Adaptive market hypothesis: Evidence from the Vietnamese stock market

Phan Tran Trung Dzung; Hung Pham Quang Article Adaptive market hypothesis: Evidence from the Vietnamese stock market Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Phan Tran Trung Dzung; Hung Pham Quang (2019) : Adaptive market hypothesis: Evidence from the Vietnamese stock market, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 12, Iss. 2, pp. 1-16, https://doi.org/10.3390/jrfm12020081 This Version is available at: https://hdl.handle.net/10419/238976 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Adaptive Market Hypothesis: Evidence from the Vietnamese Stock Market Dzung Phan Tran Trung 1and Hung Pham Quang 2,* 1Faculty of Banking & Finance, Foreign Trade University, Hanoi 100000, Vietnam; [email protected] 2Branch of PwC (Vietnam) Limited in Hanoi, Hanoi 100000, Vietnam *Correspondence: [email protected]; Tel.: +84-383-351-868 Received: 30 March 2019; Accepted: 5 May 2019; Published: 8 May 2019   Abstract: This paper aims to test the adaptive market hypothesis in the two main Vietnamese stock exchanges, namely Ho Chi Minh City Stock Exchange (HSX) and Hanoi Stock Exchange (HNX), by measuring the relationship between current stock returns and historical stock returns. In particular, the tests employed are the automatic variance ratio test (“AVR”), the automatic portmanteau test (“AP”), the generalized spectral test (“GS”), and the time-varying autoregressive (TV-AR) approach. The empirical results validate the adaptive market hypothesis in the Vietnamese stock market. Furthermore, the results suggest that the evolution of HSX has served as an important factor of the adaptive market hypothesis. Keywords: adaptive market hypothesis; market efficiency; autocorrelation 1. Introduction Efficient market hypothesis (EMH), proposed by Fama (1970), despite being well known and influential in finance theory and practice, is still controversial in the predictability of stock market return. A strong and still rising school of theory, behavioral finance, is one of the strongest opponents against the arguments of EMH. The core question to this debate is whether stock market movements are predictable. The idea for the adaptive market hypothesis (AMH) stemmed from the reasoning of Lo (2004), which claimed much of evidence of an investor’s irrationality (e.g., loss aversion, overconfidence, overreaction) is, in fact, consistent with the evolutionary model of human behaviors. Such an evolutionary model indicates that humans adaptation to the ever-changing surroundings is a result of their continuous perception and learning of the latter. This evolutionary model is further developed into the adaptive market hypothesis (Lo 2004). Lo (2005) states “Based on evolutionary principles, the adaptive market hypothesis implies that the degree of market efficiency is related to environmental factors characterizing market ecologies such as the number of competitors in the market, the magnitude of profit opportunities available, and the adaptability of the market participants.” The key point for AMH to be the harmonization of EMH and BF is investor’s ability to learn and to adapt to the market’s updated situation, as a result, financial markets can be wrong from time to time, but they learned, evolved to be right, until the next mistake. Cyclical repeats of market inefficiency are the sign of adaptation, according to AMH. Since its emergence, the AMH theory has drawn significant attention from academic researchers. 1.1. Papers Discussing Adaptive Market Hypothesis (AMH) In expansion to Lo’s work (2004), Lim and Brooks (2011) proposed two criteria to test the AMH theory: •The market efficiency should be varying through time; J. Risk Financial Manag. 2019,12, 81; doi:10.3390/jrfm12020081 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2019,12, 81 2 of 16 • The market efficiency should be dependent on market conditions (i.e., financial crises, market crashes, stock bubbles, . . . ). Most of the evidences found in existing literature conclude that the AMH theory describes the fluctuations of stock returns better that the EMH theory. Specifically, interesting findings about overtime changes of market efficiency are clearly seen in the studies of Lim et al. (2006) examining the cases of in developed and emerging countries and Ito and Sugiyama (2009) researching the phenomenon of time-varying autocorrelation of the monthly S&P 500 stock returns. Neely et al. (2009) studied the intertemporal stability of excess returns to technical trading rules in the foreign exchange market by conducting true, out-of-sample tests on previously studied rules. They found that excess returns were genuine in the 1970s and 1980s, but gradually declined in the 1990s. This result was consistent with the AMH framework. Kim et al. (2011) examined the AMH theory by testing the feasibility of stock return forecast, using the daily and weekly Dow Jones Industrial Average (DJIA) stock returns from 1900 to 2009. They discovered that in the case where current stock returns can be accurately predicted using historical prices, the market efficiency is low. Kim employed three autocorrelation tests, which were the variance ratio test, portmanteau test and the generalized spectral (GS) test to measure the significance of the autocorrelation. The results indicate that the autocorrelation significance in the data series fluctuated over time and was dependent on the market conditions. Regarding the market condition dependency, the market is efficient in times of market crashes, however, it is inefficient in times of crises. Smith (2012) tested the AMH theory in 15 stock markets of emerging European countries with those of more developed ones, such as Greece, Portugal, and United England. The variance ratio test is conducted on the time series data from February 2000 to December 2009. The results show that market efficiency changes over time to some extent, which is consistent with the AMH theory. Lim et al. (2013) found that in large US stock indices, the degree of market efficiency expressed volatility over time. The study applies autocorrelation tests on a rolling-window basis. Additionally, the bootstrapping procedure was also used to make conclusions. Their main finding is that the studied market went through multiple periods of efficiency and periods of inefficiency. Urquhart and McGroarty (2014) tested the adaptive market hypothesis through four well-known calendar anomalies in the Dow Jones Industrial Average from 1900 to 2013 using subsample analysis and rolling window analysis. The results showed that all four calendar anomalies support the AMH, with each calendar anomaly’s performance varying over time. They concluded that the AMH provides a better explanation for calendar anomalies than the EMH. Hiremath and Kumari (2014) employed linear and non-linear methods to test the cyclical phenomenon in India’s stock market. They found that with the linear method the Indian stock market showed a cyclical pattern, while this pattern was not found using the non-linear method. Besides the statistical tests using the moving window method, another approach is also employed to measure market efficiency, i.e., the time-varying model approach. Ito et al. (2014) applied a non-Bayesian time-varying vector autoregressive (TV-VAR) model to estimate the joint degree of market efficiency. Their results conclude that the international linkages and market efficiency change over time and that market behaviors correspond well to historical events of the international financial system. Ito et al. (2016) applied TV-AR model to test the evolution through time of the U.S. stock market. The main findings show that (i) the U.S. market efficiency changes over time, and (ii) efficiency were violated during recessions, consistent with the assumption of behavioral finance. Noda (2016) tested AMH within the Japanese context, the study employed a time-varying model approach, and concluded that the degree of market efficiency changes over time in the markets (TOPIX and TSE2), the evolving process of the market efficiency varies among stock markets, and the results support the AMH for the more qualified stock market in Japan. Almost all of the aforementioned papers have the same finding, which is that the AMH theory describes fluctuations in stock prices more accurately than the EMH theory. Using two criteria proposed J. Risk Financial Manag. 2019,12, 81 3 of 16 by Lim and Brooks (2011), some important aspects needed to take into consideration in this paper are: (1) the evolution through time of market, (2) applicability of AMH in different exchanges and (3) the connection between market efficiency and economic cycle. 1.2. Papers Discussing Vietnamese Stock Market Efficiency The Viet Nam Stock Market (VSM) consists of two stock exchanges: Ho Chi Minh stock exchange (HOSE) and Hanoi Stock Exchange (HNX) and the VSM performs better now than in the pre-World Trade Organization (WTO) period in terms of both initial public offerings (IPOs) and seasoned offerings (Vuong 2018). Vietnamese stock market, despite having a great opportunity to have an upgraded status from frontier market to emerging market by Morgan Stanley Capital International (MSCI) in the near future, still faces with many potential issues regarding efficiency due to its special characteristics especially in the age of digitization and globalization, listed companies worldwide particularly are facing increasing pressure to innovate, increase productivity and increase competitiveness (Vu et al. 2019). Efficiency is, therefore, of the utmost importance to firms and markets as well. With 19 years of existence, the market has gone through different phases, from almost inactive in the first 5 years, then a boom and burst in the next 3 years, and unstable status with different ups and downs afterward until the moment when speed is everything: Speed of calculation, the speed of thinking, the speed of failing (Vuong 2019). There is a constant pressure to keep up the streams of content and investors tend to make decisions based on external information of quotations on the stock market. Apart from that, the number of institutional investors and foreign investors in the Vietnamese stock market is approximately around 1% each, contributing to the strong emotional volatility of the market. Dong Loc et al. (2010) reviewed developments in the Stock Trading Centre (STC) in Ho Chi Minh City, the precedent of the current HSX to test the weak-form efficiency of the Vietnamese stock market. An important element of the investigation concerns the possible bias of the results caused by the thin trading that characterizes the STC. The main conclusion of this paper is that the STC is not efficient in the weak form. Phan and Zhou (2014) discussed the weak-form efficiency for the Vietnamese stock market. The paper tested the random walk hypothesis for weekly stock market returns employing the autocorrelation test, variance ratio test, and runs test for the period from July 2000 to July 2013. Results have strongly rejected the random walk hypothesis for the whole period and two out of the three cycles of the market. Interestingly, the third cycle alone (from February 2009 to July 2013) provided evidence supporting the random walk hypothesis in the stock index of HSX (VN-Index) showing that the efficiency of the Vietnamese stock market has gradually been improved during nearly 10 years in operation. Cuong and Jian (2014) applied Theory of Planned Behavior (TPB) to explore the impact of factors influencing individuals’ investment behavioral intention in the Vietnamese stock market. Results found in this research have supported the hypotheses that an individual investor’s investment intention is significantly affected by three factors mentioned in the original TPB model including attitude, subjective norm and perceived behavioral control. The study also found evidence that psychological factors and also gender have a significant impact on the individuals’ attitude towards investment. Vo (2015) discussed the role of foreign investors in reducing market volatility using panel data analysis and found evidence supporting this hypothesis. A suggestion for a higher ratio of foreign investor existence was also proposed. However, there was no paper mentioned AMH for the Vietnamese stock market in the literature the authors have researched, setting up a research gap for further evaluations and evidence of this market’s efficiency. Also, in line with the assumption of the varying efficiency from time to time of the AMH, there were signals from the market movements showing that Vietnamese markets might have learned to adapt through time, and became more stable after each period of turmoil, and a test for AMH in the Vietnamese market context shall contribute to answering the question of efficiency. J. Risk Financial Manag. 2019,12, 81 4 of 16 2. Methods and Data Sources 2.1. Methods This paper examines the AMH theory in two ways: a battery of autocorrelation tests and a time-varying autoregressive model. The first battery of tests includes popular serial correlation tests, i.e., the automatic variance ratio (AVR) test, automatic portmanteau (AO) test, the generalized spectral (GS) test. These tests detect the linear and non-linear relationships in a time series, from which a conclusion regarding the validation of the AMH theory can be made. The second method is involved with the construction of a time-varying autoregressive (“TV-AR”) model, i.e., an autoregressive model with the coefficients changing over time. This is a fairly new testing method, which has been introduced in the work of Ito et al. (2014). This method possesses several advantages over other testing methods. The details of the methods are presented in the following sections. 2.1.1. Autocorrelation Testing Approach This paper quantifies the market efficiency through three tests of autocorrelation to examine whether the level of market efficiency significantly changes over time, aiming at evaluating the weak-form efficiency of the stock market. The task usually requires autocorrelation tests. The common understanding is that if time series data exhibit significant autocorrelations, it will be easier to predict the stock returns using historical prices, and the investors will obtain abnormal gains more easily. In other words, the more serially correlated the time series data are, the lower the market efficiency is. Among popular autocorrelation tests, the paper adopted the following quantitative tests: •Automatic Variance Ratio (“AVR”) test; •Automatic Portmanteau (“AP”) test; and •Generalized Spectral (“GS”) test The main reason why these tests are utilized is that these tests work on data which suffer from conditional heteroskedasticity, which is a common symptom of financial time series. This is also the case of Vietnamese stock market indices—in fact, a simple plotting (i.e., Figures 1and 2) shows that both VN-INDEX data and HNX-INDEX data exhibit the conditional heteroskedasticity: J. Risk Financial Manag. 2019, 12, x FOR PEER REVIEW 4 of 16 2. Methods and Data Sources 2.1. Methods This paper examines the AMH theory in two ways: a battery of autocorrelation tests and a time-varying autoregressive model. The first battery of tests includes popular serial correlation tests, i.e., the automatic variance ratio (AVR) test, automatic portmanteau (AO) test, the generalized spectral (GS) test. These tests detect the linear and non-linear relationships in a time series, from which a conclusion regarding the validation of the AMH theory can be made. The second method is involved with the construction of a time-varying autoregressive (“TV-AR”) model, i.e., an autoregressive model with the coefficients changing over time. This is a fairly new testing method, which has been introduced in the work of Ito et al. (2014). This method possesses several advantages over other testing methods. The details of the methods are presented in the following sections. 2.1.1. Autocorrelation Testing Approach This paper quantifies the market efficiency through three tests of autocorrelation to examine whether the level of market efficiency significantly changes over time, aiming at evaluating the weak-form efficiency of the stock market. The task usually requires autocorrelation tests. The common understanding is that if time series data exhibit significant autocorrelations, it will be easier to predict the stock returns using historical prices, and the investors will obtain abnormal gains more easily. In other words, the more serially correlated the time series data are, the lower the market efficiency is. Among popular autocorrelation tests, the paper adopted the following quantitative tests: • Automatic Variance Ratio (“AVR”) test; • Automatic Portmanteau (“AP”) test; and • Generalized Spectral (“GS”) test The main reason why these tests are utilized is that these tests work on data which suffer from conditional heteroskedasticity, which is a common symptom of financial time series. This is also the case of Vietnamese stock market indices—in fact, a simple plotting (i.e., Figures 1 and 2) shows that both VN-INDEX data and HNX-INDEX data exhibit the conditional heteroskedasticity: Figure 1. The weekly returns of HSX in the period from 2005 to 2019. -0.2 -0.1 0 0.1 0.2 Weekly returns - HSX Figure 1. The weekly returns of HSX in the period from 2005 to 2019. J. Risk Financial Manag. 2019,12, 81 5 of 16 J. Risk Financial Manag. 2019, 12, x FOR PEER REVIEW 5 of 16 Figure 2. The weekly returns of Hanoi Stock Exchange (HNX) in the period from 2006 to 2019. There is visual evidence of conditional heteroskedasticity in both plots above that the series of returns since volatility clustering can be observed. Particularly, the weeks with large variances in weekly returns cluster in the period from 2006 to 2010 and the weeks with smaller variance clusters in more recent periods. Due to such symptom, the paper opts to the three tests above to analyze this type of data. On a side note, the AVR test and the GS test employ the wild bootstrapping approach, which is most suited for data of small sample size like Vietnamese stock indices. Each test possesses different statistical characteristics, which can be complementary to one another. The characteristics are as follows: • The AVR test, which is modified from the traditional variance ratio test, is the most popular test in the AMH examination. This is the primary testing method of this paper. • The AP test is an asymptotic test, which relies on the squared correlation coefficients. This method eliminates the possibility that the positive correlations and the negative correlations offset one another (Kim et al. 2011). • The GS test is an autocorrelation test that can determine the non-linear relationship in the data series. The non-linear relationship in stock data can be recognized (Lim and Brooks 2011), yet cannot be detected by popular linear tests such as the AVR test and the AP test. These tests are conducted with the “vrtest” package in R. Specifically, the test statistics were calculated on the basis of the rolling window method with a fixed 1-year window length. In fact, the window length observably had a minimal effect on the test results according to the work of Kim et al. (2011). Afterward, the quantitative results were assessed qualitatively to examine the impact of market conditions on the degree of market efficiency. The details on how to conduct each test are as follows: • The Automatic Variance Ratio (AVR) test The AVR test is developed based on the traditional variance ratio test (Lo and MacKinlay 1988), which is the most popular test of the random walk theory, according to Hoque et al. (2007). The test is based on the statistical feature that the variance of k-periods stock returns equals to k times the variance of 1-period stock returns, provided that the stock returns series follows a random walk. The variance ratio is defined to be the weighted sum of correlation coefficients in the stock returns series: 𝑉𝑅(𝑘)=    =1+2∑(1− )   𝜌, -0.2 -0.1 0 0.1 0.2 Weekly returns - HNX Figure 2. The weekly returns of Hanoi Stock Exchange (HNX) in the period from 2006 to 2019. There is visual evidence of conditional heteroskedasticity in both plots above that the series of returns since volatility clustering can be observed. Particularly, the weeks with large variances in weekly returns cluster in the period from 2006 to 2010 and the weeks with smaller variance clusters in more recent periods. Due to such symptom, the paper opts to the three tests above to analyze this type of data. On a side note, the AVR test and the GS test employ the wild bootstrapping approach, which is most suited for data of small sample size like Vietnamese stock indices. Each test possesses different statistical characteristics, which can be complementary to one another. The characteristics are as follows: • The AVR test, which is modified from the traditional variance ratio test, is the most popular test in the AMH examination. This is the primary testing method of this paper. • The AP test is an asymptotic test, which relies on the squared correlation coefficients. This method eliminates the possibility that the positive correlations and the negative correlations offset one another (Kim et al. 2011). • The GS test is an autocorrelation test that can determine the non-linear relationship in the data series. The non-linear relationship in stock data can be recognized (Lim and Brooks 2011), yet cannot be detected by popular linear tests such as the AVR test and the AP test. These tests are conducted with the “vrtest” package in R. Specifically, the test statistics were calculated on the basis of the rolling window method with a fixed 1-year window length. In fact, the window length observably had a minimal effect on the test results according to the work of Kim et al. (2011). Afterward, the quantitative results were assessed qualitatively to examine the impact of market conditions on the degree of market efficiency. The details on how to conduct each test are as follows: •The Automatic Variance Ratio (AVR) test The AVR test is developed based on the traditional variance ratio test (Lo and MacKinlay 1988), which is the most popular test of the random walk theory, according to Hoque et al. (2007). The test is based on the statistical feature that the variance of k-periods stock returns equals to ktimes the variance of 1-period stock returns, provided that the stock returns series follows a random walk. The variance ratio is defined to be the weighted sum of correlation coefficients in the stock returns series: VR(k)=σ2 k kσ2=1+2Xk−1 j=1(1−j k)ρj, in which: σ2 k is the variance of k-period stock returns, σ2 is the variance of 1-period stock returns. ρj is the j-degree correlation coefficient. The closer to 1 the VR test statistics are, the better the data series J. Risk Financial Manag. 2019,12, 81 6 of 16 follow a random walk. If the VR test statistics are greater than 1, a positive correlation is concluded, otherwise, if such statistics are less than 1, the series is undergoing a mean reversion. The VR test statistics are estimated as follows: VR(k)=σ2 k kσ2=1+2 k−1 X j=1 (1−j k)ˆ ρj in which ˆ ρjis an estimate of ρj. One major setback of this approach is that the value of kis selected based on personal perceptions. In other words, the traditional variance ratio approach is subjective and may provide irreproducible results. Choi (1999) proposed the automatic variance ratio test, which employs the data-oriented method to determine the optimal ( ˆ k ). In the AVR test, Choi assumed that the stock returns series is identical and independently distributed, and claimed that: AVRˆ k=qT ˆ khVRˆ k−1i √2 d →N(0 , 1) However, if the data suffered from conditional heteroskedasticity, the test results may not be reliable, especially when the sample size is small. In particular, the construction of confidence intervals following N(0,1) distribution may not reflect the level of uncertainty in the estimate of k. Accordingly, Kim (2006) proposed an alternative to the normal distribution approach—the wild bootstrapping approach. This approach is proved to be more suitable than the simple residual bootstrapping method in case the data exhibits heteroskedasticity. •The Automatic Portmanteau (AP) test The AP test is a popular tool to detect autocorrelations in time series data. However, one shortcoming of the traditional AP test is that when the time series data suffer from conditional heteroskedasticity, the test statistics may be inaccurate. Accordingly, Lobato et al. (2001) suggested an advanced AP test, which utilizes the following test statistics: Q∗ p=T p X i=1e ρ2 i in which: e ρ2 i=ˆ γ2 i/τ2 i ( ˆ γ2 i is the estimator for the autocovariance of the time series data of order i, and τ2 i represents the autocovariance of the squared stock returns. Similar to the selection of kin the traditional VR test, the advanced AP test also suffers from irreproducibility as the selection of lag pis based on personal judgements. In order to fix the problem, Escanciano and Lobato (2009) proposed an automatic data-driven approach which determines the optimal lag p. The test statistics, therefore, can be calculated as follows: AQ ≡Q∗ p=Te p X i=1e ρ2 i In which, e p is the optimal estimator of lag p, determined by Akaike information criterion (AIC) or Bayesian information criterion (BIC). The AP test statistics follow the Chi-squared distribution with one degree of freedom. •The Generalized Spectral (GS) test The nonlinear correlation, which is ignored in the linear correlation tests such as the AVR test and the AP test, has been commonly detected in the stock returns (Lim and Brooks 2011). J. Risk Financial Manag. 2019,12, 81 7 of 16 Accordingly, Escanciano and Velasco (2006) proposed the generalized spectral test, which can assess the nonlinear relationship in time series data. This is based on the knowledge that when the stock return follows a general martingale difference sequence, its normalized spectral density function is equal to one at all frequencies. The paper suggests that the test statistics are to be calculated as follows: D2 n= n−1 X j=1 (n−j)1 (jπ)2ZRˆ γj(x)2W(dx) The paper also involved the wild bootstrapping method, in which they obtained the p-value of the test. If the p-value corresponding with the test statistics is less than 5%, it can be concluded that the market is inefficient at that point in time. 2.1.2. The Time-Varying Autoregressive (TV-AR) Approach The time-varying autoregressive model is developed from the simple autoregressive model. The simple autoregressive model has long been used to assess the linear relationship in time series data. However, one major drawback of this model is that the coefficients are fixed, and therefore the simple model cannot handle the time series data with structural breaks, such as stock returns. Ito et al. (2014) introduced the time-varying autoregressive model as a solution for the aforementioned drawback.The detail on how to employ the model in the AMH testing is presented below. Firstly, the optimal lag order for each series is chosen using the BIC. The optimal lag orders for VN-INDEX and HNX-INDEX are 2 and 1 correspondingly. Subsequently, a regression followingthe TV-AR model is conducted, of which the theoretical framework is elaborated below. The TV-AR model is presented in the form of an equation system as follows (with the assumption that parameter dynamics restrict the parameters): xt=α0+α1,txt−1+α2,txt−2+···+αq,txt−q+ut(1) αi,t=αi,t−1+vi,t(i=1, 2, ··· ,q)(2) E(ut)=Eu2 t=E(utut−m)=0∀m Evi,t=E(v2 i,t) = E(vi,tvt−m) = 0∀m where xt represents the stock return at time t, αi is the time-varying coefficients, ut and vt are the residuals of the model. (1) and (2) form a system of simultaneous equations for the model. Denotation of matrices are deployed as follows: Xt−1= xt−1 xt−2 . . . xt−q  ;At=hα1,tα2,t··· αq,ti;Iqis an identity matrix of order q. Equation (1) can be rewritten accordingly: xt=α0+XT t−1×AT t+Ut where XT t−1,AT tare the transpose of Xt−1,Atcorrespondingly. J. Risk Financial Manag. 2019,12, 81 8 of 16 Assign a range of values (from 1 to T) to the parameter t to obtain the following equation:  x1 x2 . . . xT  = 1XT 0O1×T 1XT 1 . . .... 1O1×TXT T−1 × α0 AT 1 . . . AT T  + u1 u2 : uT  (3) Denotey= x1 x2 . . . xT  ;β= α0 AT 1 . . . AT T  ;U= u1 u2 : uT  ;M= 1XT 0O1×T 1XT 1 . . .... 1O1×TXT T−1  where O1×Tis a 1 ×Tnull matrix. Equation (3) can be simplified accordingly: y=M×β+U(4) In a similar manner, Equation (2) can be re-written as:  −AT 0 0 . . . 0  = Oq×1−IqOq×q Oq×1Iq−Iq . . . . . .. . .... Oq×1Oq×q··· −Iq  × α0 AT 1 . . . AT T  + v1 v2 : vT  Denote : Z= −AT 0 0 . . . 0  ;W= Oq×1−IqOq×q Oq×1Iq−Iq . . . . . .. . .... Oq×1Oq×q··· −Iq  ;V= v1 v2 : vT  The following equation is obtained: z=W×β+V(5) Equations (4) and (5) form the equation system of the TV-AR model in the matrix form. It can also be further deducted as below: "y z#="M W#β+"U V# The result below is obtained using ordinary least squares (OLS) regression: ˆ β="M W#T"M W# −1"M W#T"y z# The market efficiency is quantified using the following formula, which was used in the work of Noda (2016), a special case of Ito et al. (2014): MEt = Pp j=1ˆ αj,t 1−Pp j=1ˆ αj,t J. Risk Financial Manag. 2019,12, 81 15 of 16 Escanciano, J. Carlos, and Carlos Velasco. 2006. Generalized Spectral Tests for the Martingale Difference Hypothesis. Journal of Econometrics 134: 151–85. 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