Theoretical background

A large number of asset pricing models extensively studied in the literature are special cases of a general consumption-based asset pricing model, in which the marginal utility of investors is the only factor that drives asset prices [8] (pp.149-183). Specifically, some prominent models, such as the CAPM [9–11], the CCAPM [12, 13], or the Fama-French three-factor model [14], are developed by naturally imposing some constraint on the investors’ utility function or the dynamics of consumption growth. In this regard, the Fama-French three-factor model deserves special mention. According to [15], the Fama-French three-factor model constitutes a successful implementation of the intertemporal asset pricing theory, as defined by [16]. On this basis, the wealth portfolio–frequently assimilated to a broad-based portfolio, such as the value-weighted portfolio of the Tokyo Stock Exchange–does not correctly proxy the marginal utility of investors, given the forecastable nature of expected returns. This makes it necessary to introduce other state variables that allow the model to forecast shifts in the investment opportunity set of investors [8] (pp. 165–167). The Fama-French three-factor model introduces two additional state variables apart from the value-weighted market portfolio, specifically a size factor and a value factor, that allow the model to correctly price a wide range of anomaly portfolios. These factors are often referred to as SMB (the excess return of the portfolio that comprises the small minus big market value firms) and HML (the excess return of the portfolio that comprises the high minus low book-to-market equity firms) in the asset pricing literature.

Although consumption-based asset pricing models are generally presented in the form of single-factor or multifactor models, they can all be written in terms of a stochastic discount factor (hereafter, SDF) model. This model assumes that asset prices are determined by the expectation conditional on time-t information of the product of the SDF and the asset payoff, where the SDF is the marginal rate of substitution or ratio of price to probability for all states of nature [8] (pp. 6–7). The SDF model is the dominant approach in contemporary asset pricing research [17] (p. 83), as it is easily adaptable to most of the topics addressed in the asset pricing literature. Below we exploit this flexibility to allow consumption-based asset pricing models to account for conditioning information used by economic agents and, more particularly, consumer sentiment.

Although recent research on financial markets highlights the important implications of consumer sentiment on asset returns [18–21] and its strong predictive power [22–25], its role in the cross-sectional analysis of stock returns has not yet been sufficiently addressed [26]. evaluates the validity of the Arbitrage Pricing Theory (hereafter, APT) in the Japanese equity market using investor confidence as an explanatory variable for asset returns, among other factors. Although his results support evidence of the APT as opposed to the CAPM, the author proxies investor confidence by the difference between the returns of long-term corporate and government bonds, using data series that cover only ten years (from January 1975 to December 1984) [27]. use an APT model to study the extent to which different macroeconomic factors determine returns in the Japanese stock market. Although the authors ignore investor confidence into the model, they use several variables frequently used as a proxy for consumer sentiment. Their results show that money supply, inflation, exchange rate, and industrial production have a significant influence on expected returns.

On the other hand [28], studies how market indicators of investor sentiment and arbitrage constraints affect the performance of cross-country stock market anomalies. His results show that most anomaly-based strategies are poorly related to market sentiment except value strategies. Remarkably, the author uses a composite measure of market sentiment that comprises the market-level investor sentiment index suggested by [29], the State Street Investor Confidence Index (SSIC), the Sentix Economic Indices Global Aggregate Overall Index (Sentix), and the Weighted Manufacturing and Non-Manufacturing Composite Purchasing Managers’ Index (PMI).

In contrast, we use the CCI for Japan, as provided by the OECD, within the ‘Composite Leading Indicators’ dataset, as a proxy for consumer sentiment. This data series comprises a large number of components, including real economy- and credit market-oriented indicators, which make the CCI a useful instrument to capture conditioning information used by investors, as noted below. However, it should be noted that many other variables can be used for the same purpose, given their strong relationship with business cycle. This is the case of illiquidity measures [30, 31] or market volatility, as measured by indicators such as the Chicago Board Options Exchange Market Volatility Index (VIX) [32–34].

Notwithstanding the above, consumption-based asset pricing models are only a partial explanation for asset prices, since they focus exclusively on the demand-side of the economy. Conversely, production models focus on the supply-side, seeking to connect asset prices to firms’ profitability and investment. Although first production models are long-standing [35–40], recent research has considerably improved their performance, allowing some production models to correctly price a wide range of anomaly portfolios that are typically mispriced in previous research. As noted above, the Fama-French five-factor model and the q-model, developed by [6] and [7], respectively, are especially remarkable [6]. rely on the present value identity, as defined by [41], to add two investment-inspired factors to their three-factor model, namely, RMW (the excess return of the portfolio that comprises the most profitable stocks minus the least profitable) and CMA (the excess return of the portfolio that comprises firms that invest conservatively minus aggressively) [7]. adopt a similar approach to propose a four-factor model (i.e. the q-model) that explicitly ignores HML. Importantly, the q-model fits several anomalies better than other models that, as the [42] four-factor model, include a momentum factor (the excess return of the portfolio that comprises the high minus low past one-year return stocks) as an explanatory variable. In fact, the apparent convenience of including HML as a factor is currently the subject of a lively debate, to the point that [7] and [43] claim that HML and the momentum factor appear to be noisy versions of the q-factors.

Consumption models

According to the Existence Theorem, the law of one price–that is, the fact that every asset that provides the same payoff as another has the same price–guarantees the existence of a single payoff x* in the payoff space that perfectly prices all payoffs [44–46]. In this context, the following pricing function is exactly satisfied: (1) where p_t is the N-dimensional vector of asset prices at time t, E_t (·) is the expectation conditional on time-t information, and x_t+1 is the N-dimensional vector of asset payoffs. Since asset returns are investments with price one and payoff R_t+1, Expression (1) can be particularized as follows: (2) where R_t+1 is the N-dimensional vector of gross returns. Defining an excess return as the difference between two returns, then: (3) where is the N-dimensional vector of excess returns in the payoff space. Additionally to the law of one price, when the payoff space and the pricing function leave no arbitrage opportunities, at least one strictly positive SDF is guaranteed. If markets are complete, that is, if there are the same number of securities as states of nature, that ensures that is the only valid SDF and that it is positive in all states of nature. However, when markets are incomplete there are a potentially infinite number of SDFs that satisfy the pricing function and such positive SDF no longer needs to be in the payoff space. In that case, Expression (3) can be written as follows: (4) where m_t+1 denotes any strictly positive SDF. Since the expected utility theory assumes that marginal utility is always nonnegative, the Existence Theorem allows for a macroeconomic explanation of asset prices. In particular, the investor’s first-order condition allows us to write Expression (4) as follows [8] (pp. 4–5): (5) where u’ is the investor’s marginal utility and C_t denotes the consumption in levels at date t. Assuming that m_t+1 is linear in consumption growth, the SDF can be written as follows (for a complete review of the procedures that allow linearizing the SDF in Expression (5) see [8] (pp. 161–165)): (6) where ΔC_t+1 denotes consumption growth. Expression (4) combined with Expression (6) results in the CCAPM. However, many consumption-based asset pricing models overcome measurement problems tied to consumption data using broad-based portfolios as a proxy for marginal utility growth. Specifically, different assumptions about investors’ preferences, such as quadratic utility, log utility, or the combination of exponential utility with normal distributions, make the return on the wealth portfolio a valid proxy for consumption growth [8] (pp. 152–161). In that framework, Expression (4) becomes the CAPM, with the wealth portfolio typically being assimilated to the value-weighted market portfolio minus the risk-free rate (hereafter, RMRF) in empirical research. On the other hand, the Fama-French three-factor model uses three different portfolios (RMRF, SMB and HML) as consumption state variables of special hedging concern to investors, in the language of the intertemporal CAPM, as defined by [16]. In order to allow Expression (4) to encompass all of these models, we can generalize Expression (6) as follows: (7) where a_t and b_t are parameters and f_t+1 is the K-dimensional vector of fundamental factors used as a proxy for the growth in marginal utility (e.g. the consumption growth for the CCAPM, RMRF for the CAPM, and RMRF, SMB and HML for the Fama-French three-factor model). However, it is important to note that, contrary to Expression (6), Expression (7) does not assume a specific economic structure in the SDF. This means that any vector of factors whose covariances with excess returns are perfectly correlated with expected excess returns will price assets correctly. Therefore, without imposing any other restriction on the dynamics of the vector of factors f_t+1, Expression (7) is fully consistent with the mechanics of the APT [47, 48].

Although most empirical work on asset pricing uses beta models rather than the SDF model, the transformation of Expression (4) into a beta model is straightforward. Specifically, Expression (4) can be rearranged as follows: (8) or equivalently: (9) where λ_t,m and β_t,m,R^e can be interpreted as the price of risk and the risk loadings of the assets, respectively. Substituting Expression (7) into Expression (9): (10) where β_t,f,R^e is the N×K matrix of slopes of the time-series regressions of excess returns on the vector of factors f, and λ_t,f is the vector of slopes of the cross-sectional regression of expected returns on β_t,f,R^e.

When both the SDF and the excess returns are independent and identically distributed (i.i.d.) there is not difference between the conditional and unconditional versions of the model and we can remove t subscripts in the previous expressions. Specifically, in that case, the unconditional version of Expression (4), particularized to the linear SDF shown in Expression (7), can be written as follows: (11) while Expression (10) results in: (12) However, the pattern of predictability of asset returns and the time-varying nature of the equity risk premium make substantial differences emerge between conditional and unconditional asset pricing models [1, 49]. In this regard, it is important to note that a conditional asset pricing model does not necessarily imply an unconditional asset pricing model. In any case, according to [39], to determine the unconditional version of Expression (4) it is sufficient to consider that coefficients a_t and b_t in Expression (7) vary linearly with an instrument or vector of instruments observable at time t, where instruments should be forecasting variables for excess returns [2]. Remarkably [50], follow an analogous procedure to transform the conditional CAPM into an unconditional model. Consequently, in order to consider conditioning information into the model, we assume that the conditional nature of Expressions (4) and (7) results from the dependence of the coefficients a_t and b_t on an instrument z observable at time t. In this context, we can write the unconditional version of Expression (4) by parameterizing the coefficients a_t and b_t in Expression (7) as a function of z, as follows: (13) Expression (13) shows that conditioning information adds two extra explanatory variables to the SDF in addition to the fundamental factors f_t+1, namely, z_t and f_t+1 z_t. Following [2, 39], this allows us to write the following vector, which comprises the fundamental factors and the new variables that result from the information included in z_t: (14) so that Expression (10) can be written as follows: (15) In the next section we compare the performance of the unconditional and conditional versions of single-factor consumption models–i.e. the CCAPM and the CAPM–, as defined in Expressions (12) and (15), respectively, with that of the Fama-French three-factor model and the production models described below. For that purpose, we use the CCI as the instrument z in Expression (14).

Importantly, while the CAPM and the Fama-French three-factor model use traded factors as explanatory variables in the vector f, the CCAPM uses a non-traded factor–the per capita consumption growth–to explain expected returns. Consumption data is reported annually and quarterly, preventing Expressions (12) and (15) from being used at monthly or higher frequencies. However, it is always possible to determine the SDF implicit in Expressions (12) and (15) and subsequently estimate the factor-mimicking portfolio of the model as the projection of the SDF on the payoff space. Specifically, coefficients λ_f in Expression (12) allow us to determine the parameters a and b in Expression (11) straightforwardly, as follows [8] (pp. 106–107) (for Expression (15) this procedure is identical, simply by replacing f with F): (16) (17) where is the vector of demeaned factors. Importantly, Expression (16) sets a = 1 to normalize the SDF to E_t(m_t+1) = 1. In this regard, it is worth noting that the pricing function does not identify the mean of the SDF when the payoff space comprises solely excess returns, so any arbitrary value for the coefficient a will produce exactly the same fitted values.

As noted above, the factor-mimicking portfolio of the model can be estimated as the projection of m_t+1 –determined using Expressions (16) and (17)–on the payoff space, so that [8] (p. 67): (18) where proj(m_t+1|R^e) is the mimicking portfolio of the SDF (hereafter denoted as x*), R^e is the space of excess returns, and ε_t+1 is the error term. Expression (18) is equivalent to the regression of m_t+1 on the excess returns that constitute the test assets, that is: (19) where w is the vector of coefficients that result from the regression of m_t+1 on or, equivalently, the vector of portfolio weights of the factor-mimicking portfolio x*, so that: (20) Since the error term ε_t+1 is orthogonal to , the factor-mimicking portfolio carries exactly the same pricing information as m_t+1. However, importantly, depends entirely on market data and can be determined at the same frequency as excess returns. Consequently, below we use Expression (20) to test the CCAPM on a monthly basis.

Production models

While consumption models relate asset prices to the marginal utility that results from the investors’ first order condition, production models focus on the supply-side of the economy, exploiting the crucial role of investment and profitability of firms in equity returns. Specifically, production models assume that a firm chooses investments to maximize the cum-dividend value of equity, V_t, which is given by the risk-adjusted present value of dividends, according to the following Bellman equation [17] (p. 209): (21) where l_t is the investment, and D_t is the dividend paid by the firm at time t. Most production models constitute special cases of Expression (21). In particular, the Fama-French five-factor model, as defined by [6], starts from a standard dividend discount model to write the inverse of the book-to-market equity ratio (hereafter, BE/ME ratio) as follows: (22) where Π_t denotes the total equity earnings for period t, and r is the internal rate of return on expected dividends. Expression (22) suggests that a high BE/ME ratio (low ME/BE ratio) must either imply a low return on equity (ROE) or high returns over time and, consequently, ROE (i.e. profitability) should forecast future returns controlling for value. Additionally, controlling for BE_t, ME_t and expected earnings, higher expected growth in book equity (i.e. investment) implies a lower expected return [6]. identify RMW and CMA as proxies for profitability and investment, respectively. These two factors combined with the Fama-French three-factor model result in the Fama-French five-factor model, which is represented in the form of an unconditional beta model, as defined in Expression (12), with the following vector of factors: (23) On the other hand, the q-model, as defined by [7], starts from Expression (21) to fully develop a production-based asset pricing model that assumes that the dividend at time t is determined as the difference between the gross profits paid to shareholders, Π_t, and the cost of investment, Φ_t. Thus, defining: (24) (25) where a > 0 is a constant parameter, K_t is the productive capital, and the second term in the right-hand side of Expression (25) denotes the adjustment costs, then Expression (21) can be rewritten as follows, particularized to the case of two periods: (26) Expression (26) leads to the first principle of investment for firms, which says that: (27) Combining Expressions (26) and (27) and operating the result yields the investment-CAPM, which can be written as follows [7, 43]: (28) where r_t+1 is the net stock return. The beta representation that results from Expression (28) is the q-model, which assumes the following vector of factors in Expression (12): (29) where r_I/A is the difference between the return on a portfolio of low investment stocks and the return on a portfolio of high investment stocks, and r_ROE is the difference between the return on a portfolio of high ROE stocks and the return on a portfolio of low ROE stocks.

Apart from some differences in the process followed to form the portfolios comprised in the factors, investment and profitability factors shown in Expressions (23) and (29) are very similar. In particular, both CMA and r_I/A use the growth of firms’ assets as sorting variable. Regarding profitability factors, RMW sorts stocks using revenues minus cost of goods sold, minus selling, general, and administrative expenses, minus interest expense, all divided by book equity. On the other hand, r_ROE uses ROE as sorting variable, which is essentially the same ratio as that used by RMW, except for the fact that ROE is determined after taxes.

In order to compare both models in a uniform basis and better analyze the apparent redundancy of HML with investment and profitability factors, in the next section we determine the three classic Fama-French factors following the standard methodology suggested by [14], using the approach followed by [7] to determine r_I/A and r_ROE.

Data series

In order to test the models described in the previous section, we take the perspective of a Japanese representative investor, thus assuming not only the existence of a representative investor–i.e. perfect risk-sharing–, but also the national identity of this investor. Although most tests on the performance of asset pricing models in the US equity market adopt this perspective, research on international equity markets often takes the perspective of a US investor investing in a foreign market. For example [52], conduct a research on the performance of the Fama-French five-factor model on different countries and geographical areas, measuring all returns in dollars and using the one-month US Treasury bill rate as the risk-free rate. Conversely, we measure returns in local currency, using the three-month Treasury Bill rate for Japan as the risk-free rate. This helps minimize currency effects on asset returns, allowing us to focus exclusively on the dynamics of equity risk premiums from a domestic perspective.

The previous assumption largely conditions the source of data used in the paper for stock returns. Specifically, while many studies on the cross-sectional properties of stock returns use the data made publicly available by Kenneth French on his website, the Japanese representative investor assumption prevents us from using those series, as these data are provided in dollars. In order to avoid errors arising from changing data to the local currency, we compile all stock data from the Datastream database.

We test all models on different sets of portfolios, which comprise all stocks listed in the Tokyo Stock Exchange, for the period from July 1992 to June 2018 (all data is publicly available at [53]). For that purpose, we collect data for the following series, on a monthly basis: (i) total return index (RI series), which allows us to determine asset returns, (ii) market value (MV series), (iii) market-to-book equity (PTBV series), (iv) total assets (WC02999 series), (v) return on equity (WC08301 series), (vi) price-to-cash flow ratio (PC series), and (vii) dividend yield (DY series). We exclude non-common equity securities from Datastream data following the generic rules suggested by [54]. We also exclude stocks with less than twelve observations in the period from July 1992 to June 2018. Accordingly, our sample comprises a total number of 5,312 stocks. As noted, we use the three-month Japanese Treasury Bill rate, as provided by the OECD database, as a proxy for the risk-free rate.

We follow the methodology suggested by [14] to create three sets of portfolios, which constitute our test assets. Specifically: (i) 25 size-BE/ME portfolios, (ii) 20 momentum portfolios, and (iii) 25 price-to-cash flow-dividend yield portfolios (hereafter, P/CF-DY portfolios). Additionally, we follow [14] and [7] to determine market factors, namely, RMRF, SMB, HML, r_I/A and r_ROE. Importantly [55], attribute the strong explanatory power of different asset pricing models studied in the literature to the specific test assets typically used in empirical research and, in particular, to the strong factor structure of size-BE/ME portfolios. To avoid this drawback, the authors suggest using other portfolios to test models, in addition to size-BE/ME portfolios. We have followed this recommendation to configure our test assets.

We compile all macroeconomic data series from the OECD Statistics section. Regarding consumption data, although the CCAPM establishes the consumption growth in nondurable goods and services as the only explanatory variable for asset prices, we use total consumption data instead (‘Private final consumption expenditure’ series, CQR measure), following [56], which show that total consumption of households fits the cross-section of expected returns better than consumption in nondurables and services. We use the population series for Japan, as provided by the OECD, to determine per capita consumption growth.

We collect the CCI data series for Japan, for the period from July 1991 to June 2018, on a monthly basis, from the OECD Statistics section (dataset ‘Composite Leading Indicators’, MEI). This indicator comprises the following data series: (i) the inverse of the ratio of inventories to shipments, (ii) the quotient of the volume of imports and exports, (iii) the inverse of the loans to deposits ratio, (iv) the number of hours worked in the manufacturing industry, (v) the work started for dwellings, (vi) the share prices in the TOPIX index, (vii) the spread of interest rates, and (viii) the Small Business Survey on sales trend. Our data show that the annual CCI is positively correlated with the gross domestic product growth rate (ΔGDP), while it is poorly correlated with the consumer price index (CPI), with correlations of 0.46 and -0.05, respectively. Following [57], we perform all calculations below using the demeaned CCI.

Table 1 shows the main summary statistics for portfolios and factors under consideration. Panel A shows that the size effect–that is, the fact that stocks with lower market equity provide higher returns, and vice versa–works as expected, with portfolios comprising the smallest stocks providing significantly higher returns. However, the value effect–that is, the fact that stocks with higher BE/ME provide higher returns, and vice versa–exhibits a less clear pattern, which is consistent with the sharp reduction in the value premium experienced in other markets, such as the New York Stock Exchange [8] (p. 452).

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Table 1. Summary statistics.

https://doi.org/10.1371/journal.pone.0241318.t001

On the other hand, Panel B in Table 1 suggests a U-shape curve in the mean returns of momentum portfolios, with portfolios comprising stocks with the highest and lowest past one-year return providing the highest returns. Regarding P/CF-DY portfolios, Panel C in Table 1 shows that, in general, the higher the price-to-cash flow ratio, the lower the mean return, and vice versa, which is consistent with the results provided by most ratios that scale prices by some firm-specific characteristic. Conversely, the dividend yield does not deliver a clear pattern in average returns.

Table 1, Panel D shows means and standard deviations of factors used as explanatory variables. As shown, the mean of RMRF is relatively high (1.17% on a monthly basis) compared to the result achieved by [52], which provide a near-zero mean return for the period from 1990 to 2015. However, the fact that the authors measure all returns in dollars and use the US Treasury bill rate as a proxy for the risk-free rate makes these results not directly comparable. Regarding macroeconomic series, Panel D in Table 1 shows that, for the period under study, Japan exhibits an average consumption growth of 0.44% using annual consumption data, which is a relatively low rate compared to other countries. Additionally, Panel D shows that the CCI is highly volatile, with a standard deviation of 124.15%, on a monthly basis.

Table 2 summarizes the correlations between our test assets and factors. As shown, most portfolios exhibit high positive correlations with RMRF and RMW, while size-BE/ME portfolios are also significantly correlated with SMB. Remarkably, all portfolios under consideration exhibit a weak negative correlation with quarterly consumption growth. However, it is important to mention that these correlations do not condition the performance of asset pricing models. Instead, according to Expression (8), it is the correlation of expected excess returns with the covariances between excess returns and factors that determines the size of pricing errors.

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Table 2. Correlations.

https://doi.org/10.1371/journal.pone.0241318.t002

Predictive power of consumer sentiment

Any instrument useful for capturing the conditioning information used by investors must have some predictive power to forecast asset returns or macroeconomic indicators [8] (p. 135). Accordingly, Table 3 summarizes the results of different forecasting regressions, using the CCI for Japan as a predictor. Panel A shows the regression results of the AR(1) model for the CCI at different horizons. As shown, consumer sentiment is strongly persistent in the short-run, providing a slope coefficient and a R² statistic of 0.895 and 0.801, respectively, for the three-month horizon. However, that persistence declines rapidly over time, providing negligible estimates for two- and five-year horizons. These results contrast with those provided by other predictors, such as the dividend yield or the consumption-wealth ratio, which deliver one-year slope coefficients of around 0.9 and 0.6, respectively [1].

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Table 3. Predictive power analysis for the CCI.

https://doi.org/10.1371/journal.pone.0241318.t003

Panel B in Table 3 shows the results of the forecast regression analysis for the value-weighted market portfolio, RMRF, using the CCI as a predictor. Additionally, in order to study the potential shifts in the predictive power of the CCI in the presence of other macroeconomic regressors, Panel C shows the regression results using ΔGDP as a forecasting variable, while Panel D uses both the CCI and ΔGDP as predictors. As shown in Panel B, although three-month and one-year estimates are virtually negligible, the predictive power of the CCI increases as the time interval expands, which is a common feature of most asset return predictors. Specifically, although the one-year slope coefficient amounts to -0.01, it reaches -0.279 for a five-year horizon. Similarly, the R² statistic increases from 0.002 to 0.203, for those time intervals. Remarkably, the five-year slope coefficient is statistically significant, with standard errors corrected for the autocorrelation that results from overlapping returns, according to the [43] methodology.All slope coefficients shown in Table 3, Panel B, are negative with the sole exception of the three-month estimate. This is consistent with the cyclical nature of consumer sentiment, which means that the higher the CCI, the higher the market prices and, consequently, the lower the expected returns. Regarding three-month estimates, the positive slope coefficient suggests a continuation of the recent trend in stock returns, which is consistent with the momentum anomaly, as described by [14] and [42].

Panels C and D in Table 3 show that the same conclusions hold when we include ΔGDP as a predictor. Indeed, Panel C shows that, although the GDP exhibits some predictive power for the one-year forecast, in the long-run its importance declines sharply, providing a R² statistic of 0.033 for the five-year forecast. Consistently, Panel D in Table 3 shows that most of the estimates for the CCI, as well as the R² statistics, remain almost unchanged from those shown in Panel B, which is indicative of the small predictive power of ΔGDP.

In summary, although our results suggest that the predictive power of the CCI for Japan is lower than that of other predictors widely studied in the literature, the fact that consumer sentiment provides significantly better estimates as the time interval expands, together with the high volatility of the CCI, makes us expect that consumer sentiment can be a good instrument to account for conditioning information in cross-sectional consumption models.

Model results

In order to estimate the coefficients of the asset pricing models described above, we use a two-pass cross-sectional regression (CSR) methodology, following a procedure similar to that pioneered by [59] and [60]. According to this methodology, in a first stage, the time-series regressions of excess returns on factors provide the beta coefficients of the model (i.e. Expressions (12) and (15), depending on the unconditional or conditional nature of the model, respectively). In a second stage, the cross-sectional regression of average excess returns on betas produces lambda coefficients. Although this procedure does not guarantee that the model correctly prices the factors–as is the case in a purely time-series approach, where the factors are in the payoff space and λ = E(f) is assumed–, it constitutes a well-established methodology in the asset pricing literature, which allows the model to fit expected payoffs to betas and determine the prices of risk even when the factors are not traded.

Nevertheless, the two-pass CSR methodology involves some specific issues that can lead to misleading estimates. In particular, the fact that errors are typically cross-sectionally correlated at a given time, the estimated nature of betas, and the problems arising from measurement errors can severely distort model estimates [8] (pp. 247–248). Thus, while the cross-sectional correlation can result in misleading standard errors, the fact that betas are generated regressors not only can lead to biased t-statistics, but can also result in inconsistent estimators. In this regard [61, 62], examine a wide range of implications of the use of constructed variables on the properties of estimators, including their consistency, efficiency and potential to provide valid inferences, while [63] study the effects of estimated regressors in models with risk terms. Notably, the classic procedure developed by [60] (hereafter, the Fama-MacBeth procedure) helps to flexibly correct standard errors for cross-sectionally correlated errors, while other approaches, such as that suggested by [64], allow correcting standard errors for the fact that betas are estimated.

Importantly [8], (pp. 247–251) explains that, under certain conditions, the Fama-MacBeth procedure is numerically equivalent to a pooled regression, and emphasizes the benefits of mapping this approach into the generalized method of moments (GMM), developed by [65]. Notably [66], introduces a GMM estimator that relies on distance instruments to estimate panel data regression models containing errors in variables [32, 33]. use this approach to test the Fama-French five-factor model augmented with illiquidity measures, concluding that, in general, the most significant factor is RMRF, with measurement errors largely determining this result. On the other hand [34], use a distance instrument-based algorithm into a GMM framework, to study the impact of illiquidity on the returns of twelve sector portfolios. Consistently with [32, 33], the authors conclude that the only relevant factor according to their dynamic GMM approach is the market risk premium.

Given the aforementioned issues related to the two-pass CSR methodology, in order to allow comparison between our results and previous research that analyzes the effects of conditioning variables, we estimate the models described above by mapping OLS regressions into GMM, following [8] (pp. 241–242). This allows us to simultaneously estimate betas and lambdas in Expressions (12) and (15) and correct standard errors for cross-sectional autocorrelation in a manner similar to the Fama-MacBeth procedure, but additionally correcting for the fact that betas are generated regressors. Specifically, following [8] (p. 241), we define the following vector of moments: (30) where a is the vector of intercepts of the time-series regressions, and X_t is the vector of factors, that is, f_t or F_t, depending on the unconditional or conditional nature of the model, respectively. In order to allow GMM to reproduce the two-pass CSR estimates, we weight the moments in Expression (30) using the following matrix [8] (p. 242) (I denotes the identity matrix): (31) so that: (32) In order to estimate standard errors and the distribution of moments, we use the standard GMM formulation (see [8] (pp. 203–204)) and a spectral density matrix S with zero leads and lags, as follows: (33) Tables 4–6 show the regression results provided by all models under consideration, namely, the CCAPM and the CAPM, both conditional on consumer sentiment and unconditional, the Fama-French three- and five-factor models, and the q-model. In order to accurately evaluate the performance of these models, we study the anomaly portfolios described above separately. In particular, Table 4 shows the regression results achieved for size-BE/ME portfolios, while Tables 5 and 6 do the same for momentum portfolios and P/CF-DY portfolios. Each table comprises three panels, the first showing annual results and the second and third showing quarterly and monthly results, respectively.

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Table 4. Regression results for 25 portfolios size-BE/ME.

https://doi.org/10.1371/journal.pone.0241318.t004

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Table 5. Regression results for 20 momentum portfolios.

https://doi.org/10.1371/journal.pone.0241318.t005

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Table 6. Regression results for 25 portfolios P/CF-DY.

https://doi.org/10.1371/journal.pone.0241318.t006

Tables 4–6 display two rows for each model, where the first row shows the lambda estimates that result from Expressions (12) and (15), and the second row shows the t-statistics. In the case of the CCAPM, we use the factor-mimicking portfolio approach, as defined in Expressions (16), (17), (19) and (20), to transform the quarterly estimates in Panel B into the monthly results shown in Panel C.As usual in the asset pricing literature, Tables 4–6 report the adjusted OLS R² statistics for each model. In any case [67], show that R² statistics can give rise to large differences in model performance that are not statistically significant. In fact, the estimated nature of betas can lead to significant distortions in this indicator. In this regard [55], suggest complementing the OLS R² statistic with the GLS R² statistic, as the latter constitutes a stronger hurdle for asset pricing models. In particular, the authors explain that the GLS R² statistic is directly determined by the distance of the model’s factor-mimicking portfolio to the mean-variance frontier, which ultimately conditions the size of the pricing errors provided by the model. Conversely, the OLS R² statistic is weakly related to the mean-variance efficiency of factors, leading to spurious conclusions in some cases. Accordingly, Tables 4–6 show both the OLS and GLS R² statistics for each model.

Additionally, problems tied to OLS R² statistics and other test statistics make the recent asset pricing literature emphasize the convenience of supplementing these indicators with visual representations of pricing errors. Consequently, Figs 1–3 explicitly relate the fitted values provided by the main models under study to the mean excess returns of the portfolios that constitute our test assets. In particular, any model that produces a perfect fit will provide estimates lying on the 45 degrees angle. Therefore, the better the performance of the model, the more concentrated the estimates will be around the 45 degrees angle, and vice versa. Fig 1 plots the estimates that result from size-BE/ME portfolios, while Figs 2 and 3 do the same for momentum portfolios and P/CF-DY portfolios, respectively.

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Fig 1. Realized excess returns versus fitted values for 25 portfolios size-BE/ME.

The figure depicts 25 portfolios size-BE/ME using a code with two numbers, the first number being the size code (with 1 being the smallest and 5 the largest) and the second number being the BE/ME ratio code (with 1 representing a low ratio and 5 a high ratio). The closer the portfolios are to the 45 degrees axis, the better the fit produced by the model.

https://doi.org/10.1371/journal.pone.0241318.g001

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Fig 2. Realized excess returns versus fitted values for 20 momentum portfolios.

The figure depicts 20 momentum portfolios using a number from 1 to 20. Stocks with the lowest past one-year return comprise portfolio 1 and stocks with the highest past one-year return comprise portfolio 20. The closer the portfolios are to the 45 degrees axis, the better the fit produced by the model.

https://doi.org/10.1371/journal.pone.0241318.g002

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Fig 3. Realized excess returns versus fitted values for 25 portfolios P/CF-DY.

The figure depicts 25 portfolios P/CF-DY using a code with two numbers, the first number being the P/CF ratio code (with 1 representing a low ratio and 5 a high ratio) and the second number being the DY code (with 1 representing a low ratio and 5 a high ratio). The closer the portfolios are to the 45 degrees axis, the better the fit produced by the model.

https://doi.org/10.1371/journal.pone.0241318.g003

Tables 4–6 also show the mean absolute pricing error (MAE) for each model, where pricing errors are determined as the difference between the mean excess returns of the portfolios under consideration and the results provided by Expressions (12) and (15), as appropriate. The last column in Tables 4–6 comprises the results delivered by the J-test for overidentifying restrictions, as defined by [65].