Overreaction in the REITs Market: New Evidence from Quantile Autoregression Approach

Real estate investment trusts (REITs) provide portfolio diversification and tax benefits, a stable stream of income, and inflation hedging to investors. This study employs a quantile autoregression model to investigate the dependence structures of REITs’ returns across quantiles and return frequencies. This approach permits investigation of the marginal and aggregate effects of the sign and size of returns, business cycles, volatility, and REIT eras on the dependence structure of daily, weekly, and monthly REIT returns. The study documents asymmetric and misaligned dependence patterns. A bad market state is characterized by either positive or weakly negative dependence, while a good market state is generally marked by negative dependence on past returns. The results are consistent with under-reaction to good news in a bad state and overreaction to bad news in a good state. Past negative returns increase and decrease the predictability of REIT returns at lower and upper quantiles, respectively. Extreme positive returns in the lower (upper) quantiles dampen (amplify) autocorrelation of daily, weekly, and monthly REIT returns. The previous day’s REIT returns dampen autoregression more during recession periods than during non-recession periods. The marginal impact of the high volatility of daily returns supports a positive feedback trading strategy. The marginal impact of the Vintage REIT era on monthly return autocorrelation is higher than the New REIT era, suggesting that increased participation of retail and institutional investors improves market efficiency by reducing REITs’ returns predictability. Overall, the evidence supports the time-varying efficiency of the REITs markets and adaptive market hypothesis. The predictability of REIT returns is driven by the state of the market, sign, size, volatility, and frequency of returns. The results have implications for trading strategies, policies for the real estate securitization process, and investment decisions.


Introduction
Real estate investment trusts (REITs) are broadly categorized into equity REITs (EREITs) and mortgage REITs (MREITs). EREITs typically own and operate income-producing physical real estate. The total returns of EREITs consist of rental income paid by the tenants and changes in the value of properties they own. The MREITs invest in mortgage-backed securities (MBS) and engage in the provision of mortgages by real estate debt owners. The total returns of MREITs consist of interest income and changes in the present value of MBS. EREITs and MREITs significantly differ in their funding profile, business model, and asset base. These differences underlie their classifications where on all quantiles of REIT returns. The model has been applied by Baur et al. (2012); Baur (2013); and Li et al. (2016), inter alia, to investigate asymmetric autocorrelation in stock market returns. Chevapatrakul and Mascia (2019) also employ the QAR model to investigate investors' overreaction in the cryptocurrency market. We utilize the QAR model to analyze REITs' return dynamic dependence patterns on past returns, the impact of extreme past positive and negative returns, and business cycles' influence on REITs' returns autocorrelation. The study adds a new dimension to the emerging literature in REITs research by investigating the potential presence of investors' overreaction and under-reaction to macroeconomic news across different return distributions and return frequencies. Existing studies have not addressed the autocorrelation structure of daily, weekly, and monthly REIT returns. These studies employ conditional mean of the return distribution to examine the influence of a lagged REITs' return on the current REIT return.
In contrast, the present study attempts to offer a detailed description of the full return distribution conditional on a lagged return. To this end, we use the quantile autoregression (QAR) framework of Koenker and Xiao (2006), a variant of the quantile regression developed by Koenker and Bassett (1978). In the present study, the QAR model is used to study the impact of lagged REIT returns on all quantiles of REIT returns. The model has been applied by Baur et al. (2012); Baur (2013); and Li et al. (2016), inter alia, to investigate asymmetric autocorrelation in stock market returns. Chevapatrakul and Mascia (2019) also employ the QAR model to investigate investors' overreaction in the cryptocurrency market. We utilize the QAR model to analyze REITs' return dynamic dependence patterns on past returns, the impact of extreme past positive and negative returns, and business cycles' influence on REITs' returns autocorrelation. The study adds a new dimension to the emerging literature in REITs research by investigating the potential presence of investors' overreaction and under-reaction to macroeconomic news across different return distributions and return frequencies. Existing studies have not addressed the autocorrelation structure of daily, weekly, and monthly REIT returns. What causes return autocorrelation? Finance literature provides the theoretical foundation of stock returns autocorrelation. 2 Among the most prominent causes of the autocorrelation of returns is delayed price adjustment, which causes a partial price adjustment effect (PPA). There are three major perspectives on the PPA effects. First, according to Copeland (1976) and Jennings et al. (1981), sequential arrival of information in the market, followed by the sequential acquisition of information by investors (Holden and Subrahmanyam 2002), creates trends in stock price movement, thereby, generating positive autocorrelation of returns. Second, even in the presence of instantaneous arrival of information, there is a delay by the agents in processing the information and acting on it due to transaction costs and limited mental capabilities (Mech 1993). The slow price adjustments would create a trend in price formation behavior, resulting in positive autocorrelation of returns. Lastly, 2 A natural question is whether REITs are similar to non-REIT stocks. There are unique requirements which differentiate REITs from non-REIT stocks and bonds. Specifically, at least 75 % of a REIT's assets must be invested in real estate. REITs must also redistribute at least 90% of their taxable income as dividends to shareholders. These idiosyncratic and restrictive requirements ensure that the performance of REITs and the underlying assets are closely linked. Past studies have found that REITs are more similar to non-REIT equities (stocks) than bonds. Therefore, models and theories associated with non-REIT stocks can be applied to REITs. investors generally make systematic errors in interpretation of information leading to overreaction, overshooting of the asset price, and subsequent price reversals to the fundamental value, thereby generating a negative autocorrelation of returns. Overreactions to information may be initiated by the interplay of the conservatism and representativeness (Barberis et al. 1998) or overconfidence and self-attribution bias of irrational investors (Daniel et al. 1998). The existence of positive (negative) feedback trading may reinforce (dampen) price overreactions. 3 Using an intertemporal, rational expectations equilibrium model of asset pricing, Veronesi (1999) finds that investors' reaction to macro news is state-dependent. Specifically, investors tend to overreact to bad news during market booms (good state or upper quantiles) and under-react to good news during market depressions (bad state or lower quantiles). Therefore, the impact of lagged returns or degree of return persistence differs across quantiles of conditional distribution on current returns, thereby engendering inconsistent predictability of REIT returns.
The behavioral finance empirical models of Barberis et al. (1998) and Lewellen (2002) can empirically explain the price "momentum" and "reversal" phenomena, which are, respectively, attributed to under-reaction and overreaction to macro news. 4 Investors' under-reaction and overreaction occur at different market states. Therefore, different return dependence patterns are likely to emerge across conditional distributions of current returns associated with each quantile.
This study hypothesizes that investors in EREIT and MREIT markets may not act in an entirely rational way in bad and good (extreme) market states when returns are very low or very high, respectively. The presence of significant positive or negative autocorrelation (serial dependence) 5 of REIT returns at any quantile, either due to under-reaction or overreaction of the investors, coupled with an intermittent absence of autocorrelation, supports the time-varying (adaptive) market hypothesis (AMH) postulated by Lo (2004Lo ( , 2017 and Brennan and Lo (2011). The autoregression pattern will vary with the frequency of returns due to time aggregation, information, and the motives of the investors.
The study briefly documents asymmetric and misaligned dependence patterns where lower quantiles or bad market states are characterized by either positive or weakly negative dependence, while upper quantiles or good market states are generally marked by negative dependence on past returns. The results are consistent with under-reaction to good news in a bad state and overreaction to bad news in a good state. Lagged negative returns increase the predictability of REIT returns at lower quantiles (bad state) and decrease predictability in the upper quantile (good state), which is chiefly consistent with the overreaction hypothesis. Extreme positive returns in the lower (upper) quantiles dampen (amplify) autocorrelation of daily, weekly, and monthly REIT returns. The effects of the recession on the autocorrelation of returns are frequency specific. Recession induces increased autoregression for weekly returns, mainly in the lower and central quantiles but does not affect autoregression of monthly returns. For daily returns, the previous day's returns reduce autoregression more during recession periods compared to non-recession periods. Overall, the dependence structure is a function of the state of the market, sign, size, volatility and time aggregation (frequency) of returns, and the class (EREIT and MREIT) of REITs. The episodic predictability and non-predictability of REITs returns at different quantiles, and return frequencies support the REITs' markets time-varying efficiency and adaptive market hypothesis.
The study's remaining parts are organized as follows: Section 2 details data sources and its characterization. Section 3 explains the econometric methodology. Section 4 offers empirical results, while Section 5 summarizes the results and concludes. See the studies by Sentana and Wadhwani (1992); Sias and Starks (1997);or Nofsinger and Sias (1999). 4 Investors' overreaction, according to Daniel et al. (1998), is a psychological phenomenon arising from individuals' tendency to allocate excessive weight to the most recent information. Specifically, when investors acquire new information, their initial reaction is too strong, causing rapid price increase and positive return autocorrelation. However, as investors receive new information, overreaction is corrected in subsequent periods, originating price and return reversal (negative autocorrelation). Return reversals are essentially "corrections" of past mistakes. 5 The terms dependence, autocorrelation and serial correlation are used fairly interchangeably in this study.

Data Sources and Data Characterization
We cull the data from daily, weekly and monthly data from Bloomberg. The daily and weekly data are available from 01/04/1999. The monthly data are available from February 1972. The sample period for all three frequencies ends on 07/14/2020. The corresponding number of observations is N = 5617 for daily data, N = 1124 for weekly data, and N = 581 for monthly data. The use of daily, weekly, and monthly frequency data is motivated by the findings of Campbell et al. (1997) and Lo and MacKinlay (1990). Using the US stock indices returns, the authors document higher daily return autocorrelations than weekly and monthly autocorrelations. They attribute the higher daily return autocorrelations to the potential presence of either non-trading or non-synchronous trading.
The percent return, R t , of EREIT (MREIT) index at period t, is computed as the natural log difference of the closing index value at t and t − 1; that is, R t = ln(Index t /Index t−1 ) * 100. The daily close to close index is used to derive the daily returns. Weekly returns are computed from Friday close to Friday close the following week. In case there was no trading due to market closure, we utilize the last obtainable closing index value to calculate the weekly return. Monthly returns are equal to the natural log difference between the closing price indices in two successive months. The summary descriptive statistics of the three data frequencies are presented in Table 1. Results in Table 1 reveal that at every frequency, EREIT has higher average returns but a lower range of returns, volatility (standard deviation) of returns, negative asymmetric distribution (as measured by skewness except for daily frequency), leptokurtic (as measured by kurtosis) and non-normal distribution of returns (as measured by the Jarque-Bera statistic) than the MREIT returns. The average, maximum, and minimum returns and the volatility and asymmetry of returns distribution increase as the frequency decreases (time-aggregation increases). It is also apparent that the leptokurtic (non-normal) distribution of returns is decreasing in frequency. This is due to the smoothing associated with the time-aggregation of returns. While the results in Table 1 are point estimates, they suggest that EREIT and MREIT returns have different autoregressive patterns. Figure 1b illustrates the time-series of EREIT and MREIT returns at daily, weekly and monthly frequencies. We have also included recession periods (gray bars) as defined by the National Bureau of Economic Research (NBER).
Since the sample period ends in mid-July, it is also interesting to observe the impact of the COVID-19 pandemic on the performance of REITs. The sharp decline in returns occurred in 2020M03. Both categories of REITs respond differently to recession shocks. For example, the daily returns of EREIT (MREIT) troughed at 21% (27%) in 2020M03. The weekly returns of EREIT (MREIT) troughed at 30% (47%) in 2020M03, while the monthly returns troughed at 38% (79%) during the same month. Generally, returns seem more volatile during periods of economic or financial turmoil. Unsurprisingly, the minimum returns are, in absolute terms, higher than the maximum returns, especially for monthly data, which cover additional recessionary and crisis periods compared to daily and weekly data.
For example, the sharp decline in EREIT returns in October 1987 coincided with the Black Monday panic sell-off.

Econometric Model
Given a stationary return series, R t , a first-order autoregressive process, AR (1), based on the ordinary least squares (OLS) would be modeled as follows: where α 0 is a constant, β 1 is the slope which measures the mean or average persistence of returns and z t is white noise disturbances. The equivalent first-order conditional quantile autoregression-QAR (1) model is expressed as follows: In Equation (2), Q τ (•) designates the conditional quantile function at the τth quantile where τ ∈ (0, 1); π t−1 is the publicly available set to the market participants up to and including period t − 1. According to Nikolaou (2008), α z (τ) = α τ is the error term, z t , of the τ th quantile and includes the impact of unexpected shocks of different sizes and signs in period t − 1 that initiate variations in REIT returns in period t. Negative and positive shocks are associated with negative and positive market sentiments, respectively. β τ , which is the primary focus of the present study, is the coefficient estimate of the quantile-specific autoregressive parameter. The estimates of α τ and β τ in Equation (2) are derived by resolving the following minimization problem: In Equation (3), T is the total number of return observations, ρ τ (z) = z τ − I [z≤0] and I [z≤0] = 1 if z ≤ 0 and 0 otherwise. Relative to the first-order conditional mean model, AR (1), the QAR (1) 6 model presented in Equation (2) offers the advantages of investigating the return dependence patterns across the entire return distributional features. Therefore, we can gain vibrant and dynamic insights into the predictability of returns during different market states and sentiments. For example, extremely low (high) or negative (or positive) returns are generally indicative of negative (positive) market sentiments. The quantile regression is robust in the presence of outliers in the data. The median, which acts as the location parameter, is more efficient than the mean in the presence of heavy-tailed return distributions. Indeed, Baur et al. (2012) argue that each quantile can be interpreted as a state or a regime of asset return. The lower (upper) quantiles are consistent with a current bad (good) market state or regime.
Equation (1) is considered as the base QAR model. We extend and augment the model to account for the impact of the sign and size of the lagged REIT returns on dependence patterns. The model that accounts for the sign effects ("Sign Model"), in line with the Glosten et al. (1993) Generalized Autoregressive Conditional Heteroskedasticity (GJR-GARCH) model, is specified as: where the indicator variable, I NEG (R t−1 < 0) is equal to one if the REIT return in period t − 1 is negative and zero otherwise. In Equation (4), β τ measures the dependence in τ − th quantile when returns are positive while β τ + γ τ , measures the dependence in the τ − th when REITs returns are negative. According to Basu (1997), asymmetric dependence occurs whenever γ τ is statistically different from zero. 6 A lag order of one was selected using the Swartz Bayesian information criterion (BIC).
Likewise, the following "Size Model" can help in assessing the impact of extreme positive previous month returns (measured by coefficient estimate θ τ ) on the current period's REIT return.
In Equation (5), I SIZE (|R t−1 | > R q , is an indicator variable equal to one when the past period's REIT return exceeds a pre-specified threshold and zero otherwise. We then select R q at the 90th and 95th quantiles of the unconditional distribution of absolute returns of EREIT and MREIT. The coefficient estimate θ τ is used to evaluate the impact of the previous period's extreme positive or upper tail REIT returns on current REIT returns. Lastly, the degree of dependence of REIT returns may be an artifact of business cycles as defined by crisis versus non-crisis periods. Figure 1a,b show that REIT returns may differ across business cycles (recession versus non-recession periods) in the US. Therefore, it is plausible to test the marginal effects of economic recessions on REITs' returns dependence patterns. We use the following model to assess that possibility.
where DUM is a dummy variable equal to one if the period (day, week, or month) falls within a recession period (as defined by NBER) and zero otherwise. The expression, ϕ τ (R t−1 * DUM), captures the marginal effects or differences in the degree of return dependence between recession and non-recession periods. Specifically, ϕ τ measures the marginal effects during the crisis period while ϕ τ + β τ measures the aggregate effect on dependence for a specific quantile during the crisis period. If ϕ τ is not constant across all quantiles; then, the recession is likely to change the dependence structure. Following Baur (2013), we test the following two hypotheses: (1) H 0 : ϕ = 0 (there is no recession-specific change in the average degree of dependence.) Rejecting the null hypothesis implies that the degree of return dependency, on average, changes during the crisis period.
(2) H 0 : ϕ τ ϕ τ * ∀ τ and τ * with τ τ * (there is no recession-specific change in the structure of dependency or ϕ τ is not constant across all quantiles). Rejection of the null means that dependence structure changes during recession periods relative to non-recession periods.
There is a need to generate asymptotic standard errors in estimating coefficients in Equation (2) through Equation (6). To this end, we use the pair bootstrapping procedure proposed by Buchinsky (1995) to mitigate the adverse effects of potential misspecifications of the QAR model and heteroskedasticity.

Empirical Results
The results for the AR (1) and QAR (1) models are gathered in Table 2a,b for EREIT and MREIT, respectively. Consistent with the autocorrelation test based on the data-driven Portmanteau tests 7 , all the AR (1) coefficients, except for the daily returns of EREIT, are insignificant at daily, weekly, and monthly frequency, implying that the behavior of EREIT and MREIT returns at the means is characterized by a white noise process or unpredictability. This evidence partly supports the findings of Jirasakuldech and Knight (2005), who confirm the weak-form efficiency of EREIT but some predictability of MREIT returns. However, the results contradict the conclusions of Nelling and Gyourko (1998). The authors find that while the EREIT returns are predictable based on past performance, the exploitable arbitrage profits are eroded by transaction costs. Our evidence partially supports or contradicts past 7 Escanciano and Lobato (2009) developed a Portmanteau test which, unlike the Ljung and Box (1978) test, permits the data to automatically select the lag length. The test is also robust to conditional heteroskedasticity. studies due to the more extended sample periods, econometric modeling, and different frequencies covered by the present study.     (2). ***, **, and * indicates the statistical significance at 1%, 5%, and 10% level, respectively; The slope equality test was developed by Koenker and Bassett (1982). ***, **, and * indicates the statistical significance at 1%, 5%, and 10% levels, respectively.
In the QAR(1) model, the autoregressive parameter (β τ ) estimate is the focus of our attention. Results in Table 2a reveal that daily EREIT returns have significant and monotonically increasing negative dependence (downward sloping) patterns at all quantiles. The weekly returns have significant and rising negative autoregression from 25th quantile and above, suggesting white noise or weak-form efficiency in the majority of lower quantiles (5th to 20th quantiles). Although the monthly returns exhibit decreasing (increasing) positive (negative) autoregression at lower (upper) quantiles, the dependence pattern vastly contrasts that of the weekly returns. Specifically, while the monthly returns exhibit positively significant autoregression in the lower and middle (5th to 50th) quantiles, insignificant (weak-form efficiency) dependence at some middle quantiles, the parameter estimates for the upper (85th to 95th) quantiles show negative return dependence. Table 2a discloses an interesting dependence structure across different frequencies. The daily (weekly) MREIT returns exhibit independence in the lower quantiles since β τ is insignificant at 5th to 35th (50th) quantiles and negative autoregression in the middle and upper quantiles. However, the monthly returns exhibit almost an inverse dependence pattern where positive return dependence is in the lower and middle (10th to 45th) quantiles and return unpredictability (independence), or weak negative dependence is evident in the middle and upper quantiles.
Graphically, Figure 2a,b display return dependence patterns. The middle "line and symbol" represent quantile coefficient estimates. The estimates, α τ , measures the intercept (magnitude or the average daily, weekly, and monthly return per quantile) while β τ is the slope coefficients in the QAR(1) model where τ ranges between 0.05 and 0.95. The two outer lines define 95% confidence bands for all quantile estimates.
It should be noted that unlike the results in Table 2a,b, Figure 2a,b cannot disclose the statistical significance of either the magnitude and the autoregression coefficient as measured by α τ and β τ estimates, respectively, across frequencies or time aggregations.
For the daily and weekly (monthly) returns, the lower quantiles are marked by insignificantly (significantly positive) autocorrelations or slope coefficients while the middle and upper tail slope coefficients exhibit significantly negative return dependency. In contrast, the middle quantiles autoregressive coefficient estimates of monthly returns are mostly statistically immaterial and economically small. Figure 2a,b graphically confirm the downward-sloping autoregressive pattern of EREIT and MREIT daily, weekly, and monthly return frequencies. Several inferences can be made from the base results gathered in Table 2a First, according to Figure 2a,b, EREIT and MREIT returns have a similar dependence structure across different returns frequencies. However, Table 2a,b reveal that the statistical significance, sign, and size of autoregression vastly differ across return frequencies and the three states comprising the bad state (lower quantiles), normal state (central quantiles), and good state (upper quantiles) 8 . We attribute these differences to non-trading and non-synchronous trading for the daily prices, short-term arbitrage opportunities, and differing investment horizons of the investors such as day traders versus swing traders.
Second, the evidence of statistically negative and positive autoregressive and statistically insignificant coefficient estimates for EREIT and MREIT index returns supports the REITs market's adaptive market hypothesis where predictability of REITs returns is quantile-or state-dependent. Indeed, the decreasing autoregressive parameter pattern as we move from lower to upper quantiles signifies a higher degree of negative autocorrelation or predictability. The evidence suggests that the response of the current period's return on its most recent return is dependent on the state or the side of the EREIT and MREIT return distribution. It should be noted that unlike the results in Table 2a,b, Figure 2a,b cannot disclose the statistical significance of either the magnitude and the autoregression coefficient as measured by and estimates, respectively, across frequencies or time aggregations. Third, the overall evidence in Figure 2a,b implies that the previous period's positive return mostly reduces the current period's positive returns, thereby exacerbating the bad state. This is consistent with overreaction to bad news at period t−1, which either further reduces future positive returns or magnifies future negative returns. Declining positive (increasing negative) autocorrelation in the lower quantiles (bad state) is consistent with a negative momentum (price continuation) 9 or price reversals. The statistically significant negative serial correlation in the upper quantiles (good state) also evinces price reversals. It is mainly consistent with a market where positive returns are more likely to be followed by negative returns and vice versa. Another way to explain overreaction based on the negative dependence or predictability patterns of EREIT and MREIT is postulated by Baur et al. (2012) and Lehmann (1990). For example, the estimates of α τ (intercept) and β τ (slope) at the 10th quantile in Table 2a (EREIT) is −1.391 and −0.062, respectively. Both estimates are statistically material. The intercept's negative value reflects both the magnitude of the daily return at roughly 1.4% and negative market sentiments. The negative market sentiment is characterized by falling EREIT prices, causing the investors to overreact, followed by a hasty exit from the market, thereby causing further price declines and negative autoregression. This interpretation can be extended to the daily and weekly MREIT returns. At monthly frequency, EREIT and MREIT return in Table 2a,b follow a white noise process at the upper quantiles, suggesting that any mispricing during the month is arbitraged away in a good state, resulting in a weak negative autoregression or serial independence of returns. 10 In terms of investment strategy, a negative serial correlation in the good state would suggest that investors can buy EREIT and MREIT after periods with negative returns and sell after periods with positive returns. However, this is only possible if negative autocorrelations are high enough to absorb transaction costs. Positive-feedback investors (trend followers) make investment decisions by extrapolating historical price patterns and return sequences. The investors perceive price variations to have inertia. A trend strategy investor will take a positive (long) position if prices have recently exhibited momentum or positive autocorrelation and a negative or short position if REIT prices have lately declined (negative autocorrelation or return reversals).
Fourth, the autoregressive coefficient estimates increase as time aggregations increases (return frequency decreases). Specifically, the autoregressive estimates for monthly (weekly) returns are higher than those of weekly (daily) returns. This evidence is consistent with the findings of Campbell et al. (1997) and Lo and MacKinlay (1990).
The results described so far indicate a downward sloping pattern of the autoregressive coefficient estimates for EREIT and MREIT indices. We find varying statistical significance and magnitude of autoregressive parameter estimates across different quantiles of the conditional EREIT and MREIT return distributions across different return frequencies. Specifically, lower quantiles exhibit either a positive or low negative dependence on past returns than upper quantiles, which are characterized by negative dependence patterns. The results for the upper quantiles (good state) are mainly consistent with the findings of Baur et al. (2012) and Lewellen (2002), who find negative stock index return autocorrelation, especially for monthly returns. The authors attribute the negative serial correlation to investors' overreaction to macroeconomic news in a good state. 11 In line with Baur et al. (2012) and Veronesi (1999), the positive or low negative autoregression patterns in lower quantiles (bad state) point to investors' under-reaction to macroeconomic news. The rationale for under-reaction phenomena 9 As Lewellen (2002) notes, momentum is not synonymous with positive autocorrelation. While momentum is a cross-sectional result where winning stocks beat losing stocks, autocorrelation is a time-series phenomenon where a stock's or an index's past and future returns are correlated. However, Lo and MacKinlay (1990) show that indeed, momentum may be caused by serial correlation of returns, lead-lag relations among stocks (cross-serial correlation), or cross-sectional dispersion in unconditional means. Lewellen (2002), for example, shows that negative autocorrelations tend to reduce momentum profits. 10 Anderson et al. (2013) intimate that daily return autocorrelation is attributable to market microstructure biases such as nonsynchronous trading effect and bid-ask bounce, partial price adjustment (PPA), and time-varying risk premia. However, Mech (1993) argues that weekly and monthly return autocorrelations are generally caused by changing investment opportunities, which are naturally low-frequency events. The strong daily return autocorrelation is also consistent with profit taking by technical analysts and day traders. 11 Since EREIT and MREIT are well diversified portfolios, their returns and dependence patterns reflect systematic risks caused by macroeconomic factors, as opposed to individual firm-specific or idiosyncratic risk.
is associated with partial price adjustment effect (PPA) either due to "conservative" response of the investors to news, sequential arrival of information in the market, followed by the gradual acquisition of the information by investors (either due to delay in acquisition of information or limited mental ability to process the information or market frictions such as transaction costs), resulting in trend formation, positive serial correlation, and momentum in returns. 12 The evidence so far supports the narrative that a single autoregressive coefficient estimate based on the expected EREIT and MREIT returns, will often obfuscate the state-and return frequency-dependent nature of the autoregressive processes in terms of the sign, economic and statistical significance. The lagged REIT returns have a material influence on current returns across different quantiles and frequencies. Our results support evidence by Clayton and MacKinnon (2003) and Zhou and Lee (2013), who find that REIT return behavior and predictability are time-varying. Zhou and Lee (2013) further show that market conditions indeed influence return predictability.
We further investigate whether autocorrelation is asymmetric using a variation of the Wald test proposed by Koenker and Bassett (1982). The test is employed to assess the hypotheses that the slope coefficients, β τ , are identical at different quantiles. Specifically, we test the following null hypotheses: (i) β 0.05 = β 0.95 ; (ii) β 0.05 = β 0.50 ; (iii) β 0.50 = β 0.95 and (iv) β 0.05 = β 0.25 = β 0.50 = β 0.75 = β 0.95 . Table 2c gathers the F-statistics for each of the two REITs indices at different return frequencies. The results suggest that apart from the statistically insignificant difference between β τ estimated at the 5th against the 50th quantiles of daily and weekly returns for the EREIT and MREIT index, the rest of the test statistics reject each null hypothesis of equality of slopes between or among different quantiles.
Collectively, the slope equality test results provide compelling evidence that autocorrelations of REITs return vary across quantiles of the conditional return distributions of different return frequencies. This supports the evidence from Table 2a,b, where autocorrelations tend to be positive or low negative (negative) for REITs returns in the respective low (high) conditional quantiles. Therefore, the response of a representative (median) return to the previous month's return certainly differs from that observed for two successive observations from the (upper or lower) tail of return distribution.
The 'Sign Model' described in Equation (4) is used to assess the likelihood that daily, weekly, and monthly lagged positive and negative returns differentially influence the conditional return distribution of REITs.
The coefficient estimates of lagged positive and negative returns, β τ and γ τ , respectively, are gathered in Table 3a,b and graphically illustrated using Figure 3a,b. We make the following observations and inferences. First, dependence patterns seem to be state-dependent since the parameter estimates, β τ , markedly vary in magnitude, sign, and statistical significance across quantiles and frequencies.
Second, the γ τ coefficient estimates vary in sign, statistical and economic significance. Specifically, the estimates are significantly positive, insignificant, and significantly negative at the bad, normal, and good state, respectively, across all frequencies. Therefore, the previous period's negative and positive REIT returns have a state-or quantile-specific predictive power on future returns. The statistical significance of γ τ also implies that positive and negative lagged EREIT and MREIT returns asymmetrically induce changes in current returns at daily, weekly and monthly frequencies.
In cases when γ τ is insignificant, for example, γ 0.50 for daily returns, γ 0.50 and γ 0.55 , for weekly returns and γ 0.35 and γ 0.55 for monthly returns in Table 3a, past positive and negative returns symmetrically impact the current returns, while negative returns have insignificant effects on current returns. Third, for both EREIT and MREIT, evidence that β τ + γ τ > β τ (β τ + γ τ < β τ ) suggests that current REIT returns are more sensitive to past negative (positive) returns than positive (negative) returns. Take the case of daily returns in Table 3a where β 0.05 (−0.990) + γ 0.05 (1.989) = 0.999. This is greater than β 0.05 = −0.990. Furthermore, take the case of weekly returns in Table 3b where β 0.85 (0.158) + γ 0.85 (−0.584) = −0.436. This is less than β 0.85 = 0.158. The overall evidence from Table 3a,b is that in the bad (good) state (lower (upper) quantiles), past period's negative (positive) returns provide more predictive power on current EREIT and MREIT returns than vice versa. 13 This evidence is pervasive across all return frequencies. Second, the coefficient estimates vary in sign, statistical and economic significance. Specifically, the estimates are significantly positive, insignificant, and significantly negative at the bad, normal, and good state, respectively, across all frequencies. Therefore, the previous period's negative and positive REIT returns have a state-or quantile-specific predictive power on future returns. The statistical significance of also implies that positive and negative lagged EREIT and MREIT returns asymmetrically induce changes in current returns at daily, weekly and monthly frequencies. In cases when is insignificant, for example, . for daily returns, . and . , for weekly returns and . and . for monthly returns in Table 3a, past positive and negative returns symmetrically impact the current returns, while negative returns have insignificant effects on current returns. Third, for both EREIT and MREIT, evidence that + > ( + < ) suggests that current REIT returns are more sensitive to past negative (positive) returns than positive (negative) returns. Take the case of daily returns in Table 3a where . (−0.990)+ . (1.989) = 0.999. This is greater than . = −0.990. Furthermore, take the case of weekly returns in Table 3b where . (0.158)+ . (−0.584) = −0.436. This is less than . = 0.158. The overall evidence from Table 3a,b is that in the bad (good) state (lower (upper) quantiles), past period's negative (positive) returns provide more predictive power on current EREIT and MREIT returns than vice The rationale for this evidence is that in the bad (good) state, price changes are driven by negative (positive) sentiment, albeit asymmetrically, as indicated by the upward sloping intercept, which is negative in the lower quantiles and positive in the upper quantiles. Indeed, the composite effect of negative returns on the autocorrelation of returns, as measured by β τ + γ τ , supports this view. Since γ τ is positive (negative) in the lower (upper) quantiles, then the aggregate effect of negative returns at the lower (upper) quantiles is to amplify (dampen) autocorrelation of REIT returns. For example, at the 5th, 10th, and 15th quantiles of Table 3a,b, negative weekly returns increase autocorrelation or predictability of EREIT (MREIT) returns by 1.335, 0.922, and 0.616 (1.294, 0.546, and 0.380), respectively.
Conversely, in the upper quantiles such as the 85th and 90th quantiles, weekly negative returns will reduce autocorrelation of EREIT (MREIT) conditional returns by −0.548 and −0.783 (−0.436, −0.911), respectively. In summary, negative returns increase (decrease) predictability of REIT returns at lower quantiles or bad state (upper quantile or good state). The evidence of positive (negative) autocorrelation of negative returns in the lower (upper) is consistent with overreaction to macro news.
As Figure 3a,b demonstrate, the addition of lagged negative returns as an explanatory variable reverses the autoregressive pattern of the slope estimates, β τ 14 from downward sloping (as portrayed by Figure 2a,b) to upward sloping. Interestingly, the lagged negative returns now take the downward-sloping dependence pattern. Therefore, the autoregression pattern seems to be an artifact of past negative returns. The results in Table 2a,b and Figure 3a,b suggest that investors react differently to positive and negative lagged returns and that their reaction is mostly state-or quantile-dependent. As Baur et al. (2012) note, the presence of lagged positive (negative) returns in a current good (bad) state, means that the lagged returns and current states are aligned, resulting in positive autocorrelation. This is closely associated with investors' under-reaction in the preceding period. Contrastingly, misalignment of lagged returns and current states causes negative autocorrelation, which is in line with the notion of investors' overreaction in period t − 1. Investors' overreaction to macro news causes overshooting of asset price and the subsequent price reversals to the fundamental value, producing negative autocorrelation patterns. Evidence from Figure 3a,b points to investors' under-reaction to positive news (upward sloping β τ estimates of positive returns) in the good state and overreaction to negative news (downward sloping γ τ estimates of negative returns) in a bad state, which is consistent with the findings of Barberis et al. (1998);Veronesi (1999); Lewellen (2002); and Chevapatrakul and Mascia (2019).
The 'Size Model' described in Equation (5) explicitly considers the impact of the size of a previous period's return on the daily, weekly, and monthly return distribution of EREITs and MREITs.
The estimated coefficients of lagged returns (β τ ) and of the 95th percentile 15 of extreme positive returns (θ τ ), which account for the impact of lagged extreme positive returns, are summarized in Table 4a,b and graphically illustrated using Figure 4a,b. We make the following substantive observations from Table 4a,b. First, θ τ is negative (positive) in the lower (upper) quantiles. 16 Therefore, large positive lagged returns negatively (positively) and asymmetrically impact the conditional EREIT and MREIT returns at the lower (upper) quantiles.
14 In addition to testing the impact of lagged negative returns on dependence patterns, we tested the impact of extreme negative returns using 10th, 5th and 2.5th quantiles as thresholds. The results, which are available on request, mirror those presented in Table 3a,b and Figure 3a,b. 15 We also tested the effects on 90th and 97.5th percentiles of return on predictability of the conditional returns of EREITs and MREITs at different frequencies. The evidence remains qualitatively the same as shown in Table 4a,b. Results are available on request. 16 That is, 5th to 45th quantiles for the daily EREIT returns, 5th to 50th quantiles for the EREIT weekly returns and 5th to 25th quantiles for the monthly EREIT returns. The upper quantiles refer to 50th, 55th and 70th quantile and above for daily, weekly and monthly EREIT returns. Similar interpretations can be made from Table 4b.   (5). We set τ = 0.05, 0.10, 0.15, . . . , 0.95. ***, ** and * indicates the statistical significance at 1%, 5%, and 10% level, respectively.
Second, the inclusion of large lagged positive returns subsumes the predictive power of the lagged returns. Still, this effect depends mainly on the quantile, the frequency of returns, and the category of REITs. For example, relative to the results in Table 2a, results in Table 4a,b show that β τ is insignificant or unpredictable at 5th to 25th (35th to 45th) quantiles for daily returns, 20th to 30th (15th to 50th) quantiles for weekly returns, and 50th to 95th (5th to 20th) quantiles for monthly returns once large lagged positive returns are included in Equation (5).
Third, Table 4a,b demonstrate that the past extreme positive returns align with the current state since negative (positive) autoregressive coefficient estimates in the lower (upper) quantile or current bad (good) state. This indicates positive autocorrelation as supported by the upward sloping estimates of, θ τ , in Figure 4a Second, the inclusion of large lagged positive returns subsumes the predictive power of the lagged returns. Still, this effect depends mainly on the quantile, the frequency of returns, and the category of REITs. For example, relative to the results in Table 2a, results in Table 4a,b show that is insignificant or unpredictable at 5th to 25th (35th to 45th) quantiles for daily returns, 20th to 30th Lastly, the aggregate effect of substantial positive returns on autoregression patterns at each quantile is assessed using β τ + θ τ . For example, at β 0.10 and θ 0.10 in Table 4a, extreme positive returns diminish autoregression by 0.590 (0.142-0.732) and 0.299 (0.140-0.439) for the weekly and monthly returns. At β 0.90 and θ 0.90 , past extreme positive returns upsurge autoregression of weekly and monthly EREIT returns by 0.388 (−0.203 + 0.591) and 0.174 (−0.082 + 0.256), respectively. Similarly, at β 0.10 and θ 0.10 in Table 4b, extreme positive returns diminish autoregression by 0.345 (0.107-0.452), 0.498 (0.123-0.621) and 0.082(0.159-0.241) for the daily, weekly and monthly returns, respectively. At β 0.90 and θ 0.90 , past extreme positive returns improve autoregression of daily, weekly, and monthly MREIT returns by 0.743 (−0.101 + 0.844), 0.598 (−0.145 + 0.743), and 0.254 (0.070 + 0.324), respectively. Overall, extreme positive returns in the lower (upper) quantiles dampen (amplify) autocorrelation of EREIT and MREIT returns. The downward sloping dependence patterns documented in Figure 4a,b essentially mirrors those shown in Figure 2a,b based on the QAR (1) model. Therefore, we conclude that the shape of dependence across quantiles using the QAR (1) is not an artifact of the previous period extreme positive returns. The increase in confidence intervals at the lower and upper tails is attributable to increased dispersion of returns in bad and good states, respectively, and increased risk premium. Table 5a,b gather the results assessing the marginal and aggregate impact of business cycles (recession and non-recession periods) on conditional returns and dependence patterns of REIT returns. The average recession quantile coefficient estimate, ϕ τ for EREIT (MREIT) is −0.215 (−0.161), 0.092 (0.123) and 0.199 (0.077) for daily, weekly and monthly frequency, respectively. Each average is significantly different from zero. Since the average quantile estimate for daily (weekly and monthly) is negative (positive), there is a negative (positive) recession-induced change in the average degree of dependence relative to the average degree of dependence, β τ , associated with daily (weekly and monthly) REIT returns. Furthermore, we test the hypothesis that H 0 : ϕ τ ϕ τ * ∀ τ and τ * with τ τ * (there is no recession-specific change in the structure of dependence. The chi-square statistic for this null for EREIT (MREIT) was statistically significant (insignificant) at every conventional level with an F-statistic of 37.021 (24.302), 140.110 (23.200), and 41.115 (17.632) for daily, weekly, and monthly frequency, respectively. The rejection of (failure to reject) the null confirms the presence (absence) of a recession-specific change in the structure of dependence of conditional returns of EREIT (MREIT) across the return frequencies.
According to the results in Table 5a, the recession coefficient estimate, ϕ τ , for weekly and monthly frequency is significantly positive in the lower and central quantiles of EREIT returns but largely insignificant for the upper quantiles. This implies that lagged weekly and monthly EREIT returns cause a higher degree of dependence in the lower tail and central quantiles during recession periods (recession-induced autoregression) than non-recession periods.
However, lagged returns have an insignificant effect on the predictability of weekly and monthly returns during the recession, particularly in the upper quantiles. Results in Table 5b reveal that recession induces increased autoregression of weekly returns, mainly in the lower and central quantiles, but has no effect on autoregression patterns using monthly returns. In both Table 5a,b, the recession coefficient estimate, ϕ τ , is significantly negative across all quantiles based on the daily frequency of returns. This suggests that previous day returns reduce autoregression more during recession periods compared to non-recession periods. To assess the aggregate quantile-specific effect of the recession on dependence pattern, we use the sum of the coefficient estimates, β τ + ϕ τ . Anecdotally, β 0.05 + ϕ 0.05 for the daily returns in Table 5a,b is −0.217 (−0.095). Therefore, the daily autoregression coefficient estimates declined from 0.057 (0.079) to −0.217 (−0.095) for the daily returns of EREIT (MREIT) at the extreme lower tail. Likewise, β 0.95 + ϕ 0.95 in Table 5a,b is −0.392 (−0.337), suggesting that at the extreme upper tails, the autoregression coefficient estimates declined from −0.165 (0.081) to −0.392 (−0.337) for the daily 17 EREIT (MREIT) returns.   (6). We set τ = 0.05, 0.10, 0.15, . . . , 0.95. ***, ** and * indicates the statistical significance at 1%, 5%, and 10% level, respectively.
In Figure 5a,b, the general downward-sloping pattern of dependence coefficient estimate, β τ confirms investors' overreaction to the news. However, the recession coefficient estimate, ϕ τ is mainly flat for the daily returns in Figure 5a and daily, weekly, and monthly MREIT returns in Figure 5b. Furthermore, the daily and weekly returns exhibit a downward sloping pattern at the extreme upper tails coupled with an increase in the confidence interval of ϕ τ . This illustrates an increase in dispersion of returns triggered by higher risk-premium at the tails of the negative return distributions compared to the intermediate quantiles. The nearly flat, negative autoregressive structure of daily returns in Figure 5a,b suggests daily prices generally decline due to negative sentiment associated with recession. Indeed, the intercept is upward sloping in Figure 5a,b, but even when it is positive (positive sentiment), daily negative autocorrelation persists, suggesting that past negative investor sentiments dominate current positive sentiments in trading decisions. dispersion of returns triggered by higher risk-premium at the tails of the negative return distributions compared to the intermediate quantiles. The nearly flat, negative autoregressive structure of daily returns in Figure 5a,b suggests daily prices generally decline due to negative sentiment associated with recession. Indeed, the intercept is upward sloping in Figure 5a,b, but even when it is positive (positive sentiment), daily negative autocorrelation persists, suggesting that past negative investor sentiments dominate current positive sentiments in trading decisions.
(a) monthly returns. Similarly, the dummy coefficient estimates in Table 5b are insignificant across all quantiles for monthly returns.

Additional Analysis
We conduct further analysis and investigate two main issues: (i) What is the marginal effect of the low and high volatility of REIT returns on autocorrelation? (ii) Does the return autocorrelation pattern differ between the REIT eras? (Vintage REIT Era and the New REIT Era).
According to Black (1989), feedback traders base their trading decisions on the previous period's price movements. Their trading activity is driven by returns volatility. Positive feedback (trend

Additional Analysis
We conduct further analysis and investigate two main issues: (i) What is the marginal effect of the low and high volatility of REIT returns on autocorrelation? (ii) Does the return autocorrelation pattern differ between the REIT eras? (Vintage REIT Era and the New REIT Era).
According to Black (1989), feedback traders base their trading decisions on the previous period's price movements. Their trading activity is driven by returns volatility. Positive feedback (trend following) arises whenever increased volatility engenders further demand for portfolio insurance-type trading activity such as stop-loss orders. The rising demand will intensify positive feedback trading, resulting in negative or lower positive return autocorrelation when a market correction occurs. Conversely, a negative feedback trading activity will increase the return autocorrelation. Overall, an increase in volatility reduces return autocorrelation (Faff and McKenzie 2007). Campbell et al. (1993) offer a parallel view. Specifically, during periods of higher than average volatility, return autocorrelation should be relatively lower since investors know, with high precision, the aggregate level of risk aversion. In light of this evidence, we implement the following model to test the marginal impact of low and high volatility on return autocorrelation at different market states or quantiles.
In Equation (7) 18 , D L (D H ) is equal to unity if conditional volatility is low (high) and zero otherwise. Low (high) is the conditional volatility (measured using the threshold GARCH model to account for the asymmetric response of volatility to positive and negative shocks) when the volatility is below (above) the average volatility. Table 6a,b summarizes the coefficient estimates β L τ and β H τ . Table 6a,b reveal that the positive and negative feedback trading story is driven by the market state and return frequency. For example, for EREITs and MREITs, high volatility of daily returns negatively and incrementally impacts return autocorrelation in all quantiles (bad state or low quantile, central quantiles, and good state or high quantiles), and is consistent with the positive feedback trading. However, the positive marginal impact of low volatility of MREIT returns at daily and monthly frequency supports the negative feedback trading thesis in the bad state. The MREITs coefficient estimate, β L τ , for the monthly return frequency is higher than β L τ of the daily frequency at every quantile (market state) due to aggregation of information over a longer period.
As Feng et al. (2011) note, the REITs' regulatory framework has undergone several changes since the market was initiated in 1960, triggering dramatic changes in the industry's size, nature, and structure. However, the watershed point was in 1993, when the government enacted the Revenue Reconciliation Act (RRA). The RRA was stipulated in part XIII of the Omnibus Budget and Reconciliation Act (OBRA) of 1993 19 . The law eliminated the REITs ownership rule of five-or-fewer people, triggering an influx of institutional investors (such as pension funds), the rapid expansion of REITs property types, and massive capital inflow in the REIT market. Shen et al. (2020) note that the evolutionary and structural changes in the REIT market created two distinct REIT eras, namely the Vintage or old REIT Era (1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993) and the New REIT Era (1994-2020). We interact each REIT's era dummy variable with 18 The equation can be augmented with R t−1 but the marginal impact of low and high volatility remain qualitatively the same with and without the lagged returns. 19 Other notable regulatory reforms were the formation of open-ended, REITs-devoted, mutual funds in 1985; the Tax Reform Act (TRA) of 1986, which abolished the favorable tax treatment of real estate limited partnerships, permitted internal advisement and management, and established vertical integration in the REITs industry; The inception of REITs' initial public offering (IPO) in 1991; the IRS private letter ruling which paved the path for the creation of the umbrella partnership REITs (UPREIT) structure and the first UPREIT IPO in 1992; and the REIT Modernization Act (RMA) of 1999, which reduced required percentage of taxable income distributed to investors from 95% to 90% and permitted REITs to form wholly-owned, taxable REIT subsidiaries through which REITs could offer supplementary services to their tenants. lagged monthly returns to assess each era's marginal impact on REITs' return autocorrelation using the following model:   Table are based on the model shown in Equation (7). We set τ = 0.05, 0.10, 0.15, . . . , 0.95. ***, ** and * indicates the statistical significance at 1%, 5%, and 10% level, respectively.
In Equation (8), D OLD (D NEW ) is a binary variable equal to one for the OLD or vintage (NEW) REIT Era else zero. β OLD τ (β NEW τ ) measures the marginal impact of the passive (integrated) REIT market period on return autocorrelation. We expected that the reform during the New REIT Era made the REIT market more integrated with the other financial markets and increased investor participation, thereby improving market efficiency. Table 7 presents results based on monthly REIT's returns.  (7). We set τ = 0.05, 0.10, 0.15, . . . , 0.95. ***, ** and * indicates the statistical significance at 1%, 5%, and 10% level, respectively.
The results in Table 7 offer interesting evidence. For either EREITs or MREITs, β OLD τ > β NEW τ suggesting that the New REIT Era reduced autocorrelation of REIT returns. This is logical since retail and institutional investors' increased participation improved the REITs' market liquidity and efficiency, thereby decreasing predictability. Alternatively, the Vintage Era was characterized by passive and infrequently traded (illiquid) REIT assets. It is challenging to determine and observe the economic price of an illiquid asset. This results in smoothing of asset returns and understating return volatility, thereby generating increased return autocorrelation relative to a more liquid asset. It is also apparent from Table 7 that when returns are high in a good market state, and many investors are making profits, return autocorrelation decreases, and market efficiency is attained.

Summary and Conclusions
This study contributes to the burgeoning literature on autocorrelation or predictability of REIT returns. It is the first study to provide supporting evidence of investors' overreaction to REITs price movements at different quantiles of the return distribution across different return frequencies and different REIT categories. We employ a quantile autoregression model to investigate the quantile-specific predictability of daily, weekly, and monthly EREIT and MREIT returns. Different quantiles may have varying return distributions. We find that the autoregression based on the QAR model generally follows a declining pattern over the conditional return distribution quantiles. Specifically, lower quantiles, which represent a bad market state with negative returns, are characterized by either positive or weakly negative dependence.
In contrast, upper quantiles, considered a good market state with positive returns, are generally marked by negative dependence on past returns. This evidence is consistent with overreaction in the REITs markets. However, the structure of return dependency in the REITs market seems to be a function of the state of the market, frequency of returns, size, and the sign of returns, and the category of REITs (EREIT or MREIT).
Additional analysis that considers the marginal and aggregate effects of lagged negative returns demonstrates that positive and negative returns asymmetrically predict future returns. Current returns are more sensitive to the previous period's negative returns than lagged positive returns in the bad state or lower quantiles. Contrastingly, current returns are more sensitive to the past period's positive returns than negative returns in the upper quantiles or good state, although this pattern varies with the frequency of returns. The autoregressive coefficient estimates of lagged negative returns exhibit a downward sloping dependence pattern and obliterated the statistical significance of the autoregression coefficient estimates of positive returns. Therefore, the overreaction evidenced using the base model seems to be an artifact of lagged negative returns. In the presence of large positive lagged returns, the downward sloping autoregressive pattern is mostly maintained for EREIT, and MREIT returns across daily, weekly and monthly frequencies. The finding of a positive (negative) autocorrelation in the lower (upper) quantiles is consistent with investors' overreaction to macroeconomic news. However, extreme positive lagged returns exhibit positive autocorrelation of returns, supporting an under-reaction phenomena and partial price adjustment mechanism. High volatility reduces serial dependence, consistent with a positive feedback story. The reduced serial dependence at nearly all quantiles during the New REIT Era supports the view that increased integration of the REIT market with other markets as well as increased participation by retail and institutional investors improves market efficiency.
The evidence of a quantile-specific dependency structure supports the adaptive market hypothesis as suggested by Clayton and MacKinnon (2003); Zhou and Lee (2013); and Liu et al. (2019) while partly contradicting past evidence of the efficiency of the REITs markets. The results of the study should be valuable to market participants in the REITs markets, whose successful investment decisions depend on their ability to model and forecast REITs price movements. Although REITs occupy a smaller space relative to stocks and bonds, policymakers should strengthen the regulatory and trading framework of REITs to better protect investors from aggressive price surges, which could engender risk spillover to other financial assets. The evidence from this study can inspire additional studies on the predictability of REIT returns in other developed and emerging markets.
Author Contributions: C.A.M. did the introduction and the literature review. I.F.J. gathered the data, cleaned it and performed the preliminary analysis. G.M.N. (Jeff) did the econometric modeling, interpretation and discussion of the results, drew the conclusion and edited the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.