Predictability/Trading strategies

Transcription

1 Predictability/Trading strategies Bernt Arne Ødegaard 6 June 2018 Contents 1 Introduction 1 2 Predictability, Autocorrelation and related topics. 1 3 Random walks 2 4 Short-term dependence. 3 5 Tests based on autocorrelation Looking at all horizons Conclusions of the tests Can market microstructure issues explain the test results? Continuing the autocorrelation investigations Doing Lo Mackinlay tests 6 7 Longer term dependence Fama and Frech Lo Test Statistic Results Trading Strategies 12 9 Trading strategies The Conrad Kaul paper The Kandel Stambaugh paper 13 1 Introduction The purpose of the lecture is to introduce two empirical techniques which are common in finance. The first is the use of variance bounds in the time domain, particularly applied to the old issue of predictability of asset series. The second empirical technique is to construct portfolios, or alternatively, set up trading strategies, to capture the capture the issue one want to investigate. The idea is that in attempting to construct a portfolio, one can use the portfolio return to measure the economic importance of an effect. If one can construct portfolios that generates excess returns by some metric, that is an economic measure which we can relate to. 2 Predictability, Autocorrelation and related topics. In this lecture the focus is on dependencies in stock returns. The question is the possibility of forecastability of stock returns (or prices). As a background of this question recall the Fama (1970) overview of market efficiency. He defined his weak-form efficiency as saying that we could not use information in past prices to forecast expected future 1

2 returns better than the market. Note that to make this statement precise we need to specify what we mean by the market expected return. Hence a discovery of dependencies in stock return may not contradict market efficiency if this dependency is reflecting changing expectations about future returns, we can only reject specific models of future expectations. Most of the literature in this area tends to be technical in nature, using econometric time series analysis. 3 Random walks The random walk hypothesis in its various forms is one of the oldest paradigms in finance, and has been tested in a large number of ways. Summarized: Can future prices be predicted on the basis of past prices only? Question: Is the possibility of forecasting future prices evidence of inefficiency in itself? No, because to some degree we can forecast returns: Basic assumption of finance: Riskier stocks have higher expected returns. Thus, the question should not be whether we can predict returns/prices, but whether we can predict the unexpected part of future returns: r t+k E t [r t+k ] The models/tests we will consider in this lecture are not that sophisticated, they will assume either E t [r t+k ] = 0 or E t [r t+k ] = µ t, k. With that reservation, we now proceed to the classical analysis of the predictability of asset returns. Categorization: Let r t, r t+k be returns at dates t, t + k, X t = f(r t ), Y t = g(r t+k ) for arbitrary function f(), g( ). Will see that almost all versions of random walks and martingales can be specified by the orthogonality conditions cov(x t, Y t+k ) = 0 Classification scheme g( ) linear f( ) nonlinear f( ) linear Random Walk 3 Uncorrelated increments proj(r t+k r t ) = µ f( ) nonlinear E[r t+k r t ] = µ pdf(r t+k r t ) = pdf(r t+k ) Martingale/Fair Game Independent Increments Random Walk I, II. Martingale model {P t } stochastic process P t is a martingale if E[P t+k P t, P t 1, P t 2,...] = P t k > 0 or E[P t+k P t P t, P t 1, P t 2,...] = 0 k > 0 Classical work: Martingale equivalent efficient market But this forgets about risk. Modern analysis: Equivalent martingales: Prices, adjusted for risk, follow Martingales. Random Walk I: P t = P t 1 + ε t ε iid white noise (0, σ 2 ). Zero expected return,not realistic, add constant expected return. P t = µ + P t 1 + ε t = µ + (µ + P t 2 + ɛ t 1 ) + ɛ t = µt + P t i=1 ɛ i

3 E[P t P 0 ] = µt + P 0 var(p t P 0 ) = var(µ + P t 1 + ε t ) = var(µt + P 0 + Random Walk II t t ɛ i ) = var(ɛ i ) = tσ 2 i=1 i=1 P t = P t 1 + ε t ε independent, not necessarily identical. Random Walk III ε t are uncorrelated, cov(ɛ t, ɛ t+k ) = 0 k 0 Tests of the random walk hypothesis. Concentrate on RW III. Early tests of RW I (iid) Sequences and Reversals: Given return, either as likely, fraction = 1. Biased under drift. Runs: Number of consequitive positive and negative returns. Filter rules (technical trading) Measure the returns of prespecified trading rule, compare to e.g. buy and hold. 4 Short-term dependence. The main concern with short-term dependence is that this can be used to generate profitable trading rules, thus violating efficiency. Information in past prices should not be able to generate abnormal returns. Note that the existence of some dependence in returns can not be used directly to argue against market efficiency, as long as the return is not abnormal. However, if we can generate arbitrage profits it clearly violates market efficiency. This is the perspective you should have in mind when looking at papers documenting short-term dependence. 5 Tests based on autocorrelation We start by looking at Lo and MacKinlay (1988) (LM) in detail. The reason for looking at this paper is the fact that they find test statistics that are very robust. It is therefore one of the papers that is most widely cited on the issue of short-term autocorrelations. LM investigate one specific form of expected returns, namely the Random Walk hypothesis, that stock prices follow a random walk, (possibly with drift) Take logs, X t = ln (P t ) E[P t ] = ap t 1 X t = µ + X t 1 + ɛ t LM construct a statistic to test this hypothesis that is robust to a large number of deviations from standard (iid) assumptions. This statistic relies on the fact that the estimated variances of the process are linear in the sampling interval. Consider the case with homeoskedastic (iid) normal observations (ɛ t = N(0, σ 2 )). If σ a 2 = 1 2n (X t X t 1 µ) 2 2n t=1 3

4 σ 2 b = 1 2n n (X 2t X 2t 2 µ) 2 t=1 that is, σ 2 a is sampled at double the frequency of σ 2 b, then the asymptotic variance of σ2 b is twice that of σ2 a. 1 2n 2 ( σ a σ0 2 ) D N(0, 2σ 4 0 ) The test statistic relies on the difference of these two, 2n 2 ( σ b σ0 2 ) D N(0, 4σ 4 0 ) J r = σ2 b σ a 2 1 which has the following limiting distribution 2 2nJ r D N(0, 2) This can be used to construct a test statistic, we look at the sample equivalents of J r. This is the basic intuition underlying the test statistics, but it needs to be refined a bit before it can be used. First, there are two refinements of the iid normal case, we 1. Use overlapping observations. This makes estimates more efficient, we use more of the data. 2. Small sample correction. (Like when we take averages, divide by N 1 instead of N). Statistics: Normalise this, M r (q) = σ2 c (q) 1 nq Mr (q) D N z(q) = σ 2 a ( 0, nq Mr (q) 2(2q 1)(q 1) 3q a.s. 0 ) 2(2q 1)(q 1) 3q D N (0, 1) 1 On the result in (5), the variance of ˆσ 0 2. This is applying a general result on variances on moments: See (Rao, 1973, pg 437). The asymptotic variance equals: V (ˆσ 2 ) = κ 4 + 2(κ 2 ) 2, ( where κ i is the i th cumulant. In the case of the normal distribution κ 2 = σ 2 and κ i = i > 2, so V (ˆσ 2 ) = σ 2) 2 = σ 4. 2 Showing this uses a result from Hausman (1978). This paper is useful to know about, it shows how to implement very general specification tests. Consider example of how this works. Suppose, under the null, we have the usual regression model, y = Xb + ε, with E[ε X] = 0 var(ε X) = σ The efficient estimator of b in this case is the OLS one. Consider now an inefficient estimator of b from the regression y = Xb + f(x)a + ɛ, where f( ) is some transformation of the data X. Under the null, a = 0. The estimate of b from this regression is still consistent, but not efficient. To test for general misspecification we would look at the difference between the two estimates of b. Large differences would signify a misspecification of the original model. You may notice that this is what is done in eg Fama and MacBeth (1973), where they added the (own) variance to the CAPM, and tested whether the coefficient in front of the variance is significant. Using the results from Hausman, we could have tested the CAPM specification in a more robust way. 4

5 A problem with these statistics is that they rely on the iid normality of the errors. X t = µ + X t 1 + ɛ t with ɛ t N(0, σ 2 ). But we have strong evidence that the errors in stock returns are not iid, they are heteroskedastic and dependent. L&M therefore develops further their test statistics by making them robust to non-iid errors. The statistics allow for general dependencies, eg an ARCH specification. The statistics rely on the same theory we looked at when we calculated weighting matrices for the GMM estimator. 3 To apply these methods we need to make some assumptions about the dependence in the errors {ε t } T t=1. Intuitively, we need to assume that events at (say) time t die out as time passes. This is formulated as a conditions on E[ε t ε t+j ]. As j, (the difference in time between the two dates t and t + j increases), we must have E[ε t ε t+j ] 0 The mixing conditions L&M use are ways to quantify this. 4 The end result is a set of tests of the RWH that are robust to heteroskedastic and dependent error terms. The statistics used in the paper are ˆM r (q), the variance ratio, ˆM r (q) a.s. 0 and z (q), which measures the significance of any deviation of ˆMr (q)from its expected value, These are the numbers listed in the tables. 5.1 Looking at all horizons z (q) D N(0, 1) ρ k = corr(x t, x t+k ) = cov(x t, x t+k ) var(x t ) (using stationarity) Can test whether ρ i = 0 i (Sampling distribution) Can look simultaneously at all k Q m = T 3 Recall that we were calculating estimates of ( T ) t=1 V T = var xtut T This was estimated using where = 1 T T t=1 m k=1 ρ 2 k [ T 1 ] E x tu tu tx 1 t + T V T = This is similar to the way θ(q) is estimated in theorem 3. 4 See (White, 1984, Pg 44-5) for definitions of mixing. T τ=1 t=τ+1 T 1 τ= T +1 Box Pierce [ ] E x tu tu t τ x t τ + x t τ u t τ u tx t ( ) τ K ΓT (τ) S T { 1 T T ΓT (τ) = τ+1 [xtûtû t τ x t τ ] for τ 0 1 T T τ+1 [x t+τ û t+τ û tx t ] for τ < 0, 5

6 5.2 Conclusions of the tests. Strong evidence against the Random Walk Hypothesis. Positive autocorrelation in stock index returns. Negative autocorrelation in individual stock returns. 5.3 Can market microstructure issues explain the test results? One possible explanation of autocorrelation: Prices of stocks are only observed when there is a trade. If there is no trade, the observed price used to calculate the market index may not be what the stock price would have been if there had been a trade. This is the problem of stale prices. This may induce autocorrelation. The section in the paper that looks at the autocorrelations induced by non-trading does not test a model, it ask how big the probability of non-trading has to be for individual stocks before we observe the levels of autocorrelation we found using the variance ratios. For realistic probabilities of nontrading, we find that the level of autocorrelations found can not be explained. 5.4 Continuing the autocorrelation investigations. There is by now a huge literature on autocorrelations. Whitelaw (1994), but that is a bit dated by now. 6 Doing Lo Mackinlay tests A good reference is Boudoukh, Richardson, and Let us look at some examples of running the Lo and MacKinlay (1988) test. Exercise 1. Calculate the Lo-MacKinlay(1988) variance ratio tests for daily and monthly return for Norwegian stock market indices for the post 1980 period. Solution to Exercise 1. library(zoo) DailyData <- read.zoo("../../../data/norway/stock_market_indices/market_portfolio_returns_daily.txt", header=true,sep=";",format="%y%m%d"); MonthlyData <- read.zoo("../../../data/norway/stock_market_indices/market_portfolio_returns_monthly.txt", header=true,sep=";",format="%y%m%d"); library(vrtest) Lo.Mac(DailyData$EW,c(2,5,10,25)) Lo.Mac(MonthlyData$EW,c(2,5,10,25)) > Lo.Mac(DailyData$EW,c(2,5,10,25)) k= k= k= k= > Lo.Mac(MonthlyData$EW,c(2,5,10,25)) k= k= k= k=

7 Exercise 2. Calculate the Lo-MacKinlay(1988) variance ratio tests for daily returns for the following Norwegian stocks. Norsk Hydro Sydvaranger Yara Use lags 2, 5,10, 25 Solution to Exercise 2. library(zoo) SRets <- read.zoo("../../../data/norway/ose_individual_stocks/daily_rets.csv",format="%y%m%d", header=true,skip=2,sep=","); NHY <- na.omit(srets$norsk.hydro) SYD <- na.omit(srets$sydvaranger) Yara <- na.omit(srets$yara) library(vrtest) Lo.Mac(NHY,c(2,5,10,25)) Lo.Mac(SYD,c(2,5,10,25)) Lo.Mac(Yara,c(2,5,10,25)) Results in the following > Lo.Mac(NHY,c(2,5,10,25)) k= k= k= k= > Lo.Mac(SYD,c(2,5,10,25)) k= k= k= k= > Lo.Mac(Yara,c(2,5,10,25)) k= k= k= k= Exercise 3. Calculate the Lo-MacKinlay(1988) variance ratio tests for log differences of daily exchange rates. The exchange rates are against NOK CAD USD DKK 7

8 SEK GBP Use lags 2, 5,10, 25 Solution to Exercise 3. library(zoo) cur <- read.zoo("../../../data/exchange_rates/valutakurser_daglig_selected.csv", header=true,format="%m/%d/%y",sep=",") library(vrtest) CAD <- na.omit(cur$cad) Lo.Mac(as.matrix(diff(log(CAD))),c(2,5,10,25)) USD <- na.omit(cur$usd) Lo.Mac(as.matrix(diff(log(USD))),c(2,5,10,25)) DKK <- na.omit(cur$dkk) Lo.Mac(as.matrix(diff(log(DKK))),c(2,5,10,25)) SEK <- na.omit(cur$sek) Lo.Mac(as.matrix(diff(log(SEK))),c(2,5,10,25)) GBP <- na.omit(cur$gbp) Lo.Mac(as.matrix(diff(log(GBP))),c(2,5,10,25)) > Lo.Mac(as.matrix(diff(log(CAD))),c(2,5,10,25)) k= k= k= k= > Lo.Mac(as.matrix(diff(log(USD))),c(2,5,10,25)) k= k= k= k= > Lo.Mac(as.matrix(diff(log(DKK))),c(2,5,10,25)) k= k= k= k= > Lo.Mac(as.matrix(diff(log(SEK))),c(2,5,10,25)) k= k= k= k= > Lo.Mac(as.matrix(diff(log(GBP))),c(2,5,10,25)) 8

9 k= k= k= k= Longer term dependence. We next look at possible long term dependence in stock returns. Again, the existence of dependencies should not necessarily be viewed as evidence against efficient markets. It could for example be viewed as evidence of changing risk premia in the markets. Rather, the perspective you should keep in mind is that any systematic dependence in returns should have implications for our modelling of asset prices. For example, investment and savings decisions will be affected by any types of long-term dependency because this will have implications about future expected asset returns, and therefore the savings decision. Again, most of the work in this area tend to concentrate on the technical econometric issues. Problem: very few non-overlapping observations when looking at longer horizons. I have chosen two papers (out of a large number) for us to look at. 7.1 Fama and Frech The paper by Fama and French (1988) is widely cited as evidence for long-term memory in stock return. Reading the paper, it strikes you as simple, the estimation consist of estimating a linear regression in returns. R t = a + b r R t r + ε t The results show that we estiate significant coefficients b r for long term horizons, such as five years. They do some corrections based on simulations. These corrections can be viewed as a form of bootstrapping, they look at the observed biases in a set of simulated data. While this bias adjustment is a step in the right direction, it does not address a large number of technical reservations with the regression approach of Fama & French, which can be summed up as nonrobustness. 7.2 Lo For a paper that takes the econometric issues into account, we look at the Lo (1991) paper. Most of the paper is very technical, and we will not look at all the technicalities. But the idea of the paper is very similar to Lo and MacKinlay (1988). He Operationalizes a concept of long-term dependence. Shows how to design a test statistic to test this concept of dependence. The test statistic is robust to deviations from the usual assumptions about the error process. Investigates properties of the test statistic. Applies the test statistic to stock market data Test Statistic Main issue: We have strong evidence of short-term dependecies in stock returns. We therefore need to design a statistic that removes the short term dependencies, before we can test for long-term dependence. To understand what Lo is testing, the important part is found by looking at the formulation of the null hypothesis. Stock returns follow X t = µ + ε t 9

10 {X t (ω)} is the stochastic process for stock returns. It is defined on (Ω, F, P ) This is the standard way of defining a probability space. The way to read this is that Ω is the set of all possible events. For example, define Ω = {1, 2, 3} The sigma-algebra F is a set of subsets of Ω that fulfills certain conditions. In the example, one possible F is: {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, Ω} The probability measure P is a function that gives the probability of events in Ω, eg P ({1}) = 1 3, P ({2}) = 1 3, P ({3}) = 1 3, P ({1, 2}) = 2 3 Look at the definition of α( ) in equation (2.2). This gives the intuition for why the condition rules out short-term dependence. α(a, B) = sup P (A B) P (A)P (B) for A F, B F {A A,B B} Note that this should be zero if the two events A and B are independent, since this is the definition of independence: P (A B) = P (A)P (B) If all events in F are independent, then α is zero. It should therefore make sense that α measures the degree to which two events are dependent. To formulate a concept of long-range dependence, we say that there is no long range dependence if α(a, B) goes to zero as the difference in time between the two events A and B increases. Hence we define the following information set: Bs t is the information set generated by observing all the returns between times s and t, which is written formally as the sigma-algebra generated by {X s (ω),, X t (ω)}. The coefficient α k measures the dependence of observations k periods apart: ( ) α k = sup α B, j Bj+k j The definition of long term dependence in the null hypothesis then involves a condition on α k as k. (See A4). Let us now look at how we modify the statistic to take into account dependence. The original rescaled range statistic is Q n = 1 k max (X j S X k n ) min (X j X n ) n 1 k n {1 k n} where j=1 S n = 1 n the usual (Least Squares) estimate of the variance. n (X j X n ) 2 j=1 j=1 10

11 To modify the RS statistic, Lo uses the usual White (1980) style of replacing the estimate of the variance (S n ) with a variance estimator that is consistent under heteroskedastic and dependent errors: σn(q) 2 = 1 n (X j n X n ) q n ω j (q) (X i n X n )(X i j X n ) j=1 j=1 ω j (q) is a kernel. This style of adding crossterms and weighting them is the usual type of correction of heteroskedasticity and dependence. The new test statistic is then Q n = 1 ˆσ n ( max 1 k n j=1 i=j+1 k (X j X n ) min {1 k n} j=1 ) k (X j X n ) which is like the original RS statistic, the only change is the new estimate of the variance Results The modified RS statistic is applied to the usual suspects, the CRSP indexes. Conclusion: There is little evidence for long-term dependencies in stock returns. 11

12 8 Trading Strategies Have shown existence of positive autocorrelation in index returns, cross-correlations in stocks. Also find other predictable variation in cross-sectional asset returns. predictability in short-horizon returns. Important question: Does predictability have any economic significance? One way to ask that question: Does the predictability affect trading decisions? 9 Trading strategies Construct trading strategies in terms of a plan for buying and selling stocks based on current & past price behaviour to try to measure economic importance of predictability, or: Can we make a million in the stock market? How do we construct a portfolio? Example: Lehmann (1990). Given N securities over T periods. Period t: buy ω it k dollars of security i. Some strategies that have been investigated: Buy winners & sell losers (price continuation) Buy losers & sell winners (price reversals) Strategy: Reversals Choose ω it k > 0 if last return was negative Choose ω it k < 0 if last return was positive Let R it k be return on asset i in period t k, and R t k be the average return in period t k. Can write strategy as ω it k > 0 if R it k < R t k or R it k R t k < 0 ω it k > 0 if R it k > R t k or R it k R t k > 0 If we also want to buy more of the worst losers, make weights proportional to returns ω it k = (R it k R t k ) This is what Lehmann does. His strategy of re-balancing the portfolio every week has a very high profit. If he investest 100 in winners and 100 in losers, get average semiannual profits less transaction costs of Conclude: Construct portfolios in a way that seem feasible, generate large profits. Sign of market inefficiency? Open question, Again: large literature, good fight. 10 The Conrad Kaul paper Conrad and Kaul (1998) try to be more careful. Are the results robust to trading horizon? Can we explain part of the results by rational reasons? Point to one problem with the continuation strategies. Will end up buying high risk stocks and selling low risk stocks consistently. Expect on average large returns coming from the cross-sectional variation Abnormal return only what is left after this is accounted for. Doing this, find that only one horizon, (out of eight) where price continuations are important. Happen to be the one (6 months) studied in the literature. 12

13 11 The Kandel Stambaugh paper Kandel and Stambaugh (1996) considers an alternative way of asking whether the predictability of returns is economically signfifiant: Does it change the asset allocation of a typical investor? Investor allocates portfolio between bonds (ω) and stocks (1 ω). If return predictability affects this asset allocation, economic significance, otherwise no economic significance. Problem: How to quantify this relation between asset allocations and predictability? This is quantified in the context of Bayesian estimation. One measure of how predictability enters is through its effect on the posterior probability distribution of asset returns. If the posterior depends on predictability, will change from prior. Question then: Does the predictability move the posterior significantly away from the prior? Results: very simple setting, but find significant impact on trading decisions. The exact details of the Bayesian analysis of the paper is not really important for our purposes. 13

14 References Jacob Boudoukh, Matthew P Richardson, and Robert Whitelaw. A tale of three schools: Insights on autocorrelations of short-horizon stock returns. Review of Financial Studies, 7(3):534 74, Jennifer Conrad and Gautam Kaul. An anatomy of trading strategies. Review of Financial Studies, 11: , February Eugene F Fama. Efficient capital markets: A review of theory and empirical work. Journal of Finance, 25: , Eugene F Fama and Kenneth R French. Permanent and temporary components of stock prices. Journal of Political Economy, 96: , Eugene F Fama and J MacBeth. Risk, return and equilibrium, empirical tests. Journal of Political Economy, 81: , J A Hausman. Specification tests in econometrics. Econometrica, 46: , Shmuel Kandel and Robert F Stambaugh. On the predictability of stock returns: An asset allocation perspective. Journal of Finance, 51(2): , June Bruce N Lehmann. Fads, martingales and market efficiency. Quarterly Journal of Economics, 105:1 28, February Andrew W Lo. Long-term memory in stock market prices. Econometrica, 59(5): , September Andrew W Lo and A Craig MacKinlay. Stock market returns do not follow random walks: Evidence from a simple specification test. Review of Financial Studies, 1:41 60, C Radhakrisna Rao. Linear Statistical Inference and its applications. Wiley, Second edition, Halbert White. A heteroskedasticity-consistent covariance matrix estimator and a direct test of heteroskedasticity. Econometrica, 48:817 38, Halbert White. Asymptotic theory for econometricians. Academic Press,