貨幣市場與股市泡沫的動態分析

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(1)國立高雄大學應用經濟學系碩士論文. 貨幣市場與股市泡沫的動態分析 A Dynamic Analysis of Money Market and Stock Market Bubbles. 研究生：魏慶林撰指導教授：翁銘章博士. 中華民國 99 年 6 月.

(2) 謝誌誠感謝指導教授翁銘章老師在撰寫研究論文的這段時間裡，給予觀念上的啟發與編寫程式時的協助，引領學生正確的論文研究方向。也感謝口試委員柯秀欣老師和李慶男老師於口試時不吝賜教，令本論文更臻完善。另外，很感謝語言中心陳信智老師與郭虹希老師熱心、不厭其煩挑出本論文中的文法錯誤，也順便感謝周孚陽同學潤飾本論文中的文字。感謝研究所兩年期間一起打拼和研討的所有同學和學弟妹，祝福大家在工作上與學業上皆能順利。特別感謝家人在背後的支持和鼓勵，使學生能全心全意地專注在論文上。最後，謹以此文獻給我摯愛的家人及所有關心我的人。. 魏慶林謹誌 2010 年六月二十四日. i.

(3) 貨幣市場與股市泡沫的動態分析指導教授：翁銘章博士國立高雄大學應用經濟學系學生：魏慶林國立高雄大學應用經濟學系碩士班摘要由於金融市場有自我預期實現之特性，股市泡沫的增長與崩滅反映了市場對於泡沫信念之強化與削弱。由 Minsky (1991)所提出的金融市場不穩定假說認為經濟體的波動有可能是來自金融市場的不穩定，而此不穩定不需藉由非預期到的外在力量來達成。由於貨幣市場主導了股票市場籌碼的增減與信用擴張之程度對未來景氣預期的影響，如果金融市場不穩定假說可以充分描述經濟體系的不穩定，那貨幣市場便有可能主導市場對於未來泡沫的預期。由於理性預期假說無法解釋經濟體的內在波動亦無法解釋為何泡沫完全崩滅後會再度形成，本研究主要目的便是檢驗究竟在金融市場不穩定假說的架構下，市場對於泡沫的重複形成與崩滅之信念是否由貨幣市場來主導。因此，本研究在股利價格比的架構下，推導出當存在本質泡沫與只反映市場基要的共整合向量，用以過濾出市場對泡沫的信念。接著利用 Probit 模型去描述貨幣市場變數對先驗泡沫信念的影響，然後結合貝氏估計與馬可夫轉換下去估計共整合向量與 Probit 模型。藉由結合貝氏估計與馬可夫轉換，實證結果能夠顯示市場對於泡沫信念之強化與削弱的動態過程，並且能夠檢驗貨幣市場變數是否對股市泡沫的信念有顯著影響。關鍵字：金融市場不穩定假說、股市泡沫、自我預期實現、貝氏估計、馬可夫轉換. ii.

(4) A Dynamic Analysis of Money market and Stock Market Bubbles Advisor: Dr. Ming-Jang Weng Department of Applied Economics National University of Kaohsiung Student: Ching-Lin Wei Master Program, Department of Applied Economics National University of Kaohsiung ABSTRACT Since the financial market posses the feature of the self-fulfilling prophecy, the growth and collapse of stock market bubbles reflects the amplification and diminishing of the beliefs of bubbles. The financial instability hypothesis proposed by Minsky (1992) suggested that the fluctuation in the economy may be resulted from the instability of financial market and such instability could be triggered without exogenous disturbances. As the money market dominates the increases and decreases in stock market funds and the impact of credit amplification on the future expectation of the economy, the money market may be capable of dominating the expectation of bubbles in stock market if the economy system is characterized sufficiently by the financial instability hypothesis. Due to that the rational expectation hypothesis is unable to illustrate endogenous fluctuations in the economy and the reoccurrence of bubbles after complete collapse, the goal of this thesis is to examine whether the belief of repeated crash and arise on bubbles is dominated by the money market following the structure of the financial instability hypothesis. Therefore, this thesis derives cointegration vectors which represent existing intrinsic bubbles and market fundamentals. These vectors can be utilized to filter out the market participant’s belief about bubbles. By using Probit model, the influence of monetary variables on the prior belief of bubbles can be depicted. These vectors and Probit model can be estimated by combining Bayesian econometric framework and Markov Regime-switching approach. The empirical result can display the dynamic process of beliefs of repeat crash and arise on bubbles and show how money market does play a crucial role to dominate these beliefs. Keywords: financial instability hypothesis, stock market bubbles, self-fulfilling prophecy, Bayesian estimate, Markov regime-switching. iii.

(5) Contents Acknowledgment ........................................................................................................................i Chinese abstract .........................................................................................................................ii English abstract........................................................................................................................ iii 1. Introduction............................................................................................................................1 2. Literature review....................................................................................................................4 2.1 A review of the financial instability hypothesis and stock market bubbles .................4 2.2 Methodologies for testing bubble ................................................................................6 3. Econometric modeling .........................................................................................................12 3.1 The background .........................................................................................................12 3.2 Modeling framework .................................................................................................15 3.3 Estimation procedure .................................................................................................24 4. Empirical results ..................................................................................................................31 4.1 Data description and choice of the prior state variable..............................................31 4.2 Estimation outcomes and their implication................................................................33 5. Conclusions..........................................................................................................................39 References................................................................................................................................40. iv.

(6) 1. Introduction Due to the transmission lag, the discretionary monetary policy devised to mitigate macroeconomic imbalances may destabilize the economy. Because over high productivity growth is always considered a fuel for exaggerated optimism and stock market bubbles, the transmission lag may provide a hotbed for occurrence of bubbles. Except the transmission lag, other policies fail to accordance also lead bubbles trend to arise. Some historical evidences provide such examples for inadequate policy. The Japanese bubble in the second half of 1980s is an example of unduly expansionary monetary policy. The instrumental rate downed to 2.5 per cent in February 1987.In their opinion, the aim of this policy was to create more balanced foreign trade and stabilized exchange rate fluctuations. As a consequence, the Nikkei index climbed to a high of almost 40000 around in 1989. The bobbles burst soon after following years. Other example such as 1997 Asian Financial Crisis may be also contributed by inadequate policy. High savings and investment rates, robust growths, and moderate inflation had been maintained in East Asia for several decades. This economic background combined with the inadequate liberalization in the financial market and maintained attractively high interest rates to foreign investors had generated a massive foreign fund inflow. The combination of fixed exchange rates, and weaknesses in the financial systems did not ready to support ongoing liberalization in financial market. As a result, the financial systems could not bear external shocks such as the devaluation of the Chinese remnimbi and the Japanese yen and caused asset prices eventually began to collapse.. Since tighter monetary policies can discourage speculation activities, reducing and preventing bubbles, the higher interest rate or other tighter monetary policies may 1.

(7) mitigate bubbles before they lead to undesirable economic fluctuation. Nevertheless, asset prices can reflect market expectations of future economic state. If that monetary policies for alter bubbles are deviation from other macroeconomic goals and leading other imbalances is expected by market participants, the stock market will exhibit more volatile and could deteriorate the current economic condition. Even tighter monetary policies can debilitate bubbles without any problem; policy makers would not be able to identify them. When the stock market does not exhibit markedly speedy rise, bubbles can not be easily detected. If bubbles start to growth and does not be detected, the central bank may not ready to offset adverse consequences.. Hence the money market plays an important role to accommodate the stability of financial market. The dynamic process of the money market and stock market bubbles should be meticulously evaluated. To investigate the dynamic process between the money market stock market bubbles, the financial instability hypothesis proposed by Minsky (1992) is adopted. The financial instability hypothesis suggested endogenous boom and busting in the asset market, and the central bank should take more attention on the role of money market. Not only should the price and the output level be considered, the stability on the financial market should also be considered. If asset prices are endogenous fluctuation, bubbles would be able to arise after they completely crash. Therefore, the self-confirming belief of bubbles should be accurately assessed for the purpose of assessing the process of repeated crash. Since this hypothesis emphasizes the effect of credit amplification on asset prices, the money market may be capable to accommodate the stability of financial market. Hence this thesis is going to examine whether the money market is suitable to characterize the self-confirming belief of bubbles following the structure the financial instability hypothesis 2.

(8) Following the characterization of repeated crash bubbles, the critical progress of testing bubbles is to accurately characterize the self-fulfilling prophecy of bubbles. Moreover, this also implies that the uncertainty of recognition of whether bubbles exist is governed by the prior belief of bubbles. Hence the past realizations may originate form mixture distribution and the self-confirming belief of bubbles should be incorporated into the information set. Therefore, the main aim of this thesis is to develop a new econometric model in order to filter out the market participant’s beliefs about bubbles. In addition, this model can be utilized to study the connection between the self-fulfilling prophecy of bubbles and the money market. The empirical result can show how the money market does play a crucial role to govern the self-confirming belief of bubbles. The evidence provided by this thesis can test whether stock market bubbles are characterized by the financial instability hypothesis.. This thesis focuses on intrinsic bubbles. By utilizing Markov Regime-switching Model in which regime switching between bubbles exists and not exists, the belief for bubbles can be estimated by smooth probability. By allowing the transition probability contains information about monetary variable, policy makers can assess how much tighter policy has effect on the probability of bubbles existing and furthermore evaluate whether such tighter policy is deviation or benefit to other macroeconomic goals. The rest of the thesis is organized as follows. Section 2 presents relative literatures about methodologies for testing bubbles. The theory of bubbles and the concept of financial instability hypothesis are also discussed in this section. In section 3, the detail methods for developing an econometric model and estimation procedures are provided. Section 4 contains the empirical results and their implications and section 5 offers concluding remarks.. 3.

(9) 2. Literature review The first part of this section will discuss the connection between the financial instability hypothesis and stock market bubbles. The characteristic of repeated crashes bubbles is emphasized in the first part. The second part of this section will survey the methodologies adopted for testing bubbles and point out what crucial progress should be endeavored in order to study whether stock market bubbles are resulted from the instability financial system.. 2.1 A review of the financial instability hypothesis and stock market bubbles The free disposal assumption comment that stockholders can not rationally expect stock prices to diminish without bound. Hence the stock price can not be negative at a finite future date and this also rules out the negative rational bubbles. By applying free disposal assumption, Diba and Grossman (1988b) mentioned that once rational bubbles have crashed, they can not restart. Besides the free disposal assumption, the key postulation which leads to such result is the rational expectation hypothesis. As bubbles are rationally expected, any innovation in the stock price after the first date of trading will be expected equating zero. In this regard, this thesis deliberately emphasizes that bubbles can not restart after complete burst is given by the integration of several postulations which includes the rational expectation, free disposal and excluding possible multiple equilibriums in which they contain the unstable regime. Adopting that the economy is depicted by these assumptions, a researcher who want to empirically test the existence of bubbles would utilize whole sample period. But historical evidence seems to support bubbles with repeated crashes.. 4.

(10) Fukuta (1998) proposed a class of rational bubbles called incomplete bursting bubbles which have three possible states. These states are large bubbles, small bubbles and incomplete burst. When bubbles enter the incomplete burst state, bubbles heavily decline as if they completely crash. Due to incomplete bursting, this class of bubbles can restart after the incomplete crash. Also, periodically collapsing bubbles presented by Evans (1991) contain the property of repeated crashes. However, such classes of bubbles would make many econometric techniques encounter the difficulty in distinguishing between the market fundamentals and bubbles when bubbles are incomplete burst.. The financial instability hypothesis, on the contrary, takes a different perspective about bubbles. This hypothesis considers expectations of business profits determining both the cash flow of financial contracts to business and the market price of existing financial contracts. For instance, the cash flow of common stock includes dividend payout and the market price of stock. This hypothesis further considers expectations of profits depending upon investment in the future and current realized profits are settled by current investment. By taking banking as a profit-seeking, owing to that weather liabilities are validated is determined by investment, credit amplification may take place to earn more profit for banks when economy is in prosperity. As this hypothesis suggests that the reason investment takes place now is entrepreneur and their bankers expect investment to take place in the future, the sizable degree of fluctuation in asset price is endogenous could be inferred from this hypothesis. If the prosperity economy prolongs for lengthy periods, an increase in the degree of credit amplification would lead the economy enter to unstable financing regimes. Thus the first theorem of the financial instability hypothesis is that the economy has both the stable financing regimes and unstable financing regimes. The 5.

(11) implication from this hypothesis is that the economy can exhibit fluctuations even though there are no exogenous disturbances and furthermore, bubbles may arise after complete bursting. In this manner, by defining that the economy state is whether the economy is under unstable financing regimes, this thesis use the mixture contingent state to construct a newly econometric to test whether the stock market bubble is result form financial instability.. 2.2 Methodologies for testing bubbles The conception of bubbles is the part of observed real stock prices above the equilibrium level. According to the dynamic optimization problem of a representative consumer, the equilibrium level of real stock prices can be calculated from present value model. If the components of real stock prices only reflect market fundamental, the present value model can be a pricing model. However, the observations of real stock price may not always exhibit what pricing model predicts. The difference between stock price and its equilibrium level will trigger the proceeding of arbitrage. By adopting the theoretical view, due to the existence of arbitrage, real stock prices eventually converges to equilibrium level. For this reason, according to the present value model, after excluding possible non-rational trade in the stock market; meanwhile, real stock prices and dividends are not cointegrated, the stock market may have shown the evidence of rational bubbles. Therefore, the common approaches to test the null hypothesis of rational bubbles are unit-root tests to the price-dividend ratio or cointegration test between these two variables with long span data.. In that the arduously to obtain the real pricing model, the unobservable variables are not easily distinguished into fundamentals and bubbles components. Different assumptions on the Euler equation, potential switching in market beliefs and 6.

(12) types of bubbles have made a long debate on testing technique. This section will introduce the reasons. Final part of this section will briefly introduce such debate that could be solved by the conception of mixture contingent state.. The price-dividend cointegration approach was applied by Campbell and Shiller (1987); and Diba and Grossman (1988a), following early empirical work includes Timmermann (1995). However, as discussed by Diba and Grossman (1988a), the rejection of cointergation between real stock prices and real dividends would not necessarily conclude the evidence of existing rational bubbles. The present value model which they considered contained the unobservable variables that could have an effect on market fundamentals. From their present value model, if the first differences of the real dividends and unobservable variables are stationary, the rejection of the unit-root on first differences of real stock prices may be resulted from non-existence of rational bubbles. As the first differences on stock prices is stationary implies the transversality condition holds, Diba and Grossman (1988a), further derived a cointegrating vector between real stock prices and real dividends. But the cointegration test they provided requested unobservable variables in market fundamentals must stationary in levels. This is a more restrictive condition than unit-root test on the first differences of real stock prices.. These methods had opened up new avenues for empirical strategy to overcome the identification problem pointed out by Flood and Garber (1980). The problem was difficulty in distinguishing from the market fundamentals components that researcher cannot observe when using the specification tests introduced by Singleton (1979); Hansen and Sargent (1979). For instance, emphasized by Hamilton (1986), the changes in the expectations of market participants, such as future tax treatment of. 7.

(13) dividend income, a researcher may be unable to observe or to infer. Other misspecification includes adding a constant discount rate restriction on the present value model or neglecting the inter-temporal marginal rate of substitution due to the assumption of risk neutral. All of these misspecifications should not be worried as long as they are stationary in levels when adopting the method of unit-root test on first differences of the real stock price and cointegration test between real stock prices and real dividends.. Nevertheless, different restrictions on the unobservable variables in market fundamentals of these tests had generated mixed results shown by Diba and Grossman (1988a). This suggests the fact that real stock prices are stationary in first differences results from the unobservable market fundamentals following the unit-root process. Timmermann (1995) illustrated that real stock prices and real dividends did not cointegrated was caused by the assumption of constant discount rate. By allowing time-varying discount rate in the present value model, Timmermann (1995) showed that regression residuals contained linear component of real dividend. This implied that the unobservable market fundamentals are non-stationary in levels but stationary in first differences. For this reason, price-dividend ratio test provided by Craine (1989), Cochrane (1992) and log price-dividend ratio test provided by Campbell and Shiller (1988) were suitable methodologies for testing bubbles if the persistence of the discount rate was not very high. Therefore, Timmermann (1995) also investigated the impact of persistence in discount rate in present value model on cointegration test in levels and in logarithms by Monte Carlo experiment. But samples were generated by nonlinear model which followed the approach of Poterba and Summers (1986). However, that the argument of the linear present value model is suitable for the assumption of constant discount rate which had been discussed by Campbell et al., 8.

(14) 1997, (CLM) had enlightened nonlinear econometric techniques to test present value model. Manzan, 2004, also followed the approach of Poterba and Summers (1986) to derived the nonlinear present value model by allowing time-varying discount rate which is known in the literature as the dynamic Gordon model. To capture such nonlinear phenomenon, Manzan, 2004, studied the demeaned log price-dividend ratio, that is, log deviations of the price from static Gordon valuation, by using smooth transition regression (STR) approach.. Although log dividend-price ratio test implies releasing the assumption of constant discount rate, some researches had provided other reasons of nonlinear adjustment towards equilibrium. Gallagher and Taylor (2001), in their investigation of risky arbitrage and the limits of arbitrage hypothesis also utilized STR approach and the variable was a linear combination of log dividend-price ratio and discount rate. The linear combination was a cointegration vector between log dividend-price ratios and discount rates. Not only does the existence of risky arbitrage and the limits of arbitrage suggests that log dividend-price ratio displays nonlinear trajectory, irrational fads or investor sentiment and misspecification of market fundamentals had also been argued repeatedly. Coakley and Fuertes (2005) argued nonlinear effects may occasionally dominate the linear components because the log dividend-price ratio test conflates bull and bear market phases may suffer the problem of potential switching in government policy and other mispricing part which belongs to fads or waves of optimism. In their investigation of asymmetrical adjustments during bull and bear market phases, the nonlinear model they applied was a parsimonious two-regime model that belongs to the momentum threshold autoregressive (MTAR) class. The MTAR method was proposed by Enders & Granger (1998); Enders and Siklos (2001).. 9.

(15) Except nonlinearities behavior, the more serious problem of linear model argued by Evans (1991) was incorrect conclusions with respect to the presence of bubbles if periodically collapsing bubbles appears in stock market. Relying on Evans’ Monte Carlo simulations with periodically collapsing bubbles presence, these simulations results incorrectly showed the absence of bubbles when using standard unit root and cointegration tests. Base on these simulations results, Evans (1991), further argued that periodically collapsing bubbles would appear to be a stable linear autoregressive process unless the probability of bubbles collapsing was not very high.. Afterward, a nonlinear model literature to overcome the Evans critique which was proposed by Hall, Psaradakis, and Sola (1999), used Markov switching ADF test in which time series switching between explosive and stable process. Since such bubbles exhibit the characteristic of sudden collapse, standard tests may incorrect conclusion with respect to mean reversion. Hall, Psaradakis, and Sola (1999), argued if data only contain the expansion phase of bubbles, a test would more likely to find the evidence of divergence process. Basing on this criticism, Markov switching ADF test may provide more testing power of null hypothesis. Relying on Monte Carlo experiments shown by Hall, Psaradakis, and Sola (1999), the Markov switching ADF test had considerable power to detect the presence of periodically collapsing bubbles. Bohl (2002) applied momentum threshold autoregressive (MTAR) model to capture the characteristics of periodically collapsing bubbles. The results of the Bohl’s Monte Carlo studies displayed that the MTAR approach provided a sufficiently powerful test to detect the presence of periodically collapsing bubbles.. It appears nonlinear techniques is adequate to test present value model. On the contrary, a noticeable dilemma of these nonlinear models is difficulty on specifying a. 10.

(16) specific form for the sake of the testing price-dividend relation or bubbles. In that these nonlinear components are difficulty to detect whether they originations from the fundamentals or bubbles, a researcher would have perplexity on specifying. However, such indeterminate originates could be regarded as a mixture contingent state. By manipulating inter-temporal marginal rate of substitution, a partial of unobservable elements can be derived which belongs to fundamentals to solve some of identification problem. Detail econometric methodology will discuss in next section.. 11.

(17) 3. Econometric modeling This section is going to introduce the detail method for developing an econometric model. Section 3.1 discusses the background of combining Bayesian econometric framework and Markov Regime-switching model. Section 3.2 shows all detail methods about model building. The final section will briefly introduce the estimation procedure.. 3.1 The background It is important to stress the intention of the research to use nonlinear model. Specifically, that a researcher is going to test the price-dividend relation or examining the presence of bubbles has significant impact on developing a nonlinear model. Such the specification issue may have engendered the reemergence of the identification perplexity that a researcher has to confront. Precisely illustrating, that the specification of these nonlinear techniques does not divide the unobservable into fundamentals and bubbles may encounter the identification problem if there occur some of segment of the unobservable which belongs to fundamentals following the unit-root process or revealing nonlinear pattern which analogous to the periodically collapsing bubbles. Such circumstance has motivated this thesis to develop a parametric model which can be exploited to filter out the bubble component and the fundamentals with tracking the unit-root process from the unobservable. The other segment of the unobservable will be remained as an error term which appertains to nonlinear fundamentals and irrational trading.. Beside the identification issue of previous nonlinear techniques, the critical progress of the modeling direction is to accurately characterize the self-fulfilling 12.

(18) prophecy of bubbles. For the sake of investigating the financial instability hypothesis, that this hypothesis implies the endogenous boom and busting in asset prices has suggested that bubbles are be able to arise again after they completely crash. Owing to that the uncertainty of recognizing bubbles when the stock market does not exhibit unusually rise or the appearance of stable periodically collapsing bubbles, these past realizations may originate form mixture distribution. Namely, what contingent state of distribution is categorized in if the state of economy is defined as the event of whether bubbles exist? This uncertainty suggests that it is deficiently to characterize the self-fulfilling prophecy of bubbles if the information set only contains the current and past realizations variables. Consequently, the prior belief of bubbles should be included in the information set.. However, regardless the previous linear or nonlinear model, these econometric methodologies all neglect the prior belief of bubbles in the information set. This suggests the past realization of state variables should be identified in order to extract the market participant’s belief about bubbles. Therefore, econometric techniques should be able to assess bubbles in any sample unit of time period instead of full sample time period. Due to the whole sample period base, the procedures of previous linear or nonlinear models have difficulty on extracting the belief of bubbles. If the financial market exhibits endogenous fluctuations which coincident with that bubbles can begin to grow after they completely crash, these previous techniques will be insufficiently employed to test the presence of bubbles. Therefore, the main aim of this thesis is to clearly specify the testing model for purpose of working out the handicap of the previous techniques that are unable to reflect the self-confirming belief of bubbles.. 13.

(19) In short, the primary errand of this thesis is to address the theme of accurate representing the self-fulfilling prophecy bubbles. To achieve this goal, not only the identification issue should be dedicated, the self-confirming belief of bubbles should be also incorporated into the information set. Econometric techniques developed in this thesis must satisfy these two requirements in order to filter out the market participant’s belief about bubbles .This thesis will follow the econometric framework proposed by Filardo and Gordon (1998) for the sake of studying the connection between the self-fulfilling prophecy of bubbles and financial instability hypothesis.. Therefore, this thesis extends the log-linearized present value model suggested by Campbell and Shiller (1988) to show there is another specific cointegration relationship between log-stock prices and log-dividends when presenting a specific type of intrinsic bubble proposed by Froot and Obstfeld (1991). Also, by releasing restriction on the assumption of risk neutral and allowing switching in relative risk aversion level, this thesis derives there presence a miss pricing part which belongs to fundamentals. This part is a linear combination of logarithm consumption level. Precisely, there exist a specific type of fundamental cointegration vector between log dividend-price ratio and log consumption level. Based on the fact that the cointegration vector is not unique when intrinsic bubble appears, this thesis further develops nonlinear model by utilizing Markov Regime-switching approach. This model can be employed to depict the cointegration vector switching between fundamental and deviations from fundamental in the event of the existing of intrinsic bubbles. By combining Bayesian econometric framework recommended by Filardo and Gordon (1998), a researcher can update prior belief about bubbles so that this Markov Regime-switching model can reflect the self-confirming belief of bubbles.. 14.

(20) 3.2 Modeling framework To construct the object function of optimization problem, let the time varying i. discount factors for future consumption are ψ t ,0 = 1 , {ψ t ,i }i∞=1 = { Π (1 + Rt + i ) −1}i∞=1 , where j =0. Rt +i. is the discount rate in period t + i . In addition, let the consumption level which. representative consumer chooses in period t + i is denoted by Ct +i , and let the time varying utility function U t +i (•) is strictly concave, increasing and continuously differentiable. In this thesis, the utility function is assumed exhibiting two risk aversion regimes which changes with real income level. This setting will useful to derive inter-temporal marginal rate of substitution is a function of consumption level by assuming U t +i (•) follows relative risk aversion function. This idea obey Vanden(2005) in his study of volatility clustering. Vanden(2005) pointed out that if the volatility patterns of returns on a stock can be reconciled within the representative agent, then the assumption of constant relative (or absolute) risk aversion must be abandoned. Therefore, the utility function should exhibit several risk aversion regimes to reflect several volatility regimes in the stock market. To simplify the econometric model, only two risk aversion regimes are considered. Let E t is denoted conditional expectations base on the current information set. The dynamic optimization problem of the representative consumer which this thesis considers is defined below:. (3.2.1). ⎡ ∞ ⎤ E ψ t ,iU t + i (Ct + i ) ⎥ t ⎢ ∞ {Ct +i , X t +i }i =0 ⎢⎣ i = 0 ⎥⎦ Max. ∑. Subject to the sequence of budget constraints,. 15.

(21) (3.2.2). Ct + i = Yt + i + ( Pt + i + Dt + i ) X t + i − Pt + i X t + i +1 , i = 0,1, 2,...,. where Pt +i , Dt +i and X t +i is denoted real stock prices ,real dividends and quantity of shares holding in period t + i respectively. The real income denoted by Yt +i. is assumed to exogenous given.. The Euler equation for the utility maximization problem is ⎡ U ′(Ct +1 ) ⎛ Pt +1 + Dt +1 ⎞ ⎤ Pt = Et ⎢ ⎜ ⎟⎥ ⎢⎣ U ′(Ct ) ⎝ 1 + Rt +1 ⎠ ⎥⎦. (3.2.3). To obtain the well-known log-linear approximation provided by Campbell and Shiller(1988a,b), this thesis define: rt +1 ≡ ln(1 + Rt +1 ). The Euler equation can be rearranged as: (3.2.4). Et rt +1 = Et ln[U ′(Ct +1 )] − ln[U ′(Ct )] + Et ln[ Pt +1 + Dt +1 ] − ln Pt. The term Et ln[U ′(Ct +1 )] − ln[U ′(Ct )] can be interpreted a misspecification part when a researcher assumes that the representative consumer is risk neutral.. Suppose that the utility function is given by:. (3.2.5). U t + i (Ct + i ) =. Ct1+−iγ t +i 1 − γ t +i. Where the relative risk aversion level complies with following setting:. γ t + i = I t +i γ 0 + (1 − I t +i )γ 1 ⎧0, if Yt + i < ς ⎪ It +i = ⎨ ⎪⎩1, if Yt + i ≥ ς. 16.

(22) Suppose the risk aversion level in period t is γ 0 and in period t + 1 is γ 1 . As a result, the term Et ln[U ′(Ct +1 )] − ln[U ′(Ct )] can be written as (γ 0 − γ 1 ) ln Ct − γ 1Δ ln Ct +1 .. Now, by defining two constant parameters, the log-linear approximation of Et ln[ Pt +1 + Dt +1 ] − ln Pt. (3.2.6). becomes:. ln[1 + eδt +1 ] + Δpt +1 ≅ k − ρδ t +1 + dt +1 − pt = k + δ t − ρδ t +1 + Δdt +1. where dt +1 , pt +1 and δ t +1 are defined as dt +1 ≡ ln[ Dt +1 ], pt +1 ≡ ln[ Pt +1 ], δ t +1 ≡ dt +1 − pt +1. The two constant parameters are defined below:. k ≡ ln[1 + eδ ] −. δ eδ 1 + eδ. , ρ ≡ 1−. eδ 1 + eδ. , where. δ ≡d − f. , and. d. ,. f. are denoted log. average real dividends fundamental stock prices respectively. Assume that log real dividends are generated by following stochastic process:. (3.2.7). i.i.d. dt +1 = μd + dt + ξ d ,t +1 , ξ d ,t +1 ~ N (0, σ d2 ). In addition, the log fundamental stock price satisfies the transversality condition:. (3.2.8). ⎡ exp( ft +T ) ⎤ lim Et ⎢ ⎥=0 T →∞ ⎣ 1 + Rt +T ⎦. Only if the stock price consists of market fundamental, the particular solution of stochastic difference equation (3.2.3) can be a fundamental real stock price. To include possible rational bubbles, considering the stock price equates the sum of the. 17.

(23) fundamental value and a rational bubble, that is, the general solution of equation (3.2.3):. (3.2.9). Pt +i = Ft +i + Bt + i. Where Ft +i is denoted the fundamental value of stock in period t + i and Bt +i is denoted the level of a rational bubble in period t + i . Moreover, rational bubbles are any sequence of random variables such that:. (3.2.10). Bt + i = {Bt + i ∈R ++ (1 + Rt + i + i ) Et + i ( Bt + i +1 ) = Bt + i }i∞=0. Suppose the rational bubbles follow the class of intrinsic bubbles which was specified by Froot and Obstfeld (1991):. (3.2.11). Bt + i = α Dtλ+ i. The parameter α is an arbitrary constant other than zero or negative number and λ is the positive root of the following quadratic equation:. (3.2.12). λ 2σ d2 2. + λμ d − rt +1 = 0. Froot and Obstfeld (1991) pointed out that the intrinsic bubbles are inherited entirely from the fundamentals. In the case of stock pricse, only one stochastic fundamental factor, that is, the dividend process is considered. Hence intrinsic rational bubbles depend on dividends alone.. By verifying whether intrinsic bubbles depicted by equation (3.2.11) satisfies (3.2.10), it is explicit to know that even though rational expectation is performed by. 18.

(24) market participants, the stock market would be still inefficiency:. λμd + 1 1 1 λ ( μ +ξ ) Et ⎣⎡B ( Dt +1 ) ⎦⎤ = Et ⎡α Dtλ e d d ,t +1 ⎤ = (α Dtλ e ⎣ ⎦ 1 + Rt +1 1 + Rt +1 1 + Rt +1. λ 2σ d2 2. )=. 1 (α Dtλ (1 + Rt +1 )) = B ( Dt ) 1 + Rt +1. ⇒ Pt ≠ Ft ⇔ Bt > 0. However, when bubbles can arise after complete bursting, the equation (3.2.10) no longer be satisfied. In such case, this thesis assigns a regime selection function such that:. St = St ( St* ), St* ∈ (−∞, +∞) St = {st st = 1 ⇔ St* ≥ 0, st = 0 ⇔ St* < 0}. where St is denoted regime selection function, and St* is latent variable which is given by following latent regression:. i.i.d. St* = g 0 + g z′ Z t + g s st −1 + ut , ut ~ N (0,1). (3.2.12). The vector Zt g z = ( g1 , g 2 , g3 )′ M 1t. contains monetary variables and output: Zt = (mt , yt , it ) '. variables mt , yt and are defined as mt ≡ ln M1t , yt ≡ ln PIt , Where. is real M1 money stock in period t, and PIt is the real disposable personal. income in period t. it is a LIBOR rate in period t.. Furthermore, equation (3.2.11) is reset up as (3.2.13). Bt + i = α St + i Dtλ+ i. Now it is easy to show that such bubbles no longer satisfy equation (3.2.10):. 19.

(25) α Pr( St*+1 ≥ 0) λ 1 Et ⎣⎡B ( Dt +1 ) ⎦⎤ = Dt (1 + Rt +1 ) = α Pr( St*+1 ≥ 0) B( Dt ) 1 + Rt +1 1 + Rt +1. where Et +1[ B( Dt +1 )] = α Pr( St*+1 ≥ 0) Et +1[ Dtλ+1 ]. To update prior belief of bubbles (or unstable regime) , suppose the state variable St follows a Markov process with time variant, The transition probabilities are associated: ⎡ pr (0,0),t Pr( St = st St −1 = st −1 , Z t ) = ⎢ ⎢⎣1 − pr (0,0),t. 1 − pr (1,1),t ⎤ ⎥ pr (1,1),t ⎥⎦. From equation (3.2.12), transition probabilities are given by: pr (1,1),t = Pr( St = 1 st −1 = 1, zt ) = Pr(ut ≥ −[ g0 + g ′Z t + g s ]) = 1 − F (− g0 − g ′Z t − g s ) pr (0,0),t = Pr( St = 0 st −1 = 0, zt ) = Pr(ut < −[ g0 + g ′Z t ]) = F (− g0 − g ′Z t ). Hence the real stock price and dividends are originated from mixture distribution and the probability of contingent state of distribution is given by Pr( St* ≥ 0). and Pr( St* < 0) .. As equation (3.2.9) asserts that stock prices equate the sum of fundamental values and bubbles, it is clear δ t = dt − ln[ Ft + Bt ] .. The term ln[ Ft + Bt ] can be log-linear approximated as:. (3.2.14). ln[ Ft + Bt ] ≅ Λt + θt ft + (1 − θt )[ln α + λ dt ]. where Λt = st Λ, θt = stθ + (1 − st ) , and constant parameters are defined below:. Λ ≡ ln[1 + e l ] −. le l 1 + el. ,θ ≡ 1 −. el 1 + el. ,l ≡ b − f. 20.

(26) b is denoted log average bubbles level, and note that such bubbles no longer. belongs to the previously class of rational bubbles.. Let δ t f ≡ dt − ft such that lim ρ T δ t f+T = 0 , and further define following constant T →∞. parameters : η ≡ Λ + (1 − θ ) ln α , φc ≡ (γ 0 − γ 1 ) , φd ≡ (1 − λ )(1 − ρ )(1 − θ ) , φΔd ≡ ρ (1 − λ )(1 − θ ). The Euler equation (3.2.4) can be manipulated as: Et rt +1 = k + φc ct − γ 1 Et Δct +1 + Et Δdt +1 + θ t δ t f − ρ Etθt +1δ t f+1. (3.2.15). + η ( ρ Et St +1 − st ) + φd dt st − φΔd Et ( St +1 − st ) dt +1. Since St +1 is unknown at time t, to conduct the econometric model, it is helpful to guess the value. Let ct ≡ ln Ct , suppose ct also following geometric martingale:. i .i.d. ct +1 = μc + ct + ξ c ,t +1 , ξ c ,t +1 ~ N (0, σ c2 ). (3.2.16). After rearrange (3.2.15) by law of iterated expectations, there are four possible models or cointegration vectors:. Case 1, St +1 = st = 1 : ⎡1. ∞. ⎣. i =1. ⎤. ∑ρ r ⎢θ. δt f = E ⎢. i. t +i ⎥ −. ⎦⎥. H0. θ. − H 0,(1,1) −. Hc. θ. − H d ,(1,1) −. Hc. θ. ct − H d ,(1,1) dt + ε (1,1),t. where H d ,(1,1) ≡. ρφd μd φd η , H d ,(1,1) ≡ , H 0,(1,1) ≡ 2 θ (1 − ρ ) θ θ (1 − ρ ). ε (1,1),t ≡ Et. ∞. ρ {r θ∑ 1. i. i =1. t +i. ⎡1 + γ 1Δct +i − (1 − φΔd )Δdt +i } − E ⎢ ⎢⎣ θ. Case 2, St +1 = 1, st = 0 :. 21. ∞. ∑ρ r i. i =1. ⎤. t +i ⎥. ⎥⎦.

(27) ⎡. 2 ⎤ 1− ρ ⎡ 1− ρ ⎤ ⎡ 1− ρ ⎤ H0 − Hcθ ⎢ ( ρθ )i rt +i ⎥ − ⎥ + H d ,(1,0) − H 0,(1,0) − H c ⎢ ⎥ ct + H d ,(1,0) dt + ε (1,0),t ⎢⎣ i =1 ⎥⎦ 1 − ρθ ⎣ 1 − ρθ ⎦ ⎣ 1 − ρθ ⎦. δt f = E ⎢. ∞. ∑. where H d ,(1,0) ≡. φΔd μd 1 − ρθ. ε (1,0),t ≡ Et. ⎡ ρθ ⎤ φΔd ηρ − 1⎥ , H d ,(1,0) ≡ , H 0,(1,0) ≡ ⎢ − − − 1 (1 ) 1 ρθ ρθ ρθ ⎣ ⎦. ∞. ∑ ( ρθ ) {r i. t +i. i =1. ⎡ ∞ ⎤ + γ 1Δct +i − Δdt +i } − E ⎢ ( ρθ )i rt + i ⎥ ⎢⎣ i =1 ⎥⎦. ∑. Case 3, St +1 = 0, st = 1 : ⎡1. ∞. ⎣. i =1. ρ. ∑(θ ) r ⎢θ. δt f = E ⎢. i. ⎤. 2. t +i ⎥ − H 0. ⎥⎦. ⎡ 1− ρ ⎤ θ (1 − ρ ) ⎡1 − ρ ⎤ − Hc ⎢ − H d ,(0,1) − H 0,(0,1) − H c ⎢ ⎥ ct + H d ,(0,1) dt + ε (0,1),t ⎥ θ −ρ ⎣1−θ ⎦ ⎣θ − ρ ⎦. where H d ,(0,1) ≡. ρ (φd + φΔd ) μd (φ + φΔd ) θη , H d ,(0,1) ≡ d , H 0,(0,1) ≡ 2 θ −ρ θ −ρ (1 − θ ). ε (0,1),t = Et. ⎡1 ρ ( )i {rt +i + γ 1Δct +i − Δdt +i } − E ⎢ θ i =1 θ ⎣⎢ θ 1. ∞. ∑. ∞. ρ. ⎤. ∑(θ ) r i. t +i ⎥. ⎦⎥. i =1. Case 4, St +1 = st = 0 : ⎡. ∞. ∑ρ r ⎢. δt f = E ⎢. i. ⎣ i =1. ⎤. t +i ⎥ − H c. ⎦⎥. − H 0 − H c ct + ε ( 0,0),t. where Hc ≡. ρφc μc φ k , H0 ≡ , Hc ≡ c , 2 1− ρ 1− ρ (1 − ρ ). ε (0,0),t ≡ Et. ∞. ∑ i =1. ⎡. ∞. ∑ ⎢. ρ i {γ 1Δct +i − Δdt +i } − E ⎢. ⎣ i =1. ⎤. ρ i rt +i ⎥ + Et ⎦⎥. ∞. ∑ρ r i. t +i. i =1. Due to lack of prior information about St +1 , this thesis only consider the following simple model to cover case 1, case 3 and case 4:. (3.2.16). δ t f (1,3,4) = Ξ 0 − Ξ1st − H c ct (1 + hc st ) − H d st dt + ε t(1,3,4). 22.

(28) where. Ξ 0 ≡ H E + H c + H 0 , Ξ1 ≡ H 0 + H c hc + H 0 h0 + H d ∞ ⎤ ⎡1 ∞ ρ i ⎤ ⎡ ∞ ⎤⎤ 1 1 ⎡ ⎡1 ρ i rt +i ⎥ + E ⎢ ( H 0,(0,1) + H 0,(1,1) ), H E ≡ ⎢ E ⎢ ( ) rt + i ⎥ + E ⎢ ρ i rt +i ⎥ ⎥ 2 3 ⎢ ⎢⎣ θ i =1 ⎥⎦ ⎢⎣ θ i =1 θ ⎥⎦ ⎢⎣ i =1 ⎥⎦ ⎥⎦ ⎣ 2 ⎤ ⎤ ⎤ 1 ⎡ 1 1− ρ 1 ⎡ 1 ⎡1 − ρ ⎤ 1 ⎡ 1 θ (1 − ρ ) hc ≡ ⎢ + − 2⎥ − 2 ⎥ , hc ≡ ⎢ + ⎢ − 2 ⎥ , h0 ≡ ⎢ + ⎥ θ θ θ 2 ⎣θ θ − ρ 2 1 2 θ ρ − − ⎣ ⎦ ⎢⎣ ⎥⎦ ⎣ ⎦ ⎦ 1 1 H d ≡ ⎡⎣ H d ,(0,1) + H d ,(1,1) ⎤⎦ , H d ≡ ⎡⎣ H d ,(0,1) + H d ,(1,1) ⎤⎦ 2 2. ∑. H0 ≡. ∑. ∑. Once the particular solution of δ t f is obtained, the empirical model can be established.. From equation (3.2.14), the real data of log dividend-price ratio can be approximated as:. (3.2.17). δ t ≅ θ t δ t f + dt (1 − θ t )(1 − λ ) − ηt. By setting δ t f = δ t f (1,3,4) , equation (3.2.17) can be manipulated as:. (3.2.18). {. }. δ t = Ξ 0 − {Ξ1 + (1 − θ )Ξ 0 +[ Λ − (1 − θ ) ln α ]} st + H c ct st [(1 − θ )(1 + h ) − hc ] − 1 +. ⎡⎣(1 − θ )(1 − λ ) − θ H d ⎤⎦ dt st + [1 − (1 − θ ) st ]ε t(1,3,4). Therefore, the empirical model can be established below:. (3.2.19). δ t = α 0 + α1st + α 2 ct [sin( st rad ) − 1] + α 3 dt st + ε t. where the symbol "rad" is represent that the value is measured in radian.. The coefficients are defined below: 23.

(29) α 0 ≡ Ξ 0 , α1 ≡ {Ξ1 + (1 − θ )Ξ 0 +[ Λ − (1 − θ ) ln α ]} , α 2 ≡ H c , α 3 ≡ ⎡⎣ (1 − θ )(1 − λ ) − θ H d ⎤⎦. The. value. of. st [(1 − θ )(1 + h ) − hc ] − 1. is. approximately. substituted. by. sin( st rad ) − 1 .. For simplicity, (3.2.19) is rewritten in matrix form:. (3.2.17). δ = Xα + ε. where δ ≡ d − p, α ≡ (α 0 , α1 , α 2 , α 3 )′, X ≡ (1, s, d cs ≡ c. [sin(s rad ) − 1] ,. s, c s )T ×4 ,. ε X ~ N (0, σ δ2 I). c ≡ ({ct }Tt =0 )′, d ≡ ({dt }Tt =0 )′, p ≡ ({ pt }Tt =0 )′ s ≡ ({st }Tt =0 )′. Detail estimation procedures will be discussed in next section.. 3.3 Estimation procedure Since state spaces for the unobserved state variables grow with the sample size, the joint distributions of sampling for maximum likelihood are difficult to establish under the classical techniques. By combining Bayesian method and Gibbs sampler, such problems can be avoided. However, the key advantage of utilizing Bayesian method is to update the state variables, because such methodology treats all unobserved elements of the model as parameters to be estimated. The lake of capability for updating state variables is primary consideration that this thesis does not use classical maximum likelihood estimates. Up to now, the parameters of interest are. 24.

(30) Ω = {g 0 , g ′, g s , α 0 , α1 , α 2 , α 3 , σ δ2 ,{st }Tt = 0 ,{St* }Tt =1 ,{ pr (0,0),t }Tt =1 ,{ pr (1,1),t }Tt =1}. The data set is denoted by Q = {{dt }Tt =0 ,{ pt }Tt =0 ,{ct }Tt =0 ,{Z t′}Tt =0 } . All the estimation parameters are based on the posterior distribution P(Ω Q) . Owing to that there are two different structures of model, namely, the regression model depicted by (3.2.19) and binomial probit model depicted by (3.2.12), it is arduous to acquire the explicit form of P(Ω Q) . To extract the information contained by P(Ω Q) , the Gibbs sampling can generate a parameters sequence {Ω( j ) }Nj=1 to converges the distribution P(Ω Q) by drawing the conditional posterior distribution. Given the data set, prior value of {st }Tt = 0. and the initial value of g0 , g ′ , g s , the initial value for { pr (0,0),t }Tt =1 and. { pr (1,1),t }Tt =1 can also be generate by the probit model. Utilizing these initial values. denoted by the set Ω(0) , the sequence {Ω( j ) }Nj=1 are generated by following steps:. 3.3.1 Step 1 : α, σ δ2. To construct the conditional posterior distribution for α and σ δ2 , the following joint density of the likelihood function given by the data and Ω(0) should be manipulated:. (3.3.1-1). L(α, σ δ2 δ, X, Ω (0) ) = [2πσ δ2 ]. −T 2. ⎡ −(δ − Xα )′(δ − Xα ) ⎤ exp ⎢ ⎥ 2σ δ2 ⎣⎢ ⎦⎥. By defining d ≡ T − K (the degrees of freedom parameter) and further defining a multiplier A , the multiplying L(α, σ δ2 δ, X, Ω(0) ) by A can be written:. (3.3.1-2). 25.

(31) L(α, σ δ2 δ, X, Ω(0) ) ∝ d. where. [υ sδ2 ]υ +1 1 υ −υ s 2 ( 2 ) exp( 2δ )[2π ] Γ(υ + 1) σ δ σδ. −K 2. σ δ2 ( X′X) −1. −1/ 2. ⎡ −1 ⎤ exp ⎢ (α − a)′[σ δ2 ( X′X) −1 ]−1 (α − a) ⎥ ⎣2 ⎦. +1. ⎛ d 2 ⎞2 d ⎜ 2 sδ ⎟ −1/ 2 ⎠ [2π ] 2 X′X A≡ ⎝ ,υ ⎛d ⎞ Γ ⎜ + 1⎟ ⎝2 ⎠. =. d 2. , sδ2 = (T − K )σ δ2 and a = ( X′X)−1 X′δ. Let the prior conditional distribution for α is given by g (α σ δ2 ) , such that: L(α, σ δ2 δ, X, Ω(0) ) = L(α, δ, σ δ2 , X, Ω(0) ) g (α σ δ2 ). Form Bayes theorem, conditional posterior density for α is. (3.3.1-3). p(α, δ, σ δ2 , X, Ω(0) ) =. L(α, σ δ2 δ, X, Ω(0) ). ∝ L(α, δ, σ δ2 , X, Ω(0) ) g (α σ δ2 ). ∞. ∫ p(α, δ σ δ , X, Ω 2. (0). )dσ δ2. 0. Because g (α σ δ2 ) ∝ a constant, the density function in (3.3.1-3) can be written:. (3.3.1-4) p (α, δ, σ δ2 , X, Ω(0) ) ∝ h(σ δ2 )[2π ]. where. h(σ δ2 ) =. −K 2. σ δ2 ( X′X) −1. −1/ 2. ⎡ −1 ⎤ exp ⎢ (α − a)′[σ δ2 ( X′X) −1 ]−1 (α − a) ⎥ ⎣2 ⎦. [υ sδ2 ]υ +1 1 υ −υ s 2 ( 2 ) exp( 2δ ) Γ(υ + 1) σ δ σδ. This give the fact that the conditional posterior density for α is proportional to a multivariate normal distribution with mean a and covariance matrix σ δ2 ( X′X)−1 . Hence p(α, δ, σ δ2 , X, Ω(0) ) = N [a, σ δ2 ( X′X)−1 ]. Now the joint posterior distribution for α and σ δ2 can be obtained by multiplying (3.3.1-4) the prior distribution for σ δ2 which denoted by g (σ δ2 ) .. 26.

(32) (3.3.1-5). p(α, σ δ2 δ, X, Ω(0) ) = L(α, δ, σ δ2 , X, Ω(0) ) g (σ δ2 ) = p(α, δ, σ δ2 , X, Ω(0) ) g (σ δ2 ). To obtain the conditional posterior density for σ δ2 , take integration on (3.3.1-5):. (3.3.1-6). p(σ δ2 , δ, α, X, Ω(0) ) =. ∫. p(α, δ, σ δ2 , X, Ω(0) ) g (σ δ2 )dα =. α. −υσˆ 2 [υσˆδ2 ]υ +1 1 υ ( 2 ) exp( 2 δ ) Γ(υ + 1) σ δ σδ. where σˆδ2 = (α − a)′(α − a)[T − K ]−1 d σˆδ2 d ), 2 2. Hence σ δ2 ~ IG ( ,. where IG (i) is inverse-gamma distribution.. d σˆδ2 d ) 2 2. Therefore, α and σ δ2 are drawn from N [a, σ δ2 ( X′X)−1 ] and IG ( , respectively:. The drawing procedure for α is α ( j ) = a + Choleski(X′X) × v. d 2. The drawing procedure for σ δ2 is σ δ2,( j ) = σˆδ2 [rnG ]−1. The letter v is denoted a matrix of standard normal random numbers and rnG d 1 2 2. is denoted a gamma random numbers drawn from G ( , ) .. 3.3.2 Step 2 : {st }Tt =0. Given the Markovian structure of state variable, it is obvious to write:. (3.3.2-1). Pr( St +1 St , St −1 ,..., St − k ) = Pr( St +1 St ). 27.

(33) The density function of the observation is written below:. (3.3.2-2). −1 ⎡ −(δ t − xt α ) 2 ⎤ f (δ t α, σ δ2 , st , δt / , St / , Z t , Ω (0) ) = [2πσ δ2 ] 2 exp ⎢ ⎥ 2σ δ2 ⎢⎣ ⎥⎦. where δt / = {δτ τ = 0,1,..., t − 1, t + 1,...T } and St / = {sτ τ = 0,1,..., t − 1, t + 1,...T }. For drawing st , the procedure provided by Albert and Chib(1993a) is considered. To incorporate time-varying transition probabilities, modifying their formulae should be executed. Further defining δt ≡ {δτ }τt =0 and S n ≡ {sτ }τn=0 , the full conditional smoothed probability can be calculated by below equation:. (3.3.2-3) t +r. ∏ f (δ. Pr( st δt + r , Zt , St / ) ∝ Pr( st st −1 , Zt −1 , Ω(0) ) Pr( st +1 st , Zt , Ω(0) ). k. α, σ δ2 , δk −1 , S k , Zt , Ω(0) ). k =t. The value of st can be simulated from a series of Bernoulli distributions by using (3.3.2-4):. st = {0,1 st = 0 ⇔ ru < Pr( st δt + r , Z t , St / ), st = 1 ⇔ ru > Pr( st δt + r , Z t , St / )}. ru is a number generated from uniform distribution U [0,1] . 3.3.3 Step 3 : g0 , g ′, g s. The likelihood function of the latent regression (3.2.12) given by the data is T. L(s Z, g, Ω(0) ) =. ∏[F ( g. 0. + g z′ Z t + g s st −1 )]st [1 − F ( g0 + g z′ Z t + g s st −1 )]1− sr. t =0. where s = ({st }Tt =0 )′ , Ζ = (1,{Zt }Tt =0 {st −1}Tt =1 )′ and g = ( g0 , g z′ , g s )′. The Gibbs sampler for probit model pioneered by Albert and Chib(1993b) is 28.

(34) adapted.. If St* is known, then the posterior density for g is similar to (3.3.1-4). From earlier results, it follows that. p(g, s, S* , Z, Ω(0) ) = N [ g * , (Z′Z)−1 ]. (3.3.3-1). where g * = (Z′Z)−1 Z′S* and S* = ({St*}Tt =0 )′. The posterior density for S* can also be written as:. (3.3.3-2). p[ St* st = 1, g, (1, Z t , st −1 )] = N + [ g 0 + g z′ Z t + g s st −1 ,1] p[ St* st = 0, g, (1, Z t , st −1 )] = N − [ g0 + g z′ Z t + g s st −1 ,1]. where N + [•] and N − [•] are truncated normal density with positive and negative values respectively.. The drawing procedure for St* is. (3.3.3-3) St*,( j ) = g 0( j −1) + g z′( j −1) Z t + g s( j −1) st −1 + F −1[1 − (1 − U ) F ( g0( j −1) + g z′( j −1) Z t + g s( j −1) st −1 )] ∀st = 1 St*,( j ) = g 0( j −1) + g z′( j −1) Z t + g s( j −1) st −1 + F −1[U × F (− g 0( j −1) − g z′( j −1) Z t − g s( j −1) st −1 )] ∀st = 0. where U is random number drawn from U [0,1]. The drawing procedure for g is. g ( j ) = g * + Choleski(Z ′Z ) × v. 29.

(35) After running these steps for the desired frequency of iterations, the point estimate for posterior mean of coefficients and its standard errors are. αˆ =. 1 N − burn. N. ∑. α( j). j =burn +1 N. std [αˆ ] =. gˆ =. ∑. 1 [α ( j ) − αˆ ]2 N − burn − 1 j =burn +1. 1 N − burn. N. ∑. g( j ). j = burn +1 N. std [gˆ ] =. ∑. 1 [g ( j ) − gˆ ]2 N − burn − 1 j =burn +1. The point estimate of the variance is: σˆδ2 =. 1 N − burn. N. ∑. σ δ2,( j ). j =burn +1. where burn is denoted the size of burn in sample. 30.

(36) 4. Empirical results 4.1 Data description and choice of the prior state variable Monthly samples from U.S are applied in this thesis and sample period runs from February 1960 to April 2010. Monthly observations of Standard and Poors 500 Index (S&P500) and associated real dividends were collected from the Global Financial Data (GFD) base. The M1 money stock, disposable personal income, personal consumption expenditures and one month LIBOR rate also obtained from GFD base. Before using these data, removing the trend is necessary especially when variables markedly exhibit a time trend. Since M1 money stock and personal incomes always increases over time; this thesis considers removing the trend on these two variables. Due to that time series may have a polynomial trend, in this thesis, the degree of polynomial is determined by standard t-test for every individual coefficient. That is, the polynomial will be added until there at least one coefficient is insignificant. After few trying, the appropriate polynomial for M1 money stock and disposable personal income are 6 and 7 respectively:. M1. DT6M1. 3600. .16. 3200. .12. 2800 .08. 2400 2000. .04. 1600. .00. 1200. -.04. 800 -.08. 400 0 1960M02 1970M02 1980M02 1990M02 2000M02 2010M02. -.12 1960M02 1970M02 1980M02 1990M02 2000M02 2010M02. Figure 4-1-2 M1 money stock in level and in detrending M1 is level M1 money stock and DT6M1 is detrending M1 for 5 degree polynomial. 31.

(37) YY. DT7Y. 24000. .12. 20000. .08. 16000. .04. 12000. .00. 8000. -.04. 4000. -.08. 0 1960M02 1970M02 1980M02 1990M02 2000M02 2010M02. -.12 1960M02 1970M02 1980M02 1990M02 2000M02 2010M02. Figure 4-1-3 disposable personal income in level and in detrending YY is level disposable personal income and DT7Y is detrending YY for 5 degree polynomial. Finally, the prior state variable (prior belief for bubbles) is obtained by residuals from fundamental pricing model. From previous discussion, the fundamental pricing model can be rearranged as:. (4.1.1). pt − dt = β 0 + β1 ln Ct + ε f ,t. Estimate result for this regression model is: ∧. (4.1.2). pt − dt = −0.7356 + 0.6957 ln Ct (0.1279) (0.0166). If the residual is greater than zero, then state variables equal to one which represent existence of bubbles. Otherwise, state variables equal to zero which represent nonexistence of bubbles.. 32.

(38) 4.2 Estimation outcomes and their implication The estimation outcomes reported in this section are based on 50000 passes of the Gibbs sampler. To avoid the effect of the choice of starting values, the first 2000 observations were abandoned, remaining the 48000 observations to assess the posterior moments.. Up to now, the econometric models are regression model depicted by (3.2.19) and binomial probit model depicted by (3.2.12). For clearly understandable, the two models are rewritten below:. The regression model:. i.i.d. δ t = α 0 + α1st + α 2 ct [sin( st rad ) − 1] + α 3 dt st + ε t , ε t ~ N (0, σ δ2 ). where st is simulated by equation (3.3.2-3) and (3.3.2-4).. The Probit model:. St = St ( St* ), St* ∈ ( −∞, +∞ ) St = {st st = 1 ⇔ St* ≥ 0, st = 0 ⇔ St* < 0} i.i .d. St* = g 0 + g1mt + g 2 yt + g3it + g s st −1 + ut , ut ~ N (0,1) Pr( St* > 0) = Pr[ut > − g 0 − g1mt − g 2 yt − g3it − g s st −1 ] = 1 − F (− g 0 − g1mt − g 2 yt − g3it − g s st −1 ). The implication of the regression model is intuitive. Since equation (3.2.19) contains the state of bubbles crash, α1 could be positive. Coefficient α 3 represents the bubbles effect and it should be negative. If bubbles do exist, from the equation of intrinsic bubbles, additional dividends payout will lead bubbles become larger. Hence. 33.

(39) the effect of increasing in stock prices will dominate the effect of increasing in dividends on dependent variable. The sign of coefficient α 2 can be positive or negative. If α 2 is positive, the relative risk aversion level of next period is less than current period. In this case, note that the term is sin( st rad ) − 1 negative; investors will put more money on buying stock. When the market is in equilibrium state, the consumption level will equate dividends payout. Hence the effect of increasing in stock prices will dominate the effect of increasing in dividends payout on dependent variable.. Finally, the implication of probit model is associated with the financial instability hypothesis. The financial instability hypothesis suggested that the degree of credit amplification will ascend when economy is in the prosperity for lengthy periods. Therefore, the coefficient g1 and g 2 should be positive while the coefficient g3 should be negative. Also, the coefficient g s should be positive to reflect the fact that when past state was in the bubbles regime, the larger likelihood of the investor anticipating the current state is in the bubbles regime.. The estimation outcomes for regression model are reported below. The first part outcome will report the posterior moments, and second part will test whether the regression model is adequate to depict the cointegration relationship. Engle-Granger methodology is implemented in the second part.. 34.

(40) Table 1 Parameters of cointegration equation dt − pt = α 0 + α1st + α 2 ct [sin( st rad ) − 1] + α 3 dt st + ε t. Parameter. Mean. Standard errors. α0. 2.88 -5.68 -2.09 0.88. 0.26 0.24 0.15 0.03. α1 α2 α3. Results from table 1 shows that all estimated posterior means are significant at 5% level. The posterior mean of α1 and α 2 are negative and it indicates that the effect of bubble component does raise the stock price. The posterior mean of α 3 is positive and it indicates that the relative risk aversion level of next period is less than current period. Therefore, the effect of increasing in stock price due to existence of bubbles will dominate the effect of increasing in dividends payout on dependent variable. For the sake of investigating whether these variables are cointergated, Engle-Granger methodology is implemented. Results are shown in table 2. Table 2 ADF-unit root test for residuals εˆtlinear = dt − pt − αˆ 0 + αˆ3ct. ε tbubbles = dt − pt − αˆ 0 − αˆ1st − αˆ 2 ct [sin( st rad ) − 1] − αˆ3 dt st. t-Statisti. Prob. t-Statisti. Prob. -1.69. 0.44. -4.75. <0.01. Table 2 shows that the fundamental cointegration vector is rejected and it is coincident with the result from table 1. The posterior means of all bubbles components are significant at 5% level. Also, the residuals from bubbles model are stationary in level. This indicates that the bubbles can arise after complete bursting.. 35.

(41) Table 3 Parameters of transition probability equation St* = g 0 + g1mt + g 2 yt + g3it + g s st −1 + ut. Parameter. Mean. Standard errors. g0. -2.51. 0.44. g1. -2.05. 2.88. g2. -1.76. 4.61. g3. 0.12. 4.69. gs. 4.90. 0.36. From table 3, the posterior means of g z all insignificant has suggested that the role of the money market does not have momentous impact on bubbles. However, as the posterior mean of g s is significant, the self-confirming belief of bubbles does have momentous impact on bubbles.. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1960M02 1964M02 1968M02 1972M02 1976M02 1980M02 1984M02 1988M02 1992M02 1996M02 2000M02 2004M02 2008M02. smooth probability(no bubbles). February 1960 to April 2010 Figure 1 the smooth probability of no bubbles Figure 1 strongly indicates that the bubbles do arise after complete crash. The overall period seems to exhibit extremely switching between the stable regime and the unstable regime. 36.

(42) 1. 0.995. 0.99. 0.985. 0.98. 0.975. 0.97 1960M02 1964M02 1968M02 1972M02 1976M02 1980M02 1984M02 1988M02 1992M02 1996M02 2000M02 2004M02 2008M02. pr(0,0). pr(1,1). February 1960 to April 2010 Figure 2 the transition probability of staying in same regime Figure 2 shows that the economy either stays in the stable regime or unstable one and the probability of economy stating in the bubbles regime may dominate the probability of economy stating in the stable regime. The transition probability extremely higher has a policy implication that central bank should not change the monetary policy. The main achievement is to keep the interest rate and quantity of money level to remain the target level which maintains the economy always staying in stable regime.. 37.

(43) 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 1960M02 1964M02 1968M02 1972M02 1976M02 1980M02 1984M02 1988M02 1992M02 1996M02 2000M02 2004M02 2008M02. pr(1,0). pr(0,1). February 1960 to April 2010 Figure 3 the transition probability of switching to alternative regime As the figure 3 shows that the probability of switching to alternative regime is vary small, the policymaker would not worry that the stock market can switch to alternative regime sensitively with changing in monetary policies. Such results should be more deliberately examining because the probability of bubbles crash is always dominate the probability of bubbles arising.. 38.

(44) 5. Conclusions The financial instability hypothesis proposed by Minsky (1992) suggested that the fluctuation in the economy may be resulted from the instability of financial market and such instability could be triggered without exogenous disturbances. As the money market dominates the increases and decreases in stock market funds and the impact of credit amplification on the future expectation of the economy, the money market may be capable of dominating the expectation of bubbles in stock market if the economy system is characterized sufficiently by the financial instability hypothesis. Due to that the rational expectation hypothesis is unable to illustrate endogenous fluctuations in the economy and the reoccurrence of bubbles after complete collapse, the goal of this thesis is to examine whether the belief of repeated crash and arise on bubbles is dominated by the money market following the structure the financial instability hypothesis.. Therefore, the primary errand of this thesis is to address the theme of accurate representing the self-fulfilling prophecy of bubbles. The empirical result can display the dynamic process of beliefs of repeat crash and arise on bubbles and show how money market does play a crucial role to dominate these beliefs.. From estimation outcomes, although the role of money market does not have momentous impact on bubbles, the bubbles do exhibit repeated crashes. Hence the policy marker should pay more attention to concern that such repeated crash bubbles will be impacted by policy change.. 39.

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