Tests of return predictability

NES EFM 2005/6 2 Plan for today Brief review of the previous lecture Brief review of the previous lecture The efficient market hypothesis The efficient market hypothesis Tests for return predictability Tests for return predictability

NES EFM 2005/6 3 Up to now: subject of EFM 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?

NES EFM 2005/6 4 The efficient market hypothesis EMH: stock prices fully and correctly reflect all relevant information EMH: stock prices fully and correctly reflect all relevant information P t+1 = E[P t+1 |I t ] + ε t+1 R t+1 = E[R t+1 |I t ] + e t+1 –The error has zero expectation and is orthogonal to I t –E[R t+1 |I t ] is normal return or opportunity cost implied by some model

NES EFM 2005/6 5 Different forms of ME Weak: Weak: –I includes past prices Semi-strong: Semi-strong: –I includes all public info Strong: Strong: –I includes all (also private) info

NES EFM 2005/6 6 Different types of models Constant expected return: E t [R t+1 ] = μ Constant expected return: E t [R t+1 ] = μ –Tests for return predictability CAPM: E t [R i,t+1 ] – R F = β i (E t [R M,t+1 ] – R F ) CAPM: E t [R i,t+1 ] – R F = β i (E t [R M,t+1 ] – R F ) –Tests for mean-variance efficiency Multi-factor models Multi-factor models The joint hypothesis problem: The joint hypothesis problem: –We simultaneously test market efficiency and the model

NES EFM 2005/6 7 Implications of ME If the EMH is not rejected, then… the underlying model is a good description of the market, the underlying model is a good description of the market, –the fluctuations around the expected price are unforecastable, 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), or insider trading (SFE) are useless –the role of analysts limited to diversification, minimizing taxes and transaction costs or corporate policy: or corporate policy: –the choice of capital structure or dividend policy has no impact on the firm’s value (under MM assumptions) –still need to correct market imperfections (agency problem, taxes)

NES EFM 2005/6 8 Implications of ME (cont.) Perfect ME is unattainable: The Grossman-Stiglitz paradox: The Grossman-Stiglitz paradox: –there must be some strong-form inefficiency left Operational efficiency: Operational efficiency: –one cannot make profit on the basis of info, accounting for info acquisition and trading costs Relative efficiency: Relative efficiency: –one market vs the other (e.g., auction vs dealer markets)

NES EFM 2005/6 9 Different properties of the stochastic processes Martingale: E t [X t+1 ] = X t Martingale: E t [X t+1 ] = X t –First applied to stock prices, –But they must be detrended Fair game: E t [Y t+1 ] = 0 Fair game: E t [Y t+1 ] = 0 –Under EMH, applies to the unexpected stock returns: E t [R t+1 - μ t+1 ] = 0

NES EFM 2005/6 10 Tests for return predictability Simplest model: constant expected return Simplest model: constant expected return E t [R t+1 ] = μ Sufficient conditions: Sufficient conditions: –Common and constant time preference rate –Homogeneous expectations –Risk-neutrality

NES EFM 2005/6 11 The random walk hypotheses Random walk with drift: Δln(P t ) = μ + ε t Random walk with drift: Δln(P t ) = μ + ε t RW1: IID increments, ε t ~IID(0, σ 2 ) RW1: IID increments, ε t ~IID(0, σ 2 ) –Any functions of the increments are uncorrelated –E.g, geometric Brownian motion: ε t ~N(0, σ 2 ) RW2: independent increments RW2: independent increments –Allows for unconditional heteroskedasticity RW3: uncorrelated increments, cov(ε t, ε t-k ) = 0, k>0 RW3: uncorrelated increments, cov(ε t, ε t-k ) = 0, k>0

NES EFM 2005/6 12 Tests for RW1 Sequences and reversals Sequences and reversals –Examine the frequency of sequences and reversals in historical prices –Cowles&Jones (1937): compared returns to zero assuming symmetric distribution The Cowles-Jones ratio of # sequences and reversals: CJ=N s /N r The Cowles-Jones ratio of # sequences and reversals: CJ=N s /N r H 0 : CJ=1, rejected H 0 : CJ=1, rejected –Later: account for the trend and asymmetry, H 0 not rejected Runs Runs –Examine # of sequences of consecutive positive or negative returns: Mood (1940), Fama (1965) ME not rejected ME not rejected

NES EFM 2005/6 13 Tests for RW2 Technical analysis Technical analysis –Axioms of the technical analysis: The market responds to signals, which is reflected in ΔP, ΔVol The market responds to signals, which is reflected in ΔP, ΔVol Prices exhibit (bullish, bearish, or side) trend Prices exhibit (bullish, bearish, or side) trend History repeats History repeats –Examine profit from a dynamic trading strategy based on past return history Alexander (1961): filter rules give higher profit than the buy- and-hold strategy Alexander (1961): filter rules give higher profit than the buy- and-hold strategy Fama (1965): no superior profits after adjusting for trading costs Fama (1965): no superior profits after adjusting for trading costs Pesaran&Timmerman (1995): significant abnormal profits from multivariate strategies (esp in the volatile 1970s) Pesaran&Timmerman (1995): significant abnormal profits from multivariate strategies (esp in the volatile 1970s)

NES EFM 2005/6 14 Tests for RW3 Autocorrelations Autocorrelations –For a given lag Fuller (1976): asy distribution with correction for the small- sample negative bias in autocorrelation coef Fuller (1976): asy distribution with correction for the small- sample negative bias in autocorrelation coef –For all lags: Portmanteau statistics Box-Pierce (1970): Q-statistic Box-Pierce (1970): Q-statistic Ljung-Box (1978): finite-sample correction Ljung-Box (1978): finite-sample correction –Results from CLM, Table 2.4: US, CRSP stock index has positive first autocorrelation at D, W, and M frequency CRSP stock index has positive first autocorrelation at D, W, and M frequency The equal-wtd index has higher autocorrelation The equal-wtd index has higher autocorrelation Predictability declines over time Predictability declines over time

NES EFM 2005/6 15 Tests for RW3 (cont.) Variance ratios: VR(q)≡Var[r t (q)]/(qVar[r t ]) Variance ratios: VR(q)≡Var[r t (q)]/(qVar[r t ]) –H 0 : VR=1, the variance of returns is a linear function of the time interval –In general, VR is a function of autocorrelation coefficients CLM, Tables 2.5, 2.6, 2.8: US, , weekly CLM, Tables 2.5, 2.6, 2.8: US, , weekly –Indices: VR(q) goes up with time, predictability declines over time and is larger for small-caps –Individual stocks: weak negative autocorrelation –Size-sorted portfolios: sizeable positive cross- autocorrelations, large-cap stocks lead small-caps

NES EFM 2005/6 16 Tests for RW3 (cont.) Time series analysis: ARMA models Time series analysis: ARMA models –Testing for long-horizon predictability: regressions with overlapping horizons, R t+h (h)=a+bR t (h)+u t+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, Negative autocorrelation (mean reversion) for horizons from 2 to 7 years, peak b=-0.5 for 5y Negative autocorrelation (mean reversion) for horizons from 2 to 7 years, peak b=-0.5 for 5y Poterba&Summers (1988): similar results based on VR Poterba&Summers (1988): similar results based on VR –Critique: Small-sample and bias adjustments lower the significance Small-sample and bias adjustments lower the significance Results are sensitive to the sample period, largely due to (the Great Depression) Results are sensitive to the sample period, largely due to (the Great Depression)

NES EFM 2005/6 17 Interpretation Behavioral: investor overreaction Assume RW with drift, E t [R t+1 ] = μ Assume RW with drift, E t [R t+1 ] = μ There is a positive shock at time τ There is a positive shock at time τ The positive feedback (irrational) traders buying for t=[τ+1:τ+h] after observing R τ >μ The positive feedback (irrational) traders buying for t=[τ+1:τ+h] after observing R τ >μ SR (up to τ+h): positive autocorrelation, prices overreact SR (up to τ+h): positive autocorrelation, prices overreact LR (after τ+h): negative autocorrelation, prices get back to normal level LR (after τ+h): negative autocorrelation, prices get back to normal level Volatility increases Volatility increases

NES EFM 2005/6 18 Interpretation (cont.) Non-synchronous trading Low liquidity of some stocks (assuming zero returns for days with no trades) induces Low liquidity of some stocks (assuming zero returns for days with no trades) induces –negative autocorrelation (and higher volatility) for them –positive autocorrelation (and lower volatility) for indices –lead-lag cross-autocorrelations Consistent with the observed picture (small stocks are less liquid), but cannot fully explain the magnitude of the autocorrelations Consistent with the observed picture (small stocks are less liquid), but cannot fully explain the magnitude of the autocorrelations

NES EFM 2005/6 19 Interpretation (cont.) Time-varying expected returns: E t [R t+1 ] = E t [R F,t+1 ] + E t [RiskPremium t+1 ] Changing preferences / risk-free rate / risk premium Changing preferences / risk-free rate / risk premium Decline in interest rate => increase in prices Decline in interest rate => increase in prices –If temporary, then positive autocorrelation in SR, mean reversion in LR

NES EFM 2005/6 20 Conclusions Reliable evidence of return predictability at short horizon Reliable evidence of return predictability at short horizon –Mostly among small stocks, which are characterized by low liquidity and high trading costs Weak evidence of return predictability at long horizon Weak evidence of return predictability at long horizon –May be related to business cycles (i.e., time- varying returns and variances)

NES EFM 2005/6 21 Up to now: Tests for informational WFE assuming constant expected returns Tests for informational WFE assuming constant expected returns –Autocorrelations –Variance ratios –Time series analysis Why use different types of tests? Why use different types of tests?

NES EFM 2005/6 22 Plan for today: Tests for informational SSFE assuming constant expected returns Tests for informational SSFE assuming constant expected returns –Regression analysis Tests for operational SSFE Tests for operational SSFE –Analysis of profits from trading strategies

NES EFM 2005/6 23 Harvey (1991) "The world price of covariance risk" Objective: Objective: –Investigate predictability of developed countries’ stock index returns Methodology: Methodology: –Time series regressions Consider dollar-denominated excess returns Consider dollar-denominated excess returns Use global and local instruments Use global and local instruments

NES EFM 2005/6 24 Data Monthly returns on MSCI stock indices of 16 OECD countries and Hong Kong, Monthly returns on MSCI stock indices of 16 OECD countries and Hong Kong, –The indices are value-weighted and dividend-adjusted –Only investable domestic companies are included –Investment and foreign companies are excluded (to avoid double counting) Risk-free rate: US 30-day T-bill Risk-free rate: US 30-day T-bill

NES EFM 2005/6 25 Data (cont.) Common instruments: Common instruments: –Lagged world excess return –Dummy for January –Dividend yield of S&P500 –Term spread for US: 3month – 1month T-bill rates 3month – 1month T-bill rates –Default spread for US: Moody’s Baa – Aaa yields Moody’s Baa – Aaa yields

NES EFM 2005/6 26 Data (cont.) Local instruments: Local instruments: –Lagged own-country return –Country-specific dividend yield –Change in FX rate –Local short-term interest rate –Local term spread

NES EFM 2005/6 27 Results Common instruments, Table 3 Common instruments, Table 3 –Reject SSFE for most countries (F-test based on R2) 13 out of 18 at 5% level, 10 at 1% level 13 out of 18 at 5% level, 10 at 1% level –The world ptf is most predictable: R2a = 13.3% –Strongest predictors: Dividend yield: + for 11 countries Dividend yield: + for 11 countries Term spread + for 7 countries Term spread + for 7 countries Default spread + for US and world, - Austria Default spread + for US and world, - Austria January dummy + Hong Kong and Norway, - Austria (16 positive) January dummy + Hong Kong and Norway, - Austria (16 positive)

NES EFM 2005/6 28 Results (cont.) Adding local instruments to common instruments, Table 4 Adding local instruments to common instruments, Table 4 –Overall improvement in R 2 is small The largest increase in R 2 a for Norway and Austria The largest increase in R 2 a for Norway and Austria –Surprisingly small impact of FX rate and local interest rates –Most important: local return, dividend yield

NES EFM 2005/6 29 Conclusions Stock indices of developed countries are predictable Stock indices of developed countries are predictable Common information variables capture most of the predictable variation Common information variables capture most of the predictable variation Later they will be used as instruments in conditional asset pricing tests Later they will be used as instruments in conditional asset pricing tests

NES EFM 2005/6 30 Pesaran and Timmerman (1995 ) "Predictability of stock returns: Robustness & economic significance" Examine profits from trading strategies using variables predicting future stock returns Examine profits from trading strategies using variables predicting future stock returns Simulate investors’ decisions in real time using publicly available info Simulate investors’ decisions in real time using publicly available info –Estimation of the parameters –Choice of the forecasting model –Choice of the portfolio strategy Account for transaction costs Account for transaction costs

NES EFM 2005/6 31 Methodology: Recursive approach Each time t, using the data from the beginning of the sample period to t-1: Each time t, using the data from the beginning of the sample period to t-1: –Choose (the best set of regressors for) the 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 costs!) Financial: wealth / Sharpe (adjusted for transaction costs!) –Choose portfolio strategy Switching (100%) between stocks and bonds Switching (100%) between stocks and bonds –Account for transaction costs Constant, symmetric, and proportional Constant, symmetric, and proportional Zero, low (0.5% stocks / 0.1% bonds), or high (1% / 0.1%) Zero, low (0.5% stocks / 0.1% bonds), or high (1% / 0.1%)

NES EFM 2005/6 32 Data Monthly returns on S&P500 in Monthly returns on S&P500 in Forecasting variables Forecasting variables –Dividend yield, P/E ratio –1-month T-bill rate / 12-month T-bond rate –Inflation rate –Δ industrial production / money supply Adjustments: Adjustments: –12-month moving averages –2-month lag for macro variables (1m for others)

NES EFM 2005/6 33 Results Robustness of the return predictability, Fig. 1-3 Robustness of the return predictability, Fig. 1-3 –The volatility of predictions went up, esp after 1974 –The predictability was decreasing, except for 1974 Main predictors, Table 1 Main predictors, Table 1 –Most important: T-bill rate, monetary growth, dividend yield, and industrial growth –The best prediction model changed over time Predictive accuracy, Table 2 Predictive accuracy, Table 2 –The market timing test (based on % of correctly predicted signs) rejects the null Mostly driven by 1970s Mostly driven by 1970s

NES EFM 2005/6 34 Results (cont.) Performance of the trading strategy, Table 3 Performance of the trading strategy, Table 3 –Market is a benchmark: Mean return 11.4%, std 15.7%, Sharpe 0.35 Mean return 11.4%, std 15.7%, Sharpe 0.35 –Zero costs All but one criteria yield higher mean return, around 14-15% All but one criteria yield higher mean return, around 14-15% All criteria have higher Sharpe, from 0.7 to 0.8 All criteria have higher Sharpe, from 0.7 to 0.8 –High costs R2 and Akaike yield higher mean return R2 and Akaike yield higher mean return Most criteria still have higher Sharpe, from 0.5 to 0.6 Most criteria still have higher Sharpe, from 0.5 to 0.6 –Results mostly driven by 1970s Test for the joint significance of the intercepts in the market model: Test for the joint significance of the intercepts in the market model: –The null rejected, even under high trans costs

NES EFM 2005/6 35 Conclusions Return predictability could be exploited to get profit Return predictability could be exploited to get profit –Using variables related to business cycles Importance of changing economic regimes: Importance of changing economic regimes: –The set of regressors changed in various periods –Predictability 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-selection models –Returns are not explained by the market model