Tests of return Tests of return predictabilitypredictability

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Plan for todayPlan for today

Brief review of the previous Brief review of the previous lecturelecture

The efficient market hypothesisThe efficient market hypothesis Tests for return predictabilityTests for return predictability

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Up to now: subject of Up to now: subject of EFMEFM What is the specifics of financial data?What is the specifics of financial data? How to model asset prices / returns?How to model asset prices / returns? What is efficient market?What is efficient market? How to test the models?How to test the models? Can rational models explain the data?Can rational models explain the data? When do we need behavioral models?When do we need behavioral models?

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The efficient market The efficient market hypothesishypothesis EMHEMH: : stock prices fully and correctly stock prices fully and correctly

reflect all relevant informationreflect all relevant information

PPt+1t+1 = E[P = E[Pt+1t+1 |I |Itt] + ε] + εt+1t+1

RRt+1t+1 = E[R = E[Rt+1t+1 |I |Itt] + e] + et+1t+1 – The error has zero expectation and is The error has zero expectation and is

orthogonal to Iorthogonal to Itt

– E[RE[Rt+1t+1 |I |Itt] is normal return or ] is normal return or opportunity cost implied by some modelopportunity cost implied by some model

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Different forms of MEDifferent forms of ME

WeakWeak: : – I includes past pricesI includes past prices

Semi-strongSemi-strong: : – I includes all public infoI includes all public info

StrongStrong: : – I includes all (also private) infoI includes all (also private) info

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Different types of Different types of modelsmodels Constant expected return: EConstant expected return: Ett[R[Rt+1t+1] = μ] = μ

– Tests for return predictabilityTests for return predictability CAPM: ECAPM: Ett[R[Ri,t+1i,t+1] – R] – RFF = β = βii(E(Ett[R[RM,t+1M,t+1] – R] – RFF))

– Tests for mean-variance efficiencyTests for mean-variance efficiency Multi-factor modelsMulti-factor models The The joint hypothesis problemjoint hypothesis problem: :

– We simultaneously test market efficiency We simultaneously test market efficiency and the modeland the model

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Implications of MEImplications of ME

If the EMH is not rejected, then…If the EMH is not rejected, then… the underlying model is a good description of the the underlying model is a good description of the

market,market,– the fluctuations around the expected price are the fluctuations around the expected price are

unforecastable, due to randomly arriving newsunforecastable, due to randomly arriving news there is no place for active ptf management…there is no place for active ptf management…

– technical analysis (WFE), fundamental analysis (SSFE), technical analysis (WFE), fundamental analysis (SSFE), or insider trading (SFE) are uselessor insider trading (SFE) are useless

– the role of analysts limited to diversification, minimizing the role of analysts limited to diversification, minimizing taxes and transaction coststaxes and transaction costs

or corporate policy:or corporate policy:– the choice of capital structure or dividend policy has no the choice of capital structure or dividend policy has no

impact on the firm’s value (under MM assumptions)impact on the firm’s value (under MM assumptions)– still need to correct market imperfections (agency still need to correct market imperfections (agency

problem, taxes)problem, taxes)

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Implications of ME Implications of ME (cont.)(cont.)Perfect ME is unattainable:Perfect ME is unattainable: The Grossman-Stiglitz paradox: The Grossman-Stiglitz paradox:

– there must be some strong-form inefficiency leftthere must be some strong-form inefficiency left Operational Operational efficiency: efficiency:

– one cannot make profit on the basis of info, one cannot make profit on the basis of info, accounting for info acquisition and trading accounting for info acquisition and trading costscosts

Relative Relative efficiency: efficiency: – one market vs the other (e.g., auction vs dealer one market vs the other (e.g., auction vs dealer

markets)markets)

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Different properties of Different properties of the stochastic the stochastic processesprocesses MartingaleMartingale: E: Ett[X[Xt+1t+1] = X] = Xtt

– First applied to stock prices,First applied to stock prices,– But they must be detrendedBut they must be detrended

Fair gameFair game: E: Ett[Y[Yt+1t+1] = 0] = 0– Under EMH, applies to the Under EMH, applies to the

unexpected stock returns: Eunexpected stock returns: Ett[R[Rt+1t+1 - - μμt+1t+1] = 0] = 0

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Tests for return Tests for return predictabilitypredictability Simplest model: constant Simplest model: constant

expected returnexpected return

EEtt[R[Rt+1t+1] = ] = μμ Sufficient conditions:Sufficient conditions:

– Common and constant time Common and constant time preference ratepreference rate

– Homogeneous expectationsHomogeneous expectations– Risk-neutralityRisk-neutrality

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The random walk The random walk hypotheseshypotheses Random walk Random walk with drift: with drift: ΔΔln(Pln(Ptt) = ) = μμ + + εεtt

RW1: IID increments, εRW1: IID increments, εtt~IID(0, σ~IID(0, σ22))– Any functions of the increments are Any functions of the increments are

uncorrelateduncorrelated– E.g, geometric Brownian motion: εE.g, geometric Brownian motion: εtt~N(0, ~N(0,

σσ22)) RW2: independent incrementsRW2: independent increments

– Allows for unconditional Allows for unconditional heteroskedasticityheteroskedasticity

RW3: uncorrelated increments, RW3: uncorrelated increments, cov(εcov(εtt, ε, εt-kt-k) = 0, k>0) = 0, k>0

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Tests for RW1Tests for RW1

Sequences and reversalsSequences and reversals– Examine the frequency of sequences and reversals in Examine the frequency of sequences and reversals in

historical priceshistorical prices– Cowles&Jones (1937): compared returns to zero Cowles&Jones (1937): compared returns to zero

assuming symmetric distribution assuming symmetric distribution The Cowles-Jones ratio of # sequences and reversals: The Cowles-Jones ratio of # sequences and reversals:

CJ=NCJ=Nss/N/Nrr

HH00: CJ=1, rejected : CJ=1, rejected

– Later: account for the trend and asymmetry, HLater: account for the trend and asymmetry, H00 not not rejectedrejected

RunsRuns– Examine # of sequences of consecutive positive or Examine # of sequences of consecutive positive or

negative returns: Mood (1940), Fama (1965)negative returns: Mood (1940), Fama (1965) ME not rejectedME not rejected

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Tests for RW2Tests for RW2

Technical analysisTechnical analysis– Axioms of the technical analysis:Axioms of the technical analysis:

The market responds to signals, which is reflected The market responds to signals, which is reflected in ΔP, ΔVolin ΔP, ΔVol

Prices exhibit (bullish, bearish, or side) trendPrices exhibit (bullish, bearish, or side) trend History repeatsHistory repeats

– Examine profit from a dynamic trading Examine profit from a dynamic trading strategy based on past return historystrategy based on past return history

Alexander (1961): filter rules give higher profit than Alexander (1961): filter rules give higher profit than the buy-and-hold strategythe buy-and-hold strategy

Fama (1965): no superior profits after adjusting for Fama (1965): no superior profits after adjusting for trading coststrading costs

Pesaran&Timmerman (1995): significant abnormal Pesaran&Timmerman (1995): significant abnormal profits from multivariate strategies (esp in the profits from multivariate strategies (esp in the volatile 1970s)volatile 1970s)

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Tests for RW3Tests for RW3

AutocorrelationsAutocorrelations– For a given lagFor a given lag

Fuller (1976): asy distribution with correction for Fuller (1976): asy distribution with correction for the small-sample negative bias in autocorrelation the small-sample negative bias in autocorrelation coefcoef

– For all lags: Portmanteau statisticsFor all lags: Portmanteau statistics Box-Pierce (1970): Q-statisticBox-Pierce (1970): Q-statistic Ljung-Box (1978): finite-sample correctionLjung-Box (1978): finite-sample correction

– Results from CLM, Table 2.4: US, 1962-Results from CLM, Table 2.4: US, 1962-19941994

CRSP stock index has positive first CRSP stock index has positive first autocorrelation at D, W, and M frequencyautocorrelation at D, W, and M frequency

The equal-wtd index has higher autocorrelationThe equal-wtd index has higher autocorrelation Predictability declines over timePredictability declines over time

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Tests for RW3 (cont.)Tests for RW3 (cont.)

Variance ratiosVariance ratios: VR(q)≡Var[r: VR(q)≡Var[rtt(q)]/(qVar[r(q)]/(qVar[rtt])])– HH00: VR=1, the variance of returns is a linear : VR=1, the variance of returns is a linear

function of the time intervalfunction of the time interval– In general, VR is a function of autocorrelation In general, VR is a function of autocorrelation

coefficientscoefficients CLM, Tables 2.5, 2.6, 2.8: US, 1962-94, CLM, Tables 2.5, 2.6, 2.8: US, 1962-94,

weeklyweekly– Indices: VR(q) goes up with time, predictability Indices: VR(q) goes up with time, predictability

declines over time and is larger for small-capsdeclines over time and is larger for small-caps– Individual stocks: weak negative autocorrelationIndividual stocks: weak negative autocorrelation– Size-sorted portfolios: sizeable positive cross-Size-sorted portfolios: sizeable positive cross-

autocorrelations, large-cap stocks lead small-capsautocorrelations, large-cap stocks lead small-caps

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Tests for RW3 (cont.)Tests for RW3 (cont.)

Time series analysis: Time series analysis: ARMA modelsARMA models– Testing for long-horizon predictability: Testing for long-horizon predictability:

regressions with overlapping horizons, regressions with overlapping horizons, RRt+ht+h(h)=a+bR(h)=a+bRtt(h)+u(h)+ut+ht+h, ,

Serial correlation: ρ(k)=h-k => use HAC s.e.Serial correlation: ρ(k)=h-k => use HAC s.e.– Results from Fama&French (1988): US, 1926-Results from Fama&French (1988): US, 1926-

19851985 Negative autocorrelation (mean reversion) for horizons Negative autocorrelation (mean reversion) for horizons

from 2 to 7 years, peak b=-0.5 for 5yfrom 2 to 7 years, peak b=-0.5 for 5y Poterba&Summers (1988): similar results based on VRPoterba&Summers (1988): similar results based on VR

– Critique:Critique: Small-sample and bias adjustments lower the Small-sample and bias adjustments lower the

significancesignificance Results are sensitive to the sample period, largely due Results are sensitive to the sample period, largely due

to 1926-1936 (the Great Depression)to 1926-1936 (the Great Depression)

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InterpretationInterpretation

Behavioral: Behavioral: investor overreactioninvestor overreaction Assume RW with drift, EAssume RW with drift, Ett[R[Rt+1t+1] = μ] = μ There is a positive shock at time τThere is a positive shock at time τ The positive feedback (irrational) traders The positive feedback (irrational) traders

buying for t=[τ+1:τ+h] after observing buying for t=[τ+1:τ+h] after observing RRττ>μ>μ

SR (up to τ+h): positive autocorrelation, SR (up to τ+h): positive autocorrelation, prices overreactprices overreact

LR (after τ+h): negative autocorrelation, LR (after τ+h): negative autocorrelation, prices get back to normal levelprices get back to normal level

Volatility increasesVolatility increases

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Interpretation (cont.)Interpretation (cont.)

Non-synchronous tradingNon-synchronous trading Low liquidity of some stocks (assuming zero Low liquidity of some stocks (assuming zero

returns for days with no trades) induces returns for days with no trades) induces – negative autocorrelation (and higher volatility) negative autocorrelation (and higher volatility)

for themfor them– positive autocorrelation (and lower volatility) for positive autocorrelation (and lower volatility) for

indicesindices– lead-lag cross-autocorrelationslead-lag cross-autocorrelations

Consistent with the observed picture (small Consistent with the observed picture (small stocks are less liquid), but cannot fully stocks are less liquid), but cannot fully explain the magnitude of the explain the magnitude of the autocorrelationsautocorrelations

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Interpretation (cont.)Interpretation (cont.)

Time-varying expected returnsTime-varying expected returns: :

EEtt[R[Rt+1t+1] = E] = Ett[R[RF,t+1F,t+1] + E] + Ett[RiskPremium[RiskPremiumt+1t+1]] Changing preferences / risk-free Changing preferences / risk-free

rate / risk premiumrate / risk premium Decline in interest rate => increase Decline in interest rate => increase

in pricesin prices– If temporary, then positive If temporary, then positive

autocorrelation in SR, mean reversion in autocorrelation in SR, mean reversion in LRLR

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ConclusionsConclusions

Reliable evidence of return Reliable evidence of return predictability at short horizonpredictability at short horizon– Mostly among small stocks, which are Mostly among small stocks, which are

characterized by low liquidity and characterized by low liquidity and high trading costshigh trading costs

Weak evidence of return Weak evidence of return predictability at long horizonpredictability at long horizon– May be related to business cycles May be related to business cycles

(i.e., time-varying returns and (i.e., time-varying returns and variances)variances)

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Up to now:Up to now:

Tests for informational WFE Tests for informational WFE assuming constant expected assuming constant expected returnsreturns– AutocorrelationsAutocorrelations– Variance ratiosVariance ratios– Time series analysisTime series analysis

Why use different types of tests?Why use different types of tests?

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Plan for today:Plan for today:

Tests for informational SSFE Tests for informational SSFE assuming constant expected assuming constant expected returnsreturns– Regression analysisRegression analysis

Tests for operational SSFETests for operational SSFE– Analysis of profits from trading Analysis of profits from trading

strategiesstrategies

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Harvey (1991)Harvey (1991)

"The world price of covariance risk"The world price of covariance risk"" Objective:Objective:

– Investigate predictability of developed Investigate predictability of developed countries’ stock index returnscountries’ stock index returns

Methodology:Methodology:– Time series regressionsTime series regressions

Consider dollar-denominated excess returnsConsider dollar-denominated excess returns Use global and local instrumentsUse global and local instruments

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DataData

Monthly returns on MSCI stock Monthly returns on MSCI stock indices of 16 OECD countries and indices of 16 OECD countries and Hong Kong, 1969-1989Hong Kong, 1969-1989– The indices are value-weighted and The indices are value-weighted and

dividend-adjusted dividend-adjusted – Only investable domestic companies are Only investable domestic companies are

includedincluded– Investment and foreign companies are Investment and foreign companies are

excluded (to avoid double counting)excluded (to avoid double counting) Risk-free rate: US 30-day T-billRisk-free rate: US 30-day T-bill

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Data (cont.)Data (cont.)

Common instruments:Common instruments:– Lagged world excess returnLagged world excess return– Dummy for JanuaryDummy for January– Dividend yield of S&P500Dividend yield of S&P500– Term spread for US: Term spread for US:

3month – 1month T-bill rates3month – 1month T-bill rates

– Default spread for US: Default spread for US: Moody’s Baa – Aaa yieldsMoody’s Baa – Aaa yields

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Data (cont.)Data (cont.)

Local instruments:Local instruments:– Lagged own-country returnLagged own-country return– Country-specific dividend yieldCountry-specific dividend yield– Change in FX rateChange in FX rate– Local short-term interest rateLocal short-term interest rate

– Local term spreadLocal term spread

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ResultsResults

Common instruments, Table 3Common instruments, Table 3– Reject SSFE for most countries (F-test Reject SSFE for most countries (F-test

based on R2)based on R2) 13 out of 18 at 5% level, 10 at 1% level13 out of 18 at 5% level, 10 at 1% level

– The world ptf is most predictable: R2a = The world ptf is most predictable: R2a = 13.3%13.3%

– Strongest predictors:Strongest predictors: Dividend yield: + for 11 countriesDividend yield: + for 11 countries Term spread + for 7 countriesTerm spread + for 7 countries Default spread + for US and world, - AustriaDefault spread + for US and world, - Austria January dummy + Hong Kong and Norway, - January dummy + Hong Kong and Norway, -

Austria (16 positive)Austria (16 positive)

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Results (cont.)Results (cont.)

Adding local instruments to Adding local instruments to common instruments, Table 4common instruments, Table 4– Overall improvement in ROverall improvement in R22 is small is small

The largest increase in RThe largest increase in R22a a for Norway for Norway

and Austriaand Austria

– Surprisingly small impact of FX rate Surprisingly small impact of FX rate and local interest ratesand local interest rates

– Most important: local return, Most important: local return, dividend yielddividend yield

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ConclusionsConclusions

Stock indices of developed Stock indices of developed countries are predictablecountries are predictable

Common information variables Common information variables capture most of the predictable capture most of the predictable variationvariation

Later they will be used as Later they will be used as instruments in conditional asset instruments in conditional asset pricing testspricing tests

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Pesaran and Pesaran and Timmerman (1995Timmerman (1995 ))"Predictability of stock returns: "Predictability of stock returns:

Robustness & economic significanceRobustness & economic significance"" Examine profits from trading strategies Examine profits from trading strategies

using variables predicting future stock using variables predicting future stock returnsreturns

Simulate investors’ decisions in Simulate investors’ decisions in real time real time using using publicly available infopublicly available info– Estimation of the parametersEstimation of the parameters– Choice of the forecasting modelChoice of the forecasting model– Choice of the portfolio strategyChoice of the portfolio strategy

Account for transaction costsAccount for transaction costs

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Methodology: Methodology: Recursive approachRecursive approach Each time t, using the data from the Each time t, using the data from the

beginning of the sample period to t-1:beginning of the sample period to t-1:– Choose (the best set of regressors for) the Choose (the best set of regressors for) the

forecasting model using one of the criteria:forecasting model using one of the criteria: Statistical: Akaike / Schwarz (Bayes) / R2 / sign Statistical: Akaike / Schwarz (Bayes) / R2 / sign Financial: wealth / Sharpe (adjusted for transaction Financial: wealth / Sharpe (adjusted for transaction

costs!)costs!)

– Choose portfolio strategy Choose portfolio strategy Switching (100%) between stocks and bonds Switching (100%) between stocks and bonds

– Account for transaction costsAccount for transaction costs Constant, symmetric, and proportionalConstant, symmetric, and proportional Zero, low (0.5% stocks / 0.1% bonds), or high (1% / Zero, low (0.5% stocks / 0.1% bonds), or high (1% /

0.1%)0.1%)

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DataData

Monthly returns on S&P500 in 1954-1992Monthly returns on S&P500 in 1954-1992 Forecasting variablesForecasting variables

– Dividend yield, P/E ratioDividend yield, P/E ratio– 1-month T-bill rate / 12-month T-bond rate1-month T-bill rate / 12-month T-bond rate– Inflation rateInflation rate– Δ industrial production / money supplyΔ industrial production / money supply

Adjustments:Adjustments:– 12-month moving averages12-month moving averages– 2-month lag for macro variables (1m for 2-month lag for macro variables (1m for

others)others)

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ResultsResults

Robustness of the return predictability, Fig. Robustness of the return predictability, Fig. 1-31-3– The volatility of predictions went up, esp after The volatility of predictions went up, esp after

19741974– The predictability was decreasing, except for The predictability was decreasing, except for

19741974 Main predictors, Table 1Main predictors, Table 1

– Most important: T-bill rate, monetary growth, Most important: T-bill rate, monetary growth, dividend yield, and industrial growthdividend yield, and industrial growth

– The best prediction model changed over timeThe best prediction model changed over time Predictive accuracy, Table 2Predictive accuracy, Table 2

– The market timing test (based on % of correctly The market timing test (based on % of correctly predicted signs) rejects the nullpredicted signs) rejects the null

Mostly driven by 1970sMostly driven by 1970s

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Results (cont.)Results (cont.)

Performance of the trading strategy, Table 3Performance of the trading strategy, Table 3– Market is a benchmark: Market is a benchmark:

Mean return 11.4%, std 15.7%, Sharpe 0.35Mean return 11.4%, std 15.7%, Sharpe 0.35– Zero costsZero costs

All but one criteria yield higher mean return, around All but one criteria yield higher mean return, around 14-15%14-15%

All criteria have higher Sharpe, from 0.7 to 0.8All criteria have higher Sharpe, from 0.7 to 0.8– High costsHigh costs

R2 and Akaike yield higher mean returnR2 and Akaike yield higher mean return Most criteria still have higher Sharpe, from 0.5 to 0.6Most criteria still have higher Sharpe, from 0.5 to 0.6

– Results mostly driven by 1970sResults mostly driven by 1970s Test for the joint significance of the Test for the joint significance of the

intercepts in the market model:intercepts in the market model:– The null rejected, even under high trans costsThe null rejected, even under high trans costs

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ConclusionsConclusions

Return predictability could be exploited to Return predictability could be exploited to get profitget profit– Using variables related to business cyclesUsing variables related to business cycles

Importance of changing economic regimes:Importance of changing economic regimes:– The set of regressors changed in various periodsThe set of regressors changed in various periods– Predictability was higher in the volatile 1970sPredictability was higher in the volatile 1970s

Incomplete learning after the shock?Incomplete learning after the shock? Results seem robust:Results seem robust:

– Similar evidence for the all-variable and hyper-Similar evidence for the all-variable and hyper-selection modelsselection models

– Returns are not explained by the market modelReturns are not explained by the market model