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 L10: Market Efficiency Tests

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* L10: Market Efficiency Tests

Martingale Hypothesis E[Pt+1|Pt, Pt-1,…]=Pt or, equivalently, E[Pt+1-Pt|Pt, Pt-1,…]=0 If Pt represents one’s 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 period’s 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 L10: Market Efficiency Tests

L10: Market Efficiency Tests 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. L10: Market Efficiency Tests

L10: Market Efficiency Tests Testing Weak-form EMH Which of the following does weak-form EMH imply? f(rt+k| rt, It ) = f(rt+k|It), or Cov[g(rt+k),h(rk)] = 0 for any g, h E(rt+k|rt) = u, or Cov[rt+k, h(rt)] = 0 for any h Or a simple put as Cov(rt+k, rt) = 0 Alternatively, consider stock price Pt+1 = u + Pt + et+1 Random Walk 1 (iid increments): et iid (strongest) Random Walk 2 (independent increments): Cov[g(et+1), h(et)]=0 Or (weaker) Cov[et+1, h(et)] = 0 3. Random Walk 3 (uncorrelated increments): Cov(et+1, et)=0 (weakest), but Cov(et+12, et2) ne 0 L10: Market Efficiency Tests

Early Nonparametric Tests Early tests (for iid): Spearman rank correlation test, Speamn’s footrule test, Kendall τ 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) L10: Market Efficiency Tests

Sequences and Reversals L10: Market Efficiency Tests

L10: Market Efficiency Tests Runs Use the number of consecutive positive and negative returns 1001110100 versus 0000011111 L10: Market Efficiency Tests

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 L10: Market Efficiency Tests

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). L10: Market Efficiency Tests

Autocorrelation Coefficients With a covariance-stationary time series of continuously compounded returns, we can define the kth order autocovariance, γ(k) kth order autocorrelation, ρ(k) Sample counterparts: L10: Market Efficiency Tests

Sampling Theory for Autocorrelations If rt is iid (RW1), and finite first 6 moments, Negative bias (E(ρ) 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. L10: Market Efficiency Tests

Asymptotic Distribution If rt is iid (RW1), and finite first 6 moments, sample autocorrelations are asymptotically ( T  ∞ ) normal: Joint tests: Box-Pierce Statistic Ljung-Box Statistic Can be extended beyond RW1 L10: Market Efficiency Tests

L10: Market Efficiency Tests Variance ratio test Intuition Under the RW null VR(2) = 1 With positive (negative) first-order autocorrelation VR > (<) 1. To Generalize, Why? VR(q) is a particular linear combination of ρ(k) Linearly declining weights Under all three RW nulls, VR(q) = 1, but the asymptotic distributions for sample VR(q) are different L10: Market Efficiency Tests

L10: Market Efficiency Tests Under RW1 We estimate Variance ( ) estimated using non-overlapping data: Asymptotic distributions for sample variances: Question: how about asymptotic distributions for L10: Market Efficiency Tests

Results from Hausman’s Specification Test θe : asymptotically efficient estimator; θc : consistent estimator Among all consistent estimators, the efficient estimator has the lowest variance Hauseman (1978): Cov [θe , θc - θe ] = 0 Otherwise, let Cov [θe , θc - θe ] = γ, there exists w such that, Var [ θe + w (θc - θe )] < Var (θe)  contradicts efficiency of θe Applied to L10: Market Efficiency Tests

L10: Market Efficiency Tests Delta Method How about ? Take 1st order Taylor expansion: Therefore, Delta method is discussed on page 118, Greene (2000) L10: Market Efficiency Tests

Generalization: VD(q) and VR(q) Data is nq+1 observations of log prices {p0,…,pqn) where q is an integer greater than 1. Consider the following estimators: Asymptotic distributions under RW1: L10: Market Efficiency Tests

L10: Market Efficiency Tests Refinements Using overlapping observations to estimate q-period variance: Bias adjustment: (nq)1/2VD(q)  N( 0, 2(2q-1)(q-1)/(3q) σ4 ) (nq)1/2 [VR(q) -1] N( 0, 2(2q-1)(q-1)/(3q) ) L10: Market Efficiency Tests

L10: Market Efficiency Tests 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) L10: Market Efficiency Tests

L10: Market Efficiency Tests Long-Horizon Returns L10: Market Efficiency Tests

L10: Market Efficiency Tests 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 L10: Market Efficiency Tests

L10: Market Efficiency Tests Empirical Evidence Variance Ratios As the autocorrelations suggest the variance ratios are greater than one. The equal-weighted index has VR’s that are highly significant, larger in the 1st half of the sample (a common pattern). VR’s increase in q suggesting positive serial correlation for multiperiod returns. VR’s 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. L10: Market Efficiency Tests

L10: Market Efficiency Tests 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. L10: Market Efficiency Tests

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: Γ(k) = Cov[ Rt-k , Rt ] k-th order autocorrelation matrix: Ŷ(k) L10: Market Efficiency Tests

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 L10: Market Efficiency Tests

Economic-Trick Review (1) Asymptotic Distributions L10: Market Efficiency Tests

(2) Maximum Likelihood Estimator L10: Market Efficiency Tests

L10: Market Efficiency Tests MLE Example L10: Market Efficiency Tests

L10: Market Efficiency Tests Properties of MLE L10: Market Efficiency Tests

(3) Consistent Estimator: MOM L10: Market Efficiency Tests

L10: Market Efficiency Tests MOM L10: Market Efficiency Tests

L10: Market Efficiency Tests MOM Estimator of N(μ,σ2) L10: Market Efficiency Tests

(4) Assumptions of Linear Regression Models L10: Market Efficiency Tests

(5) Least Square Estimation L10: Market Efficiency Tests

Generalized Least Squares When Ω 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 L10: Market Efficiency Tests

L10: Market Efficiency Tests 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 L10: Market Efficiency Tests