series that display the sample quantiles against the theoretical quantiles of a normal distribution. On each horizontal axis are the normal quantiles, on each vertical axis are the quantiles of the input sample. The observations diverge from the normal distribution towards the tails. This corresponds with the assumption of fat tails in ﬁnancial series. The Student’s t distribution has the ability to control for the fat tails and account for them using the degrees of freedom parameter ν.

3.3 Model selection

This section describes the stochastic volatility models that explain the characteristics of the return series most accurately in order to derive the filtered residuals ˆ_t, filterered conditional variances ˆ_σ_t2 and resulting standardized residuals ˆ_z_t. Two models are derived, one for the four stock returns and one for the three market index returns. Table 7 in the Appendix shows the AIC values for each return series per combination of parameters. The filter that yields the lowest sum of the combined AIC values is selected in both cases highlights. The

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included to control for the fat fails associated with ﬁnancial time series. The optimal number of autoregressive lags is 1. ARCH and GARCH lags are examined up to order 3, although including third lags only increaes the AIC values and does not carry an improvement.

Next, Table 8 in the Appendix compares these optimal models by the test statistics of the Ljung-Box Q-test and ARCH-LM tests. Judging by the p−values, two models are selected that control for most autocorrelation and heteroskedasticity. The model that best ﬁts the return series of the stocks r1is the AR(1)-GJR-GARCH(2, 1) − t. ﬁlter. The model contains

7 parameters and is given by

rt= μ + φrt−1+ t

with ν degrees of freedom and t= σtztwith ztan independent and identically t−distributed

process with ν degrees of freedom and

σt2= κ + γ1σt−12 + γ2σt−22 + α12t−1+ ξ12t−1I[t−1<0]

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The model that best ﬁts the return series of the market indexes r2 is the

ARMA(1,1)-GJR-GARCH(1, 2) − t ﬁlter. The model contains 8 parameters and is given by

rt= μ + φrt−1+ t+ θt−1

with ν degrees of freedom and t= σtztwith ztan independent and identically t−distributed

process with ν degrees of freedom and

σt2 = κ + γ2σt−12 + α22t−2+ ξ12t−1I[t−1<0]+ ξ22t−2I[t−2<0]

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The table below displays for all series and the two ﬁlters their estimated parameter values, including standard errors and signiﬁcance levels.

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Table 4: Parameters of the estimated models

PCL PCH RYN WY SP500 SP400 R2000

Conditional mean equations

μ 0.0004∗ 0.0000 0.0005∗∗ 0.0002 _μ 0.0002∗ 0.0008∗∗∗ 0.0009∗∗∗ (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) φ -0.092∗∗∗ -0.092∗∗∗ -0.066∗∗∗ -0.046∗∗ _φ 0.710∗∗∗ -0.444∗∗ -0.499∗∗ (0.014) (0.014) (0.013) (0.013) (0.135) (0.201) (0.216) θ -0.743∗∗∗ 0.493∗∗ 0.541∗∗ (0.128) (0.195) (0.210)

Conditional variance equations

κ 0.000∗∗ 0.000∗∗ 0.000∗∗ 0.000∗ _κ 0.000∗∗∗ 0.000∗∗∗ 0.000∗∗ (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) γ1 0.285∗∗ 0.425∗∗ 0.352∗∗ 0.564 γ1 0.895∗∗∗ 0.888∗∗∗ 0.891∗∗∗ (0.098) (0.165) (0.127) (0.297) (0.009) (0.010) (0.009) γ2 0.582∗∗∗ 0.472∗∗ 0.532∗∗∗ 0.378 (0.092) (0.156) (0.120) (0.285) α1 0.089∗∗∗ 0.074∗∗∗ 0.077∗∗∗ 0.031∗∗ (0.014) (0.014) (0.014) (0.010) α2 0.010 0.020 0.040∗ (0.025) (0.032) (0.019) ξ1 0.059∗∗ 0.041∗ 0.052∗∗ 0.051∗∗ ξ1 0.115∗∗∗ 0.144∗∗∗ 0.163∗∗∗ (0.020) (0.017) (0.019) (0.016) (0.033) (0.032) (0.032) ξ2 0.041 0.001 -0.040 (0.033) (0.032) (0.032) Degrees of Freedom ν 5.123 6.336 5.042 6.369 7.906 13.422 12.624 (0.357) (0.442) (0.291) (0.585) (0.820) (2.030) (1.730)

Note: Signiﬁcance of the parameters is denoted by *, ** and *** for the 10%, 5% and 1% level, respectively. The numbers in parentheses are standard errors. AR(1)-GJR-GARCH(2,1)-t for the stock returns. ARMA(1,1)-GJR-GARCH(1,2)-t for the market index returns

Figure 10 in the Appendix displays the ﬁltered conditinal volatility ˆ_σ_t for the three market indexes, estimated by the ARMA(1,1)-GJR(GARCH(1,2)−t model. These values are later used as independent variables to determine their impact on the copula dependence structure.

3.4 Marginal distributions

The filtered residuals ˆ_t and filtered conditional standard deviations ˆ_σ_t for Plum Creek Timber are shown in the figure below.

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Figure 4: Plum Creek Timber (PCL)

Notice a similar pattern between larger residuals and increased volatility. The residuals are standardized by the corresponding conditional volatility, representing the underlying zero-mean, unit-variance, i.i.d. series. The ACF plots below reveal that the standardized residuals of PCL are now approximately i.i.d., and ready for copula modeling.

Figure 5: Plum Creek Timber (PCL)

The table below displays summary statistics for all residual series. Notice that all se-ries exhibit the characteristics of a standard normal distributed sese-ries, since all means are approximately zero, standard deviation approximately 1 and the number of observations outside 95% conﬁdence interval [−1.96, 1.96] of the standard normal distribution is close to 5%. Most series are still negatively skewed and exhibit (mild) excess kurtosis. For the

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hypothesis tests, the results are as follows. The Jarque-Bera test statistics are still highly sig-nificant, indicating rejection of the null hypotheses of normality of the residual series. Since a t−distribution is assumed, this result is unsurprising. The ARCH LM test statistics are no longer highly significant. Most p−values are greater than 0.05, indicating no significant ARCH effects at the 5% significance level. The remaining p−values are close to 0.01, which is an acceptable result and a great improvement compared to the test statistics of the return series. Similarly, the Ljung-Box Q-test statistics are now no longer significant, indicating that there is not enough evidence to reject the null hypothesis that the filtered residuals are not autocorrelated. Figure 11 in the Appendix shows scatter plots of the standardized residuals of the stocks and market indexes. Finally, the empirical cdfs are transformed to U(0, 1) series u and v, which form the input for the copula models.

Table 5: Summary statistics of the standardized residual series

PCL PCH RYN WY SP500 SP400 R20000 Mean -0.018 -0.003 -0.013 -0.004 -0.034 -0.020 -0.034 Std. deviation 0.997 1.010 1.028 0.985 0.997 1.000 1.000 Skewness -0.143 -1.044 -1.019 0.054 -0.466 -0.376 -0.425 Kurtosis 5.871 23.337 20.912 4.538 4.417 3.705 3.865 Outside 95% CI (%) 5.102 5.230 5.010 4.900 5.486 5.065 5.138 Jarque-Bera test 1896.3 95244.4 74058.3 542.0 655.3 242.3 335.1 [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] Ljung-Box Q-test 0.030 0.059 0.221 2.022 1.052 0.000 0.001 [0.863] [0.808] [0.638] [0.155] [0.305] [0.987] [0.974] ARCH LM test 7.060 0.099 0.183 0.352 6.142 7.233 1.710 [0.008] [0.752] [0.669] [0.553] [0.013] [0.007] [0.191]

Note: The numbers in brackets are p−values

3.5 Constant copulas

The Kolmogorov-Smirnow test fails to reject the null hypotheses ofU(0, 1) series, which is a satisfying result, shown in Table 9 in the Appendix. Marginal distributions u and v are now qualiﬁed for copula modeling.

Table 10 in the Appendix displays for each constant copulas their optimized dependence parameter and corresponding tail dependence. The results are given for all combinations of stocks and market indexes. Notice that all SJC copulas exhibit a greater lower tail

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3.6 Copula performance

Table 11 in the Appendix ranks the constant copulas based on their log-likelihood. This should give an indication of the relative performance. However, since the models are non-nested, Vuong’s method is applied. The values of test statistic Z are in Table 12 to Table 15 in the Appendix. The Student’s t copula outperforms all other copulas, for every combination of stock and market index, proving to be the most accurate copula for the data. However, the Student’s t copula does not signiﬁcantly outperform the Plackett copula, as the values of Z range from−0.163 to −2.373. The Plackett copula, despite exhibiting zero tail dependence, is second-best every time. The next four copulas interchange the third to the sixth position: the Frank copula, rotated Gumbel copula, Normal copula and SJC copula. The bottom three positions reveal that the Gumbel, Clayton and rotated Clayton copula perform worst. The Clayton copulas perform poorly, even with lower tail dependence, suggesting that it is just a poor parameterisation for these ﬁnancial series. All values are irrespective of the choice of m.

3.7 Time-varying copulas

The choice of parameter m does have an impact on the time-varying copulas. It follows from evaluating both the log-likelihood values and the size of the standard errors of the estimated parameters for diﬀerent chioces of m. Table 16 in the Appendix compares the diﬀerent choices for m by the resulting likelihood. Although m = 63 and m = 84 often exhibit slightly better log-likelihoods, they shoot themselves in the foot with much higher standard errors.

The time-varying dependence structures ρt, ζt, τtU and τtL are computed for the

Nor-mal, rotated Gumbel and Symmetrized Joe-Clayton copula, respectively. Table 17 in the Appendix compares the relative performance of the Student’s t copula and the three time-varying copulas using Vuong’s likelihood ratio test. The values of Z are highly signiﬁcant (except two), indicating that all time-varying copulas outperform the Student’s t copula, for every combination of stock and market index.

The ﬁgure below displays the evolution of the dependence parameters over time with

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Figure 6: Time varying copulas display the dependence between Plum Creek Timber (PCL) and three market indexes for m = 42

The relative performance of the three time-varying copulas is again examined by Vuong’s likelihood ratio test. The table below displays these values of test statistic Z. Recall that under the null hypothesis of equal performance, Z converges to a standard normal distri-bution. Positive values indicate that the copula on the ith row outperforms the copula on the jth _{column at a certain signiﬁcance level, if any, and vice versa. Notice that only the}

SJC copula is outperformed in every case, although not always significantly, whereas the Normal and rotated Gumbel’s performance in do not significantly differ. An explanation for the poor performance of the SJC copula could lie in the higher number of parameters.

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Table 6: Relative performance of three time-varying copulas

PCL PCH RYN WY

Index Copula Rot.Gumbel SJC Rot.Gumbel SJC Rot.Gumbel SJC Rot.Gumbel SJC

SP500 Normal -0.766 0.774 0.806 2.295 0.691 1.686 0.355 1.400 Rot.Gumbel 2.230 2.112 1.304 1.301 SP400 Normal 0.200 0.410 1.258 2.382 0.740 1.488 -0.224 1.026 Rot.Gumbel 0.237 1.213 0.847 1.614 R2000 Normal 0.980 1.863 1.130 2.837 1.547 2.560 0.474 2.465 Rot.Gumbel 0.988 2.092 1.269 2.768

Note: The numbers are values of Z

Table 18 to 21 in the Appendix display for the four companies separately the estimated parameters θi for i = 1, ..., 4 including standard errors for time-varying dependence

struc-tures ρt, ζt, τtU and τtL. The tables include the estimated parameters θi,b and θi,a for

i = 1, ..., 4, representing the parameters before and after REIT conversion. These results

will be discussed in Section 3.8.

The results show that all time-varying dependence structures have parameter θ1 signif-icantly greater than zero for all combinations of stock and market index, except for two. Evidence is found that parameter θ2 is non-negative for ˆρt, non-positive for ˆζt and ˆτtL, and

neither for ˆ_τU_{. Parameter θ}₃, which includes the eﬀect of the market volatility, is strictly positive for ˆ_ζ_t and non-negative for the other dependence structures with most values sig-niﬁcantly positive. This indicates a positive relation between the market volatility and the dependence structures and measures of tail dependence. Parameter θ4 is strictly positive for

all ˆ_ρ_t. However, the parameter is negative for any other depedence structure. Parameter

θ4 has particularly very negative values for the SJC parameters ˆτtU and ˆτtL, often highly

siginiﬁcant.

3.8 Structural break

Table 22 in the Appendix shows the time-varying dependence structures including a struc-tural break improve the performance of all copulas, judging by the highlighted, lower AIC-values. The AIC penalizes these models for having twice the number of parameters, but still they deliver better results. Tables 18 to 21 display the estimated parameters θi,b and θi,a

for i = 1, ..., 4. In most cases either θi,b < θi < θi,a or θi,a < θi < θi,b. Despite the preferred

AIC values, the models including a structural break fail to deliver more accurate estimates, judging by the high standard errors. Therefore, for answering the main research question,

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the time-varying dependence structures without a structural break are considered.

3.9 Relation between dependence structures and market volatility

The first and third line in the figure below display scatterplots of the market volatilities (×100) on the horizontal axis and the corresponding values of the time-varying dependence structures on the vertical axis. All 5,469 observations are included. The second and fourth line display the same scatterplots, but without the 161 observations during the peak of the 2008 financial crisis. Comparison of the scatterplots reveals that the excluded observations mostly compose the values in the upper-right quantiles. Hence, the impact of the volatility on the dependence structures should be greatly reduced by omitting these 161 observations. The HAC estimates are shown in Table 23 in the Appendix.

Figure 7: Scatter plots the three market volatilities and the four time-varying dependence structures for Plum Creek Timber. The lighter scatterplots omit 161 observations during the 2008 ﬁnancial crisis

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that link the market volatility to the dependence structures. Observe that for all Λ−1( ˆ_ρ_t), Ψ−1(ˆ_ζ_t), Φ−1(ˆ_τ_tU) and Φ−1(ˆ_τ_tL), which include all 5,469 observations, the values of ˆ_{β are} strictly positive and highly signiﬁcant.

However, after excluding 161 observations during the peak of the 2008 financial crisis, observe that all ˆ_{β estimates for Plum Creek Timber, Potlatch and Weyerhaeuser and adjusted} models Ψ−1(˜_ζ_t), Φ−1(˜_τ_tU) and Φ−1(˜_τ_tL) have become insignificant. Hence, the remaining 5,308 observations fail to expose significant tail dependence. This indicates that the positive relation between market volatility and the tail dependences is mainly driven by the crisis. For Rayonier, however, the ˆ_{β estimates remain strictly positive. This is a convenient result,} which comes as no surprise, judging by the dynamics of the relative price movement in Figure 2. It is not investigated if there exists an amount of observations to exclude greater than 161 that would cause the ˆ_{β estimates of Rayonier to become insignificant as well. Furthermore,} the Normal copula, which does not capture tail dependence, still has significantly positive values for ˆ_β.

4 Conclusion

This study provides clear results to answer the research questions. The ﬁlters that perform best for estimating the marginal distributions are the AR(1)-GJR-GARCH(2,1)−t for the stocks and the ARMA(1,1)-GJR-GARCH(1,2)−t for the market indexes. Autocorrelation and heteroskedasticity are succesfully ﬁltered out, leading to i.i.d. series. The Student’s

t−copula consistently delivers the best performance of the family constant copulas.

Un-surprisingly, the three copulas with time-varying dependence structures perform better in describing the dependence between stock and market index over 20 years of daily observa-tions. Allowing for a structural break in the parameters of the dependence structure at the REIT conversion date does improve the models, but fails to deliver accuracy and is therefore disregarded.

Evidence is found of a highly signiﬁcant, positive relationship between the dependence parameters and market volatility. For three of the companies, the measures of upper and lower tail dependence melt away when a series of 161 observations from September 2008 to May 2009 is excluded from the data. This indicates that investments in these companies provide a certain stability against extreme events of market volatility in all periods aside

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from the crisis.

All of the above results are robust to large cap, mid cap and small cap market indexes.

5 Further research

The biological growth of a tree is highly predictable. However, the demand for timber may still be aﬀected by macroeconomic trends, such as the demand for housing construction. This causes the returns of the timber companies to be more correlated with the overall market than one may initially expect. Further research could investigate the returns of large food-processing companies such as Tyson Foods, Inc. (TSN) applying the same model used for this study. The demand for food suggests more stability than the demand for timber and the returns of these companies may indicate an even lower correlation with the overall market.

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6 Bibliography

Bollerslev, Tim (1986). Generalized autoregressive conditional heteroskedasticity. Journal

of econometrics 31 (3), 307–327.

Caulﬁeld, JP and WA Flick (2000). Prospects and challenges with securitized timberland.

University of Georgia, Athens, GA.

Caulﬁeld, Jon P (1998). Timberland return drivers and investing styles for an asset that has come of age. Real Estate Finance 14, 65–78.

Conroy, Robert and Mike Miles (1989). Commercial forestland in the pension portfolio: The biological beta. Financial Analysts Journal 45 (5), 46–54.

Embrechts, Paul, Alexander McNeil, and Daniel Straumann (2002). Correlation and depen-dence in risk management: properties and pitfalls. Risk management: value at risk and

beyond , 176–223.

Ghysels, Eric, Pedro Santa-Clara, and Rossen Valkanov (2006). Predicting volatility: get-ting the most out of return data sampled at diﬀerent frequencies. Journal of

Economet-rics 131 (1), 59–95.

Giacomini, Enzo, Wolfgang H¨ardle, and Vladimir Spokoiny (2012). Inhomogeneous depen-dence modeling with time-varying copulae. Journal of Business & Economic Statistics. Glosten, Lawrence R, Ravi Jagannathan, and David E Runkle (1993). On the relation

between the expected value and the volatility of the nominal excess return on stocks. The

journal of ﬁnance 48 (5), 1779–1801.

Joe, Harry (1997). Multivariate models and multivariate dependence concepts. CRC Press. Joe, Harry (2014). Dependence modeling with copulas. CRC Press.

Joe, Harry and James J Xu (1996). The estimation method of inference functions for margins for multivariate models. Technical report, Technical report.

La, Le and Bin Mei (2015). Portfolio diversiﬁcation through timber real estate investment trusts: A cointegration analysis. Forest Policy and Economics 50, 269–274.

MathWorks, Inc. (2015). MATLAB version 8.5.0.197613 (R2015a). Natick, Massachusetts: MathWorks, Inc.

Mei, Bin, Michael L Clutter, and Thomas G Harris (2013). Timberland return drivers and timberland returns and risks: A simulation approach. Southern Journal of Applied

Forestry 37 (1), 18–25.

(38)

Patton, Andrew et al. (2012). Copula methods for forecasting multivariate time series.

Handbook of economic forecasting 2, 899–960.

Patton, Andrew J (2001). Modelling time-varying exchange rate dependence using the con-ditional copula.

Patton, Andrew J (2006). Modelling asymmetric exchange rate dependence*. International

economic review 47 (2), 527–556.

Ruppert, David (2011). Statistics and data analysis for ﬁnancial engineering. Springer. Scholtens, Bert and Laura Spierdijk (2010). Does money grow on trees? the diversiﬁcation

properties of us timberland investments. Land Economics 86 (3), 514–529.

Sklar, M (1959). Fonctions de r´epartition `a n dimensions et leurs marges. Universit´e Paris 8.

Sun, Changyou (2013a). Extreme risk of public timber reits during the global ﬁnancial crisis.

Journal of Real Estate Portfolio Management 19 (1), 73–88.

Sun, Changyou (2013b). On the market risk of securitized timberlands. Journal of Forest

Economics 19 (2), 110–127.

Vuong, Quang H (1989). Likelihood ratio tests for model selection and non-nested hypothe-ses. Econometrica: Journal of the Econometric Society, 307–333.

Wan, Yang, Michael L Clutter, Bin Mei, and Jacek P Siry (2015). Assessing the role of us timberland assets in a mixed portfolio under the mean-conditional value at risk framework.

Forest Policy and Economics 50, 118–126.

Wan, Yang, Bin Mei, Michael L Clutter, and Jacek P Siry (2013). Assessing the inﬂation hedging ability of timberland assets in the united states. Forest Science 59 (1), 93–104. Wang, Kehluh, Yi-Hsuan Chen, and Szu-Wei Huang (2011). The dynamic dependence

between the chinese market and other international stock markets: A time-varying copula approach. International Review of Economics & Finance 20 (4), 654–664.

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7 Appendix

Figure 8: Return scatter plots of the stocks and market indexes

Do trees read the Wall Street Journal?

Universiteit van Amsterdam

Do Trees read the Wall Street Journal?

Contents

1

Introduction

2

Methods

3

Results

4

Conclusion

5

Further research

6

Bibliography

7

Appendix