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CHAPTER 15 Forecasting Funds of Hedge Funds Performance: A Markov Regime-Switching Approach Szabolcs Blazsek School of Business, Universidad Francisco Marroquı ´n, Guatemala Chapter Outline 15.1. Introduction 229 15.2. Data 231 15.2.1. FoHFs and the Hedge Fund Database 231 15.2.2. Data Description 231 15.2.3. FoHFs Return Drivers 233 15.3. Forecasting Models 235 15.3.1. AR(p) Model 235 15.3.2. AR(p)-GARCH(1,1) Model 235 15.3.3. MS-AR(p) Model 237 15.3.4. MS-AR(p)-GARCH(1,1) Model 237 15.3.5. MS-AR(p)-MS-GARCH(1,1) Model 238 15.4. Results 238 15.4.1. Out-of-sample Forecasting Procedure and Model Diagnostics 238 15.4.2. Out-of-Sample Forecasting Precision 239 15.4.3. Out-of-Sample Forecast Accuracy Test for FoHFs 247 15.4.4. In-Sample FoHFs Regime Determinants 247 Conclusion 251 Appendix 252 References 257 15.1. INTRODUCTION Several papers in the finance literature have investigated the performance of funds of hedge funds (FoHFs). It has been reported that, at least, three advan- tages of investing in FoHFs are as follows: n They are diversified since they represent investments in several funds. n Due diligence: a detailed review of their operation and management is offered to investors. n They require lower initial investments as hedge funds. Reconsidering Funds of Hedge Funds. http://dx.doi.org/10.1016/B978-0-12-401699-6.00015-0 Copyright Ó 2013 Elsevier Inc. All rights reserved. 229

Reconsidering Funds of Hedge Funds || Forecasting Funds of Hedge Funds Performance

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Reconsidering Funds of Hedge Funds. http://d

Copyright � 2013 Elsevier Inc. All rights reserve

CHAPTER 15

Forecasting Funds of HedgeFunds Performance: A MarkovRegime-Switching Approach

229

Szabolcs BlazsekSchool of Business, Universidad Francisco Marroquın, Guatemala

Chapter Outline

15.1. Introduction 229

15.2. Data 231

15.2.1. FoHFs and the

Hedge Fund

Database 231

15.2.2. Data Description 231

15.2.3. FoHFs Return

Drivers 233

15.3. Forecasting Models 235

15.3.1. AR(p) Model 235

15.3.2. AR(p)-GARCH(1,1)

Model 235

15.3.3. MS-AR(p) Model 237

15.3.4. MS-AR(p)-GARCH(1,1)

Model 237

15.3.5. MS-AR(p)-MS-GARCH(1,1)

Model 238

15.4. Results 238

x.doi.org/10.1016/B978

d.

15.4.1. Out-of-sample

Forecasting

Procedure and Model

Diagnostics 238

15.4.2. Out-of-Sample

Forecasting

Precision 239

15.4.3. Out-of-Sample

Forecast Accuracy

Test for FoHFs 247

15.4.4. In-Sample FoHFs

Regime

Determinants 247

Conclusion 251

Appendix 252

References 257

15.1. INTRODUCTIONSeveral papers in the finance literature have investigated the performance offunds of hedge funds (FoHFs). It has been reported that, at least, three advan-tages of investing in FoHFs are as follows:

n They are diversified since they represent investments in several funds.n Due diligence: a detailed review of their operation and management is

offered to investors.n They require lower initial investments as hedge funds.

-0-12-401699-6.00015-0

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SECTION 3 Performance230

One of the drawbacks of FoFHs, however, is that they may have high manage-ment and performance fees, decreasing realized returns.1 Moreover, it has beenreported that returns on FoHFs and hedge funds may be persistent, and thereforehistorical returns may provide information to forecast future fund performance(e.g., Gregoriou, 2003; Gregoriou et al., 2007; Abdou and Nasereddin, 2011;Jordan and Simlai, 2011; Laube et al., 2011).

In this chapter, we focus on forecasting the performance of FoHFs indices. Wecompare out-of-sample and multistep ahead forecasts of the performance offive FoHFs indices and 15 hedge fund indices over the period January2000eMarch 2012. Index performance is defined as excess return above theMSCI Global Equity market index. Out-of-sample forecasts are computed usingdata collected from Hedge Fund Research (HFR) for the period January1990eMarch 2012. To forecast, we use a large number of competing econo-metric specifications. These models are: AR(p), AR(p)-GARCH(1,1), MS-AR(p),MS-AR(p)-GARCH(1,1), and MS-AR(p)-MS-GARCH(1,1) specifications withp¼ 1, ., 12 lags. Consideration of the AR (autoregressive) model with severallags is motivated by Pesaran and Timmermann (2004), who show that ARmodels frequently produce smaller forecast errors than models with morecomplicated non-linear dynamics or with additional explanatory variables.The MS (Markov-switching) formulation is motivated by Billio et al. (2006,2009) and Blazsek and Downarowicz (2012), who find significant regime-switching dynamics in hedge fund returns. We use the GARCH (generalizedautoregressive conditional heteroscedasticity) model, since it is widely appliedfor modeling volatility in the literature (e.g., Donaldson and Kamstra, 1997;Dunis et al., 2003; Hansen and Lunde, 2005; Preminger et al., 2006; Muzzioli,2010). Out-of-sample forecasts are estimated by a moving time windowapproach. The first time window is set for January 1990eDecember 1999.Then, we extend the time window by adding one more observation and re-estimate all models to forecast fund performance for the subsequent months of2000. We repeat this approach until out-of-sample forecasts are obtained foreach month over the period January 2000eMarch 2012. Multistep aheadforecasts are provided for the 1-, 3-, 6-, 9-, and 12-month time horizons. Thenumber of specifications used for forecasting is 20 indices� 5 models� 12 ARlags¼ 1200. We perform diagnostic tests for each model and we forecast onlyby correctly specified models. Since for each specification there are fivedifferent forecast horizons, there may be 1200� 5 horizons¼ 6000 out-of-sample forecasts. Forecast accuracy is measured by the root mean squared error(RMSE) loss function. We break down the full forecasting period of January2000eMarch 2012 to pre-crisis, crisis, and post-crisis subperiods. RMSEs ofdifferent indices are compared in the full forecasting period, as well as in thethree subperiods. Moreover, the RMSE of each index is compared among the

1See Gregoriou and Rouah (2002), Gregoriou (2003), Gregoriou et al. (2007), Brown et al. (2008),

Abdou and Nasereddin (2011), Jordan and Simlai (2011), Laube et al. (2011), and Brown et al.(2012).

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 231

three subperiods. We apply the DieboldeMariano (DM) (Diebold andMariano, 1995) of predictive accuracy to see if forecast precision differences aresignificant.

We find that (i) FoHFs have significant regime-switching dynamics and (ii) thelowest RMSE model is an MS specification for several FoHFs. These resultsmotivate the following question that may be interesting for practitioners: whatdrives the regime-switching dynamics of FoHFs? To answer, we apply anendogenous switching AR model, where the transition probability of regimes isformulated according to a probit specification. The estimation results of thismodel identify specific factors driving FoHFs regimes. To the best of ourknowledge, MS models with endogenous switching have not been applied yet inthe hedge fund literature.

The remaining part of this chapter is organized as follows. Section 15.2 reviewsthe database applied, provides a data description, and analyzes FoHFs excessreturn drivers. Section 15.3 gives a brief overview of the econometric modelsapplied. Section 15.4 presents out-of-sample forecasting and model diagnosticprocedures, out-of-sample forecasting precision results, out-of-sample forecastaccuracy tests, and in-sample determinants of FoHFs regimes. This is followed bythe conclusion. Additional figures showing the probability of regimes forexogenous switching and endogenous switching models are presented inFigure 15.A1 in the Appendix.

15.2. DATA15.2.1. FoHFs and the Hedge Fund DatabaseSimilarly to Agarwal and Naik (2000), Jagannathan et al. (2010), and Blazsekand Downarowicz (2012), we use FoHFs and hedge fund index data obtainedfrom HFR. HFR Monthly Indices (HFRI) are equally weighted performanceindices, which are used as benchmarks in the hedge fund industry. As HFR datacontain information on when funds actually joined the database, they are cor-rected for backfill bias. See Ackermann et al. (1999) and Brown et al. (1999)about backfill bias. Moreover, the HFR database is constructed so as to accountfor survivorship bias, since if a fund closes, then the fund’s last reportedperformance will be included in the HFRI. Fung and Hsieh (1997) providea discussion on survivorship bias. Finally, Fung and Hsieh (2000) suggest thatdata biases may be less severe for FoHF data. They suggest using data on FoHFs,arguing that FoHFs returns are a more accurate representation of the returnsearned by hedge fund investors than hedge fund returns. For example, see alsoFung and Hsieh (1999), Hedges (2005), Goodworth and Jones (2007), Funget al. (2008), Bollen and Pool (2009), and Straumann (2009).

15.2.2. Data DescriptionThe dataset includes T¼ 267 observations of monthly returns of five FoHFsindices and 15 hedge fund indices over the period January 1990eMarch 2012.

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SECTION 3 Performance232

The indices used represent FoHFs and hedge fund performance at various levels.We classify them as follows:

n HFRI Fund of Funds Composite Index (FF).n Specific FoHFs strategies: HFRI FOF Strategic Index (FF1); HFRI FOF Market

Defensive Index (FF2); HFRI FOF Diversified Index (FF3); HFRI FOFConservative Index (FF4).

n HFRI Fund-Weighted Composite Index (HF).n Directional funds: HFRI Emerging Markets: Asia ex-Japan Index (D1); HFRI

Emerging Markets (Total) Index (D2); HFRI Equity Hedge (Total) Index(D3); HFRI EH: Short Bias Index (D4); HFRI Macro: Systematic Diversi-fied Index (D5); HFRI Macro (Total) Index (D6); HFRI EH: QuantitativeDirectional (D7).

n Non-directional funds: HFRI RV: Fixed Income-Corporate Index (ND1);HFRI RV: Fixed Income-Convertible Arbitrage Index (ND2); HFRI EH:Equity Market Neutral Index (ND3); HFRI Relative Value (Total) Index(ND4); HFRI Event-Driven (Total) Index (ND5); HFRI ED: Distressed/Restructuring Index (ND6); HFRI ED: Merger Arbitrage Index (ND7).

This classification is similar to that for the hedge fund groups defined in severalprevious studies (e.g., Agarwal and Naik, 2000; Ineichen, 2003; Gregoriou et al.,2007). Description of FoHFs and hedge fund indices is provided by HFR (seehttp://www.hedgefundresearch.com). See also Chan et al. (2005), who providedescriptions of specific hedge fund strategies.

In addition, monthly MSCI Global Equity market index data, obtained fromBloomberg, are used to compute excess returns on FoHFs and hedge fundindices over the MSCI index. MSCI data are collected for the same time span asFoHFs and hedge fund data. We denote the monthly excess returns on FoHFsand hedge fund indices by yt. The MSCI Global Equity market factor is used fortwo reasons. (i) Several papers evidence that the equity market index isa common risk factor of hedge fund indices (e.g., Agarwal and Naik, 1999, 2004;Liang, 1999; Chan et al., 2005; Billio et al., 2006). (ii) The MSCI index is a globalindex. Gregoriou (2003) notes that a great majority of FoHFs are geographicallydiversified and FoHFs managers use the MSCI Global Equity market index toprovide a benchmark for investors. As the hedge fund indices provided by HFRcorrespond to geographically diversified portfolios, we use the MSCI index asa benchmark for hedge fund indices as well.

Figure 15.1 shows the evolution of FoHFs composite and MSCI indices over theperiod 1990e2012. The initial value of both indices is normalized to 100. Thefigure shows that there is a significant comovement between the two indices.

Descriptive statistics of yt are presented in Table 15.1, which shows the mean,maximum, minimum, standard deviation (SD), skewness, and kurtosis esti-mates for each FoHF and hedge fund index. Table 15.1 also shows theaugmented DickeyeFuller (ADF) unit root test statistic (Dickey and Fuller,1979). The ADF specification estimated is with a constant term and without

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0 Jan-90

Jan-91

Jan-92

Jan-93

Jan-94

Jan-95

Jan-96

Jan-97

Jan-98

Jan-99

Jan-00

Jan-01

Jan-02

Jan-03

Jan-04

Jan-05

Jan-06

Jan-07

Jan-08

Jan-09

Jan-10

Jan-11

Jan-12

100

200

300

400

500

600(FF) HFRI Fund of Funds Composite Index

Correlation coefficient = 86.8%

MSCI Global Equity Market Index

FIGURE 15.1Evolution of FoHFs Composite and MSCI indices over the period 1990 to 2012.

Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 233

a linear time trend. The ADF statistics indicate that the unit root nullhypothesis is rejected for all FoHFs and hedge fund indices at the 1% level ofsignificance (i.e., all time series are covariance stationary). Table 15.1 alsoshows the LjungeBox (LB) test statistic computed for 12th-order serial corre-lation (Ljung and Box, 1978). The LB test is performed since several papers inthe hedge fund literature report significant serial correlation of hedge fundreturns (e.g., Gregoriou, 2003; Gregoriou et al., 2007; Fuss et al., 2007; Abdouand Nasereddin, 2011; Jordan and Simlai, 2011; Kang et al., 2010; Laube et al.,2011). The LB statistics show that the absence of serial correlation nullhypothesis is rejected for four directional hedge funds: D1, D2, D5, and D6,while it cannot be rejected for other indices. This result coincides with that ofHarri and Brorsen (2004), who show that some hedge fund strategies are morepersistent than others.

15.2.3. FoHFs Return DriversIn this section, we evaluate the dynamic relationships among FoHFs returns andsome factors that may be considered as FoHFs return drivers. We study somehedge fund return drivers reported in the literature (e.g., Agarwal and Naik,1999, 2004; Liang, 1999; Chan et al., 2005; Billio et al., 2006; Racicot andTheoret, 2007). The excess return drivers considered are: three factors of theFamaeFrench (1993) model (i.e., market risk premium factor (Rme Rf), smallminus big (SMB) factor, and high minus low (HML) factor); 1-month USTreasury bill (T-bill) rate; percentage change of monthly EUR/US$ exchangerate; percentage change of monthly crude oil per barrel price; percentage changeof monthly volatility VIX index. Return driver data are collected for the sametime span as FoHFs and hedge fund data.

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Table 15.1 Descriptive Statistics of yt

Mean Maximum Minimum SD Skewness Kurtosis ADF LB(12)

FoHFs(FF) HFRI FOF Composite Index 0.002 0.136 e0.109 0.039 0.284 4.101 e15.420*** 11.162(FF1) HFRI FOF: Strategic Index 0.004 0.134 e0.115 0.038 0.298 4.366 e14.850*** 13.932(FF2) HFRI FOF: MarketDefensive Index

0.002 0.205 e0.132 0.046 0.602 4.976 e14.360*** 12.728

(FF3) HFRI FOF: Diversified Index 0.002 0.133 e0.109 0.039 0.303 4.109 e15.279*** 11.176(FF4) HFRI FOF: ConservativeIndex

0.001 0.132 e0.106 0.040 0.328 3.636 e15.798*** 9.921

Hedge Funds(HF) HFRI Fund-WeightedComposite Index

0.005 0.122 e0.091 0.033 0.269 3.888 e15.803*** 15.156

Directional funds(D1) HFRI EmergingMarkets: Asiaex-Japan Index

0.004 0.107 e0.097 0.035 0.181 3.139 e13.137*** 28.164***

(D2) HFRI Emerging Markets(Total) Index

0.007 0.101 e0.083 0.034 e0.055 2.929 e12.215*** 53.042***

(D3) HFRI Equity Hedge (Total)Index

0.006 0.124 e0.083 0.032 0.600 4.432 e15.686*** 14.383

(D4) HFRI EH: Short Bias Index e0.003 0.329 e0.253 0.092 0.397 3.675 e14.763*** 7.269(D5) HFRI Macro: SystematicDiversified Index

0.005 0.249 e0.147 0.041 0.999 8.915 e14.015*** 31.954***

(D6) HFRI Macro (Total) Index 0.006 0.207 e0.122 0.043 0.506 4.952 e14.129*** 25.469**(D7) HFRI EH: QuantitativeDirectional

0.006 0.127 e0.113 0.030 0.369 4.951 e15.240*** 11.075

Non-directional funds(ND1) HFRI RV: Fixed Income-Corporate Index

0.002 0.102 e0.153 0.039 e0.035 3.883 e16.934*** 11.694

(ND2) HFRI RV: Fixed Income-Convertible Arbitrage Index

0.003 0.137 e0.107 0.039 0.483 3.808 e16.587*** 11.267

(ND3) HFRI EH: Equity MarketNeutral Index

0.001 0.185 e0.113 0.044 0.494 4.064 e15.069*** 10.629

(ND4) HFRI Relative Value (Total)Index

0.004 0.118 e0.101 0.040 0.314 3.305 e16.285*** 8.676

(ND5) HFRI Event-Driven (Total)Index

0.005 0.111 e0.107 0.035 0.111 3.674 e16.802*** 13.905

(ND6) HFRI ED: Distressed/Restructuring Index

0.006 0.111 e0.120 0.039 0.093 3.482 e16.212*** 18.165

(ND7) HFRI ED: Merger ArbitrageIndex

0.003 0.166 e0.101 0.041 0.362 3.661 e15.743*** 7.945

Asterisks denote test statistically significant at the **5% and ***1% levels, respectively. Data source: HFR.

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 235

Table 15.2 presents the dynamic correlation coefficients between each returndriver and the excess return on each FoHFs index. Correlation coefficients arecomputed for the first lag, contemporaneous value, and first lead of each returndriver. Bold numbers in Table 15.2 show that, for all FoHFs indices, significantcontemporaneous correlation is evidenced for the market risk premium,percentage change of the VIX index, and percentage change of the EUR/US$exchange rate.2 Furthermore, we find significant correlation between all FoHFsindices and the first lead of market risk premium, SMB factor, and percentagechange of the VIX index. Finally, Table 15.2 shows that some FoHFs are corre-lated with the T-bill rate and the HML factor as well.

15.3. FORECASTING MODELSIn this section, five econometric models used for forecasting purposes are pre-sented. For each model, we summarize the econometric specification, condi-tions of covariance stationarity, parameter estimation procedures, and multistepahead forecasting formulas.

15.3.1. AR(p) Model

yt ¼ cþ f1yt�1 þ.þ fpyt�p þ uut ; (15.1)

where ut is an N(0,1) distributed i.i.d. (independent and identically distributed)error term. The AR(p) structure of the mean equation is motivated by possibleserial correlation of returns reported in the existing literature (see the referencesin Section 15.2.2). We consider several specifications of this model by choosingdifferent values for p¼ 1, ., 12.3 Conditions of covariance stationarity arereported in Hamilton (1994). The parameters of the model are estimated by themaximum likelihood method (see Hamilton, 1994). The formula of n-stepahead forecast, E[yt þ njFt], is reported in Hamilton (1994). Ft e 1¼ (y1, ., yt)denotes the information set available at time t.

15.3.2. AR(p)-GARCH(1,1) Model

yt ¼ cþ f1yt�1 þ.þ fpyt�p þ εt; (15.2)

where εt is specified according to the GARCH(1,1) model of Bollerslev (1986)and Taylor (1986). The GARCH(1,1) specification is considered for two reasons.(i) The GARCH(1,1) model is widely applied in the finance literature (e.g.,Donaldson and Kamstra, 1997; Dunis et al. 2003; Hansen and Lunde, 2005;Preminger et al., 2006; Muzzioli, 2010). (ii) We have found that the forecastingperformance of models with GARCH(1,1) error specification is superior tothat of alternative GARCH models with more complicated lag structure.

2The EUR/US$ data used represent X EUR¼ 1 US$ (i.e., an increasing exchange rate implies

a stronger US$).3We consider p¼ 1, ., 12 for all models in this chapter.

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Table 15.2 Dynamic Correlation Coefficients of FoHFs and Hedge Fund Returns with Return Drivers

Return Drivers (FF) (FF1) (FF2) (FF3) (FF4)

RmeRf (FamaeFrench factor 1), lag 1 e0.019 0.006 e0.089 e0.016 e0.033SMB (FamaeFrench factor 2), lag 1 0.015 0.033 e0.028 0.023 e0.002HML (FamaeFrench factor 3), lag 1 0.047 0.019 0.092 0.039 0.063Rf (1-month US T-bill rate), lag 1 0.094 0.129** 0.094 0.087 0.088EUR/US$ % change, lag 1 e0.058 e0.053 e0.048 e0.052 e0.069Crude oil price % change, lag 1 0.026 0.021 0.011 0.020 0.025VIX index % change, lag 1 0.065 0.042 0.134*** 0.063 0.076RmeRf (FamaeFrench factor 1) e0.808*** e0.678*** e0.859*** e0.805*** e0.864***SMB (FamaeFrench factor 2) e0.011 0.098 e0.102 0.000 e0.096HML (FamaeFrench factor 3) 0.078 e0.023 0.143** 0.070 0.130**Rf (1-month US T-bill rate) 0.076 0.113* 0.079 0.068 0.069EUR/US$ % change 0.262*** 0.277*** 0.211*** 0.267*** 0.243***Crude oil price % change 0.051 0.048 0.024 0.053 0.032VIX index % change 0.441*** 0.351*** 0.490*** 0.441*** 0.491***RmeRf (FamaeFrench factor 1), lead 1 e0.119* e0.106* e0.149** e0.128** e0.107*SMB (FamaeFrench factor 2), lead 1 e0.129** e0.113* e0.146** e0.122** e0.137**HML (FamaeFrench factor 3), lead 1 e0.104 e0.095 e0.134** e0.103 e0.112*Rf (1-month US T-bill rate), lead 1 0.103 0.143** 0.093 0.095 0.092EUR/US$ % change, lead 1 0.096 0.118* 0.042 0.108* 0.085Crude oil price % change, lead 1 e0.070 e0.083 e0.089 e0.064 e0.096VIX index % change, lead 1 0.207*** 0.199*** 0.220*** 0.210*** 0.195***

HFRI Fund of Funds Composite Index (FF); HFRI FOF Strategic Index (FF1); HFRI FOF Market Defensive Index (FF2); HFRI FOF Diversified Index (FF3); HFRI FOF Conservative

Index (FF4). Asterisks denote test statistically significant at the *10%, **5% and ***1% levels, respectively. Data sources of return drivers: Bloomberg, Reuters, and DataStream.

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 237

Conditions of covariance stationarity for the GARCH(1,1) model are reported inBollerslev (1986). The parameters of the model are estimated by the maximumlikelihood method (see Hamilton, 1994). The n-step ahead forecasting formulais presented in Hamilton (1994).

15.3.3. MS-AR(p) Model

yt ¼ cðstÞ þ f1ðstÞyt�1 þ � � � þ fpðstÞyt�p þ uðstÞut ; (15.3)

where ut is an N(0,1) distributed i.i.d. error term. MS models have been devel-oped as a way of allowing data to arise from a combination of two or moredistinct data-generating processes (Hamilton, 1989; Kim and Nelson, 1999). Ateach time t, the actual process generating the data is determined by the reali-zation of a latent random discrete variable denoted by st, which is called a statevariable or regime. In the MS models, st over t¼ 1, ., T is assumed to forma Markov process. We consider a two-state MS model (i.e., st¼ 1 or 2), where stforms a Markov chain with a 2� 2 transition probability matrix, P¼ {hij}. Theelements of P are given by:

Pr½st ¼ 1jst�1 ¼ 1� ¼ h11Pr½st ¼ 1jst�1 ¼ 2� ¼ h12Pr½st ¼ 2jst�1 ¼ 1� ¼ h21Pr½st ¼ 2jst�1 ¼ 2� ¼ h22

(15.4)

where h11þ h21¼1 and h12þ h22¼1 are parameters of the MS model. Weassume that P is constant over time; in other words, we consider an exogenousswitching model.4 Francq and Zakoian (2001) give stationarity conditions forthe MS-AR(p) model. The parameters of the model are estimated by themaximum likelihood method (see Kim and Nelson, 1999). The n-step aheadforecast is computed by:

E½ytþnjFt � ¼ E½ytþnjFt ; stþn ¼ 1� � Pr½stþn ¼ 1jFt� þ E½YtþnjFt ; stþn

¼ 2� � Pr½stþn ¼ 2jFt � (15.5)

For regime i, E[yt þ njFt, st þ n¼ i] is computed according to Hamilton (1994) andPr[st þ n¼ ijFt] is computed according to Kim and Nelson (1999).

15.3.4. MS-AR(p)-GARCH(1,1) Model

yt ¼ cðstÞ þ f1ðstÞyt�1 þ � � � þ fpðstÞyt�p þ εt ; (15.6)

where εt is specified according to a single regime GARCH(1,1) model. Theparameters of the model are estimated by the maximum likelihood method (seeKim and Nelson, 1999). The n-step ahead forecast is computed by equation(15.5).

4In Section 15.4.4, we extend this model and we consider an endogenous switching AR model.

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SECTION 3 Performance238

15.3.5. MS-AR(p)-MS-GARCH(1,1) Model

yt ¼ cðstÞ þ f1ðstÞyt�1 þ � � � þ fpðstÞyt�p þ εtðstÞ; (15.7)

where εt(st) is specified according to the MS-GARCH(1,1) model of Klaassen(2002). Abramson and Cohen (2007) give stationarity conditions for Klaas-sen’s MS-GARCH model. The parameters of the model are estimated by themaximum likelihood method (see Klaassen, 2002). The n-step ahead forecastis computed by equation (15.5). Consideration of MS parameters in thevolatility equation is motivated, for example, by Diebold (1986), who notesthat the GARCH specification can be improved by including regime dummyvariables for the conditional variance intercept. Moreover, Friedman andLaibson (1989) note that the GARCHmodel does not differentiate between thepersistence of large and small shocks. We apply the non-path-dependent MSvolatility model of Klaassen (2002) for two reasons. (i) The MS-GARCHmodelof Klaassen (2002) can be estimated more rapidly than path-dependentMS-GARCH models (e.g., Dueker, 1997; Bauwens et al., 2010; Henneke et al.,2011). Therefore, it is more appropriate for repeated out-of sample forecastingpurposes. (ii) We have found that the forecasting performance of modelswith Klaassen’s volatility formulation is superior to alternative MS-ARCH(e.g., Hamilton and Susmel, 1994; Cai, 1994) and other non-path-dependentMS-GARCH models (e.g., Gray, 1996; Haas et al., 2004).

15.4. RESULTS15.4.1. Out-of-sample Forecasting Procedureand Model DiagnosticsOut-of-sample forecasts of yt are derived by dividing the full sample period(1990e2012) into two subsamples. The first subsample contains 120 obser-vations from January 1990 to December 1999, which are used to estimate theparameters of competing econometric specifications to produce out-of-sample forecasts of yt for the year 2000. Next, the dataset is updated by addingthe first month of the year 2000 to the previous subsample and the param-eters are re-estimated to produce the out-of-sample forecasts for the subse-quent months of the year 2000. This procedure is repeated until out-of-sample forecasts are obtained for each month over the period January2000eMarch 2012. Multistep ahead forecasts are estimated for 1, 3, 6, 9, and12 months. More formally, n-step ahead forecasts of yt þ n with n¼ 1, 3, 6, 9,and 12 are computed.

In the forecasting procedure, diagnostic tests are performed for each model infour steps. (i) The residuals corresponding to the i.i.d. error terms arecomputed and the following properties are verified: the mean of residuals iszero and there is no significant autocorrelation among the residuals. Weforecast by specifications where these two properties are satisfied. (ii)Stationarity of the dynamic models is verified. We find that the AR(p) and

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 239

MS-AR(p) specifications are stationary for all indices and for all models.Furthermore, we find that the GARCH(1,1) and MS-GARCH(1,1) models arestationary for all indices and for all models. (iii) When volatility is modeledaccording to the GARCH(1,1) formulation, we check the significance of thecoefficients corresponding to the dynamic terms of the volatility equation. Weonly consider a specification for forecasting purposes when the dynamicparameters of GARCH are significant. For the MS-GARCH model, in somecases, we find that the dynamic coefficients are significant for one regime, butthey are non-significant for the other regime. In these cases, we forecast withthe model, since it implies dynamic volatility for one regime and constantvolatility for the other regime. (iv) For the MS specifications, we study thesignificance of parameters in the transition probability matrix (i.e., h11 andh22) to see whether two different regimes of the return process exist. For theMS-AR(p) model, we find significant regime-switching dynamics for all FoHFsindices (i.e., FF, FF1, FF2, FF3, and FF4). Nevertheless, we find significantregime-switching only for the following six hedge fund indices: D3, D4, D5,D6, D7, and ND3 (i.e., most directional hedge funds are regime switching,but most non-directional hedge funds are not regime switching). Whenmore complicated MS-AR(p)-GARCH(1,1) and MS-AR(p)-MS-GARCH(1,1)models are considered, we still find significant MS dynamics for all FoHFsindices, but we find regime switching only for the following two hedge funds:D6 and ND3. From these results, it seems that FoHFs are more sensitive toswitches in the state variable, st, than most hedge funds. These results mayimply that MS models may produce more accurate forecasts of yt forFoHFs (see Sections 15.4.2 and 15.4.3). Furthermore, practitioners may beinterested in the following question: what drives the regime-switchingdynamics of different FoHFs indices? We investigate this question in Section15.4.4.

15.4.2. Out-of-Sample Forecasting PrecisionThe n-step ahead forecasting performance of models is verified using the RMSEloss function. The RMSE is the square root of the MSE, which is given by

MSEn ¼ ð1=nÞXtþn

s¼ tþ1

fys � E½ysjFt �g2; (15.8)

Table 15.3 reports the RMSE and the best forecasting specification for each indexfor the period January 2000eMarch 2012. Table 15.3 shows that, in most cases,an MS specification dominates forecasting performance for FoHFs excessreturns. The only exception is the FoHFs Conservative Index (FF4), where theAR(p) and AR(p)-GARCH specifications dominate forecast accuracy. We can alsosee that, in several cases, the MS-AR-GARCH and MS-AR-MS-GARCH specifica-tions are the best forecasters of FoHFs excess returns. However, these models arenot the best predictors of hedge fund performance in most cases. Table 15.3also shows that the single-regime AR and AR-GARCH specifications dominate

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Table 15.3 RMSE (%) for Full Forecasting Period, January 2000eMarch 2012

Out-of-Sample Forecast Horizon

1 Month 3 Months 6 Months 9 Months 12 Months

RMSEBestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel

FoHFs(FF) HFRI Fund of FundsComposite Index

3.803 M3(2) 3.926 M4(1) 3.912 M3(2) 3.914 M4(3) 3.869 M4(7)

(FF1) HFRI FOF: StrategicIndex

3.590 M4(2) 3.567 M4(2) 3.598 M3(7) 3.650 M1(3) 3.525 M2(8)

(FF2) HFRI FOF: MarketDefensive Index

4.873 M1(3) 4.881 M1(1) 4.869 M4(8) 4.697 M4(12) 4.720 M4(8)

(FF3) HFRI FOF: DiversifiedIndex

4.009 M3(2) 4.017 M4(1) 4.070 M5(1) 4.032 M4(3) 3.949 M1(8)

(FF4) HFRI FOF:Conservative Index

4.193 M3(3) 4.214 M1(1) 4.238 M2(1) 4.157 M2(11) 4.151 M1(8)

Hedge funds(HF) HFRI Fund WeightedComposite Index

3.416 M2(2) 3.416 M2(1) 3.461 M3(1) 3.394 M2(11) 3.370 M2(8)

Directional funds(D1) HFRI EmergingMarkets: Asia ex-JapanIndex

3.134 M1(1) 3.221 M1(3) 3.270 M1(3) 3.369 M1(8) 3.171 M1(8)

(D2) HFRI Emerging Markets(Total) Index

2.903 M2(1) 3.000 M2(1) 2.985 M2(1) 3.141 M2(3) 2.835 M1(8)

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(D3) HFRI Equity Hedge(Total) Index

3.017 M2(2) 3.009 M2(1) 3.050 M3(1) 3.034 M2(3) 2.982 M2(8)

(D4) HFRI EH: Short BiasIndex

9.389 M2(1) 9.335 M2(1) 9.317 M3(4) 9.275 M3(10) 9.320 M3(9)

(D5) HFRI Macro:Systematic Diversified Index

4.597 M1(6) 4.631 M1(1) 4.723 M2(3) 4.627 M2(6) 4.646 M2(1)

(D6) HFRI Macro (Total)Index

4.545 M4(4) 4.623 M2(1) 4.788 M2(7) 4.712 M4(3) 4.635 M2(2)

(D7) HFRI EH: QuantitativeDirectional

2.757 M2(4) 2.749 M2(1) 2.734 M3(7) 2.772 M3(7) 2.739 M2(4)

Non-directional funds(ND1) HFRI RV: FixedIncome-Corporate Index

3.960 M2(2) 4.003 M3(1) 3.999 M1(1) 3.884 M2(11) 3.954 M1(8)

(ND2) HFRI RV: FixedIncome-ConvertibleArbitrage Index

4.087 M2(1) 4.062 M2(5) 4.110 M2(1) 3.911 M2(11) 4.022 M1(8)

(ND3) HFRI EH: EquityMarket Neutral Index

4.697 M1(3) 4.704 M1(1) 4.761 M2(1) 4.713 M4(1) 4.669 M3(6)

(ND4) HFRI Relative Value(Total) Index

4.208 M1(1) 4.208 M1(1) 4.231 M1(1) 4.119 M2(9) 4.131 M1(5)

(ND5) HFRI Event-Driven(Total) Index

3.510 M2(2) 3.553 M1(1) 3.511 M2(1) 3.433 M2(11) 3.462 M1(8)

(ND6) HFRI ED: Distressed/Restructuring Index

3.859 M2(2) 3.903 M3(1) 3.898 M2(1) 3.777 M2(11) 3.810 M1(8)

(ND7) HFRI ED: MergerArbitrage Index

4.401 M1(1) 4.409 M1(1) 4.392 M2(1) 4.328 M2(1) 4.335 M1(8)

M1¼ AR(p); M2¼ AR(p)-GARCH(1,1); M3¼MS-AR(p); M4¼MS-AR(p)-GARCH(1,1); M5¼MS-AR(p)-MS-GARCH(1,1). The value of p is indicated in parentheses. The

lowest RMSE is given in bold; the highest RMSE is given in italic.

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Jan-00

Jul-00Jan-01

Jul-01Jan-02

Jul-02Jan-03

Jul-03Jan-04

Jul-04Jan-05

Jul-05Jan-06

Jul-06Jan-07

Jul-07Jul-08

Jan-08

Jul-09Jan-09

Jul-10Jan-10

Jul-11Jan-12

Jan-11

600

900

1200

1500

FIGURE 15.2Evolution of the S&P 500 Index over the period 2000 to 2012.

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forecasting performance for hedge fund indices. The exception is the Short Biasindex (D4), where an MS model dominates predictive accuracy.

We break down the full forecasted period of 2000e2012 to subperiods before,during, and after the US financial crisis to see how forecast accuracy of differentmodels changes in different subperiods. The pre-crisis period is from January2000 to September 2007; the crisis period is from October 2007 to February2009; the post-crisis period is fromMarch 2009 to March 2012. We present thesesubperiods on Figure 15.2, where the evolution of the S&P 500 index is shownover the period January 2000eMarch 2012.

Tables 15.4, 15.5, and 15.6 report the RMSE and the best forecasting speci-fication for the subperiods before, during, and after the financial crisis,respectively, for all FoHFs and hedge fund indices. In addition to comparingthe forecasts of different FoHFs and hedge fund indices, these tables also helpto compare forecasting performance among different subperiods. For allFoHFs and hedge fund indices, the results show that forecast accuracy is thelowest during the crisis period. We also find that, although the predictiveaccuracy improves after the financial crisis, the RMSE is higher in the post-crisis period than in the pre-crisis period for all indices. Tables 15.4, 15.5, and15.6 also show that the MS specifications dominate forecasting performancefor most FoHFs in all subperiods. The only exception is the FoHFs Conser-vative Index (FF4) for the subperiods before and during the financial crisis,where the AR(p) model dominates predictive accuracy (see Tables 15.4 and15.5). In the after crisis subperiod, MS specifications dominate forecastaccuracy for all FoHFs indices (see Table 15.6). Furthermore, Tables 15.4 and

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Table 15.4 RMSE (%) for Pre-crisis Period, January 2000eSeptember 2007

Out-of-Sample Forecast Horizon

1 Month 3 Months 6 Months 9 Months 12 Months

RMSEBestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel

FoHFs(FF) HFRI Fund of FundsComposite Index

3.063 M3(2) 3.275 M4(1) 3.151 M3(2) 3.217 M4(3) 3.184 M1(8)

(FF1) HFRI FOF: Strategic Index 2.955 M3(8) 2.978 M3(3) 2.928 M3(8) 2.989 M2(1) 2.886 M2(7)(FF2) HFRI FOF: Market DefensiveIndex

4.022 M5(3) 4.047 M3(2) 4.051 M1(1) 3.913 M4(12) 3.934 M5(7)

(FF3) HFRI FOF: Diversified Index 3.332 M3(2) 3.329 M3(2) 3.330 M3(1) 3.301 M4(3) 3.217 M1(8)(FF4) HFRI FOF: ConservativeIndex

3.472 M3(3) 3.515 M1(1) 3.529 M1(1) 3.440 M5(4) 3.413 M1(8)

Hedge funds(HF) HFRI Fund WeightedComposite Index

2.772 M1(8) 2.820 M2(2) 2.835 M1(8) 2.759 M2(6) 2.782 M1(8)

Directional funds(D1) HFRI Emerging Markets: Asiaex-Japan Index

2.943 M1(1) 3.060 M1(4) 2.921 M1(3) 3.221 M2(9) 3.031 M1(12)

(D2) HFRI Emerging Markets(Total) Index

2.650 M2(1) 2.801 M2(4) 2.551 M2(3) 2.963 M2(4) 2.665 M2(8)

(D3) HFRI Equity Hedge (Total)Index

2.694 M3(12) 2.682 M3(2) 2.713 M3(1) 2.680 M3(3) 2.703 M2(8)

(Continued)

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Table 15.4 RMSE (%) for Pre-crisis Period, January 2000eSeptember 2007dcont’d

Out-of-Sample Forecast Horizon

1 Month 3 Months 6 Months 9 Months 12 Months

RMSEBestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel

(D4) HFRI EH: Short Bias Index 9.061 M2(1) 8.981 M3(1) 8.747 M3(4) 8.801 M2(11) 8.871 M3(4)(D5) HFRI Macro: SystematicDiversified Index

2.726 M3(6) 2.810 M1(1) 2.821 M1(1) 2.780 M1(1) 2.792 M1(1)

(D6) HFRI Macro (Total) Index 3.641 M4(4) 3.682 M2(3) 3.697 M2(12) 3.678 M4(3) 3.688 M2(1)(D7) HFRI EH: QuantitativeDirectional

2.140 M3(1) 2.141 M3(1) 2.057 M2(11) 2.122 M3(2) 2.126 M3(1)

Non-directional funds(ND1) HFRI RV: Fixed Income-Corporate Index

3.503 M2(2) 3.506 M2(5) 3.594 M3(1) 3.444 M2(11) 3.511 M2(1)

(ND2) HFRI RV: Fixed Income-Convertible Arbitrage Index

3.935 M2(1) 3.924 M2(5) 3.989 M2(3) 3.747 M2(11) 3.833 M1(8)

(ND3) HFRI EH: Equity MarketNeutral Index

3.984 M4(2) 4.013 M3(1) 4.014 M3(1) 3.941 M4(1) 3.920 M3(5)

(ND4) HFRI Relative Value (Total)Index

3.708 M3(1) 3.708 M3(1) 3.759 M1(1) 3.617 M2(10) 3.611 M1(7)

(ND5) HFRI Event-Driven (Total)Index

2.947 M2(2) 3.025 M3(1) 3.021 M3(1) 2.896 M2(11) 2.936 M1(8)

(ND6) HFRI ED: Distressed/Restructuring Index

3.422 M2(2) 3.451 M3(1) 3.452 M1(1) 3.362 M2(10) 3.350 M1(8)

(ND7) HFRI ED: Merger ArbitrageIndex

3.576 M2(1) 3.665 M1(1) 3.669 M3(1) 3.524 M2(1) 3.563 M1(8)

M1¼ AR(p); M2¼ AR(p)-GARCH(1,1); M3¼MS-AR(p); M4¼MS-AR(p)-GARCH(1,1); M5¼MS-AR(p)-MS-GARCH(1,1). The value of p is indicated in parentheses. The

lowest RMSE is given in bold; the highest RMSE is given in italic.

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Table 15.5 RMSE (%) for Crisis period, October 2007eFebruary 2009

Out-of-Sample Forecast Horizon

1 Month 3 Months 6 Months 9 Months 12 Months

RMSEBestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel

FoHFs(FF) HFRI Fund of Funds Composite Index 5.182 M3(2) 5.314 M3(7) 5.391 M3(4) 5.251 M3(2) 5.290 M3(2)(FF1) HFRI FOF: Strategic Index 4.652 M1(3) 4.547 M3(7) 4.694 M3(4) 4.812 M1(3) 4.680 M3(3)(FF2) HFRI FOF: Market Defensive Index 6.090 M4(8) 6.328 M4(7) 6.984 M4(8) 7.022 M4(8) 6.648 M4(8)(FF3) HFRI FOF: Diversified Index 5.349 M3(2) 5.415 M3(1) 5.585 M3(4) 5.405 M3(2) 5.418 M3(2)(FF4) HFRI FOF: Conservative Index 5.724 M1(8) 5.806 M3(6) 5.860 M1(7) 5.703 M5(3) 5.835 M1(3)Hedge funds(HF) HFRI Fund WeightedComposite Index

4.820 M1(8) 4.888 M3(1) 4.891 M1(7) 4.909 M3(2) 4.899 M3(1)

Directional funds(D1) HFRI Emerging Markets:Asia ex-Japan Index

3.712 M1(4) 3.750 M1(3) 4.227 M1(12) 3.961 M1(12) 3.781 M1(8)

(D2) HFRI Emerging Markets (Total) Index 3.289 M2(11) 3.593 M2(1) 3.776 M2(12) 3.441 M2(12) 3.354 M1(4)(D3) HFRI Equity Hedge (Total) Index 3.860 M3(2) 3.860 M2(6) 3.855 M3(4) 3.882 M3(11) 3.825 M3(5)(D4) HFRI EH: Short Bias Index 11.034 M3(1) 11.115 M3(1) 10.930 M3(11) 10.982 M3(11) 11.016 M3(4)(D5) HFRI Macro: SystematicDiversified Index

7.763 M1(12) 8.162 M3(1) 7.931 M3(2) 8.331 M3(1) 7.977 M3(7)

(D6) HFRI Macro (Total) Index 6.719 M2(11) 6.953 M3(8) 7.187 M3(3) 7.030 M3(4) 7.128 M3(2)(D7) HFRI EH: Quantitative Directional 3.381 M2(10) 3.495 M2(1) 3.451 M3(6) 3.368 M3(6) 3.332 M3(11)Non-directional funds(ND1) HFRI RV:Fixed Income-Corporate Index

4.985 M1(8) 5.062 M1(6) 5.145 M1(1) 5.140 M3(2) 5.119 M1(4)

(ND2) HFRI RV: Fixed Income-ConvertibleArbitrage Index

5.159 M1(12) 5.127 M1(6) 5.260 M3(1) 5.151 M1(7) 5.284 M3(1)

(ND3) HFRI EH: Equity MarketNeutral Index

6.671 M3(3) 6.815 M3(1) 6.906 M3(8) 6.777 M3(3) 6.741 M3(2)

(ND4) HFRI Relative Value (Total) Index 5.320 M1(8) 5.323 M1(6) 5.467 M1(7) 5.483 M1(2) 5.464 M1(4)(ND5) HFRI Event-Driven (Total) Index 4.710 M1(8) 4.865 M1(1) 4.776 M3(1) 4.761 M3(2) 4.798 M1(4)(ND6) HFRI ED:Distressed/Restructuring Index

4.872 M1(8) 5.173 M3(1) 5.192 M3(1) 5.143 M3(2) 5.154 M1(2)

(ND7) HFRI ED: Merger Arbitrage Index 6.138 M3(3) 6.214 M1(9) 5.763 M3(3) 5.990 M3(3) 5.612 M3(3)

M1¼ AR(p); M2¼ AR(p)-GARCH(1,1); M3¼MS-AR(p); M4 ¼MS-AR(p)-GARCH(1,1); M5¼MS-AR(p)-MS-GARCH(1,1). The value of p is indicated in parentheses. The

lowest RMSE is given in bold; the highest RMSE is given in italic.

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Table 15.6 RMSE (%) for Post-Crisis Period, March 2009eMarch 2012

Out-of-Sample Forecast Horizon

1 Month 3 Months 6 Months 9 Months 12 Months

RMSEBestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel RMSE

BestModel

FoHFs(FF) HFRI Fund of FundsComposite Index

4.489 M5(1) 4.480 M2(4) 4.542 M5(1) 4.490 M4(11) 4.351 M4(8)

(FF1) HFRI FOF: Strategic Index 4.245 M2(1) 4.081 M4(4) 4.415 M3(7) 4.401 M3(8) 4.106 M4(9)(FF2) HFRI FOF: Market Defensive Index 5.136 M4(2) 5.111 M2(1) 5.045 M4(8) 4.879 M4(12) 4.718 M4(12)(FF3) HFRI FOF: Diversified Index 4.501 M5(3) 4.581 M2(4) 4.542 M5(3) 4.585 M2(11) 4.437 M4(8)(FF4) HFRI FOF: Conservative Index 4.796 M5(4) 4.796 M5(3) 4.799 M5(4) 4.676 M5(4) 4.722 M2(8)Hedge funds(HF) HFRI Fund WeightedComposite Index

3.880 M2(2) 3.871 M2(1) 3.950 M2(2) 3.798 M2(11) 3.694 M2(8)

Directional funds(D1) HFRI Emerging Markets: Asia ex-Japan Index

3.252 M2(1) 3.286 M3(2) 3.489 M2(8) 3.257 M1(8) 3.127 M2(8)

(D2) HFRI Emerging Markets (Total) Index 3.101 M1(9) 3.109 M2(1) 3.384 M2(1) 3.096 M1(9) 2.934 M1(4)(D3) HFRI Equity Hedge (Total) Index 3.188 M2(3) 3.102 M2(4) 3.228 M2(4) 3.162 M2(11) 3.036 M2(8)(D4) HFRI EH: Short Bias Index 9.184 M2(6) 8.994 M2(6) 9.518 M3(10) 9.457 M3(10) 9.126 M2(2)(D5) HFRI Macro: Systematic DiversifiedIndex

5.642 M2(2) 5.624 M2(1) 5.703 M2(2) 5.630 M2(2) 5.565 M2(2)

(D6) HFRI Macro (Total) Index 5.112 M4(4) 5.119 M4(1) 5.494 M4(1) 5.366 M2(2) 5.098 M2(2)(D7) HFRI EH: Quantitative Directional 3.532 M2(5) 3.501 M2(3) 3.562 M3(7) 3.554 M3(7) 3.464 M2(9)Non-directional funds(ND1) HFRI RV: FixedIncome-Corporate Index

4.275 M2(2) 4.360 M2(1) 4.281 M2(1) 4.105 M2(11) 4.191 M1(9)

(ND2) HFRI RV: FixedIncome-Convertible Arbitrage Index

3.684 M3(4) 3.760 M2(2) 3.742 M2(1) 3.618 M2(9) 3.674 M2(10)

(ND3) HFRI EH: Equity Market NeutralIndex

5.015 M2(3) 5.004 M4(1) 5.044 M4(2) 5.020 M2(11) 4.880 M2(8)

(ND4) HFRI Relative Value (Total) Index 4.564 M2(2) 4.591 M2(1) 4.518 M2(2) 4.456 M2(11) 4.458 M2(8)(ND5) HFRI Event-Driven (Total) Index 3.934 M2(2) 3.989 M2(1) 3.864 M2(2) 3.777 M2(11) 3.846 M1(9)(ND6) HFRI ED:Distressed/Restructuring Index

4.112 M2(6) 4.248 M2(4) 4.165 M2(1) 3.904 M2(11) 4.115 M1(9)

(ND7) HFRI ED: Merger Arbitrage Index 4.993 M2(2) 5.001 M2(1) 4.849 M2(2) 4.836 M2(11) 4.844 M2(10)

M1¼ AR(p); M2¼ AR(p)-GARCH(1,1); M3¼MS-AR(p); M4¼MS-AR(p)-GARCH(1,1); M5¼MS-AR(p)-MS-GARCH(1,1). The value of p is indicated in parentheses. The

lowest RMSE is given in bold; the highest RMSE is given in italic.

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15.6 show that, in most cases, the AR(p) and AR(p)-GARCH models arethe most accurate forecasters for hedge fund indices in the pre-crisis andpost-crisis subperiods. Nevertheless, Table 15.5 shows that for most hedgefund indices the MS models dominate forecasting performance during thecrisis period. Forecasting benefits provided by the MS specifications duringthe financial crisis are especially important for the following directional hedgefund indices: D3, D4, D5, D6, and D7. Moreover, the MS models alsoperform well during the same period for the next non-directional hedge fundindices: ND3, ND6, and ND7.

15.4.3. Out-of-Sample Forecast Accuracy Test for FoHFsThe RMSE metric cannot determine if a given forecasting framework is, in fact,significantly better than another. To evaluate the difference between alternativeforecasting models, we perform the predictive accuracy test of Diebold andMariano (1995). The null hypothesis of this test is that the forecast accuracy oftwo competing models is equal. The DM statistic is computed for the differenceof squared forecast errors of two competing specifications. Positive values ofthe DM statistic indicate superior forecasting performance of the secondspecification, while negative values reflect better forecasting performance of thefirst specification. The DM test statistics for the full forecasting period, pre-crisisperiod, crisis period, and post-crisis period are presented in Table 15.7. (i)Table 15.7 presents the difference between the predictive accuracy of FoHFscomposite and hedge fund composite indices. According to the results, thehedge fund Composite Index can be forecasted significantly more preciselythan the FoHFs Composite Index for most forecast horizons. (ii) Table 15.7compares the forecasting performance among all specific FoHFs indices. TheDM test results exhibit that the most precise forecasts are obtained for FF1,followed by FF3, FF4, and FF2 for the full forecasting period, as well as for thethree subperiods.

15.4.4. In-Sample FoHFs Regime DeterminantsThe results reported so far have shown that: (i) the evolution of FoHFs excessreturns can be modeled by regime-switching models (Section 15.4.1) and (ii)in many cases the most accurate out-of-sample forecasting model of FoHFsexcess returns is a regime-switching specification (Section 15.4.2). It may beinteresting for practitioners to investigate what are the determinants of FoHFsregime-switching dynamics. To answer this question, we present in-sampleestimation results for FoHFs data, over the period 1990e2012, for thefollowing econometric models: MS-AR(1) model with exogenous switching(Section 15.3.3) and MS-AR(1) model with endogenous switching. Theendogenous switching model estimated is an extension of the MS-AR(1) modelpresented in Section 15.3.3. The conditional mean equation of both models isthe same: equation (15.3) with p¼ 1. Nevertheless, the transition probabilitymatrix, P, of these models is different. In the exogenous switching model, P isgiven by equation (15.4) (i.e., the elements of P are constant parameters).

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Table 15.7 DieboldeMariano (1995) Predictive Accuracy Test

Time horizon

1 Month 3 Months 6 Months 9 Months 12 Months

DM p-value DM p-value DM p-value DM p-value DM p-value

Full forecast period(FF)e(HF) 3.215 0.001 6.519 0.000 4.265 0.000 4.824 0.000 5.442 0.000(FF1)e(FF2) e4.545 0.000 e4.521 0.000 e3.873 0.000 e3.895 0.000 e4.655 0.000(FF1)e(FF3) e3.857 0.000 e4.427 0.000 e3.611 0.000 e3.532 0.000 e3.611 0.000(FF1)e(FF4) e4.225 0.000 e5.109 0.000 e4.397 0.000 e3.591 0.000 e4.922 0.000(FF2)e(FF3) 3.773 0.000 3.888 0.000 3.360 0.001 3.271 0.001 3.557 0.000(FF2)e(FF4) 3.141 0.002 3.007 0.003 2.684 0.007 2.834 0.005 2.758 0.006(FF3)e(FF4) e2.209 0.027 e4.124 0.000 e3.079 0.002 e1.518 0.129 e4.109 0.000Pre-crisis period(FF)e(HF) 1.527 0.127 4.285 0.000 2.117 0.034 3.871 0.000 4.093 0.000(FF1)e(FF2) e3.649 0.000 e4.221 0.000 e3.993 0.000 e2.982 0.003 e3.160 0.002(FF1)e(FF3) e2.586 0.010 e2.475 0.013 e2.577 0.010 e2.181 0.029 e2.101 0.036(FF1)e(FF4) e2.347 0.019 e2.838 0.005 e2.812 0.005 e1.932 0.053 e3.032 0.002(FF2)e(FF3) 3.695 0.000 3.731 0.000 4.249 0.000 2.851 0.004 2.782 0.005(FF2)e(FF4) 3.087 0.002 2.636 0.008 3.453 0.001 2.405 0.016 2.360 0.018(FF3)e(FF4) e1.005 0.315 e1.838 0.066 e2.023 0.043 e0.995 0.320 e2.385 0.017Crisis period(FF)e(HF) 1.494 0.135 2.033 0.042 2.520 0.012 3.118 0.002 2.589 0.010(FF1)e(FF2) e1.905 0.057 e2.464 0.014 e1.636 0.102 e2.721 0.007 e1.902 0.057(FF1)e(FF3) e2.228 0.026 e2.192 0.028 e2.334 0.020 e1.977 0.048 e2.481 0.013(FF1)e(FF4) e2.119 0.034 e2.756 0.006 e2.622 0.009 e1.665 0.096 e2.823 0.005(FF2)e(FF3) 1.073 0.283 1.834 0.067 1.314 0.189 2.687 0.007 1.473 0.141(FF2)e(FF4) 0.484 0.629 1.149 0.251 1.129 0.259 2.376 0.017 1.090 0.276(FF3)e(FF4) e1.328 0.184 e2.790 0.005 e1.832 0.067 e0.899 0.368 e3.257 0.001Post-crisis period(FF)e(HF) 3.369 0.001 4.006 0.000 3.960 0.000 3.844 0.000 3.442 0.001(FF1)e(FF2) e2.905 0.004 e3.216 0.001 e1.155 0.248 e1.338 0.181 e2.187 0.029(FF1)e(FF3) e1.630 0.103 e4.149 0.000 e0.763 0.445 e1.310 0.190 e1.920 0.055(FF1)e(FF4) e2.367 0.018 e3.821 0.000 e1.912 0.056 e1.326 0.185 e3.052 0.002(FF2)e(FF3) 2.340 0.019 2.168 0.030 1.141 0.254 0.825 0.409 0.945 0.345(FF2)e(FF4) 1.158 0.247 1.313 0.189 0.579 0.563 0.473 0.636 e0.014 0.989(FF3)e(FF4) e2.666 0.008 e1.858 0.063 e3.334 0.001 e0.522 0.602 e4.042 0.000

HFRI Fund of Funds Composite Index (FF); HFRI FOF Strategic Index (FF1); HFRI FOF Market Defensive Index (FF2); HFRI FOF Diversified Index (FF3); HFRI FOF Conservative

Index (FF4).

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 249

In the endogenous switching model, we use a probit specification for st, and theelements of P are modeled as:

Pr½st ¼ 1jst�1 ¼ 1� ¼ F½aðst�1 ¼ 1Þ þ Z0tbðst�1 ¼ 1Þ�

Pr½st ¼ 1jst�1 ¼ 2� ¼ F½aðst�1 ¼ 2Þ þ Z0tbðst�1 ¼ 2Þ�

Pr½st ¼ 2jst�1 ¼1� ¼ 1� F½aðst�1 ¼ 1Þ þ Z0tbðst�1 ¼ 1Þ�

Pr½st ¼ 2jst�1 ¼ 2� ¼ 1� F½aðst�1 ¼ 2Þ þ Z0tbðst�1 ¼ 2Þ�;

(15.9)

where F is the distribution function of the N(0,1) distribution, Zt is a vector ofexplanatory variables, a(st e 1) is a regime-switching parameter (constant in eachregime), and b(st e 1) is regime-switching vector of parameters capturing theimpact of Zt on the transition probabilities. We choose the variables in Zt based onthe results reported in Section 15.2.3. As there are significant contemporaneouscorrelations between each FoHF’s excess return and themarket risk premium (b1),percentage change in the EUR/US$ exchange rate (b2), and percentage change inthe VIX index (b3), we consider these variables in Zt. We estimate the endogenousswitching MS-AR(1) model for all possible combinations of these variablesincluded in Zt. The parameter estimates are obtained by the maximum likelihoodmethod summarized in Kim et al. (2008). The best performing specificationis selected by using the Bayesian Information Criterion (BIC).

For the exogenous switching model and the best-performing endogenousswitching model, Table 15.8 reports the parameter estimates and the followinglikelihood-based model performance metrics: log likelihood (LL), BIC, andlikelihood ratio (LR) test statistic. For some FoHFs indices, the BIC metricsuggests better model performance of the exogenous switching model. Never-theless, the difference between the BIC measures may not be statisticallysignificant. To validate the use of the endogenous switching model, we applytwo tests. (i) We implement the t-test suggested by Kim et al. (2008). Theysuggest testing the significance of the correlation coefficient, r, which capturesthe correlation between the error terms of the mean equation and the probitequation. They propose this test since if r is non-significant, then the endoge-nous switching model will reduce to the exogenous switching model. Table 15.8shows that r is significant at the 1% level for all FoHFs indices. This supports theendogenous switching specification. (ii) We apply the LR test to see if there isa significant difference between the LL of the two models. Since the two MSspecifications are non-nested models, the non-nested LR approach of Vuong(1989) is used to evaluate the significance of the LR statistic; see Table 15.8. Forall FoHFs indices, the LR test statistic is significant, at least, at the 10% level,supporting the endogenous switching model.

For all FoHFs indices and bothmodels, the figures showing the estimated filteredprobability of the first regime, Pr[st¼ 1jy1, ., yt e 1], are presented inFigure 15.A1A in theAppendix.Notice that, in all figures, the first regime (st¼ 1) isthe ‘low-volatility regime,’ while the second regime (st¼ 2) is the ‘high-volatilityregime.’ This idea helps to interpret the b(st e 1) coefficients of the best endoge-nous switching specification (see Table 15.8). (i) b1(st e 1)measures the impact of

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Table 15.8 In-sample Estimation Results of Exogenous Switching and Endogenous Switching AR(1) Models

Exogenous Switching AR(1) Model Endogenous Switching AR(1) Model

(FF) (FF1) (FF2) (FF3) (FF4) (FF) (FF1) (FF2) (FF3) (FF4)

c1 e0.004* e0.002 e0.006** e0.004 e0.004* c1 e0.001 0.003 0.003 e0.003 e0.004

c2 0.004 0.008 0.010 0.006 0.004 c2 e0.005 e0.008 e0.005 e0.001 0.003

41 e0.243* 0.002 e0.227** e0.110 e0.154 41 e0.158** 0.044 0.188*** 0.067* e0.216***

42 0.073 0.081 0.192* 0.078 0.053 42 0.023 0.034 e0.213*** e0.114*** 0.145***

u1 0.018*** 0.018*** 0.025*** 0.021*** 0.023*** u1 0.021 0.019 0.064* 0.062* 0.028

u2 0.045*** 0.047*** 0.062*** 0.049*** 0.049*** u2 0.056* 0.052* 0.026 0.023 0.060*

h11 0.982*** 0.944*** 0.934*** 0.964*** 0.984*** a(1) 3.651*** 1.434*** 1.526*** 0.951*** 3.746***

h22 0.960*** 0.929*** 0.947*** 0.956*** 0.973*** a(2) e0.927*** e0.749*** e1.695*** e6.291*** e1.344***

b1(1) 4.377*** 1.536*** NA 0.739*** NA

b1(2) e0.965*** e1.216*** NA e9.152*** NA

b2(1) NA NA NA NA NA

b2(2) NA NA NA NA NA

b3(1) e0.440*** NA NA NA e9.453***

b3(2) 0.057*** NA NA NA 0.871***

r 0.702*** 0.713*** 0.513*** 0.766*** 0.703***

Pr 0.000 0.000 0.000 0.000 0.000

Model performance metrics

LL 519.280 526.719 478.093 518.834 502.718 534.980 538.646 480.846 535.589 512.960

BIC e993.863 e1008.740 e911.488 e992.971 e960.739 e991.738 e1010.245 e905.819 e1004.130 e958.874

LR 31.400*** 23.854*** 5.506* 33.510*** 20.484***

HFRI Fund of Funds Composite Index (FF); HFRI FOF Strategic Index (FF1); HFRI FOF Market Defensive Index (FF2); HFRI FOF Diversified Index (FF3); HFRI FOF Conservative

Index (FF4). Pr is a parameter capturing the probability of the first regime in the initial period (see Kim et al., 2008, p. 266). Asterisks denote significance at the *10%,

***5%, and ***1% levels, respectively.

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Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 251

the market risk premium on the transition probabilities of st. The market riskpremium is included in the probit equation for the FoHFsComposite (FF), FoHFsStrategic (FF1), and FoHFs Diversified (FF3) indices. We find that b1(st e 1¼1) ispositive and b1(st e 1¼ 2) is negative for the three indices. We can interpret theseestimates based on equation (15.9) as follows. If the market risk premiumincreases, then the probability of staying in the same regime will increase.Moreover, if themarket risk premiumdecreases, then the probability of changingto the other regime will increase. The lowest b1(st e 1¼ 2) coefficient is observedfor the FoHFsDiversified Index (FF3). This shows that if themarket risk premiumdecreases, then the FoHFs Diversified Index will change from the high-volatilityregime to the low-volatility regime with a relatively high probability. (ii) We donot report the estimates of b2(st e 1) in Table 15.8, since the best performingspecification never includes the EUR/US$ exchange rate. (iii) b3(st e 1) capturesthe impact of the percentage change in the VIX volatility index variable on thetransition probabilities. The VIX index is included in the probit equation forFoHFs Composite (FF) and FoHFs Conservative (FF4) indices. We find thatb3(st e 1¼1) is negative and b3(st e 1¼ 2) is positive for both indices. According tothis result, if market volatility increases, then the probability of changing to theother regime will increase. Furthermore, if market volatility decreases, then theprobability of staying in the same regime will increase. Finally, the lowestb3(st e 1¼1) parameter is found for the FoHFs Conservative Index (FF4). Thisshows that if the VIX volatility index increases, then the FoHFsConservative Indexwill change from the low-volatility regime to the high-volatility regime witha relatively high probability.

CONCLUSIONIn this chapter we focus on forecasting the performance of FoHFs indices. Wecompare the predictive accuracy of out-of-sample and multistep ahead fore-casts of the performance of five FoHFs indices and 15 hedge fund indices. Theperformance of these indices is measured by their excess rate of return abovethe MSCI Global Equity market index. We use data obtained from HFR for theperiod January 1990eJanuary 2012. We apply a large number of AR and MS-AR specifications for the conditional mean of excess returns and we useGARCH and MS-GARCH specifications for the conditional volatility of excessreturns. We perform diagnostic tests for each model and implement only thecorrectly specified models for forecasting purposes. The full out-of-sampleforecasting period is from January 2000 to March 2012. This period is brokendown to three subperiods: before, during, and after the financial crisis. Wecompare the predictive accuracy of competing models among these subpe-riods. Out-of-sample forecasts are obtained using a moving time windowapproach. Multistep ahead forecasts are estimated for the 1, 3, 6, 9, and 12months time horizons. Forecast precision is measured by the RMSE lossfunction.

The estimation results show that forecast accuracy is the lowest during the financialcrisis for all indices. We also find that, although the predictive accuracy improves

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FIGURE 15.A1Filtered probabilities of FoH

SECTION 3 Performance252

after the financial crisis, the RMSE is significantly higher in the post-crisis periodthan in thepre-crisisperiod for all indices.Weuse theDMpredictive accuracy test toshowsignificantdifferencesof forecast errors: betweenFoHFscomposite andhedgefund composite indices, and among all specific FoHFs indices.

We provide the following contributions:

n We find that excess returns on all FoHFs indices have regime-switchingdynamics.

n We find that, in most cases, an MS specification dominates the predictiveaccuracy for FoHFs indices.

n We investigate the determinants of FoHFs regimes by using an endogenousswitching AR model. The estimation results show that the market riskpremium and the VIX volatility index are significant determinants of FoHFsregimes, but their importance depends on the specific FoHFs index.

APPENDIX

Fs for exogenous switching (a, c, e, g, and i) and endogenous switching (b, d, f, h, and j) models.

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FIGURE 15.A1Continued.

Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 253

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FIGURE 15.A1Continued.

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FIGURE 15.A1Continued.

Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 255

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FIGURE 15.A1Continued.

SECTION 3 Performance256

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FIGURE 15.A1Continued.

Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 257

ReferencesAbdou, K., & Nasereddin, M. (2011). The Persistence of Hedge Fund Strategies in Different

Economic Periods: A Support Vector Machine Approach. Journal of Derivatives and Hedge Funds,

17(1), 2e15.

Abramson, A., & Cohen, I. (2007). On the Stationarity of Markov Switching GARCH Processes.Econometric Theory, 23(3), 485e500.

Ackermann, C., McEnally, R., & Ravenscraft, D. (1999). The Performance of Hedge Funds: Risk,

Return and Incentives. The Journal of Finance, 54(3), 833e874.

Agarwal, V., & Naik, N. Y. (1999). On Taking the ‘Alternative’ Route: Risks, Rewards and Perfor-mance Persistence of Hedge Funds. The Journal of Alternative Investments, 2(4), 6e23.

Agarwal, V., & Naik, N. Y. (2000). Multi-Period Performance Persistence Analysis of Hedge Funds.

Journal of Financial and Quantitative Analysis, 35(3), 327e342.Agarwal, V., & Naik, N. Y. (2004). Risk and Portfolio Decisions Involving Hedge Funds. Review of

Financial Studies, 17(1), 63e98.

Bauwens, L., Preminger, A., & Romboust, J. V. K. (2010). Theory and Inference for a Markov

Switching GARCH Model. Econometrics Journal, 13(2), 218e244.Billio, M., Getmansky, M., & Pelizzon, L. (2006). Phase-Locking and Switching Volatility in Hedge

Funds. In Working Paper 54/WP/2006, Department of Economics. Venice: Ca’ Foscari University of

Venice.

Billio, M., Getmansky, M., & Pelizzon, L. (2009). Crises and Hedge Fund Risk. New Haven, CT: YaleSchool of Management Working Paper AMZ2561, Yale School of Management.

Page 30: Reconsidering Funds of Hedge Funds || Forecasting Funds of Hedge Funds Performance

SECTION 3 Performance258

Blazsek, S., & Downarowicz, A. (2012). Forecasting Hedge Fund Volatility: A Markov Regime-Switching

Approach. The European Journal of Finance. http://www.tandfonline.com/doi/abs/10.1080/1351847X.2011.653576.

Bollen, N. P. B., & Pool, V. K. (2009). Do Hedge Fund Managers Misreport Returns? Evidence from

the Pooled Distribution. The Journal of Finance, 63(5), 2257e2288.

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econo-metrics, 31(3), 307e327.

Brown, S. J., Goetzmann, W. N., & Ibbotson, R. G. (1999). Offshore Hedge Funds: Survival and

Performance 1989-1995. Journal of Business, 72(1), 91e118.

Brown, S. J., Fraser, T., & Liang, B. (2008). Hedge Fund Due Diligence: A Source of Alpha in a HedgeFund Portfolio Strategy. Journal of Investment Management, 6(4), 22e33.

Brown, S. J., Gregoriou, G. N., & Pascalau, R. (2012). Diversification in Funds of Hedge Funds: Is It

Possible to Overdiversify? The Review of Asset Pricing Studies, 2(1), 89e110.

Cai, J. (1994). A Markov Model of Switching-Regime ARCH. Journal of Business and EconomicStatistics, 12(3), 309e316.

Chan, N., Getmansky, M., Hass, S. M., & Lo, A. W. (2005). Systemic Risk and Hedge Funds. Working

Paper 11200. Cambridge, MA: NBER.Dickey, D. A., & Fuller, W. A. (1979). Distribution of the Estimators for Autoregressive Time Series

with a Unit Root. Journal of the American Statistical Association, 74(366), 427e431.

Diebold, F. X. (1986). Modeling the Persistence of Conditional Variances: A Comment. Econometric

Reviews, 5(1), 51e56.Diebold, F. X., & Mariano, R. S. (1995). Comparing Predictive Accuracy. Journal of Business and

Economic Statistics, 13(3), 253e263.

Donaldson, R. G., & Kamstra, M. (1997). An Artificial Neural Network-GARCH Model for Inter-

national Stock Return Volatility. Journal of Empirical Finance, 4(1), 17e46.Dueker, M. J. (1997). Markov Switching in GARCH Processes and Mean-Reverting Stock-Market

Volatility. Journal of Business and Economic Statistics, 15(1), 26e34.

Dunis, C. L., Laws, J., & Chauvin, S. (2003). FX Volatility Forecasts and the Informational Content ofMarket Data for Volatility. The European Journal of Finance, 9(3), 242e272.

Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds.

Journal of Financial Economics, 33(1), 3e56.

Francq, C., & Zakoian, J.-M. (2001). Stationarity of Multivariate Markov-Switching ARMA Models.Journal of Econometrics, 102(2), 339e364.

Friedman, B. M., & Laibson, D. I. (1989). Economic Implications of Extraordinary Movements in

Stock Prices. Brookings Papers on Economic Activity, 2(2), 137e189.

Fung, W., & Hsieh, D. A. (1997). Empirical Characteristics of Dynamic Trading Strategies: The Caseof Hedge Funds. The Review of Financial Studies, 10(2), 275e302.

Fung, W., & Hsieh, D. A. (1999). A Primer on Hedge Funds. Journal of Empirical Finance, 6(3),

309e331.Fung, W., & Hsieh, D. A. (2000). Performance Characteristics of Hedge Funds and CTA Funds:

Natural Versus Spurious Biases. Journal of Financial and Quantitative Analysis, 35(3),

291e307.

Fung, W., Hsieh, D. A., Naik, N. Y., & Ramadorai, T. (2008). Hedge funds Performance, Risk, andCapital Formation. The Journal of Finance, 63(4), 1777e1803.

Fuss, R., Kaiser, D. G., & Adams, Z. (2007). Value at Risk, GARCH Modelling and the Forecasting of

Hedge Fund Return Volatility. Journal of Derivatives and Hedge Funds, 13(1), 2e25.

Goodworth, T. R. J., & Jones, C. M. (2007). Factor-Based, Non-Parametric Risk MeasurementFramework for Hedge Funds and Fund-of-Funds. The European Journal of Finance, 13(7), 645e655.

Gray, S. (1996). Modeling the Conditional Distribution of Interest Rates as a Regime-Switching

Process. Journal of Financial Economics, 42(1), 27e62.

Gregoriou, G. N. (2003). Performance Evaluation of Funds of Hedge Funds Using ConditionalAlphas and Betas. Derivatives, Use, Trading and Regulation, 8(4), 324e344.

Gregoriou, G. N., & Rouah, F. (2002). Pitfalls to Avoid when Constructing a Fund of Hedge Funds.

Derivatives, Use, Trading and Regulation, 8(1), 59e65.

Page 31: Reconsidering Funds of Hedge Funds || Forecasting Funds of Hedge Funds Performance

Forecasting FoHFs Performance: A Markov Regime CHAPTER 15 259

Gregoriou, G. N., Hubner, G., Papageorgiou, N., & Rouah, F. D. (2007). Funds of Funds versus

Simple Portfolios of Hedge Funds: A Comparative Study of Persistence in Performance. Journalof Derivatives and Hedge Funds, 13(2), 88e106.

Haas, M., Mittink, S., & Paolella, M. S. (2004). A New Approach to Markov-Switching GARCH

Models. Journal of Financial Econometrics, 2(4), 493e530.

Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Seriesand the Business Cycle. Econometrica, 57(2), 357e384.

Hamilton, J. D. (1994). Time Series Analysis. Princeton, NJ: Princeton University Press.

Hamilton, J. D., & Susmel, R. (1994). Autoregressive Conditional Heteroskedasticity and Changes in

Regime. Journal of Econometrics, 64(1-2), 307e333.Hansen, P. R., & Lunde, A. (2005). A Forecast Comparison of Volatility Models: Does Anything Beat

a GARCH(1,1)? Journal of Applied Econometrics, 20(7), 873e889.

Harri, A., & Brorsen, B. W. (2004). Performance Persistence and the Source of Returns for Hedge

Funds. Applied Financial Economics, 14(2), 131e141.Hedges, I. V. J. R. (2005). Hedge Fund Transparency. The European Journal of Finance, 11(5), 411e417.

Henneke, J. S., Rachev, S. T., Fabozzi, F. J., & Nikolov, M. (2011). MCMC-Based Estimation of Markov

Switching ARMA-GARCH Models. Applied Economics, 43(3), 259e271.Ineichen, A. M. (2003). Absolute Returns e The Risk and Opportunities of Hedge Fund Investing. Lon-

don: Wiley.

Jagannathan, R., Malakhov, A., & Novikov, D. (2010). Do Hot Hands Exist Among Hedge Fund

Managers? An Empirical Evaluation. The Journal of Finance, 65(1), 217e255.Jordan, A. E., & Simlai, P. (2011). Risk Characterization, Stale Pricing and the Attributes of Hedge

Funds Performance. Journal of Derivatives and Hedge Funds, 17(1), 16e33.

Kang, B. U., In, F., Kim, G., & Kim, T. S. (2010). A Longer Look at the Asymmetric Dependence

between Hedge Funds and the Equity Market. Journal of Financial and Quantitative Analysis,45(3), 763e789.

Kim, C. J., & Nelson, C. R. (1999). State-Space Models with Regime Switching. Cambridge, MA: MIT

Press.Kim, C. J., Piger, J., & Startz, R. (2008). Estimation of Markov Regime-Switching Models with

Endogenous Switching. Journal of Econometrics, 143(2), 263e273.

Klaassen, F. (2002). Improving GARCH Volatility Forecasts with Regime-Switching GARCH.

Empirical Economics, 27(2), 363e394.Laube, F., Schiltz, J., & Terraza, V. (2011). On the Efficiency of Risk Measures for Funds of Hedge

Funds. Journal of Derivatives and Hedge Funds, 17(1), 63e84.

Liang, B. (1999). On Performance of HFs. Financial Analysts Journal, 55(4), 72e85.

Ljung, G., & Box, G. (1978). On a Measure of Lack of Fit in Time-Series Models. Biometrika, 65(2),297e303.

Muzzioli, S. (2010). Option-Based Forecasts of Volatility: An Empirical Study in the DAX-Index

Options Market. The European Journal of Finance, 16(6), 561e586.Pesaran, M. H., & Timmermann, A. (2004). How Costly is it to Ignore Breaks when Forecasting the

Duration of a Time Series? International Journal of Forecasting, 20(3), 411e425.

Preminger, A., Ben-Zion, U., & Wettstein, D. (2006). Extended Switching Regression Models with

Time-Varying Probabilities for Combining Forecasts. The European Journal of Finance, 12(6-7),455e472.

Racicot, F. E, & Theoret, R. (2007). The Beta Puzzle Revisited: A Panel Study of Hedge Fund Returns.

Journal of Derivatives and Hedge Funds, 13(2), 125e146.

Straumann, D. (2009). Measuring the Quality of Hedge Fund Data. The Journal of AlternativeInvestments, 12(2), 26e40.

Taylor, S. J. (1986). Modelling Financial Time Series. Chichester: Wiley.

Vuong, Q. H. (1989). Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses.

Econometrica, 57(2), 303e333.