RATIONAL SPECULATIVE BUBBLES: Theory and Empirics in Tunisian Stock Market

Adel BOUBAKER♣ Associate Professor of Finance, Faculty of Economics, Tunis, Tunisia E-mail: [email protected]

Duc Khuong NGUYEN Professor of Finance, ISC Paris School of Management, France E-mail: [email protected]

Imen TAOUNI Researcher, Faculty of Economics, Tunis, Tunisia

Abstract This article aims to evidence the theory of rational bubbles by focusing on fundamental value and major models which describe the formulation of this so-called phenomenon. We first review the related literature both at the theoretical and empirical levels. Our empirical strategy for detecting the presence of rational bubbles is then applied to financial data of the Tunisian stock market over the 1971-2005 period. While the statistical properties of the bubbles are examined through the use of indirect tests such as stationarity and cointegration tests, two direct tests previously developed by Froot and Obstfeld (1991), and Artus and Kaabi (1994) are adopted to prove the existence of the intrinsic and state bubbles respectively. Overall, the empirical results show that the Tunisian stock market is affected by an intrinsic bubble whose dynamics depend only on dividend patterns. Key words: fundamental value, intrinsic bubbles, state bubbles JEL Classifications: G10, G15

♣

The participants of the 4th International Finance Conference (March 15-17, 2007, Hammamet, Tunisia) are gratefully acknowledged for helpful comments. Of course, all remaining errors are ours.

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1. Introduction Under the hypothesis of market efficiency, stock prices instantaneously reflect all available information. It then follows that the observed quotation of any security is a good estimate of its fundamental value. However, this hypothesis has been subject to a number of critics due to the advent of various events against its validation such as the world stock market crash occurred on October 1987. One of the stylized facts is that stock prices are found to be disconnected from their fundamentals, which contrasts with the efficiency as well as rationality requirements. This situation particularly leads to ask the question of how we can explain the deviations of stock price from its fundamental value. The theory of rational bubbles constitutes one of the major approaches to explaining the differences between a security’s price and its fundamental value within a context of rational expectations. Blanchard and Watson (1982) formally define a rational bubble in financial markets and suggest that the existence of bubbles does not contradict the market equilibrium model. In other words, the rationality of economic behaviors and expectations does not imply that market value is necessarily equal to the fundamental value; each of them can deviate from the other and the observed difference corresponds to what is called a rational bubble. Contrary to popular economic analysis, Blanchard and Watson (1982) show that rational bubbles and perfectly rational investors might exist at the same time. This paper presents an empirical analysis of rational bubbles and their formulation in Tunisian stock market. Section 2 discusses the theory of rational bubbles. Section 3 proposes a methodological framework to detect the presence of bubbles on the Tunisian stock market. Section 4 describes data used in empirical analysis. Section 5 reports and interprets empirical results. Concluding remarks are given in Section 6.

2. Theoretical aspects of rational bubbles A significant number of works in finance literature has been devoted to explaining the divergence of stock prices from their fundamental value (e.g., Flood and Garber, 1980; Blanchard and Watson, 1982; Gilles and Leroy, 1992; Chan and al., 1998; Sampson, 2003; Bohl, 2003; Cunado and al., 2005; Koustas and Serletis, 2005; Hong and al., 2006; Caballero and Krishnamurthy, 2006; Engsted, 2006; McMillan, 2007). A common conclusion that emerges from the majority of related studies showed that rational deviations remain substantial even with investor’s rational expectations and behaviors.

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Initially introduced by Flood and Garber (1980) and extended thereafter by, among others, Blanchard and Watson (1982), and Tirole (1982), the stream of research which deals with the speculative bubbles, has managed to establish a quite influential theory for analyzing various market phenomena of which the stock market crashes are of greatest interest. As we have mentioned previously, a market bubble is, according to Blanchard and Watson (1982), the difference between the stock prices observed on the market and their fundamental value. This definition has two advantages. First, it introduces the concept of fundamental value. Second, it witnesses the difficulty of detecting market bubbles when the fundamental value is uncertain. Gilles and Leroy (1992) add that a bubble typically indicates an important increase of stock prices in response to the promise of future dividends announced by the companies. A dramatic drop of stock prices is then probably due to the non-realization of these promises. The empirical analysis presented in this paper will rely on both definitions. In what follows we discuss the fundamental value, the rational bubbles and their formulation.

2.1 Fundamental value and bubbles To price financial securities, it is common to use one of the following models: CAPM, APT and DCF. The DCF model is often selected by the researchers when determining the fundamental value because it offers a wide possibility of developments and modeling. Formally, it is developed on the basis of the no-arbitrage condition at the equilibrium which imposes the returns on any security to be equal to those of alternative investments and of four principal assumptions: efficient market, constant discount rate, risk-neutral operators and rational expectations of stock price based on expected utility maximization. To get further insights, consider the return of a listed firm’s stock:

Rt =

Pt +1 − Pt + Dt Pt

(1)

In the formulation (1), Pt +1 and Pt are the stock prices at times (t+1) and t respectively. Dt refers to the amount of dividend paid to shareholders between t and (t+1) period. Recall that all investors are assumed to be risk neutral agents and possess the same set of information. In addition, they do not mind either holding this asset or a risk free asset which provides an assumed constant interest rate r, because the no-arbitrage condition at the equilibrium implies the following equality1: E ( Rt I t ) = r

(2)

1

The relationship described by Eq. (2) is sometimes called efficiency condition since expected return on risky assets is just equal to the interest rate.

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E (. I t ) is referred as the expected return conditional on the set of available information.

By replacing Rt by its value in Eq. (1) and rearranging the Eq. (2), we obtain: E ( Rt I t ) = E (

Pt +1 − Pt + Dt It ) = r Pt

Pt = δ [E ( Dt I t ) + E ( Pt +1 I t )]

(3) (4)

δ = (1 + r ) −1 is none other than the so-called discounted factor and the Eq. (4) is known as the formula of Euler that can be solved by using the law of iterated expectations applied to the following formula: E [E (. I t +i ) I t ] = E (. I t ) with ∀i ≥ 0

The result of iterations over n periods, also called the approximate fundamental value of the considered stock, is given by: n

Pt = ∑ δ i +1 E ( Dt +i I t ) + δ n E ( Pt +n I t )

(5)

i =1

Note that in Eq. (5), n is the number of years wherein the stock is held and Pt+n is the price at which the stock is sold out after n periods. The theoretical fundamental value of the stock can be computed from taking the limit of stock prices with respect to the length of the holding period n, that is: n

Pt = ∑ δ i+1 E ( Dt +i I t ) + lim δ n E ( Pt +n I t ) n→∞

i =1

(6)

If n tends to infinity, the limit lim δ n E ( Pt +n I t ) tends towards zero: n→∞

lim δ n E ( Pt +n I t ) = 0 n→∞

(7)

We are then able to deduct the fundamental value of the stock as follows: ∞

Pt* = ∑ δ i +1 E ( Dt +i I t )

(8)

i =1

Eq. (8) shows that the price of a stock held up to infinity is equal to the sum of the present values of all expected future dividend flows. It is important to stress that Eq. (7) serves as the transversality

condition which limits the solutions of the problem in Eq. (4) to a unique solution presented in Eq. (8). Without imposing such condition, Blanchard (1979) demonstrates that Eq. (4) will admit a more general solution of the following form: 4

∞

Pt = ∑ δ i+1 E ( Dt +i I t ) + Bt

(9a)

i =1

Pt = Pt* + Bt

(9b)

Where Bt is defined as a bubble such as: E ( Bt +1 I t ) = δ −1 Bt = (1 + r ) Bt

(10)

Eq. (10) points out that the difference between stock price and its fundamental value can exist even when the arbitrage condition at the equilibrium is satisfied. Moreover, Bt is described as a rational bubble because it arises from the present value model in which investors’ expectations are assumed to be rational.

2.2. Brief overview of rational-bubbles models According to previous literature, rational-bubbles models can be broadly classified into two main classes: models for exogenous bubbles and endogenous bubbles. While the first class of model treats bubbles independently from the fundamental value fluctuations, the second does take into account the impact of stock fundamentals both on its fundamental value and bubble. Exogenous bubbles are in general divided into deterministic, stochastic and periodically collapsing bubbles. Initially analyzed in Blanchard and Watson (1982), the deterministic bubble is simply modeled using an exponential function of time as in Eq. (11):

B t + i = B t (1 + r ) i

(11)

Under this deterministic structure, the bubble is greatly amplified through time, leading to an explosive divergence between stock price and its fundamental value. However, the model is unrealistic in the sense that it implicitly supposes a perpetual growth of stock prices. This drawback leads Blanchard and Watson (1982) to introduce stochastic bubbles with a probability of bursting:

⎧ (1 + r ) Bt + et +1 with the probability p ⎪ Bt +1 = ⎨ p ⎪e with the probability (1 - p) ⎩ t +1

(12)

The innovation part of the bubble et+1 is such that E(et +1 I t ) = 0 . According to this specification, once the bubbles exist, they have an exponential growth, but they are likely to burst from one period to another. If they burst at a given time, their reformulation will not be possible. More importantly, these rational bubbles can only exist if they do at the time of stock issuance. They grow, then col5

lapse and finally disappear (see, Diba and Grossman, 1988). In a related study, Evans (1991) introduce a new class of bubbles, called periodically collapsing bubbles, which can deflate without bursting and grow again thereafter. His model takes the following form:

if Bt ≤ α ⎧Bt +1 = (1 + r ) Bt μt +1 ⎨ −1 −1 ⎩Bt +1 = δ + p (1 + r )θt +1 × ( Bt − (1 + r ) δ ) μt +1 if Bt > α

[

]

(13)

Where δ and α are positive parameters which satisfy 0 p δ p (1 + r )α . μ t + 1 is an exogenous independently and identically distributed positive random variable with Et (μt+1) =1. θ t + 1 refers to an exogenous independently and identically distributed Bernoulli process defined as follows2:

⎧θ t +1 = 1 with the probability p, 0 < p ≤ 1 ⎨ ⎩θ t +1 = 0 with the probability (1 − p) By construction, the periodically collapsing bubbles can have multiple regimes. Precisely, Bt ≤α implies that the bubble grows with a mean growth rate of (1 + r ) until it reaches the given value, α. If α eventually becomes inferior to Bt , the bubble enters into a period wherein it can either grow at a faster mean rate of (1+ r) p−1 , or collapse with a probability (1− p) per period. In the later case, it falls to a positive mean value of δ and the process begins again. The scale of bubbles and the length of time before collapse strictly depend on how important are the values of the parameters δ, α and p. Another category of bubbles are endogenous bubbles which essentially include intrinsic and state bubbles. The stream of research on testing for endogenous bubbles in asset returns and prices argue in essence that market fundamentals have significant effects both on the stock’s fundamental value and the formulation of price bubbles. First, as regards the intrinsic bubbles initially examined by Froot and Obstfeld (1991) and extended further by, among others, Driffill and Sola (1998), this form of bubbles is modeled as a function of market fundamentals in a nonlinear framework. Just like the other rational bubbles, the intrinsic-bubbles model is also developed from the no-arbitrage condition as follows:

Pt = e − r E t ( Dt + Pt +1 )

(14)

Here, Et(.) denotes the market’s expectation conditional on the set information available to investors at the time t and Dt refers to the dividends per share paid out over the period t. Using recursive method to solve the formula of Euler in Eq. (14), we obtain: Pt = 2

∞

∑e

− r ( s − t + 1)

s=t

E t ( D s ) + e − rs E t ( Ps )

(15)

See Evans (1991) for detailed developments.

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By applying the transversality condition Eq. (15), that is, lim e − rs Et ( Ps ) = 0 , we get the present vals →∞

ue or fundamental value of the stock price:

Ft =

∞

∑e

− r ( s − t + 1)

s=t

Et (Ds )

(16)

However, the solution as in Eq. (16) is only a particular solution to the problem in Eq. (14) and it is usually interpreted by popular economic theory as a unique equilibrium price of the asset. Other possible solutions of Eq. (14) can be expressed by the sum of the present value of the stock price and a rational bubble Bt: Pt = Ft + Bt

(17)

Note that in Eq. (17), a rational bubble is defined such that Bt = e−r Et (Bt +1 ) . By assuming that the log dividends follow a geometric random walk and the period-t dividends are known when Pt is set, Froot and Obstfeld (1991) obtain the following basic stock price equation: Pt = Ft + B ( D t ) = kD t + cD tλ σ2 μ+

Where k = (e −e r

(18)

) with r > μ +σ2 / 2, λ is the solution of the quadratic equation λ2σ 2 / 2 + λμ − r = 0

2 −1

and c is an arbitrary constant. Under these specifications, the probability of bursting associated with intrinsic bubbles is not as high as that of previous bubbles due to eventual fluctuations of the log dividends. When market fundamentals change, stock price can overreact because the bubble term amplifies the price movement. In addition, the intrinsic bubbles can generate important divergences as well as remain stable over certain periods depending on the firm’s dividend policy. Obviously, the intrinsic-bubbles model offers a possibility to explain why stock prices are highly volatile compared with the dividends as pointed out by Shiller (1981). Second, Artus and Kaabi (1994) develop a state-bubbles model in discrete time which differs from Froot and Obstfeld (1991)’s model in the sense that stock price bubbles depend both on time and dividends paid out. To do so, they suppose that the log dividends follow a random walk and show the following formula for stock’s fundamental value: F (D,t) =

D (t ) 1− e

1 −r + μ + σ 2 2

(19)

If the interest rate is higher than the dividends’ growth rate, the fundamental value in Eq. (19) can be approximated by that in Eq. (20):

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F (D, t) ≈

1 D (t ) with r > μ + σ 2 1 2 2 r − (μ + σ ) 2

(20)

Then, the bubble is defined as the product of a fundamental factor and a temporal factor: B ( D , t ) = AD tλ e β t

(21)

)

(

Where λi = − μ ± μ 2 − 2σ 2 ( β − r ) / σ 2 refers to the roots of the following quadratic equation

σ 2λ2 / 2 + μλ + β − r = 0 . And finally, stock price can be expressed by the following formula: D

F ( Dt , t ) = 1− e

t 1 −r+ μ + σ 2 2

+ A1 D tλ1 e β t + A2 D tλ2 e β t

(22)

With A1≥0, A2≥0 and β < r + μ 2 / 2σ 2 . If β = r and λi = 0 , the development of the bubbles depends only on time. If eventually β = 0 , intrinsic bubbles are observed in asset prices.

3. Empirical strategy for detection of bubbles in Tunisian stock market The goal of this article is to test whether rational bubbles exist in Tunisian stock market. To do so, two sets of tests will be applied to financial data of the market under consideration. They include both direct and indirect tests. Indirect tests As far as the indirect tests are concerned, the stationarity test developed by Dickey and Fuller (1981) and by Philips and Perron (1988), and the cointegration test in the sense of Engle and Granger (1987) are used to detect asset bubbles. The same procedure was employed by, among others, Diba and Grossman (1987, 1988), and Hamilton and Whiteman (1985). The stationarity test is useful in that it provides an easy framework to test the presence of asset bubbles through simply checking the stationary property of the price and dividend series. If both series of interest follow a stationary process after their 1-order differentiation, rational asset bubbles can not be confirmed. By contrast, if asset bubbles are effectively present, stock prices would be more explosive than dividends. Diba and Grossman (1988) show that, in the absence of asset bubbles, dividend and stock price series are co-integrated. It then follows that the cointegration technique can be used to prove the existence of bubbles if they do exist. Since the cointegration technique has become a common tool in finance literature, it only seems convenient to recall that two non-stationary series are said to be co8

integrated if a linear combination of them is stationary and the presence of their cointegration is tested using a two-stage procedure as described in Engle and Granger (1987). Direct tests Unlike the indirect tests pointed out previously, the use of indirect tests in detecting asset bubbles requires a complete specification and estimation of economic parameters. Using direct tests, past studies suggest that market bubbles do exist when asset prices are not in line with the real economic conditions (see, e.g., Flood and Garber, 1980; Shiller, 1981; West, 1987). As a result, models of stock market bubbles related to this research stream tend to compare the observed stock prices with the prices that should be, based on the fundamentals. If a significant difference is observed, the hypothesis of asset bubbles can not be rejected3. In this paper, we seek to test for the intrinsic and state bubbles in Tunis stock exchange using the methodologies of Froot and Obstfeld (1991), and Artus and Kaabi (1994) respectively. Precisely, the presence of intrinsic bubbles in the spirit of Froot and Obstfeld (1991) is investigated by estimating the following model: Pt = c 0 t + cD tλ −1 − η t + ε t Dt

⎛ r μ+σ Where c0 = k = ⎜ e − e 2 ⎜ ⎝

2

−1

⎞ 2 2 ⎟ and λ = − μ ± μ + 2 rσ . μ and σ i ⎟ σ2 ⎠

2

are estimates of the log div-

idend variable which is assumed to behave according to a random walk process, dt +1 = μ + dt + ε t +1 with ε t +1 → N (0; σ 2 ) . Given these conditions, the null hypothesis of absence of intrinsic bubbles refers and its alternative can be formulated as follows:

H 0 : c 0 = k and c = 0 H 1 : c 0 = k and c ≠ 0 Under the null hypothesis, stock price is a linear function of the log dividends and the price to dividend ratio equals k. Note that the linear regression of the intrinsic-bubble model is carried out by OLS method that corrects for heteroscedasticity if it is present in estimated residuals through the use of the covariance matrix of Newey and West (1987).

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Since direct tests commonly examine only one specified class of asset bubbles such as intrinsic bubble and periodically state bubble, the non-existence of the bubble under consideration does not necessarily imply the absence of other categories of bubbles (Hamilton and Whiteman, 1985). That is why in this paper we combine direct tests with indirect tests to avoid the methodological problem inherent in direct tests.

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Testing for the state bubbles according to Artus and Kaabi (1994)’s methodology consists of estimating the following state-bubble model: bt = c + λ d t + β t This model is derived from assuming that the dividend yield process is governed by a random walk and from taking the log of the state-bubble formulation Bt = cDtλ e βt , where Bt expresses the bubble defined by the difference between the observed stock price and the stock’s fundamental value. Accordingly, the case where β ≠ 0 and λ = 0 corresponds to an exogenous bubble (i.e., a bubble whose development is only function of time) while an intrinsic bubble is formulated if β = 0 .

4. Data The current study tests for the existence of asset bubbles using annual frequency data from ten firms listed in Tunis stock exchange over a 35-year period from 1971 to 2005. Only firms with more than 90% of available observations are put in our sample data. Discontinued data problem in newly emerging markets like Tunisia is mostly reduced by replacing any missing value by the last observation available. Clearly, the adopted strategy is justified by the random walk hypothesis of stock prices. The sample price and dividend indices are then defined as follows: 10

10

i =1

i =1

Pt = ∑ Wit Pit and Dt = ∑ Wit Dit , t = 1,...,35

Where Pit and D it are stock price and dividend yield of firm i at time t. W it represents the proportion of the firm i’s market capitalization in the total market capitalization of ten selected firms.

5. Results and interpretations This section reports and discusses the obtained results from both indirect and direct tests. Recall that the first group of tests examines whether an asset bubble exists based on the application of stationarity and co-integration tests to stock price and dividend series whereas the second focuses on the detection of intrinsic and state bubbles. Empirical evidence from indirect tests Since the selection of model specifications and the number of lagged factors has a great impact on the power properties of a stationarity test, we adopt the following strategy to select the most suitable 10

models and their appropriate number of lags. For the model selection, we first estimate model with drift and stochastic trend, then model with only drift and finally model without drift and stochastic trend. The appropriate number of lags is determined by performing a correlogram test and refers to the last partial autocorrelation which significantly differs from zero. Additionally, according to our discussions in Section 3, Augmented Dickey-Fuller and Philips-Perron tests are applied to price and dividend series expressed both in the level and first difference. Table 1: Augmented Dickey-Fuller and Philips-Perron tests of price and dividend series Treatments of price and dividend series

Price variable

Series in level

-2.609

Series in the first difference

-3.652**

ADF Dividend Critical values variable 1% -3.642 -1.138 5% -2.953 10% -2.615 1% -3.649 5% -2.956 -3.649* 10% -2.616

Price variable

PP Dividend variable

-1.039

-1.865

-4.426**

-4.037**

Critical values 1% 5% 10% 1% 5% 10%

-3.635 -2.949 -2.613 -3.642 -2.953 -2.615

Notes: ADF and PP refer to the empirical values of the Dickey-Fuller and Philips-Perron test statistics. (*) and (**) indicate that the null hypothesis of presence of unit root is rejected at 5% and 1% levels of significance respectively. The appropriate model for price series is a two-lag model without drift and constant, while a one-lag model without drift and constant is suitable for dividend series. For the shake of space, we do not report the results of selection procedure here, but they are available upon request

As the empirical values of ADF and PP statistics are higher than critical values at conventional levels of significance (see, Table 1), it is clear that both of price and dividend series in levels have a unit root. The results indicate, however, no unit root in their first differences. Accordingly, price and dividend series in the first difference are, without loss of generality, said to be integrated of order one at 5% level of significance. Table 2: Augmented Dickey-Fuller and Philips-Perron tests of the logarithms of price and dividend series Treatments of price and dividend series

Log of price variable

Series in level

-2.603

Series in the first difference

-3.654**

ADF Log of dividend Critical values variable 1% -3.642 -0.976 5% -2.953 10% -2.615 1% -3.649 5% -2.956 -4.040** 10% -2.616

Log of price variable

PP Log of dividend variable

-2.449

-0.955

-5.044**

-4.967**

Critical values 1% 5% 10% 1% 5% 10%

-3.635 -2.949 -2.613 -3.642 -2.953 -2.615

Notes: ADF and PP refer to the empirical values of the Dickey-Fuller and Philips-Perron test statistics. (*) and (**) indicate that the null hypothesis of presence of unit root is rejected at 5% and 1% levels of significance respectively. A onelag model without drift and constant is suitable both for the logarithm of price and dividend series. For the shake of space, we do not report the results of selection procedure here, but they are available upon request.

We also apply the Augmented Dickey-Fuller and Philips-Perron tests to the logarithms of price and dividend series. The results reported in Table 2 also confirm the stationary features of the first dif11

ference of the considered variables, or in other words the logarithms of price and dividend variables are integrated of order one. Therefore, speculative asset bubbles are not present in Tunis stock market according to unit root tests since in all cases stock prices are not more explosive than dividends. In Section 3, we have argued that, if both stock prices and dividend series are integrated of order one and their first difference series are co-integrated, there is then no rational asset bubble because the observed price movements are conducted by fundamentals. To examine this proposition, Engle and Granger (1987) test of cointegration is applied to stock price and dividend series both in levels and logarithms. In practical terms, we test the null hypothesis H0 of unit root against the alternative hypothesis H1 of no unit root in estimated residuals. The rejection of H1 will imply the existence of the cointegration relationship between stock price and dividend. Table 3: Test of cointegration between price and dividend series Testing relation of the cointegration test: Pt = a + bDt + ε t Variable

Estimated coefficients 7.503 aˆ -0.096 bˆ Test of unit root in estimated residuals Empirical value of the ADF statistic -3.096**

t-statistics 7.081 -0.729

p-value 0.000 0.474

Critical values 1% -3.707 5% -2.979 10% -2.629

Notes: ADF and PP refer to the empirical values of the Dickey-Fuller and Philips-Perron test statistics. (**) indicates that the null hypothesis of presence of unit root is rejected at 5% level of significance.

Table 3 reports the results from the test of cointegration between raw price and dividend series. It is demonstrated that estimated residuals are non-stationary at 1% level. In fact, empirical value of the ADF statistic of -3.09 is higher than the associated MacKinnon’s critical value at 1% (-3.707). This result typically leads us to reject the cointegration relationship between price and dividend price and dividend series, and therefore suggests the presence of asset bubbles in Tunis stock exchange. The rationale is that price movements do not follow those of fundamentals. When applying the cointegration test to the logarithms of price and dividend series, we find similar results (see, Table 4). Table 4: Test of cointegration between price and dividends in logarithms Testing relation of the cointegration test: log(Pt ) = a + b log(Dt ) + ε t Variable

Estimated coefficients 2.002 -0.010 Test of unit root in estimated residuals Empirical value of the ADF statistic aˆ bˆ

-3.322**

t-statistics 31.253 -0.642

p-value 0.000 0.525

Critical values 1% -3.707 5% -2.979 10% -2.629

Notes: ADF and PP refer to the empirical values of the Dickey-Fuller and Philips-Perron test statistics. (**) indicates that the null hypothesis of presence of unit root is rejected at 5% level of significance.

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Until then, indirect tests provide mix results regarding the existence of rational speculative bubbles in Tunis stock exchange. The divergence of obtained results may be explained by the fact that the stationarity test only enables to detect asset bubbles that cause a durable effect, but not those with limited effects. To further explore this issue in Tunisian stock market, we now turn to present the empirical results from direct tests: test of intrinsic bubbles and test of state bubble. In testing for intrinsic bubbles, the initial step is to determine the values of μ and σ from estimating the logarithms of dividend process dt +1 = μ + dt + ε t +1 . Estimates of μ and σ are equal to 0.034 and 0.098 respectively. The values of λ are deduced from the expression λi = (−μ ± μ2 + 2rσ 2 ) / σ 2 where r is the average stock returns over the studying period. Thus, we obtain λ1 = 2.876 and λ2 = -9.733. The theoretical value of k is equal to 9.101. The next step is to estimate the intrinsic-bubble model using OLS method which corrects for heteroscedasticity. The results are presented in Table 5. Table 5: Estimates of intrinsic-bubble model

c0 c

λ1 = 2.876 Parameters t-statistics 15.269*** 58.829 *** -3.389 -20.495

p-value 0.000 0.000

c0 c

R 2 = 0.927 and R 2 = 0.925

λ2 = -9.733 Parameters t-statistics 8.295*** 21.887 *** 0.047 7.644

p-value 0.000 0.000

R 2 = 0.639 and R 2 = 0.628

Notes: The intrinsic-bubble model Pt / Dt = c0 + cDtλ −1 − ηt is estimated using OLS method over the 1971-2005 period. (***) indicates that the associated coefficients are statistically significant at 1% level.

From the results revealed by Table 5, we observe that the coefficient indicating the presence of intrinsic bubble c is negative (-3.389) and significant at 1% level when λ1 = 2.876. The high score of the adjusted R-squared typically indicates the good fit of the model specification. When λ2 = -9.733, the coefficient c is still significant, but the adjusted R-squared decreases to 62.81%. We also perform a comparison test between the estimate of constant term and its theoretical value k and find no significant difference. Based on these results, we are not able to reject the alternative hypothesis and intrinsic bubble is effectively present in Tunisian stock market. Table 6: Estimates of state-bubble model Parameters c Α Β Model’s explanatory power

Estimates -2.895*** -6.290*** -0.001

t-statistics -11.732 -11.434 -0.214

p-value 0.000 0.000 0.839

R 2 = 0.849 and R 2 = 0.836

Notes: The state-bubble model bt = c + α d t + β t is estimated using OLS method over the 1971-2005 period. (***) indicates that the associated coefficients are statistically significant at 1% level.

Table 6 reports the empirical results from estimating of the state-bubble model. It is shown that all estimated coefficients are statistically significant at 1% level, except for the time trend coefficient. As a result, one can straightforwardly deduce that state-bubbles are not present in Tunisian stock 13

market, but they do exist in the form of intrinsic bubbles whose evolution depends only on dividend yield. This finding reinforces empirical results revealed by our intrinsic-bubble model. It is however, essential to note that the explanatory power of the state-bubble model is largely higher than that of the intrinsic-bubble model (83.6% against 62.9%).

6. Concluding remarks At theoretical level, a rational bubble is defined by the difference between the observed stock price and its fundamental value or the sum of its expected future dividends. At empirical level, a rational bubble might emerge even when the rationality of economic behaviors and investor expectations exists and various forms of market bubbles are detected in previous literature. The aim of this article was to examine whether speculative rational bubbles are present in the Tunisian stock market using two sets of empirical tests. In the first stage, we apply direct tests including stationarity and cointegration tests to the aggregate data of ten firms listed in Tunis stock exchange. Over the 1971 to 2005 period, the Augmented Dickey-Fuller and Philips-Perron stationarity tests reject the presence of asset bubbles, while the cointegration test in the sense of Engle and Granger (1987) favors the presence of asset bubble. In the second stage, we examined whether intrinsic and state rational bubbles exist in Tunisian market by employing the empirical methodologies of Froot and Obstfeld (1991), and of Artus and Kaabi (1994) respectively. Overall, the empirical results provided evidence on the presence of intrinsic bubbles in the considered market, which is useful in explaining why stock prices overreact to fundamental information.

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References Artus P., Kaabi M. (1994), « Bulles intrinsèques, bulles d’état : théorie et résultats empiriques dans le cas du marché boursier Français », Finance, Vol. 15, n°1, pp 7-34. Blanchard O.J. (1979), « Speculative bubbles, crashes and rational expectations », Economics Letter, Vol. 2, pp 387-389. Blanchard O.J., Watson M.W. (1982), « Bubbles, rational expectations and financial market », NBER Working Paper, n°945. Blanchard O.J., Watson M.W. (1984), « Bulles, anticipations rationnelles et marchés financiers », Les Anales de l’Insee, Vol. 54, pp 79-101. Bohl M.T. (2003), « Periodically collapsing bubbles in the US stock market? », International Review of Economics and Finance, Vol. 12, no3, pp 385-397. Caballero R.J., Krishnamurthy A. (2006), « Bubbles and capital flows volatility: Causes and risk management », Journal of Monetary Economics, Vol. 53, pp 35-53. Chan K., McQueen G., Thorley S. (1998), « Are there rational speculative bubbles in Asian stock markets? », Pacific-Basin Finance Journal, Vol. 6, no1-2, pp 125-151. Cunado J., Gil-Alana L.A., Perez de Gracia F. (2005), « A test for rational bubbles in the NASDAQ stock index: A fractionally integrated approach », Journal of Banking & Finance, Vol. 29, n°10, pp 2633-2654. Diba B., Grossman H.L. (1987), « On the inception of rational bubbles », Quarterly Journal of Economics, Vol. 102, pp 697-700. Diba B., Grossman H.L. (1988), « Explosive rational bubbles in stock prices », American Economic Review, Vol. 78, n°3, pp 520-530. Dickey D.A., Fuller W.A. (1981), « Likelihood ratio statistics for autoregressive time series with a unit root », Econometrica, Vol. 49, pp 1057-1072. Driffill (J.) et Sola (M.) (1998) « Intrinsic bubbles and regime-switching », Journal of Monetary Economics, Vol. 42, pp 357-373. Engle R.F., Granger C.W.J (1987), « Cointegration and error correction: representations, estimations and testing », Econometrica, Vol. 55, n°2, pp 251-276. Engsted T. (2006), « Explosive bubbles in the Cointegrated VAR Model », Finance Research Letters, Vol. 3, no2, pp 154-162. Evans G. (1991), « Pitfalls in testing for explosive bubbles in asset prices », American Economic Review, Vol. 81, n°4, pp 922-930. Flood R.P., Garber P.M. (1980), « Market fundamentals versus prices level bubbles: the first test », Journal of Political Economy, Vol. 88, n°4, pp 745-770. Froot K., Obstfeld M. (1991), « Intrinsic bubbles: the case of stock prices », American Economic Review, Vol. 81, no5, pp 1189-1214. Gilles G., Leroy S. (1992), « Arbitrage, martingale and bubbles », Economic Letters, Vol. 60, pp 357-368. Hamilton J., Whiteman C. (1985), « The observable implications of self-fulfilling expectations», Journal of Monetary Economics, Vol. 16, pp 353-373. Hong H., Scheinkman J., Xiong W. (2006), « Asset float and speculative bubbles », Journal of Finance, Vol. 61, no3, pp 1073-1117. 15

Koustas Z., Serletis A. (2005), « Rational bubbles or persistent deviations from market fundamentals », Journal of Banking & Finance, Vol. 29, n°10, pp 2523-2539. McMillan D.G. (2007), « Bubbles in the dividend-price ratio? Evidence from an asymmetric exponential smooth-transition mode », Journal of Banking and Finance, Vol. 31, no3, pp 787-804. Newey W., West K. (1987), « A simple positive, heterosedasticity and autocorrelation consistent covariance matrix », Econometrica, Vol. 55, n°3, pp 703-708. Phillips P.C.B., Perron P. (1988), « Testing for a unit root in time series regressions », Biometrika, Vol. 75, pp. 335-346. Sampson M. (2003), « New eras and stock market bubbles », Structural Change and Economic Dynamics, Vol. 14, no3, pp 297-315. Shiller R. (1981), « Do stock prices move too much to be justified by subsequent changes in dividends? », American Economic Review, Vol. 71, pp 421-436. Tirole J. (1982), « On the possibility of speculation under rational expectations », Econometrica, Vol. 50, pp 1163-1181. West K.D. (1987), « A specification test for speculative bubbles », Quarterly Journal of Economics, Vol. 102, no3, pp 553-580.

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