Testing the Martingale Difference Hypothesis in the European Emerging Unit-linked Insurance Markets

Available online at www.sciencedirect.com

Procedia Economics and Finance 3 (2012) 49 – 54

Emerging Markets Queries in Finance and Business

Testing the martingale difference hypothesis in the European emerging unit-linked insurance markets Diana-Maria Chi a,* a

Department of Finance, University, Str. Teodor Mihali, Nr. 58-60, Cluj Napoca 400591 , Romania

-Bolyai

Abstract

This study examines the martingale difference hypothesis (MDH) for the European emerging unit-linked insurance markets, using the automatic portmanteau (AQ) test of Escanciano and Lobato, 2009 for the three sub-periods of pre-crisis, crisis, and post-crisis. The martingale difference sequence is called conditional mean independence in the statistical literature, implying that the asset return is purely non-predictable from its own past. This study proposes a data-driven Box-Pierce for serial correlation. The automatic Portmanteau test for serial correlation test presents higher power in simulations than the existing ones for models commonly employed in empirical finance: the researcher does not need to specify the order of the autocorrelation tested, since the test automatically chooses this number; its asymptotic null distribution is chi-square with one degree of freedom, so there is no need of using a bootstrap procedure to estimate the critical values and the test is robust to the presence of conditional heteroskedasticity of unknown form. This paper examines return predictability of the daily ING unit-linked funds prices and aims at monitoring any improvement in the degree of efficiency in time. © 2012 The©Authors. Published byby Elsevier Ltd. Ltd. Selection and/or peer-review under responsibility of 2012 Published Elsevier Selection and peer review under responsibility Emerging Markets Queries in Finance and Business local organization. Markets Queries in Finance of and Business local organization.

the Emerging

Keywords: martingale difference hypothesis, unit-linked insurance markets, serial correlation, automatic Portmanteau test, Global financial crisis;

*

Corresponding author. Tel.: +40 742 255 114. E-mail address: [email protected].

2212-6716 © 2012 The Authors. Published by Elsevier Ltd. Selection and peer review under responsibility of Emerging Markets Queries in Finance and Business local organization. doi:10.1016/S2212-5671(12)00119-0

50

Diana-Maria Chiş / Procedia Economics and Finance 3 (2012) 49 – 54

1. Introduction and literature review Testing for the martingale difference hypothesis (MDH) is central in many economic and finance studies, such as market efficiency, rational expectations, and optimal consumption smoothing Kim et al., 2010. The majority of the efficient market hypothesis studies (EMH) on financial markets are tested for the weak-form efficiency through the martingale difference hypothesis (MDH),where the current price is the best predictor of the future price and the returns are independent (or uncorrelated) with the past values Charles et al., 2010. The MDH is called conditional mean independence in the statistical literature, and it means that past and current information are of no use to forecasting future values of a martingale difference sequence Escanciano and Lobato, 2007. Notable recent contributions to the category of MDH tests, based on linear and nonlinear measures of dependence, include: the automatic portmanteau (AQ) test of Escanciano and Lobato, 2009, automatic variance ratio (AVR) test of Kim, 2009 extending the earlier work of Choi, 1999, the generalized spectral (GS) test of Escanciano and Velasco, 2006 and the consistent tests of Dominguez and Lobato, 2003; Kim et al., 2010. Testing for serial correlation has held a central role in the statistical analysis of economic time series since its inception Yule, 1926. Peter Robinson highlighted the importance of establishing tests for serial correlation, which took into account two features of economic time series that were largely ignored before: the first feature regarded the existence of nonlinear dependence, especially present in financial time series in the form of conditional heteroskedasticity and the second feature regarded the existence of strong dependence present in many macroeconomic time series. Escanciano and Lobato, 2009 introduces a data-driven Box-Pierce test for serial correlation, that is very attractive compared to the existing ones Escanciano and Lobato, 2009. The martingale properties of financial returns have been studied previously by many authors leading to mixed conclusions: Escanciano and Velasco, 2006; Escanciano and Lobato, 2007; 2009, Charles et al., 2010; Kim et al., 2010; Dominquez and Lobato, 2003; Kuan and Lee, 2004; LeBaron, 1999; Levich and Thomas , 1993; Diebold and Nason, 1990, etc. The global financial crisis had a significant impact on the CEE unit-linked factor of market inefficiency. This is because investors are generally swamped by panic in that chaotic financial environment, and thus causing a decrease in the degree of efficiency. In the financial literature there are only few studies investigating the impact of the global financial crisis on the potential of predictability and implicitly on the degree of efficiency of financial markets: Holden et al., 2005; Kim and Shamsuddin, 2008; Lim et al., 2008; Cheong et al., 2007 examined the weak-form efficiency of Asian stock markets using variance ratio tests before, during and after the Asian crisis Lim and Brooks, 2011; Kim et al., 2007. Lazar and Todea, 2011 investigated the effects of the Global crisis on the relative efficiency of ten CEE stock markets. The results of the Generalized Spectral test of Escanciano and Velasco, 2006 show a decrease of predictability and an improvement of relative efficiency for seven of the ten investigated markets in the crisis period. The objective of this study is to test the martingale difference hypothesis (MDH) for the European emerging unit-linked insurance markets, using the automatic portmanteau (AQ) test of Escanciano and Lobato , 2009 for the three sub-periods of pre-crisis, crisis, and post-crisis. This study contributes to the existing literature on EMH with several distinct features: this paper provides a systematic review of the weak-form market efficiency literature that examines return predictability from past price changes, with an exclusive focus on the unit-linked insurance markets. Moreover, the article aims at monitoring any improvement in the degree of efficiency in time and also examines the relative efficiency of markets in pre-crisis and crisis periods. The structure of the article is as follows. Section 1 discusses some previous research on the issue. Section 2 and 3 describe the sample data and provides descriptive statistics and outlines the methodology. Empirical results are presented in Section 4. The conclusions are drawn in Section 5.

51

Diana-Maria Chiş / Procedia Economics and Finance 3 (2012) 49 – 54

2. Data Description This study examines four Central and Eastern European unit-linked insurance markets and the closing values of ING unit-linked funds at daily frequency for these markets are collected as follows: Romania (Bond Fund, Mixt 25 Fund), Hungary (Bond Unit Fund, Balanced Unit Fund ), Poland (Bonds Sub-Fund, Balanced Sub-Fund), Czech Republic (Bond Fund, Junior Fund). The ING Bond funds invest 100% in domestic instruments with fixed income. The ING mixed funds invest 15-50% in shares issued by companies from the European Union, in particular Romania, Hungary, Poland and the Czech Republic, and instruments with fixed income denominated in national currency. The daily data is taken from July 21, 1999 to June 1, 2012. Table 1 provides the descriptive statistics for the continuously compounded percentage returns, computed as Yt=ln(Pt)-ln(Pt-1), where Pt and Pt-1 denote the closing price of the unit-linked fund on two consecutive trading days, and prices are denominated in their respective local currency units. Table 1. Descriptive statistics of returns for pre-crisis and crisis periods Market

Romania

Hungary

Czech Republic

Poland

Pre-crisis period

Bond

Mixt 25

Bond

Balanced

Bond

Junior

Bond

Balanced

Mean

0.000568

0.000608

0.000302

0.000335

0.000130

0.000162

0.000287

0.000394

St. Dev.

0.001620

0.004417

0.003245

0.006215

0.001520

0.002213

0.001619

0.006526

Skewness

6.569553

2.298665

-1.451530

-0.248741

0.245899

-0.473242

0.271131

-0.242505

Excess Kurtosis

58.933100

42.661400

27.274590

5.988092

8.999322

7.620678

9.712003

5.576193

Crisis period

Bond

Mixt 25

Bond

Balanced

Bond

Junior

Bond

Balanced

Mean

0.000249

0.000223

0.000238

0.000028

0.000102

-0.000040

0.000217

-0.000094

St. Dev.

0.000929

0.004205

0.006695

0.009926

0.002441

0.002962

0.001926

0.008618

Skewness

-2.857740

-0.236268

0.241441

-0.127180

-1.29497

-1.040282

-0.421537

-0.268895

Excess

73.990120

8.012124

13.345840

7.439937

17.54965

14.505240

8.823117

5.535143

Test is performed by the author using software Eviews

The Global crisis had a significant impact on volatility, the standard deviation of returns series recording an increase from pre-crisis to crisis period, except for Romanian funds. The highest increases of volatility are found in the case of Hungary Bond Unit Fund, Hungary Balanced Unit Fund and Poland Balance Sub-Fund. During the crisis period, the daily returns are negatively skewed, which means that large negative returns are more probable than the higher positive returns. The excess kurtosis is positive for all indexes which indicate an increased number of returns around the average, comparative to the normal law . 3. Methodology. An automatic Portmanteau test for serial correlation of J. Carlos Escanciano and Ignacio N. Lobato (2009) Escanciano and Lobato, 2009 propose an automatic test to modify the robustified Portmanteau test statistic by allowing the data to automatically select the number of autocorrelations employed. The proposed test statistic is the maximized value of the robustified Portmanteau statistic corrected by a penalty term that is an increasing function of the included number of autocorrelations. The test allows the data to select whether AIC or BIC is employed as the penalty function. The ideal test procedure would employ the BIC when the evidence points to the null hypothesis (the autocorrelations appear to be small), and would employ the AIC when the evidence points to the alternative (there is some correlation that appears to be large). 2 The test statistic, which asymptotically follows the 1 distribution, is written as:

52

Diana-Maria Chiş / Procedia Economics and Finance 3 (2012) 49 – 54

AQ Q*~p where ~ p min p : 1 p d ; L p

Lh , h 1,2,

, d , Lp

Q*~p

(1) (2)

( p, n, q)

d is a fixed upper bound, and q is some fixed positive number, q=2.4

p log n, if max n ~ j

( p, n, q) is a penalty term that takes the form:

( p, n, q)

1 j d

2 p, if max n ~ j 1 j d

q log n (3)

q log n

The automatic version of the Box Pierce Portmanteau test for serial correlation presents additional interesting advantages over the existing ones. First, it is very simple to implement since the test statistic is easy to compute and the asymptotic null distribution is chi-square with one degree of freedom. The researcher does not need to introduce any bandwidth number or any block length number to calculate neither the test statistic nor the relevant critical values. Second, the test presents more empirical power in finite samples than the rival ones. Finally, the test is especially suitable for financial data since it is robust to conditional heteroskedasticity. Also, Deo, 2000 proposes a serial correlation test that is robust to conditional heteroskedasticity and is consistent Escanciano and Lobato, 2009. 4. Empirical results The empirical p-values for AQ, QP (p=1,5,10,15 and 20) and Dn are reported in Table 2. The upper bound d is chosen to be d=75, as it was employed in the simulations. The automatic Portmanteau (AQ) test of Escanciano and Lobato (2009) rejects the null hypothesis at 5% and overcomes the problem of the choice of p. Bold letters indicate significant value at a 5% level of significance. The results show that over the entire period, the martingale hypothesis is clearly rejected on the vast majority of markets, except for ING Poland Bonds Sub-Fund and ING Poland Balanced Sub-Fund, at 0.05 significant level. For the ING Romania Bond Fund, ING Czech Republic Bond Fund and ING Hungary Balanced Unit Fund, the robustified Box Pierce test rejects the null hypothesis at 5% when p=1. The high degree of predictability and implicitly of inefficiency of the four CEE unit-linked insurance markets is observed also in the two sub periods for the majority of ING funds. In pre-crisis period results show that ING Poland Bonds Sub-Fund returns are serially uncorrelated. We observe conflicting results for the classical Portmanteau test for different values of p: for ING Romania Bond Fund the test does not reject the null at 5% when p=5,10,15,20; for ING Poland Balanced Sub-Fund the test does not reject the null at 5% when p=15,20; for ING Hungary Balanced Unit Fund the test does not reject the null at 5% when p=5,10,15,20 and for ING Czech Republic Junior Fund the test does not reject the null at 5% when p=10,15,20. In crisis period results show that ING Poland Balanced Sub-Fund returns are serially uncorrelated. We again observe conflicting results for different values of p: for ING Romania Bond Fund and ING Hungary Bond Unit Fund the test does not reject the null at 5% when p=10,15,20; for ING Czech Republic Bond Fund and ING Hungary Balanced Unit Fund the test does not reject the null at 5% when p=5,10,15,20 and for ING Poland Bonds Sub-Fund the test does not reject the null at 5% when p=20. Deo's test statistic Dn does not reject the null at 5% for ING Romania Bond Fund and for ING Poland Bonds Sub-Fund in the precrisis period. For automatic portmanteau test construction the author made an implementation of algorithms that describe the whole process in C + + programming language.

53

Diana-Maria Chiş / Procedia Economics and Finance 3 (2012) 49 – 54

Table 2. The p-values for the automatic portmanteau (AQ) test of Escanciano and Lobato (2009) The entire period

Pre-crisis period

Crisis period

The entire period

ING Romania Bond Fund

Pre-crisis period

Crisis period

ING Hungary Bond Unit Fund

7/21/99-6/1/12

7/21/99-10/7/08

10/8/08-6/1/12

7/21/99-6/1/12

7/21/99-7/31/08

8/1/08-6/1/12

AQ

0.009298

0.025515

0.006891

0.009818

0.000061

0.034692

Q1

0.009298

0.025515

0.006891

0.009818

0.000061

0.034692

Q5

1.705700

4.065036

0.009141

0.010106

0.000510

0.038299

Q10

3.161625

6.367446

0.058051

0.013482

0.000557

0.056704

Q15

4.499784

8.220616

0.109538

0.016019

0.004228

0.069846

Q20

5.174603

9.603863

0.122144

0.016288

0.005184

0.070039

Dn

0.026788

0.057975

0.001188

0.001074

0.000089

0.003970

ING Czech Republic Bond Fund

ING Poland Bonds Sub-Fund

7/21/99-6/1/12

7/21/99-9/17/08

9/18/08-6/1/12

7/21/99-6/1/12

7/21/99-2/4/08

2/5/08-6/1/12

AQ

0.018866

0.005213

0.022931

0.308724

0.500991

0.035014

Q1

0.018866

0.005213

0.022931

0.308724

0.500991

0.035014

Q5

0.057248

0.013049

0.102461

0.320489

0.543099

0.037218

Q10

0.059494

0.020609

0.108626

0.329263

0.570417

0.038111

Q15

0.059706

0.032165

0.111336

0.344283

0.650223

0.047295

Q20

0.065067

0.046701

0.115626

0.375517

0.689568

0.058772

Dn

0.002869

0.000716

0.004490

0.031510

0.051713

0.003745

ING Romania Mixt25 Fund

ING Hungary Balanced Unit Fund

7/21/99-6/1/12

7/21/99-10/7/08

10/8/08-6/1/12

7/21/99-6/1/12

7/21/99-7/31/08

8/1/08-6/1/12

AQ

0.011374

0.005992

0.002564

0.044830

0.033705

0.048622

Q1

0.011374

0.005992

0.002564

0.044830

0.033705

0.048622

Q5

0.012717

0.006777

0.006456

0.052309

0.055017

0.051874

Q10

0.013988

0.008569

0.007286

0.061717

0.060135

0.070189

Q15

0.021860

0.011634

0.034416

0.062407

0.091194

0.097976

Q20

0.028761

0.013864

0.040797

0.068145

0.096410

0.104676

Dn

0.001216

0.000690

0.000495

0.004698

0.003674

0.005334

ING Poland Balanced Sub-Fund

ING Czech Republic Junior Fund

7/21/99-6/1/12

7/21/99-2/4/08

2/5/08-6/1/12

7/21/99-6/1/12

7/21/99-9/17/08

9/18/08-6/1/12

AQ

0.084911

0.030309

0.068969

0.001456

0.000020

0.014501

Q1

0.084911

0.030309

0.068969

0.001456

0.000020

0.014501

Q5

0.085803

0.037584

0.073519

0.005934

0.005257

0.024154

Q10

0.086873

0.040836

0.076702

0.014065

0.053345

0.030219

Q15

0.094285

0.107425

0.076825

0.014300

0.056308

0.031272

Q20

0.095355

0.108238

0.082264

0.023763

0.060834

0.044691

Dn

0.008744

0.003263

0.007309

0.000273

0.000224

0.001732

54

Diana-Maria Chiş / Procedia Economics and Finance 3 (2012) 49 – 54

5. Conclusions Most of the efficient market hypothesis (EMH) studies on financial markets are tested for the weak-form efficiency through the martingale difference hypothesis (MDH), where the current price is the best predictor of the future price and the returns are independent (or uncorrelated) with the past values. If the price follows a martingale difference sequence (MDS thereafter), then the market is weak-form efficient, and hence not predictable, and you cannot expect to earn an abnormally high return, a return greater than the equilibrium return. This study investigates an issue which is surprisingly understudied in the literature, it tests the martingale difference hypothesis (MDH) for the European emerging unit-linked insurance markets, using the automatic portmanteau (AQ) test of Escanciano and Lobato, 2009 for the three sub-periods of pre-crisis, crisis, and postcrisis. The proposed test is very attractive compared to the existing ones: first, the researcher does not need to specify the order of the autocorrelation tested, since the test automatically chooses this number; second, its asymptotic null distribution is chi-square with one degree of freedom, so there is no need of using a bootstrap procedure to estimate the critical values. In addition, the test is robust to the presence of conditional heteroskedasticity of unknown form Escanciano and Lobato, 2009. Using the automatic portmanteau (AQ) test of Escanciano and Lobato, 2009 through time, due to the existence of linear and nonlinear dependence in financial time series. In crisis period automatic portmanteau (AQ) test results show that ING Poland Balanced Sub-Fund returns are serially uncorrelated. We observe conflicting results for the classical portmanteau test for different values of p, but surprisingly the classical and automatic portmanteau (AQ) test results suggest that the Global crisis led to a decrease of predictability and hence to an improvement of relative efficiency for four of the eight ING funds. for ING Romania Bond and Mixt25 Fund, ING Poland Bonds Sub-Fund and ING Czech Republic Junior Fund. References Charles,A, Darné,O, Kim, J. H. ,2010,. "Small Sample Properties of Alternative Tests for Martingale Difference Hypothesis," Working Papers 2010.07, School of Economics, La Trobe University Charles, A, Darne, O, Fouilloux, J,.2010 ,"Testing the Martingale Difference Hypothesis in the EU ETS Markets for the CO2 Emission Allowances: Evidence from Phase I and Phase II". Working Papers (2010) Palgrave Handbook of Escanciano, J.C., Lobato, I.N., 2007. Econometrics, MacMillan, Palgrave , Journal of Econometrics, 2009, 151(2), p. Escanciano J.C., Lobato, I.N., 2009, 140-149. , Journal of Lim, K.P., Brooks, R. ,2011, economic surveys 1, p.69-108. International Review of Financial Analysis 17 p.571-591. , Todea, A, Lazar, D. 2011, Working Papers (2011). http://www.ing.com