2.5.1. Fama–French Three-Factor Model

Merton (1973)'s ICAPM and Ross's APT (1976), Fama and French (1993) first documented that a Three-Factor Model could be established to explain stock returns, which is more significant than ICAPM and APT. The model considers that the excess return of a portfolio (including a single stock) can be explained by its exposure to three factors: market risk premium (*RMRF*), market value factor (*SMB*, Small market capitalization Minus Big market capitalization), book-to-market ratio factor (*HML*, High book-to-market ratio Minus Low book-to-market ratio). Based on monthly data of Pakistan financial and non-financial firms from 2002 to 2016, Ali et al. (2018) showed that the Pakistani stock market is satisfactorily explained by the Three-Factor Model, especially with the addition of SMB and HML.

#### 2.5.2. Carhart Four-Factor Model

There are limitations in the Fama–French Three-Factor Model, as factors like short-term reversal, medium-term momentum, volatility, skewness, gambling, and others are not considered or included. Based on the Fama–French Three-Factor Model, Carhart (1997) developed an extended version which includes a momentum factor for asset pricing in stock markets. The extra consideration is *PR1YR*, which is the return for the one-year momentum in stock returns. By applying this Four-Factor Model, Carhart (1997) claimed that it helps to explain a significant amount of variations in time series. Furthermore, the high average returns of *SMB*, *HML*, and *PR1YR* indicate that these three factors could explain the large cross-sectional variation of the average returns of stock portfolios.

#### 2.5.3. Fama–French Five-Factor Asset-Pricing Model

Although Carhart's Four-Factor Model helped develop the Fama–French Three-Factor Model, Fama and French (2015) figured out a new augmentation of the Three-Factor Model by considering profitability and investment factors, which is called the Five-Factor Asset-Pricing Model, to fix more anomaly variables that cause problems to the Three-Factor Model. In addition, Fama and French (2015) concluded that the ability of the Five-Factor Model on capturing mean stock returns performs better than the Three-Factor Model.

However, the Five-Factor Model did have its drawbacks. For those low mean returns on small stocks, just like returns of those firms invest heavily regardless of low profitability, it fails to seize, and it was tested in North America, Europe, and Asia-Pacific stock markets by Fama and French (2017). Furthermore, the performance of the model is insensitive to the way in which the factors are defined. With the increase of profitability and investment factors, the value factors of the Three-Factor Model become superfluous for describing the mean return in the samples that Fama and French (2015) test.

#### 2.5.4. Factor Models in Chinese Markets

Given the development of the factor model itself, Fama–French Three-Factor Model remains the benchmark in US stock markets for many years. Yet many studies copying Fama–French three factors in China's A-share market, have not achieved very satisfactory results until Liu et al.'s (2019) Chinese version Three-Factor Model. Given the strict IPO regulation rules in China's A share market, which consists of the fact that most of the smallest listed firms are targeted as potential, and reflecting the value of, shells, in order to develop this model, Liu, Stambaugh and Yuan deleted 30% of stocks at the bottom to avoid small listed firms whose values are contaminated by shell-values. Furthermore, they use EP (earning-to-price ratio) to replace BM (book-to-market ratio) due to the former capturing the value e ffect more accurately in China in the sense that it is the most significant factor.

Using this Chinese version of the Three-Factor Model, most reported anomalies in China's A-share market are well explained, where profitability and volatility anomalies are included. Furthermore, Liu et al. (2019) add the turnover factor PMO (Pessimistic minus Optimistic) into the model to make it a Four-Factor Model and help explain reversal and turnover anomalies. Same on the studies of China stock markets, based on the Fama–French Five-Factor Model, Li et al. (2019) developed a Seven-Factor Model by adding trading volume and turnover rates factors.

When additional factors are included, the Chinese version of the Seven-Factor Model performs well in empirical testing of herding behavior in China's A-share market, especially during three famous stock market crash periods.

Table 2 briefly summarizes factor models that explain market e fficiency.


**Table 2.** Explaining market efficiency by factor models.

Apart from above models, there are methods measuring or explaining liquidity in stock markets. For low-frequency data, there are spread proxies, including Roll's (1984) spread (ROLL), Hasbrouck's (2009) Gibbs estimate (HASB), LOT measure provided by Lesmond et al.'s (1999) study, and others, while there are also price impact proxies, including the AMIHUD, a measure came up by Amihud (2002), the Amivest measure, or AMIVEST set up by Cooper et al. (1985), and the PASTOR estimate put up by Pástor and Stambaugh (2003). Ahn et al. (2018) studied tick data of 1183 stocks of 21 emerging markets and proved that LOT measure and AMIHUD are the best proxy among the three, respectively.

#### *2.6. Explaining Market E*ffi*ciency in Subdividing Areas*

There is substantial evidence to show that markets are inefficient (see, for example, Jensen 1978; Lehmann 1990; Fama 1998; Chordia et al. 2008). Loughran and Ritter (2000) suggested that multifactor models and time-series regressions should not be used to test for market efficiency.

Market efficiency has withstood the challenge of the long-term return anomaly literature. Consistent with the assumption that anomalies in market efficiency are accidental outcomes, apparent overreactions to information are as common as underreactions, and the after-event continuation of abnormal returns before an event is as frequent as the reversal after an event. Most importantly, the obvious anomalies can be methodological, and most long-term outliers, tend to disappear as technology makes sense, as market efficiency predicts (Fama 1998). Dimson and Mussavian (1998) recorded a large number of studies showing that abnormal behaviors seem inconsistent with market efficiency at first glance.

The review and analysis of market efficiency above are not the whole stories due to continuous studies are ongoing. There are supportive academics showing that EMH is significant in subdividing areas. Jena et al. (2019) proved that both volumes Put–Call Ratio and open interest Put–Call Ratio could be sufficient predictor of market returns under certain conditions. Chang et al. (2019) pointed out that the use of financial constraints has a significant positive impact on the long-term performance of companies after issuing convertible bonds.

Based on a Structure Equation Model (SEM), Li and Zhao (2019) claimed that the comprehensive effect of housing on stock investment is positive under the background of Chinese cities. Chiang (2019) studied monthly data from 15 stock markets, along with economic policy uncertainty (EPU) as a news variable, and found that news is able to predict stock market future returns. Ehigiamusoe and Lean (2019) pointed out that financial development has a long-term positive impact on economic growth, while the real exchange rate and its volatility weaken this impact. The development of the financial sector will not bring ideal economic benefits unless it is accompanied by the decline and stability of the real exchange rate.

Moreover, in some simulation cases, possibilities are found to help explain market efficiency. For example, in the experiment of Zhang and Li (2019), AR and TAR models with gamma random errors were tested on empirical volatility data of 30 stocks, with 33% of them being very suitable, indicating that the model may be a supplement to the current Gaussian random error model with appropriate adaptability. In addition, in some cases, scholars will examine and compare several methods on market efficiency explanation, before applicable practical prediction. Guo et al. (2017b) used first-order stochastic dominance and the Omega ratio in market efficiency examination and applied the theory they put forward to test the relationship between the scale of assets and real-estate investment in Hong Kong.

It is possible that scholars will find conflicts again. When studying impacts of exchange rates on stock markets, Ferreira et al. (2019) found that the exchange rate has a significant impact on Indian stock market, while there is no significant impact on European stock market. In the research of Lee and Baek (2018), although changes in oil prices do have a significant and positive impact on renewable energy stock prices in an asymmetric manner, it is a short-term impact only. Although in some of the cases, the existing models could not be considered, or quantitative analysis and modeling are still in progress, it is reasonable to postulate that, if the abnormal returns underlying anomalies are well explained, the market will become efficient with arbitrage.

Many scholars hypothesize that stock prices that are determined in efficient markets cannot be cointegrated. Nonetheless, Dwyer and Wallace (1992) showed that market efficiency or the existence of arbitrage opportunities are not related to cointegration. However, Guo et al. (2017b) developed statistics that academics and practitioners can use to test whether the market is efficient, whether or not there are any arbitrage opportunities, and whether there is any anomaly. Stein (2009) examined whether the market is efficient by both crowding and leverage factors for sophisticated investors.

Chen et al. (2011) applied cointegration and the error-correction method to obtain an interdependence relationship between the Dollar/Euro exchange rate and economic fundamentals. They found that both price stickiness and secular growth affect the exchange rate path. Clark et al. (2011) applied stochastic dominance to ge<sup>t</sup> efficient portfolio from an inefficient market. By applying a stochastic dominance test, Clark et al. (2016) examined the Taiwan stock and futures markets,

Lean et al. (2015) examined the oil spot and futures markets; Qiao and Wong (2015) examined the Hong Kong residential property market; and Hoang et al. (2015b) examined the Shanghai gold market. Each of these papers concluded that the markets are efficient. However, Tsang et al. (2016) examined the Hong Kong property market by using the rental yield, and concluded that the market is not efficient and there exist arbitrage opportunities in the Hong Kong property market.
