Lecture 10: Testing Market Efficiency

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Lecture 10 Testing Market Efficiency

• The following topics will be covered
• Different forms of MEH
• Random walk tests
• Variance ratio tests
• Autocorrelation
• Also, review economic-tricks on
• Asymptotic distribution
• Maximum likelihood estimator efficient estimator
• Method of moment estimator consistent estimator
• Least square estimator

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Efficient Market Hypothesis

• Reference Fama (1970, 1991), CLM Ch 1.5
• Definition asset prices fully reflect available
  information, to the extent that no economic
  profits can be made by trading on the information
  (see CLM page 20)
• Three forms
• Past price, return, or volume
• Sequences and reversals, runs, variance ratio,
  technical analysis, momentum and contrarian
• Publicly announced news
• Event studies, accounting stock-selection models
• Private information
• Insider trading, mutual/hedge fund performance

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Martingale Hypothesis

• EPt1Pt, Pt-1,Pt or, equivalently,
  EPt1-PtPt, Pt-1,0
• If Pt represents ones cumulative wealth at date
  t from playing some game, then a fair game is one
  for which the expected wealth next period is
  simply equal to this periods wealth.
• Another aspect is that nonoverlapping price
  changes are uncorrelated at all leads and lags.
• Martingale is considered as a necessary condition
  for an efficient market
• Does the hypothesis consider risk?
• No
• By considering risk, asset returns should be
  positive. Thus the martingale property is not
  necessary nor sufficient
• Risk-adjusted Martingale

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Issues

• Joint Hypothesis Problem
• any test of market efficiency must assume an
  asset pricing paradigm. If we assume a wrong
  asset pricing model, it may lead to false
  rejection of acceptance of market efficiency.
  Alternatively, the rejection of a
  joint-hypothesis test may either be due to market
  inefficiency or a wrong asset pricing model used.

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Testing Weak-form EMH

• Which of the following does weak-form EMH imply?
• f(rtk rt, It ) f(rtkIt), or
  Covg(rtk),h(rk) 0 for any g, h
• E(rtkrt) u, or Covrtk, h(rt) 0 for any h
  
• Or a simple put as Cov(rtk, rt) 0
• Alternatively, consider stock price Pt1 u Pt
  et1
• Random Walk 1 (iid increments) et iid
  (strongest)
• Random Walk 2 (independent increments)
• Covg(et1), h(et)0
• Or (weaker) Covet1, h(et) 0
• 3. Random Walk 3 (uncorrelated increments)
  Cov(et1, et)0
• (weakest), but Cov(et12, et2) ne 0

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Early Nonparametric Tests

• Early tests (for iid)
• Spearman rank correlation test, Speamns footrule
  test, Kendall t correlation test
• Sequences and Reversals
• Runs
• See CLM 2.2
• Nonparametric tests, using signs of returns, no
  distributional assumption for returns required
• Can be used to test both RW1(iid) and RW2
  (independence)

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Sequences and Reversals
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Runs

• Use the number of consecutive positive and
  negative returns
• 1001110100 versus 0000011111

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Tests of RW2 Independent Increments

• Testing for independence without assuming
  identical distributions is quite difficult.
• Filter rule
• An asset is purchased when its price increases by
  x, and short (short) when its price drops by x
• Compare the profit of this dynamic trading
  portfolio with that of a buy-and-hold portfolio
• Need consider transaction costs
• Technical analysis/charting
• Filter rule is an example
• Trading on patterns

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Test of Serial Correlations (RW3)

• Under RW3, the increments of the random walk are
  uncorrelated at all leads and lags.
• Therefore, to test RW3, look at the returns and
  construct tests based on
• Autocorrelations at a given order
• Joint test of autocorrelations at multiple orders
  (Box-Pierce test, Ljung-Box test).
• Variance ratios (linear combinations of the
  autocorrelations).

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Autocorrelation Coefficients

• With a covariance-stationary time series of
  continuously compounded returns, we can define
  the
• kth order autocovariance, ?(k)
• kth order autocorrelation, ?(k)
• Sample counterparts

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Sampling Theory for Autocorrelations

• If rt is iid (RW1), and finite first 6 moments,
• Negative bias (E(?) is negative) in sample
  autocorrelations
• This is follows because of the estimation
  procedure.
• You have to estimate the sum of the cross
  products of deviations from a mean (that is
  itself estimated).
• Deviations from the sample mean are zero by
  construction so positive deviations must
  eventually be followed by negative deviations.
• When you multiply these deviations together, the
  result is a negative bias.

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Asymptotic Distribution

• If rt is iid (RW1), and finite first 6 moments,
  sample autocorrelations are asymptotically ( T ?
  8 ) normal
• Joint tests
• Box-Pierce Statistic
• Ljung-Box Statistic
• Can be extended beyond RW1

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Variance ratio test

• Intuition
• Under the RW null VR(2) 1
• With positive (negative) first-order
  autocorrelation VR gt (lt) 1.
• To Generalize,
• Why?
• VR(q) is a particular linear combination of ?(k)
• Linearly declining weights
• Under all three RW nulls, VR(q) 1, but the
  asymptotic distributions for sample VR(q) are
  different

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Under RW1

• We estimate
• Variance ( ) estimated using
  non-overlapping data
• Asymptotic distributions for sample variances
• Question how about asymptotic distributions for

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Results from Hausmans Specification Test

• ?e asymptotically efficient estimator ?c
  consistent estimator
• Among all consistent estimators, the efficient
  estimator has the lowest variance
• Hauseman (1978) Cov ?e , ?c - ?e 0
• Otherwise, let Cov ?e , ?c - ?e ?, there
  exists w such that,
• Var ?e w (?c - ?e ) lt Var (?e) ?
  contradicts efficiency of ?e
• Applied to

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Delta Method

• How about ?
• Take 1st order Taylor expansion
• Therefore,
• Delta method is discussed on page 118, Greene
  (2000)

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Generalization VD(q) and VR(q)

• Data is nq1 observations of log prices
  p0,,pqn) where q is an integer greater than 1.
  Consider the following estimators
• Asymptotic distributions under RW1

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Refinements

• Using overlapping observations to estimate
  q-period variance
• Bias adjustment
• (nq)1/2VD(q) ? N( 0, 2(2q-1)(q-1)/(3q) s4 )
• (nq)1/2 VR(q) -1? N( 0, 2(2q-1)(q-1)/(3q) )

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Testing RW3

• Under RW3, rt no longer iid. ?heteroskedasticity.
  
• Properties that still hold
• VD(q) ? 0, VR(q) ? 1
• And,
• Further, sample autocorrelations at different
  orders are uncorrelated.
• Therefore, variance of VR(q) remains of the form
• Properties that no longer hold
• Asymptotic variances of sample
  autocorrelations
• Asymptotic variances of VR(q)

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Long-Horizon Returns
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Empirical Evidence

• Autocorrelations
• Daily (1962-1994) equal-weighted CRSP index has a
  first-order autocorrelation of 35.0 (with a
  standard error of 1.11). Implies that 12.3 of
  the daily variation is explainable by lagged
  return (page 66 CLM).
• Box-Pierce Q statistic for 5 autocorrelations has
  value 263.3. The 99.5-percentile for 25 is
  16.7.
• Weekly and monthly returns exhibit similar
  patterns for the indexes

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Empirical Evidence

• Variance Ratios
• As the autocorrelations suggest the variance
  ratios are greater than one.
• The equal-weighted index has VRs that are highly
  significant, larger in the 1st half of the sample
  (a common pattern). VRs increase in q
  suggesting positive serial correlation for
  multiperiod returns.
• VRs of the value-weighted index are greater than
  one but insignificant in full sample and both
  subsamples. Suggests that firm size is an
  interesting issue.
• Rejection of RW stronger for smaller firms.
  Their returns more serially correlated.

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Empirical Evidence

• Individual Securities
• Variance ratios suggest small negative serial
  correlations.
• Insignificance likely due to fact that with so
  much nonsystematic risk any predictable
  components are hard to find.

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Evidence of Cross-Correlation

• The contrast with the indexes is suggestive
  large positive cross-autocorrelations across
  individual securities across time
• In addition to evidence of significant
  autocorrelations, there are also evidence of
  significant cross-autocorrelations (account for a
  half of the return predictability). This is
  another source of return predictability.
• Lo and MacKinlay (1990) argue that
  cross-autocorrelation is the main source of
  profits for short-term contrarian strategies.
  Therefore, contrarian profits may not necessarily
  be evidence of market overreaction.
• Notations
• Rt vector of returns E( Rt ) u
• k-th order autocovariance Matrix G(k) Cov
  Rt-k , Rt
• k-th order autocorrelation matrix Y(k)

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Evidence from Long-Horizon Returns

• Negative serial correlation in multi-year index
  returns
• Fama and French (1988), Poterba and Summers
  (1988)
• There is a substantial mean revision in stock
  market prices at longer horizons
• Caveat small sample size makes inference less
  reliable
• Only 12 nonoverlapping five-year returns

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Economic-Trick Review (1) Asymptotic
Distributions
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(2) Maximum Likelihood Estimator
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MLE Example
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Properties of MLE
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(3) Consistent Estimator MOM
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MOM
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MOM Estimator of N(µ,s2)
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(4) Assumptions of Linear Regression Models
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(5) Least Square Estimation
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Generalized Least Squares
When ? is unknown, feasible generalized least
squares (FGLS) approach can be used. To be
specific, we can assume a specific form of
variance-covariance matrix, either
autocorrelation or heteroscedasticity, then
estimate it. See page 465 to 470 of Greene
(2000). There are other ways to estimate beta
here, such MLE and MOM. SAS Procedure Proc Model
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Exercises

• 2.4 2.5 CLM
• Use monthly data to make Table 2.8 and 2.9, page
  75, CLM
• Exercises regarding MLE, MOM and GLS