Asset Pricing with Disequilibrium Price Adjustment: Theory and Empirical Evidence

1
Asset Pricing with Disequilibrium Price
Adjustment Theory and Empirical Evidence

• Dr. Cheng Few Lee
• Distinguished Professor of Finance
• Rutgers, The State University of New
  JerseyEditor of Review of Quantitative Finance
  and Accounting
• Editor of Review of Pacific Basin Financial
  Markets and Policies

2
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Another way to analyze the demand and supply
  functions derived in the previous section is to
  reexamine equations (9 and 12), considering the
  possibilities of market in disequilibrium.
  Disequilibrium models have a very long history.
  All the partial adjustment models are in fact
  disequilibrium models. Much of the literature
  concerning the structure of disequilibrium
  markets focus on the commercial loan market and
  the labor market. For the commercial loan
  market, the structure of disequilibrium is
  frequently due to the governments credit
  rationing for economic policies. For the labor
  market the structural disequilibrium is
  frequently due to a rigid wage. The theory of
  credit rationing is first developed by Jaffee
  (1971) for a commercial loan market. One of the
  reasons for credit rationing is the existence of
  bankruptcy costs, as proposed by Miller (1977).
  Given that bankruptcy costs rise when firms fail,
  banks thus choose a lower amount of loan
  offerings than they would have if there were no
  bankruptcy costs. As a result, some firms will
  not receive the loan regardless of the rate they
  are willing to pay.
• In this section, we discuss and
  develop a model and methodology similar to these
  is sues regarding commercial loan markets. Early
  studies of the disequilibrium model of commercial
  loan markets include Jaffee (1971), Maddala and
  Nelson (1974) and Sealey (1979). One recent
  follow up study is Nehls and Schmidt (2003). They
  use the disequilibrium methodology similar to
  Sealy to evaluate whether loans are constrained
  by demand or supply. In fact, one can see the
  disequilibrium model as a special case of
  simultaneous equation models. Thus, here a
  similar demand and supply schedule is derived but
  solved simultaneously to reexamine the price
  adjustment behavior by assuming that market is in
  disequilibrium.

3
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• All disequilibrium models share the feature
  that prices do not fully adjust to the market
  clearing level. The model used throughout this
  section is a basic model first proposed by Fair
  and Jaffee (1972) and Amemiya (1974) and modified
  as model C in Quandt (1988). This model consists
  of the following equations5
• (13) QDt a1 Pt ß1X1t µt,
  
• (14) QSt a2 Pt ß2X2t ?t ,
  
• (15) Qt min (QDt, QSt ),
  
• (16) ?Pt Pt - Pt-1 ?(QDt - QSt
  ),

4
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• where the QDt and QSt are the quantities of
  securities demanded and supplied, respectively
  Qt is the actual or observed quantity of
  securities in the market Pt is the observed
  return rate of securities X1t and X2t are
  vectors of exogenous or predetermined variables
  including the lagged Pt-1 a1 and a2 are unknown
  parameters for Pt ß1 and ß2 are vectors of
  unknown parameters for exogenous variables ? is
  an unknown positive scalar parameter and µt and
  ?t are disturbance terms and assumed to be
  jointly normal and independent over time with
  distributions N(0, s2µ) and N(0, s2?)
  respectively. The difficulty comes in estimating
  a1, a2, ß1, ß2, ?, s2µ, and s2? with observations
  of X1t, X2t, Qt and Pt for t 1, 2, , T.

5
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Some assumptions should to be made to deal
  with the relationships between Qt, QDt QSt and
  the price adjustment process. A basic assumption
  is reflected in equation (15), which shows that
  when demand exceeds supply, the observed quantity
  lies on the supply schedule, and the market is
  characterized by the conditions of excess demand.
  This assumption is often referred to as voluntary
  exchange. That is, in the presence of excess
  demand, seller cannot be forced supply more than
  they wish to supply and in the presence of
  excess supply, purchasers cannot be forced to buy
  more than they wish to buy. Another assumption in
  this model is that the price adjustment is
  proportional to excess demand, which is shown by
  the last equation (16) in the above system. The
  model is also assumed to be identified by
  different sets of exogenous variables (i.e., X1t
  and X2t.)
• Clearly, the equation system, equations (13) to
  (16), is a special case of simultaneous equation
  models. If there is no equation (15) and if the
  system is identified, one can consistently
  estimate a1, a2, ß1, ß2, ?, s2µ, and s2? by using
  methodologies of simultaneous equation. Since we
  have equation (15) therefore we need to introduce
  equation (16) into the system for estimation of
  a1, a2, ß1, ß2, ?, s2µ, and s2?. However, one
  primary problem exists in this disequilibrium
  model, which is that QDt and QSt are not
  observable variables in the absence of the market
  clearing condition.

6
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• One can reformulate the above model by
  considering periods of rising prices ?Pt 0 and
  periods of falling prices ?Pt in the period with rising prices, the supply
  function (14) can be estimated using the observed
  quantity, Qt, as the dependent variable since
  there will be excess demand and thus Qt will
  equal to QSt. The details of constructing the
  econometric procedures is discussed in next
  section.
• The last topic of this section is to
  incorporate the demand and supply schedules
  developed in the previous section into this
  disequilibrium equation system. The demand and
  supply schedule in equations (9) and (12) can be
  restated and presented as equations (17) and (18)
  as part of the disequilibrium system as6

7
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• (17)
• QDt1 cS-1 EtPt1 - cS-1(1 r)Pt
  cS-1EtDt1 µ1t
• (18)
• QSt1 QSt A-1B Pt - A-1Et Dt1 µ2t
  
• (19)
• Qt1 min (QDt1, QSt1 )
  
• (20)
• ?Pt ?(QDt1 - QSt1 ).

8
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• From the above equation system, it is clear
  that some conditions in equation (17) and (18)
  are different from the basic disequilibrium
  equation system, particularly QSt in the supply
  schedule. These problems are dealt with, before
  the empirical studies, by imposing more
  assumptions or by using alternative
  specifications in econometric methodologies in
  the next section.

9
IV. ALTERNATIVE METHODS OF ESTIMATING ASSET
PRICING MODEL WITH DISEQUILIBRIUM EFFECT

• In this section, we first reformulate
  the disequilibrium asset pricing model to allow
  for empirical study. Then we discuss the
  alternative methods of estimating and testing
  price adjustment process in capital asset
  pricing.
• A. Reformulation of the Disequilibrium Model
• B. Estimation Methods and Hypothesis of Testing
  Price Adjustment Process
• 1. 2SLS Estimator
• 2. Maximum Likelihood Estimator
• 3. Testing of the Price Adjustment Process

10
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• A. Reformulation of the Disequilibrium Model
• To estimate a1, a2, ß1, ß2, ?, s2µ, and s2? with
  observations of X1t, X2t, Qt and Pt for t 1, 2,
  , T in equations (13), (14), (15), and (16). It
  is clear that the ordinary least squares will
  produce inconsistent estimators. Following
  Amemiya (1974) and Quandt (1988), we discuss two
  estimation methods to obtain consistent
  estimators.. The first method is the two stage
  least square (2SLS) estimator, and the other is
  the maximum likelihood estimator (MLE).

11
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Before constructing the estimating procedures,
  the model can be reformulated by considering as
  two different periods, the period of rising
  prices in which ?Pt 0 and the period of falling
  prices, ?Pt prices, there exists excess demand, QDt ? QSt, so
  the quantity observed equals the supply (i.e., Qt
  QSt). As a result, the supply schedule in
  equation (14) now can be estimated by using the
  observed quantity, Qt, as the dependent variable.
  Equation (16) becomes ?Pt ? (QDt - Qt ). or
  QDt ?Pt/? Qt (?Pt is greater than zero here).
  That is, the demand schedule can be rewritten as
• (21)

12
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Similarly, for periods of falling prices, there
  exists excess supply, and the supply schedule can
  be rewritten as
• (22)
• Or, the system of equations determining the
  endogenous variables can be summarized as
• when QDt ? QSt,
• (23a)
• (23b) Qt a 2 Pt ß2X2t ?t.

13
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Or when QDt
• (24a) Qt a1 Pt ß1X1t µt,
• (24b)

14
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• By defining the following artificial variables,
  the equations (23) and (24) can be summarized as
• (25a)
• (25b)
• where

15
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• B. Estimation Methods and Hypothesis of Testing
  Price Adjustment Process
• In this section, we first discuss two
  alternative methods (2SLS and MLE) for estimating
  disequilibrium model, then the null hypothesis of
  testing price adjustment process is developed.
• 1. 2SLS Estimator
• The equations system shown in (25) contains
  the jointly dependent variables Qt, Pt, , and .
  The parameters in the modified model seem can be
  consistently estimated by conventional two-stage
  least squares (2SLS) method. This can be briefly
  described as the following two stages. In the
  first stage, regress and on
  all the exogenous variables, X1t and X2t to
  obtain the estimations of and
  , then, in the second stage, regress Qt on X1t
  and in equation (25a), and regress Qt on
  X2t and in equation (25b). However, the
  estimators of , , and
  are not asymptotically efficient in this model,
  though could be consistent if the predictions of
  the endogenous variables are used for all
  observations7. The reasons are, first, there is
  no imposed restriction to force the same ? to
  appear in both equations and, second,
  and are not, strictly speaking, linear
  functions of the X1t and X2t.

16
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• 2. Maximum Likelihood Estimator
• To estimate the parameters of a model as (20),
  Quandt (1988) employs an appropriately formulated
  full-information maximum likelihood technique.
  Let the joint density of the endogenous variables
  Qt and Pt be denoted by ?t (Qt, Pt Xt ) where Xt
  is a vector of the exogenous variables in the
  model.
• The joint density function ?t can be derived from
  the joint density of the structure disturbances
  ut (µt, ?t). By assuming that the distribution
  of disturbance terms is joint normal, i.i.d.
  distributed with N (0, O), the density ?t becomes

17
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• (26)
• where the J is the Jacobian determinant
  of the transformation from the disturbances, ut,
  to (Q, Pt ) O is the covariance matrix of the
  structural disturbances and ut is the vector of
  disturbances (µt, ?t). To complete the joint
  density ?t, the Jacobian of the transformation
  must be evaluated separately for?Pt 0and ?Pt 0. In either case, the absolute value of the
  Jacobian is a2a11/?. Therefore, equation (26)
  implied that the log likelihood function is
• (27)
• where

18
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Alternatively, there is another way to derive the
  likelihood function. Amemiya (1974) shows the
  following iterative method of obtaining the
  maximum likelihood estimator.8 Since in period A
  when QDt QSt, the conditional density of ?Pt
  given Qt is N ?(Qt -a2Pt -ß2X2t ), ?2s2?, and
  in period B when QDt density of ?Pt given Qt is N ?(a1Pt ß1X1t
  -Qt), ?2s2µ, then, the log likelihood function
  is
• (28)

19
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• Thus, the maximum likelihood estimators can be
  obtained by solving the following equations
  simultaneously
• (29)
• (a)
• (b)
• (c)
• (d)
• (e)

20
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• That is, the ML estimators of a and ß are the
  same as LS estimators given ? applied to equation
  (25a) and (25b), respectively. The equations for
  s2µ and s2? (equations c and d) are the residual
  sums of squares of equations (25a) and (25b),
  given ?, divided by T as for the usual ML
  estimators. Equation (d) is a quadratic function
  in ?. Amemiya (1974) suggests the above
  parameters can be solved by using the following
  iterative procedures
• Step 1 Use 2LSL estimates of a, ß, s2µ and s2?
  as the initial estimates.
• Step 2 Substitute , , and
  into (e) and solve for the positive root of ?,
  .
• Step 3 Use in (25) to obtain least squares
  estimates of a, ß, s2µ and s2?.
• The iteration repeats step 2 and 3 until the
  solutions converge.

21
III. DEVELOPMENT OF DISEQUILIBRIUM MODEL
FOR ASSET PRICING

• 3. Testing of the Price Adjustment Process
• Comparing with equilibrium model, the parameter
  of most interest in the disequilibrium model is
  the market adjustment parameter, ?. In the case
  of continuous time, the limiting values of ? are
  zero and infinity. If ? 0, then there is no
  price adjustment in response to an excess demand,
  and if ? is infinity, it indicates instantaneous
  adjustment. In other words, if one assumes there
  is price rigidity in response to an excess
  demand, then the value of ? should be equal to
  zero. That is, the most important test of the
  disequilibrium model is to test the hypothesis
  that the price adjustment parameter is zero. The
  null hypothesis can be stated as if there is no
  price adjustment mechanism in response to an
  excess demand, the value of ? will be equal to
  zero. Or, can be stated as
• H0 ? 0 vs. H1 ? ? 0.
• This hypothesis will be empirically tested in the
  following section.

22
IV. ALTERNATIVE METHODS OF ESTIMATING ASSET
PRICING MODEL WITH DISEQUILIBRIUM EFFECT

• Now that we have our disequilibrium asset
  pricing model for empirical study, we test for
  the price adjustment mechanism by examining the
  market adjustment parameter, ?, as stated in the
  previous section. First in this section, we
  describe our empirical data.
• A. Data Description
• 1. International Equity Markets Country
  Indices
• 2. United States Equity Markets
• B. Testing the Existence of the Price Adjustment
  Process

23
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• A. Data Description
• Our data consists of two different types of
  markets--the international equity market and the
  U.S. domestic stock market, which we examine here
  in terms of summary return statistics and key
  profitability financial ratios. In addition, we
  also analyze 30 firms of the Dow Jones Index.
  Most details of the model, the methodologies, and
  the hypotheses for empirical tests are discussed
  in previous sections. We first examine
  international asset pricing by looking at summary
  statistics for our international country indices,
  and then we look at our data for the U.S.
  domestic stock market with portfolios formed from
  the SP 500 and also the 30 companies used to
  compile the Dow Jones Industrial Average.

24
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• 1. International Equity Markets Country Indices
• The data come from two different sources. One is
  the Global Financial Data (GFD) from the
  databases of Rutgers Libraries, and the second
  set of dataset is the MSCI (Morgan Stanley
  Capital International, Inc.) equity indices.
  Mainly we focus on the Global Financial Data,
  with the MSCI indices used for some comparisons.
  We use both datasets to perform the
  Granger-causality test. The monthly (GFD) dataset
  for February 1988 to March 2004 consists of the
  index, dividend yield, price earnings ratio, and
  capitalization for each equity market. Sixteen
  country indices and two world indices are used to
  do the empirical study, as listed in Table 1. For
  all country indices, dividends and earnings are
  converted into U.S. dollar denominations. The
  exchange rate data also comes from Global
  Financial Data.
• In Table 2, Panel A shows the first four moments
  of monthly returns and the Jarque-Berra
  statistics for testing normality for the two
  world indices and the seven indices of
  G7countries, and Panel B provides the same
  summary information for the indices of nine
  emerging markets. As can be seen in the mean and
  standard deviation of the monthly returns, the
  emerging markets tend to be more volatile than
  developed markets though they may yield
  opportunity of higher return. The average of
  monthly variance of return in emerging markets is
  0.166, while the average of monthly variance of
  return in developed countries is 0.042.

25
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• 2. United States Equity Markets
• Three hundred companies are selected from the SP
  500 and grouped into ten portfolios by their
  payout ratios, with equal numbers of thirty
  companies in each portfolio. The data are
  obtained from the COMTUSTAT North America
  industrial quarterly data. The data starts from
  the first quarter of 1981 to the last quarter of
  2002. The companies selected satisfy the
  following two criteria. First, the company
  appears on the SP500 at some time period during
  1981 through 2002. Second, the company must have
  complete data available--including price,
  dividend, earnings per share and shares
  outstanding--during the 88 quarters (22 years).
  Firms are eliminated from the sample list if
  either .their reported earnings are either
  trivial or negative or their reported dividends
  are trivial.
• Three hundred fourteen firms remain after these
  adjustments. Finally excluding those seven
  companies with highest and lowest average payout
  ratio, the remaining 300 firms are grouped into
  ten portfolios by the payout ratio. Each
  portfolio contains 30 companies. Figure 1 shows
  the comparison of SP 500 index and the
  value-weighted index (M) of the 300 firms
  selected. Figure 1 shows that the trend is
  similar to each other before the 3rd quarter of
  1999. However, the two follow noticeable
  different paths after the 3rd quarter of 1999.

26
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• To group these 300 firms, the payout ratio for
  each firm in each year is determined by dividing
  the sum of four quarters dividends by the sum of
  four quarters earnings then, the yearly ratios
  are further averaged over the 22-year period. The
  first 30 firms with highest payout ratio
  comprises portfolio one, and so on. Then, the
  value-weighted average of the price, dividend,
  and earnings of each portfolio are also computed.
  Characteristics and summary statistics of these
  10 portfolios are presented in Table 3 and Table
  4, respectively. Table 3 presents information of
  the return, payout ratio, size, and beta for the
  10 portfolios. There appears to exist some
  inverse relationship between mean return and
  payout ratio. However, the relationship between
  payout ratio and beta is not so clear. This
  finding is similar to that of Fama and French
  (1992).
• Table 4 shows the first four moments of quarterly
  returns of the market portfolio and 10
  portfolios. The coefficients of skewness,
  kurtosis, and Jarque-Berra statistics show that
  one can not reject the hypothesis that log return
  of most portfolios is normal. The kurtosis
  statistics for most sample portfolios are close
  to three, which indicates that heavy tails is not
  an issue. Additionally, Jarque-Berra coefficients
  illustrate that the hypotheses of Gaussian
  distribution for most portfolios are not
  rejected. It seems to be unnecessary to consider
  the problem of heteroskedasticity in estimating
  domestic stock market if the quarterly data are
  used.
• Finally, we use quarterly data of thirty
  Dow-Jones companies to test the existence of
  disequilibrium adjustment process for asset
  pricing. The sample period of this set of data
  is from first quarter of 1981 to fourth quarter
  of 2002.

27
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• B. Testing the Existence of the Price Adjustment
  Process
• Another way to evaluate the demand and supply
  schedules derived in Section II is to reexamine
  these equations, considering the possibilities of
  market in disequilibrium. In fact, one can see
  the disequilibrium model as a special case of
  simultaneous equation models. In this section, a
  similar demand and supply schedule will be
  derived individually but solved simultaneously to
  reexamine the price adjustment behavior by
  assuming that market is in disequilibrium. Recall
  that the disequilibrium model derived in Section
  III can be represented as

28
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• (17)
• QDt1 cS-1 EtPt1 - cS-1(1 r)Pt cS-1EtDt1
  µ1t,
• (18) QSt1 QSt A-1B Pt - A-1Et Dt1 µ2t,
  
• (19) Qt1 min (QDt1, QSt1 ),
  
• (20) ?Pt ?(QDt1 - QSt1 ).

29
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• In the above system, one need to estimate the
  coefficients of EtPt1, Pt and EtDt1 in demand
  schedule (i.e., a1 and ß1), the coefficients of
  QSt, Pt and EtDt1 in the supply schedule (a2,
  and ß2) and the coefficient of excess demand, ?,
  with the observations on EtPt1, Pt, EtDt1 and
  QSt for t 1, 2, , T. µt and ?t are disturbance
  terms and assumed to be jointly normal and
  independent over time with distributions N(0,
  s2µ) and N(0, s2?), respectively. It is clear
  that some terms in equation (17) and (18) are
  different from the basic disequilibrium equation
  system proposed. Some assumptions or adjustments
  are imposed in empirical studies. Please refer to
  Appendix A1 for details. The quantity
  observation, Qt, for each index comes from the
  capitalization data. Since the capitalization for
  each index is denominated in U.S. dollars, we
  first divide the capitalization by its own
  countrys index and then adjust by the currency
  rate with U.S. dollar.
• The maximum likelihood estimators are computed
  from the derivation in Section IV. First, use the
  2SLS approach to find the initial values for the
  estimates, and then the maximum likelihood
  estimate is obtained from the calculation of the
  log likelihood function described as equation
  (27).

30
V. DATA AND TESTING THE EXISTING OF PRICE
ADJUSTMENT PROCESS

• The results of sixteen country indexes are
  summarized in Table 5. Fifteen out of sixteen,
  the maximum likelihood estimates of ? are
  significant different from zero at the 1
  significance level.9 The results in terms of 10
  portfolios are summarized in Table 6. There are
  six portfolios, including market portfolio, with
  a maximum likelihood estimates of ? statistically
  significantly different from zero. For example,
  for portfolios 1, 2, 4, and 7, ? is significantly
  different from zero at the 1 significance level.
  Portfolio 5 and the market portfolio are
  significance level of 5 , and portfolio 10 is
  significant at a 10 level. We cannot reject the
  null hypothesis that ? equals to zero for three
  portfolios -- 3, 6, 8, and 9. The results imply
  some but less than complete price adjustment
  during each quarter in the U.S. stock markets.
• Table 7 shows the results of thirty companies
  listed in the Dow Jones Index. The price
  adjustment factor is significantly different from
  zero at the 5 level in twenty-two companies out
  of twenty-eight companies. On average, an
  individual company has a higher estimated value
  of ? than the individual portfolio and individual
  portfolio has a higher value than market
  portfolio. For example, IBMs ? is 0.0308, which
  indicates that an excess demand of 32.47 million
  shares is required to cause a change in the price
  of 1 dollar, whereas 476 million shares is
  required to cause one unit price change for
  market portfolio since its ? is only 0.0021.10
• We use four datasets (two international indexes
  and two US equity) to test the existence of the
  disequilibrium adjustment process in terms of the
  disequilibrium model defined in equations 13
  through16. We find that there exists a
  disequilibrium adjustment process for
  international indexes, ten portfolios from SP
  500, and thirty companies of Dow Jones index.
  These results imply that asset pricing with
  disequilibrium price adjustment maybe important
  for investigating asset pricing in security
  analysis and portfolio management.

31
VI. SUMMARY

• In this paper, we first theoretically review
  and extend Blacks CAPM to allow for a price
  adjustment process. Next, we derive the
  disequilibrium model for asset pricing in terms
  of the disequilibrium model developed by Fair and
  Jaffe (1972), Amemiya (1974), Quandt (1988), and
  others. MLE and 2SLS are our two methods of
  estimating our asset pricing model with
  disequilibrium price adjustment effect. Using
  three data sets of price per share, dividend per
  share and volume data, we test the existence of
  price disequilibrium adjustment process with
  international index data, US equity data, and the
  thirty firms of the Dow Jones Index. We find
  that there exist disequilibrium price adjustment
  process. Our results support Lo and Wangs
  (2000) findings that trading volume is one of the
  important factors in determining capital asset
  pricing.

32
Table 1. World Indices and Country Indices List

• I. World Indices

II. Country Indices
33
Table 1. (Cont.)

• II. Country Indices

34
Table 2 Summary Statistics of Monthly Return1, 2

• Panel A G7 and World Indices

35
Table 2 Summary Statistics of Monthly Return1, 2

• Panel B Emerging Markets

Notes 1 The monthly returns from Feb. 1988 to
March 2004 for international markets. 2 and
denote statistical significance at the 5 and
1, respectively.
36
Table 3 Characteristics of Ten Portfolios
Notes 1The first 30 firms with highest payout
ratio comprises portfolio one, and so on. 2The
price, dividend and earnings of each portfolio
are computed by value-weighted of the 30 firms
included in the same category. 3The payout ratio
for each firm in each year is found by dividing
the sum of four quarters dividends by the sum of
four quarters earnings, then, the yearly ratios
are then computed from the quarterly data over
the 22-year period.
37
Table 4. Summary Statistics of Portfolio
Quarterly Returns1
Notes 1Quarterly returns from 1981Q1to 2002Q4
are calculated. 2 and denote
statistical significance at the 5 and 1 level,
respectively.
38
Table 5. Price Adjustment Factor, ?, for 16
International Indexes 1, 2

• Notes
• 1 Sample periods used are other than 19882 to
  20043
• 2 Null hypothesis if there is no price
  adjustment mechanism in response to an excess
  demand, the value of ? will be equal to zero.
  (H0 ? 0).
• 3 , and denote statistical
  significance at the 10, 5, and 1 level,
  respectively.

39
Table 6. Price Adjustment Factor, ?, for 10
Portfolios from SP 500 1

• Notes 1 Null hypothesis if there is no price
  adjustment mechanism in response to an excess
  demand, the value of ? will be equal to zero.
  (H0 ? 0).
• 2 , and denote statistical
  significance at the 10, 5, and 1 level,
  respectively.

40
Table 7. Price Adjustment Factor, ? , Dow Jones
30 1, 2
41
Table 7. (Cont.)

• Notes 1 Null hypothesis if there is no price
  adjustment mechanism in response to an excess
  demand, the value of ? will be equal to zero.
  (H0 ? 0).
• 2 Microsoft and Intel are not in the list
  since their dividends paid are trivial during the
  period analyzed here.
• 3 , and denote statistical
  significance at the 10, 5 and 1 level,
  respectively.

42
Figure 1Comparison of SP500 and Market portfolio
43
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• A.1 The Disequilibrium Equation System
• A.2 How to Solve MLE Estimates in the
  Disequilibrium Model
• A.3 Estimation of the Disequilibrium Model

44
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• In this appendix we will describe the procedure
  for estimating the disequilibrium adjustment
  parameter, ?. First we define the equation
  system, then we will discuss how to solve MLE
  estimate of the disequilibrium model. Finally,
  the procedure of Estimating the disequilibrium
  model is discussed in detail.
• A.1 The Disequilibrium Equation System
• The last topic of this section is to
  incorporate the demand and supply schedules
  developed in the previous section into this
  disequilibrium equation system. The demand and
  supply schedule in equations (9) and (12) can be
  restated as

45
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• (17)
• QDt1 cS-1 EtPt1 - cS-1(1 r)Pt
  cS-1EtDt1 µ1t
• (18) QSt1 QSt A-1B Pt - A-1Et Dt1 µ2t
  
• (19) Qt1 min (QDt1, QSt1 )
  
• (20) ?Pt ?(QDt1 - QSt1 ).

46
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• From the above equation system, it is clear that
  some conditions in equation (17) and (18) are
  different from the basic disequilibrium equation
  system (e.g., QSt in the supply schedule and the
  expectation term of price in the demand
  function). These problems will be dealt with,
  before the empirical studies, by imposing some
  more assumptions or by using alternative
  specification in econometric methodologies in the
  next section.
• Since the purpose of this study is to understand
  the price adjustment in response to an excess
  demand therefore, all expectation terms in the
  above equations are replaced by the actual or
  real observations. In addition, the original
  model derives the supply schedule based on the
  assumption that there exist quadratic costs to
  retire or issue securities (i.e., there is a
  quadratic cost on ?QSt). In this study, the cost
  is assumed to be restricted to a deviation from
  the previous observation of the quantities
  issued. We also assume that the expectation of
  the adjustment in dividend can be forecasted and
  computed from the adaptive expectation model by
  utilizing the past dividend and earnings
  information. As a result, the disequilibrium
  equation system can be restated as followings

47
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• (A.1.1) QDt a1 Pt-1a2Pt a3Dt u1t
• (A.1.2) QSt Qt-1 ß1Pt-1 ß2 Dt u2t
• (A.1.3) Qt min (QDt, QSt )
• (A.1.4) ?Pt Pt -Pt-1 ?(QDt - QSt ).

48
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• A.2 How to Solve MLE Estimates in the
  Disequilibrium Model
• In order to estimate the disequilibrium model
  described as equation (A.1.1) to (A.1.4) in
  section A.1, we follow the method presented
  Amemiya (1974) and Quandt (1988).
• Recall that the disequilibrium equation system
  shown as equation (13) to (16) can be
  reformulated and summarized as equation (25a) and
  (25b). That is,
• (A.2.1a)
• (A.2.1b)
• where

49
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• Xi,t are vectors of predetermined variables
  including the lagging price Pt-1. Qt is the
  observed quantity, and ?Pt Pt-1 - Pt-1.
• If one assumes u1t and u2t are serially and
  contemporaneously independent with distributions
  and N0, su12 and N0, su22, and since in
  period A when QDt QSt, the conditional density
  of ?Pt given Qt is N?(Qt -ßX2t ), ?2su22, and
  in period B when QDt density of ?Pt given Qt is N?(aX1t -Qt),
  ?2su12, Amemiya (1974) shows that the log
  likelihood function can be solved as
• (A.2.2)

50
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• Amemiya (1974) suggests that the maximum
  likelihood estimator can be obtained by
  simultaneously solving the following equations
• (A.2.3)
• (A.2.4)
• (A.2.5)

51
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• (A.2.6)
• (A.2.7)
• That is, one can substitute the
  two-stage least squares (2SLS) estimates of a, ß,
  su12and su22 into (A.2.7) and solve for ?
  (choosing a positive root), then use the estimate
  of ? thus obtained to find new 2SLS estimates of
  a, ß, in (A.2.1a) and (A.2.1b), and find new
  estimates of su12 and su22 in (A.2.5) and
  (A.2.6). One can repeat this process until the
  solution converge.

52
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• That is, the parameters can be solved by using
  the following iterative procedure
• Step 1 Use 2LSL estimates of a, ß, s2µ and s2?
  as the initial estimates.
• Step 2 Substitute,,andinto (A.2.7) and solve for
  the positive root of ?,.
• Step 3 Use in (25) to obtain 2SLS estimates of
  a, ß, su12 and su22.
• Step 4 Repeat step 2 and 3 until the solutions
  converge.
• Quandt (1988) suggests an alternative way to
  derive the log-likelihood function. The
  disequilibrium equation system in equations
  (A.2.1a) and (A.2.1b) is intrinsically the same
  as the Model C in Chapter 2 suggested by Quandt
  (1988), though there are some different notions
  for the coefficients. After modifying the
  different notions, Quandts log-likelihood
  function is as followings

53
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• (A.2.8)
• where
• In Quandts specification, is
  assumed to be i.i.d. with N(0, S). In other
  words, this is the same as what we assume u1t and
  u2t are serially and contemporaneously
  independent with distributions and N0, su12
  and N0, su22.

54
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• A.3 Estimation of the Disequilibrium Model
• From section A.2, the disequilibrium equation
  system of (A.1.1) to (A.1.4) discussed in section
  A.1 can be reformulated as
• (A.3.1a)
• (A.3.1b)
• where

55
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• And, from equation (A.2.8), we can derive the
  log-likelihood function for this empirical study.
  That is, we need to estimate the following
  equations simultaneously.
• (A.3.2)
• (A.3.3)
• (A.3.4)

56
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• The procedures and the related code in
  Eviews-package are as followings. For step 2, w
  use the Marquardt procedure implemented in the
  Eviews-package to find out the ML estimates of a,
  ß, s2µ and s2? in equations (A.3.2) to (A.3.4).
  The order of evaluation is set to evaluate the
  specification by observation. The tolerance level
  of convergence, tol, is set as 1e-5.
• Step 1 Use 2LSL estimates of a, ß, s2µ and s2?
  in equation (A.3.1) as the initial estimates.
• Step 2 Substitute , , and into
  equation (A.3.2) to (A.3.4) and solve for the
  MLE of , , , and
  simultaneously.

57
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• Code
• 'Aassume zero correlation between demand and
  supply error
• Define delta(p) and delta(p)- in (A.3.1)
• series dp_pos (d(p1)0)d(p1)
• series dp_neg (d(p1)
• Estimate 2SLS for Equation (A.3.1a) and (A.3.1b)
• Equation (A.3.1a)
• equation eqa.tsls q p(-1) dv p dp_pos _at_ p(-1)
  p(-2) dv(-1) dv(-2)
• alpha eqa._at_coefs
• sigma(1) eqa._at_se
• show eqa.output
• Equation (A.3.1b)
• equation eqb.tsls q p(-1) dv q(-1) dp_neg _at_ p(-2)
  dv(-1) q(-2) dv(-2)
• beta eqb._at_coefs
• sigma(2) eqb._at_se
• show eqb.output
• mu(2) -1/eqb.c(4)

58
APPENDIX A ESTIMATING DISEQUILIBRIUM ADJUSTMENT
PARAMETER, G

• 'Setup log likelihood as in (A.3.2) to (A.3.4)
• logl ll1
• ll1.append _at_logl logl1
• Equation (A.3.3)
• ll1.append u1 q-alpha(1)p(-1)-alpha(2)dv-alpha
  (3)pdp_pos/gamma(1)
• Equation (A.3.4)
• ll1.append u2 q-beta(1)p(-1)-beta(2)dv
  -q(-1)dp_neg/gamma(1)
• Equation (A.3.2)
• ll1.append logl1 -log(2!pi) -log(_at_abs(beta(1)-a
  lpha(1)1/(gamma(1))))
• - log(sigma(1)) -log(sigma(2))
• - u12/sigma(1)2/2 u22/sigma(2)2/2
• 'Do MLE
• ll1.ml(showopts, m1000, c1e-5)

59
ENDNOTES

• 1 This dynamic asset pricing model is different
  from Mertons (1973) intertemporal asset pricing
  model in two key aspects. First, Blacks model is
  derived in the form of simultaneous equations.
  Second, Blacks model is derived in terms of
  price change, and Mertons model is derived in
  terms of rates of return.
• 2 It should be noted that Lo and Wangs model
  does not explicitly introduce the supply equation
  in asset pricing determination. Also, one can
  identify the hedging portfolio using volume data
  in the Lo and Wang model setting.
• 3 The basic assumptions are 1) a single period
  moving horizon for all investors 2) no
  transactions costs or taxes on individuals 3)
  the existence of a risk-free asset with rate of
  return, r 4) evaluation of the uncertain
  returns from investments in term of expected
  return and variance of end of period wealth and
  5) unlimited short sales or borrowing of the
  risk-free asset.

60
ENDNOTES

• 4 Theories as to why taxes and penalties affect
  capital structure are first proposed by
  Modigliani and Miller (1958) and then Miller
  (1977). Another market imperfection, prohibition
  on short sales of securities, can generate
  shadow risk premiums, and thus, provide further
  incentives for firms to reduce the cost of
  capital by diversifying their securities.
• 5 There are four major models and some
  alternative specifications in constructing
  disequilibrium issues (see Quandt (1988), though
  the time period notation is slightly different
  from the models here).
• 6 While there is a slight difference in the
  notation of time period, the essence of model is
  still remained.
• 7 Amemiya show that the 2SLS estimators proposed
  by Fair and Jaffee are not consistent since the
  expected value of error in first equation, Eµt
  given t belonging to period B, is not zero, or,
  according to Quandt, the plim Xa µa/T is not
  zero (see Amemiya (1974) and Quandt (1988)).

61
ENDNOTES

• 8 Quandt (1988) points out that the alternative
  way to derive likelihood function proposed by
  Amemiya can be obtained by, first, finding the
  joint density of Pt and Qt from equations (23)
  and (24), denoted by f1(Qt, Pt) and f2(Qt, Pt)
  respectively, and then, summing up the
  log-likelihood function for two different
  periods. One can have the appropriate
  log-likelihood function as following, which is,
  in fact, the same as (27).
• 9 The procedure of estimating disequilibrium
  adjustment parameter, is presented in the
  Appendix A.
• 10 According to equation (27), ?Pt ? (QDt1 -
  QSt1 ), the amount of excess demand needed to
  cause a dollar change in price can be calculated
  by 1/0.0308.