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ABSTRACT
The objective of this paper is to analyze the present value model between prices and dividends with time-varying discount rates considering the company as the unit of analysis rather than the aggregate traditionally represented by a stock index. Thus, panel data with respective unit root and cointegration tests have been applied, theoretically called first-generation models. This allows for events such as rational speculative bubbles and prices mean reverting processes, which generate an order of fractional cointegration, to be considered in a more robust econometric approach. All firms composing the Sao Paulo Stock Exchange Index (IBOVESPA) have been analyzed, comprising the period from December 1994 to December 2008 on a quarterly basis. Results obtained allow full validation of the present value model at the firm level, yielding evidence of cointegration between real prices and dividends for the Brazilian market.
Keywords: market efficiency; present value model; nonstationary panel unit root and cointegration tests.
RESUMO
O objetivo deste artigo analisar o modelo de valor presente entre preos e dividendos com taxa de desconto variante no tempo considerando a empresa como a unidade de anlise ao invs de agregao tradicionalmente representada por um ndice de bolsa de valores. Desta forma, dados em painel com respectivos testes de cointegrao foram aplicados, teoricamente denominados de modelos de primeira gerao. Este procedimento permite que eventos como bolhas especulativas e processos de reverso mdia dos preos acionrios, que geram uma ordem fracionria de cointegrao, sejam considerados em uma abordagem economtrica mais robusta. Todas as firmas componentes do ndice da Bolsa de Valores de So Paulo (IBOVESPA) foram analisadas, compreendendo o perodo de dezembro de 1994 a dezembro de 2008 em uma base trimestral. Os resultados permitem inteira validao do modelo de valor presente ao nvel da firma, fornecendo evidncia de cointegrao entre preos e dividendos reais para o mercado brasileiro.
Palavas-chave: eficincia de mercado; modelo de valor presente; testes de raiz unitria e cointegrao em painel no-estacionrio.
rea ANPEC: Microeconomia, mtodos quantitativos e finanas.
Classificao JEL: C10; G10; G14.
INTRODUCTION
One of the main principles in finance theory is that current security prices are function of discounted future dividends, in which the discount rate equals the required rate of return. However, empirical evidence remains scarce in relation to the long-run relationship between stock prices and dividends. Starting in the decade of 1980, the present value model of stock prices assuming rational expectations (RE) has been tested for U.S. data and many studies indicate that stock prices were more volatile than the present value model theoretically implied. As stock prices assumed higher values due to the rapid development of the stock market throughout the decade of 1990, researchers question the present value model validity as well as the effects of interest rates on the stock price-dividend relation.
General consensus is that fundamentals have basically remained unchanged throughout the period considering the large rise in stock prices at the end of last century and their subsequent fall, which implies that the market became significantly overvalued and fundamentals subsequently reasserted themselves. Among the most prominent proponents of this view is Campbell and Shiller (2001). Nevertheless, an alternative view was that stock prices reflected investors permanently revised expectations of higher future earnings and dividends due to productivity gains originating from technological change.
As Goddard et al. (2008) observe, the empirical analysis of the present value models have been mainly based upon two approaches. First, assuming a constant discount rate, real prices and real dividends should cointegrate, that is, present a stable long-run relationship. Second, if allowance is made for a time-varying discount rate, the difference between log dividends and log prices should exhibit QUOTE stationarity. However, empirical evidence is mixed and the cointegration tests results between prices and dividends are ambiguous with several studies evidencing QUOTE non-stationarity.
The examination of individual firms is unusual, since most time-series studies adopt the S&P 500 index as the benchmark of analysis. Thus, Campbell and Shiller (1987, 1988), Lee (1995), Sung and Urrutia (1995), Timmermann (1995), and Crowder and Wohar (1998) estimate the present value relation on aggregate level over a significant length of time, in accordance to the concept stated by Stoja and Tucker (2004) that the power of unit root and cointegration tests is based on the length of time period rather than the number of observations.
At the same time, it is recognized that the application of firm-level data allows for observation of patterns and relationships that may not be evidenced through stock market index analysis, since an index application may smooth noise or volatility from individual firms as observed Nasseh and Strauss (2004). For instance, Cohen, Polk and Vuolteenaho (2001) and Jung and Shiller (2005) suggest that the present value model is more likely to hold at the individual firm level than through a stock market index.
Therefore, traditional unit roots and cointegration tests have reduced power, since the existence of speculative bubbles and the mean reverting processes involve a fractional order of integration that is not detected by traditional tests. These facts may be considered and the present value model can be validated when applying the first generation models for panel data with integration and unit root in which the firm is the unit of analysis. Hence, data on stock prices and dividends from all companies that compose the Sao Paulo Stock Exchange Index (IBOVESPA) have been employed and analyzed, comprising the period from 1994 to 2008 on a quarterly basis. This study is divided into the following three parts: 1) a review on the discussion of the present value model and the efficient markets hypothesis; 2) the econometric methodology; and 3) results and discussion.
THEORETICAL FRAMEWORK
2.1 Market Efficiency
The research and study of the returns generation process of securities, as noted by French (1980), have been one of the most popular issues in finance and has its modern origin back with the publication of the thesis by Bachelier in 1900. The genesis, therefore, confounds the Theory of Finance with this concern of using information to obtain extraordinary gains and its relationship with the returns generation process in financial markets, especially of speculative character.
Empirical evidence, notably starting in the decade of 1960, have been lavish to observe a series of stylized facts that gave rise to a vast literature in finance, such as volatility clusters, non-normality of returns, negative skewness, excess kurtosis, stochastic volatility, auto-regressivity of returns and volatility, market anomalies related to seasonality or to the functioning of markets, market anomalies related to the size of the company and its capital structure, mean reverting processes and extreme values. Parallel to these findings, a number of theories have been developed, especially of economic nature, on the non-linear characteristic of data, such as: fashions, mannerisms and panic, and rational speculative bubbles.
Bachelier (1900) considered that prices follow a (arithmetic) Brownian motion. As a result, changes in price follow the laws of probability, the returns are normal and its variances are proportional to the square root of the time interval. Current prices would be function of its future value, which is to state a Martingale dependency would exist. As a result, the returns would follow a "fair game" and the speculators, on average, would obtain zero cumulative return.
The term market efficiency, as observed by Beechey, Gruen and Vickery (2000), was introduced by Fama, Fisher, Jensen and Roll (1969) and originally referred to the speed at which the market (prices) would adjust to new information. In the version of Fama (1970) the concept of efficiency refers to the fact that prices incorporate in full (fully reflect) the set of data ( EMBED Equation.3 ). Given that new information are unpredictable, therefore, future prices also behave the same way, and the best estimates of future price would be the present price, with increases (returns) purely random. It could be ideally concluded that, as a consequence, given the flow of new information, having no lag and zero cost, are processed immediately by rational economic agents. In this sense, no agent could derive a rule based on this information in order to obtain extraordinary profit.
According to LeRoy (1989), the literature of the time spread the concept that the prices depended on the foundations and they orbited around their intrinsic value. Initially, the empirical evidence of effectiveness based on the fact that professionals would not be able to perform forecasts and suggest the acquisition of undervalued shares and the selling of overvalued shares, as fundamentalists would observe. Cowles (1933, 1944) documented the inability of most market professionals and companies at that time to conduct predictions about price changes and, therefore, to select stocks whose price variation was substantially greater than the variation of the index. As a result, the use of services by information professionals to carry out economic and financial forecasts in relation to the foundations of finance, once prices follow a random walk, would be a waste of resources.
Subsequently, the concept and test efficiency were identified with statistical evidence of random walk. According to Andreou, Pitti and Spanos (2001), a random walk is observed when the returns are independent EMBED Equation.3 identically distributed (iid):
QUOTE , QUOTE (1)
QUOTE , QUOTE (2)
Therefore, the price would have a Markovian dependency and would be nonstationary. Implicitly, the literature of the time, in the continuous case, assumed the normality of returns ( EMBED Equation.3 ), according to Andreou et al. As in Spanos (1999), the assumption of independence can be replaced by other less restrictive assumption of no correlation and the premise of identical distribution can be replaced by the existence of the first two moments, resulting in a normal or non-normal white noise process.
However, Granger and Morgenstern (1963) and Godfrey, Granger and Morgenstern (1964), found that the random walk model was appropriate for lower frequencies (daily, weekly or monthly) and inappropriate, on the other hand, for higher frequencies.
Mandelbrot (1963, 1964), presenting empirical evidence contrary to the random walk model, found that, in speculative markets, the distribution of returns was leptokurtic and presented conglomerates of volatility. Mandelbrot (1966) then suggests the Martingale model, demonstrating its superiority in relation to the random walk:
QUOTE (3)
QUOTE (4)
QUOTE , QUOTE (5)
Similarly, Samuelson (1965) established the evidence and arguments in favor of random fluctuation of prices, assuming that agents are neutral to risk, have homogeneous expectations and preferences are common and constant over time. As a result, there would be an equilibrium in financial markets and prices would behave as a Martingale and its variation would be a Martingale difference ("fair game"). Therefore, the price would be the present value of dividends.
To obtain eq. 6, Samuelson used the law of iterated expectations in which todays expectation with respect to tomorrows expectation of future gain equals todays expectation of these future gains, once the set of information at t+1 is more informative than at t:
QUOTE (6)
The Martingale model assumes that there is only the first moment, independently from the distribution, that the process is stationary of first order and has a dependency of the set of information in the previous moment ( EMBED Equation.3 ):
QUOTE , QUOTE (7)
QUOTE , QUOTE (8)
More specifically, natural logarithm of stock prices behave as a Martingale difference with respect to a filtration. This means that the expected value of the excess rate of return is on average equal to zero, considering a probability measure that discounts the likelihood of the risk premium, given a set of information (historical, public or private).
Allen, Morris and Shin (2003) showed that average expectations do not the law iterated expectations when there is asymmetry of information, especially when they follow a pattern of public information.
LeRoy (1973, 1982, 1989) considered that prices do not behave like a Martingale, but rather prices and dividends jointly and that a rigorous proof of this fact solely occurs under the condition of risk neutrality of agents. If agents are averse to risk, the Martingale property is only an approximation. It is recalled that the random walk is a special case of Martingale process, in which in the latter the conditional variance process can follow a self-repressive process. However, the serial dependence of the conditional variance is irrelevant to economic agents because of the risk neutrality and this contradicts the known models of pricing and equilibrium.
Fama (1970), using the taxonomy of H. Roberts (1967) and contributions from Mandelbrot (1966) and Samuelson (1965) on fair game efficient markets model describes three relevant subsets of information to tests of efficiency: efficiency and tests relative to weak, semi-strong and strong form. In this study the efficiency is defined as:
QUOTE , QUOTE (9)
As noted by LeRoy (1973, 1989), this definition is tautological and follows the definition of return itself, and have an anthropomorphic characterization of the market besides ignoring the asymmetry of information. With respect to subsets and respective sub-tests, Pesaran (2005), based on Fama (1991), considered redundant the distinction between the weak and semi-strong form of efficiency.
According to LeRoy (1989) the relevant information is one that provides comparative advantage for negotiations. As Grossman & Stiglitz (1980), if in fact the markets were efficient in the strong form and as a consequence, prices of risk assets reflected all relevant information, then there would be no incentive to assemble them and proceed to the exchange of assets between players with asymmetric information. The differential of information that an economic agent has over another gives him a comparative advantage and can provide the exchange between both. The availability of cost information, of informed and not informed agents with the same utility function, implies a problem of adverse selection for the non-informed. The prospect that prices partly reflect the information of informed agents, via price system spreading more information to the non-informed would lead the state of equilibrium that would be Pareto optimal. Failure to verify the efficiency in the strong form implies in equilibrium, in asymmetry of information, in the figure of the insider and the existence of noise.
Fama (1970), as mentioned, lists as one of the definitions of market efficiency the comparison between an observed price and an expected price of an asset by an equilibrium model. As it necessarily comes from a theoretical model, there is the problem of joint hypothesis, that is, tests of efficiency are jointly or simultaneously tests for efficiency and market equilibrium model. It follows necessarily from this that, if the theoretical model is inadequate, the additional implication is the problem of poor model in the tests of efficiency (FAMA, 1991). It occurs, however, that the concept of efficiency brings together a set of auxiliary assumptions that define its contours, such as the rationality of economic agents, their preferences with regard to risk, or non-homogeneity of expectations, the existence of informational asymmetry, problems of microstructure, etc. Consequently, as in Fama (1991), the assumption of market efficiency, especially due to the problem of joint hypothesis, would not be testable
In relation to the "random walk" model, where prices fluctuate randomly, LeRoy (1989) stated that market forces (supply and demand) do not determine the price of shares, but their randomness. With respect to the fundamentalist vision, in which the price orbits around the intrinsic value, he questioned why the forces of the market (new entrants) again would not eliminate the opportunity for gain.
Furthermore, according to LeRoy (1973) and Lucas (1978) (apud Lo & MacKinlay, 1999) equilibrium prices under rational expectations do not need to behave as a Martingale sequence in order for markets to be efficient.
Fama (1976) recasts the concept and believes that the market is efficient when the observed return is equal to the balance of expected return, which coincides with the market assessment:
QUOTE (10)
LeRoy (1989) states that the tests of efficiency in fact test the rationality of agents and the assumptions of the Martingale model.
According to Jensen (1978), any business strategy that produces consistently economic gains, already discounted the risk, for a sufficiently long period, considering the transaction cost, represents an evidence against market efficiency. Malkiel (2003) defines efficient markets as those where it is impossible for agents to obtain above average returns without taking above-average risks.
Efficiency Tests
Bollerlev and Hodrick (1992) separate the tests of efficiency in function of the frequency of data. If less than one year, it would be considered the short-term; if longer, it would be considered the long-term. The short-term basically refers to the shape of the distributions of return and their independence.
As in LeRoy (1990), the eq. 6 holds if and only if eq. 8 occurs, given the neutrality of the agents to risk. However, if this is true, the price perfectly foreseen (perfect-foresight) or ex-post rational price ( EMBED Equation.3 ) is composed of the expected price plus a forecast error, orthogonal to this ( EMBED Equation.3 ). This implies that EMBED Equation.3 , that is, the variation of ex post rational price is greater or equal to the expected variation, depending on the variation of error prediction. Schiller (1979, 1981) and LeRoy and Porter (1981) were the first to apply variance limit tests (variance bounds) to assess the value of this model. The authors noted that the observed volatility of prices (p) was much larger than expected ( EMBED Equation.3 ), calculated recursively, which contradicts the model of market efficiency. This is true even considering the interest rate variable and dividend shocks. For Schiller (1981), the explanation would be the existence of fads or market psychology and for LeRoy and Porter (1981), it would be an anomaly to be explained.
Flavin (1983) considers that this violation can occur by the presence of small-sample bias that makes more likely the rejection of the null hypothesis of efficiency. Kleidon (1986) argues that equality of variances ( EMBED Equation.3 ) occurs mainly by considering the various states of nature in a given time, that is, assumes a cross sectional and not longitudinal cut. It is worth stating that these variances should be unconditional for the equality to be satisfied. The low variability of EMBED Equation.3 can be explained: first, because the ex-post rational price ( EMBED Equation.3 ) is calculated recursively, considering a single state of nature in the studies cited; and second, because in the calculation of ( EMBED Equation.3 ) it was used the dividend ex-post (observed). Therefore, the expected change has no informational content and its variance is unexpectedly absent. Hence, the variation of ex post rational price would only be derived from capital gains. However, assuming that these variances were calculated in conditional terms, its equality would occur from the mechanism of dividends generation. In this case, they are valid if the series are not stationary. If the variances are unconditional and the series are not stationary their equality is not defined.
Campbell and Schiller (1987), employing data referring to index of nominal shares and dividends from 1871 to 1986, use VAR methodology with the following variables: spread ( EMBED Equation.3 ) resulting from the difference between price and perpetuity of dividends and changes in dividends ( EMBED Equation.3 ). If prices and dividends are non stationary I(1) but the spread is stationary I(0), dividends and prices are cointegrated. Accordingly, if the growth rate of dividends equals zero, perpetuity is equal to the price and the spread is equal to zero. If the growth rate of dividends is positive, the spread will depend on the variation of dividends. If the representation is correct in order for the spread not to be a linear combination of dividends, it must Granger cause the change of dividends, that is, the spread will be a good estimate of the weighted values of change in dividends. The authors rejected the hypothesis of cointegration; however, this rejection could be reduced depending on the period considered and the discount rate. It occurs that the interest rate was considered constant and the absolute variation of dividends ( EMBED Equation.3 ) is not necessarily stationary.
Campbell and Schiller (1988) linearized the relationships above, setting the log of the return of the dividend as a function of the present value of the difference between the return and the growth rate of dividends. This model allows the interest rate to be variable, without compromising the cointegration between prices and dividends. The authors found little evidence that the return meets the present value model rational expectations).
The work by Fama and French (1988) includes the monthly return for the period from 1925 to 1985 in the USA. As these authors noted, the behavior of long-term returns allows a better observation of the reverting process. This temporary component can induce a process of strong negative autocorrelation in the long-term, which is not, however, observed in short-term tests of market efficiency.
Cecchetto, Lam and Mark (1990) demonstrated that the mean reverting process is compatible with the concept of general equilibrium. However, Bonomo and Garcia (1994) showed the amount of negative autocorrelation from the mean reverting process derived from a poor specification of the Markovian conversion process and from small sample bias. Once correctly specified, the mean reverting process disappears.
Kim, Morley and Nelson (2001) found that a mean reverting process can be interpreted as a misalignment of the expected returns of equilibrium because of a systematic reaction exacerbated by investors in relation to information on the fundamentals, that is, an intertemporal trade-off between risk and return.
Mankiw, Romer and Shapiro (1991), with data for the U.S. market from 1871 to 1988, used the present value model. The authors verified excess price volatility in relation to the fundamentals, but the rejection of market efficiency was moderate with little significance. Thus, predictability is consistent with the hypothesis of rationality of agents and informational efficiency.
Timmerman (1995), using the same data as Campbell and Schiller (1987), considers that the failure to observe a process of cointegration between stock prices and dividends (dividend yield) and thus the excessive variation in prices not explained by changes in fundamentals (dividend yield), is due to the persistence of these. The problem is aggravated if the persistence is high and the sample is reduced. The variance of the residuals of the cointegration process, in this case, will oscillate around trends. However, using the discount rate variable and the logarithm of the differences of endogenous and exogenous variables, the problem has been mitigated. Timmermann (1995) states that the combination of two stylized facts, mean reverting of aggregate income and theory of permanent income implies the anticipation of cyclical variation in expected rate of return.
Dupuis and Tessier (2003), with data for the U.S. market from 1973 to 2002, used the SVECM model (structural vector-error-correction model). According to these authors, if the price is the function of fundamentals, this will be the result of the permanent component of changes in dividends and the discount rate. The transient component would result in uncertainty of short-term shocks that could disalign prices and fundamentals. The authors found that: 1) the variability of prices was greater than the dividends, 2) there is an important transitional component that exerts short-term influence, 3) there is a long-term component that explains the variability in prices in relation to its fundamentals.
Manzan (2004), using the ESTAR model (Exponential Smooth Transition AR) with data for the U.S. market from 1871 to 2003, notes the presence of two regimes. In a pricing regime, prices persistently deviates from their fundamental values and, in the other regime, prices converge to their intrinsic value through a mean reverting process. The presence of bubbles is captured by this non-linear adjustment process.
Su, Chang and Chen (2007) with data from the Taiwan Stock Exchange over the period 1991 to 2005, used the M-TAR model (Momentum Threshold Autoregressive) from Enders and Granger (1998), assuming a time-varying discount rate, did not find contrary evidences from the equilibrium between prices and dividends.
In relation to the Brazilian stock market, Anchite and Issler (2001), using data on a quarterly basis for the period from 1986-1 to 1998-4, have validated the present value model, especially with time-varying the rate of return and Morales (2006) using quarterly data from 1995-1 to 2005-4 obtains similar results.
The references cited earlier in this paper analyzed the price-dividend relationship for the index of shares, that is, aggregate prices and dividends have been considered at constant rates for different share indexes. As in Jung and Schiller (2001), the use of panel data to test the present value model may be more appropriate per individual company rather than in aggregate. For economic agents, it would be appropriate to observe and predict the impact that macroeconomic factors (e.g., monetary policy and output growth) and microeconomic ones (e.g. degree of competition and performance of competitors) would have on the cash flow and prospects of company growth, rather than simply observing the relationships between macroeconomic aggregates. In addition, the value of this model may be more effective if applied to businesses that are in the maturity of their lifecycle. Information obtained at the firm level may be more operationally useful to agents who work on a stock exchange.
In this sense, this work focuses on analyzing the price-dividend model at the company level. Practically, the literature is still scarce with respect to the individual analysis, and it should noted the two references mentioned below.
Nasseh and Strauss (2004), using panel with cointegration in the present value model and unit root tests with non-aggregate data of firms between 1979 and 1999 in the U.S. market, have found a cointegration process between prices and dividends. About 40% to 50% of the variation of dividends and 90% of long-term innovation may be explained by prices. The results pointed that prices Granger cause dividends. The decline in interest rates, short term and optimistic forecasts explain eventual overvalued prices of some shares.
Goddard, McMillan and Wilson (2008) tested the present value model using panel data with unit root and cointegration with disaggregated data of annual frequency for 104 companies in the period from 1970 to 2003. Although the results do not support in full the present value model, the evidence is stronger than those found with respect to aggregate data from other studies.
ECONOMETRIC METHODOLOGY
The present value model relates the present value of expected dividends and the share price, under the implied condition of rational expectations, as follows:
QUOTE (11)
Campbell and Shiller (1987) demonstrated that, under the condition of transversality, there will be only one possible price in order to exclude the presence of bubbles and, consequently, the possibility of many solutions to the price equation above. Assuming, therefore, the validity of the model under this assumption, Campbell and Schiller (1987) showed that prices and real dividends are cointegrated and the cointegration vector equal to (1, R-1), according to the following equation:
QUOTE (12)
The methodology used by Campbell and Schiller (1988) in order to circumvent the non-stationarity of the price and dividends series and admit the possibility of time-varying discount rates, suggested the logarithmic transformation of the variables [d = ln(D), p = (ln(P) and r = ln(1+R)] and the following formula:
QUOTE (13)
where k = - ln () + (1- ) ln (-1) and = 1/[exp(d-p)]. Again, under the condition of transversality, the above equation is rewritten as:
QUOTE (14)
If the variation of the dividend and the discount rate are stationary, the spread (h= ln(Pt/Dt) = EMBED Equation.3 ) will be stationary and therefore the logarithm of prices and dividends will be cointegrated with the cointegration vector equal to (1,-1). Therefore, it would be sufficient to prove that the spread is I(0) to validate the present value model. The spread and variation of dividends can be modeled with an Autoregressive Vector, with the restriction that the returns are unpredictable ( EMBED Equation.3 ). As a result, the spread should Granger cause the dividends.
Basically, three types of criticism can be made regarding the above procedure. First, it would be to oppose empirical evidence contrary to the results above (FROOT and OBSTFELD, 1991); second, it would be to examine the assumptions of the model and verify, for example, the existence of bubbles (EVANS, 1991); and the third would be the adequacy of the tests in relation to facts such as mean reversion (CAPORALE and GIL-ALANA, 1999). Long processes of mean reversion and persistent shocks imply fractional order of cointegration, making traditional test results inconclusive, although it remains a long-term equilibrium relationship between prices and dividends. It is worth stating that the present value model is not incompatible with the existence of bubbles and mean reversion. However, the econometric approach should be modified to consider these facts.
One way to consider the existence of bubbles and the process of mean reversion is the use of panel data with unit root and cointegration. This work was performed by applying panel unit root tests by Levin, Lin and Chu (2002) and Im, Pesaran and Shin (2005) as well as panel cointegration techniques following Pedroni (1999, 2004), Kao (1999) and Larsson, Lyhagen and Lthgren (2001).
As noted by Goddard et al. (2008) these two tests can be described by an autoregressive ADF process:
QUOTE (15)
Both tests assume that the individual series are independent in each instant t. In the LLC test, LLC ci = c (homogeneous), which is configured as a restriction to this model, and Ho: c = 0 (unit root) and H1 c < 0. In the IPS test, c may be heterogeneous, and Ho: ci = 0 for all i and H1: c < 0, for some i. The t-statistic is the average of individual t and the panel should be balanced.
Kao (1999) proposed DF and ADF tests for panel in which the cointegration vectors are homogeneous and self-regressive, but without multiple exogenous variables. If more than one cointegration vector exists, it cannot be identified. The null hypothesis corresponds to the absence of cointegration.
QUOTE , QUOTE (16)
Pedroni (1999, 2004) proposed a test of cointegration, assuming trend and multiple regressors.
QUOTE , QUOTE (17)
H0: EMBED Equation.3 = 1 for all i, against H1: EMBED Equation.3 = EMBED Equation.3 for all i. Seven different statistics have been set up to test the hypothesis, considering the estimated residual for each i and t (the within dimension) and for each i (the "between" dimension). It is assumed, however, the existence of only one cointegration vector.
Larsson, Lyhagen and Lthgren (2001), based on Johansen (1988), assume that the data can be generated by a process of error correction with multiple cointegration vectors:
QUOTE (18)
This equation is estimated by maximum likelihood for each i, yielding the average LRiT and trace statistics LRNT, whose null hypothesis H0: rank (i) = ri d" r for all and H1: rank (i) = p for all i, with p = number of possible cointegrated variables. It is utilized the same sequential procedure as in Johansen (1988) for the hypothesis test. A detailed description of the unit root and cointegration tests with panel can be found in Baltagi (2000).
RESULTS
Data on stock prices, payments of dividends and interest on equity have been collected from Economatica platform, comprising the period from December 1994 to December 2008 with quarterly frequency. Firm information on stock prices and dividends must follow the underlying criteria: i) to belong to the theoretical IBOVESPA portfolio at any point of time considered in the analysis; ii) continuous information availability from start to end date in this study; iii) real variables adjusted to inflation and in logarithm form. The starting sample period coincides with the implementation of the Real Plan, named after the new currency, whose exchange rate system would be key to keeping low inflation. Hence, this event has been a milestone to a more stable economy nationwide which makes sample period adequate for finance fundamentals analysis with its sensibility to macroeconomic variables, even though a considerable part of the power of the unit root and cointegration tests applied is a function of T rather than N in panel data, as Stoja and Tucker (2004) point out.
The precondition for procedures for panel data estimation is that the variables have unit root and are cointegrated. Thus, it has been applied unit root tests for panel data following Levin, Lin and Chu (2002), Im, Pesaran and Shin (1997) and Fisher/ADF from Larsson, Lyhagen and Lthgren (2001) in order to assess whether the variables related to the logarithms of real prices, dividends and price-dividend ratio are integrated. Goddard et al. (2008), apply the LLC and IPS tests based on two specifications: individual intercept and intercept with deterministic trend. Their results are based on a range of increased lags of P =0 and P = 3 (inclusive), defined according to the Akaike Information Criterion (AIC).
Table 1 Panel Unit Root Tests Results
Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
Analyzing the logarithm of real price for the Brazilian market, in the model with intercept only, the LLC test fail to reject the null hypothesis of non-stationarity I(1) to the 5% significance level. In the model with intercept and trend, the null hypothesis is not rejected. Overall, similar to the pattern of results obtained by Goddard et al. (2008), the inclusion of time trends specific to firms in the LCC test directs the results to rejection of the null hypothesis of non-stationarity I(1).
Similarly to the standards established by Goddard et al. (2008), the IPS and Fisher/ADF tests fail to reject the null hypothesis in the first case with intercept only, but directs to the rejection of the null hypothesis of non-stationarity I(1) in the case with deterministic trend. Thus, although there is the expected ambiguity found in the results of Goddard et al (2008), since results do not fail to reject the null hypothesis in both cases of inclusion and exclusion of deterministic trends, probabilities obtained indicate the failure to reject the null hypothesis of non-stationarity I(1) to the 5% significance when trend and intercept are considered particularly on LLC test. Thus, in accordance to results verified by Goddard et al. (2008), overall evidence fails to reject the null hypothesis that the series of the logarithm of prices is non-stationary I(1) for all sample firms in the Brazilian market.
Analyzing the series of real dividends in the Brazilian market, similarly to the results of Goddard et al. (2008), the diagnoses of non-stationarity I(1) and stationarity I(0) present the ambiguity regarding the sensitivity of inclusion or exclusion the deterministic trend as well as the unit root test applied. For models with intercept only and trend with intercept, the panel test LLC rejects the null hypothesis of non-stationarity I(1) for the first and fails to reject the same statistical hypothesis for the latter. On the other hand, IPS and Fisher/ADF reject the null hypothesis of non-stationarity I(1) for the series of dividends in the Brazilian market for all firms in the two models with intercept only and intercept and trend considered.
Analyzing the series of the logarithm of the price-dividend ratio for the Brazilian market, the LLC test rejects the null hypothesis of non-stationarity I(1) when including intercept only and, following the pattern of the previous variable analyzed, it fails to reject the null when intercept and trend are included. Differently from the pattern observed in the LLC test, the IPS and Fisher/ADF tests reject the null hypothesis of non-stationarity I(1) at the 1% level both when intercept only and intercept and trend are considered as in Goddard et al. (2008).
Although expected ambiguities might be verified within the application of each test in function of inclusion or exclusion of deterministic trend in the model both for full sample and selected mature firms, overall, results point to the presence of unit root in each series and the fact that a cointegrating equilibrium cannot be rejected, evidencing a long-run relation between stock prices and dividends as observed in Goddard et al. (2008) and Nasseh and Strauss (2004).
We proceed with Pedronis panel cointegration tests, applied to residuals with intercept and trend. The null hypothesis of no cointegration is rejected in the first case (no trend) in all tests applied and, in the second case (with trend), all tests reject the null hypothesis of no cointegration as well. Because of this, it can be stated that full evidence of cointegration exists between prices and dividends at IBOVESPA, especially for stocks individually.
Table 2 Pedroni Panel Cointegration Test Results
Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
Regarding the Kao panel test, the hypothesis of no cointegration is rejected.
Table 3 Kao Panel Cointegration Test Results
Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
Regarding the Larsson et al. (2001) test in the restricted version, its general statistics rejects the hypothesis of no cointegration between 1% and 10% levels.
Table 4 Fisher Panel Cointegration Test Results
Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
Finally, the Granger causality test results reject the hypothesis that prices do not Granger cause dividends. Timmermann (1994) presents evidence that stock prices appear to Granger-cause dividends. As mentioned by this author, two ways of rationalizing this feedback basically exist. One possibility is that the feedback reflects the effect of stock prices on dividends through the cost of capital restriction faced by the firm; the other possibility is that stock prices may summarize private information in a context of asymmetric information. As expected, it is not rejected that dividends do not Granger cause prices. Depending on the magnitude of the coefficient associated with dividends and prices in each equation, it may be stated that the shares are overvalued. Thus, data are consistent with the present-value model, in which price is the expectation of future dividend movements.
Table 5 Pairwise Granger Causality Test Results
Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
Once dividend innovations are subject to a moving average process or entail a relatively large temporary component relative to stock prices, the regression of dividends on prices QUOTE is applied. As observed in Nasseh and Strauss (2004), past lags must be insignificant when prices incorporate all current innovations on dividend and is forward-looking. Since present-value holds theoretically, this regression yields coefficient estimate equal to one for full sample: QUOTE . If we define overvaluation as stock price movements neither backed nor justified by dividend movements, then stocks would be overvalued when QUOTE .
Interest rate effects which generally exert an inverse relationship on the stock market should also be observed. It is widely believed that Federal Reserve/Central Bank interest rate reductions can spark stock market rallies. One reason is that stocks and bonds are substitutes for investing funds in the short run. Second, the nominal interest rate includes an inflation component, and reduced interest rates may reflect lower anticipated inflation. Therefore, inflation declines can lead to increases in stock prices, since inflation proxies for economic activity. For instance, Geske and Roll (1983) and Nasseh and Strauss (2004) demonstrate the negative relation between expected inflation and stock returns that is due to the relationship between prices and economic activity. In order to allow for a time-varying discount rate, interest rate effects have been modeled QUOTE according to Nasseh and Strauss (2004).
Hence, the POLS (Pooled Ordinary Least Squares), DOLS (Dynamic Ordinary Least Squares) and FMOLS (Fully Modified Ordinary Least Squares) procedures in panel cointegration estimation have been undertaken as in Pedroni (2000, 2001) and Nasseh and Strauss (2004).
Table 6 Panel Cointegration Estimation Procedures
POLSDOLSFMOLSPanel A1.01441-0.004657-0.000475t-stat4.986*-9.764579*-1.165658*Panel B0.842722-0.0035850.000508t-stat3.802*-0.000001*1.131694*-1.34847-0.0304730.032722t-stat-1.943***0.000000*2.515386*Note: *, ** and *** denote the test statistic significant to the 1%, 5% and 10% levels, respectively.
Source: Elaborated by the authors.
For Panel A, results above indicate a one-for-one relationship between stock prices and dividends in POLS panel estimator and respective coefficient significant to the 1% level. For Panel B, POLS estimator suggests stock prices may be overvaluated by 15% throughout the length of time analyzed when QUOTE is considered. However, interest rates may not be accounted significantly for dividend innovations or overvaluation in stock prices given its relatively lower level of significance.
As Johnston and DiNardo (1997) recall, POLS estimators ignore the panel structure of the data, treat observations as being serially uncorrelated for a given individual, with homoscedastic errors across individuals and time periods. As Caporale and Cerrato (2004) point out, the advantage of recently developed DOLS and FMOLS estimators over the former is that they deal with possible autocorrelation and heteroskedasticity of the residuals, potential endogeneity of the regressors, take into account the presence of nuisance parameters and are asymptotically unbiased. The FMOLS estimator corrects the POLS estimator non-parametrically, while the DOLS estimator uses information from past and future leads and lags of all variables. Both yield similar results asymptotically with minor small-size sample distortion in small samples.
Since the time length considered corresponds to a period of increasing interest rates due to inflation control and unprecedented development of the Brazilian stock market, interest rates and stock prices assume ascending paths throughout the sample period, which may be responsible for an amount of the coefficient sensibility interest rates capture leading to a lower price coefficient. Such results pointing to the close to zero coefficient and non-significance of interest rates on dividend innovation and stock price overvaluations in U.S. stock market observed by Nasseh and Strauss (2004) is thus also applicable for the Brazilian stock market, suggesting an approximate one-for-one relationship between stock prices and dividends.
CONCLUSION
The objective of this study was to examine the present value model of dividends and prices with time-varying discount rate, considering the company as the unit of analysis rather than aggregate data, traditionally represented by a stock index. In this sense, it has been used panel data with unit root and cointegration, theoretically called first-generation models. This allowed for facts such as bubbles and prices mean reverting process that generate an order of fractional cointegration to be considered in a more robust econometric approach. Covering the period from 1994 to 2008, all stocks that compose the theoretical portfolio of Sao Paulo Stock Exchange Index (IBOVESPA) have been analyzed on a quarterly basis. The results obtained from this study allow full validation of the present value model at the firm level.
Possible explanations for the unit root conflicting results may be attributed to the power of the tests that is a function not of the number of observations, but the length of time under consideration instead. In this sense, given the circumstances of the Brazilian economy, the length of time analyzed is relatively smaller than that of traditional studies for developed economies, starting in the same year as the implementation of the Real Plan. Another applicable explanation is the presence of heteroskedasticity that tends to reject the unit root hypothesis as also verified in previous works. However, data are consistent with the present-value model and all cointegration tests applied evidence cointegration between prices and dividends.
Long processes of mean reversion and persistent shocks imply fractional order of cointegration, making traditional test results inconclusive, although it remains a long-term equilibrium relationship between prices and dividends. It is worth stating that the present value model is not incompatible with the existence of bubbles and mean reversion. However, the econometric approach should be modified to consider these facts. Results indicate that, throughout the period analyzed, there is an approximately one-for-run relationship between stock prices and dividends for firms composing IBOVESPA, attributing full validity to the present value model.
Granger causality test results reject the hypothesis that prices do not Granger cause dividends. As expected, it is not rejected that dividends do not Granger cause prices. Thus, data are consistent with the present-value model, in which price is the expectation of future dividend movements. Lastly, in panel cointegration estimation procedures, when real interest rate effects are considered, it is estimated an overvaluation of 15% in stock prices. However, given its relatively lower level of significance for full sample and similar paths stock prices and interest rates assume throughout the period considered for the Brazilian economy, interest rates may not be accounted significantly for dividend innovations nor stock prices overvaluation, as observed in Nasseh and Strauss (2004).
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