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Empirical Finance
Lecture 2: Tests of the EMH
Module Leader: Dr Stuart Fraser
stuart.fraser@wbs.ac.uk
留学生论文网Room D1.18 (Social Studies)Warwick Business School 2
Last time…
Price/return series (e.g., FTSE 250,) are not well modelled with the RWM:
Prices are non-stationary but…
Returns are notindependent:
Volatility clustering ⇒non-linear dependence.
Are they linearly independent? (today’s topic).
Returns are (very) non-Gaussian.Warwick Business School 3
Today…
A less restrictive version of the RWM.
The martingale model (MM).
Tests of return predictability.
ACFs
Regression based tests
Variance ratio tests
The interpretation of evidence for return predictability.
Critical appraisal of the relationship between the RWM/MM and EMH.Warwick Business School 4
Martingale Model
There are 2 main objections to the RWM as a DGP for financial data:
Assumption of independence of returns.
Assumption of normality of returns.
The MM is similar to the RWM but assumes only that returns are linearly independent.
The MM therefore provides a better description of price movements under the EMH.Warwick Business School 5
Martingale model (Cuthbertsonand Nitzsche3.3)
is a martingale process if:i) (the mean is bounded)ii) A martingale is a model of a fair game()∞<txE{}tx()0 ,>=Ω+hxxEttht()0=Ω−+tthtxxE()0>Ω−+tthtxxEMartingale propertyThe expected h-period return is zero.Example of a fair game: A game of tossing an unbiased coin: win £1 for a head; lose £1 for a tailThe expected return is £0 per play:E(r)=1×0.5 −1 ×0.5=0The process is a sub-martingale if It is a super-martingale if ()0<Ω−+tthtxxEWarwick Business School 6
Martingale model
We can write an equation for x which looks rather like the RWM:
where εis a martingale difference (or increment).
The fair game property means that the best guess of a future increment is that it equals zero.
However there is no assumption that the increments are
1. Independent (fair game property ⇒linear independence only).
2. Normally distributed.
Indeed neither independence nor a Gaussian distribution is required for returns under the EMH (only linear independence).
tttxxε+=−1()0 ,0>=Ω+hEthtε
If the process is a sub-or super-martingale then
Include a drift term in the equationtttxxεμ++=−1Warwick Business School 7
MM and the EMH (Cuthbertson& Nitzsche3.1-3.4)EMH states that asset prices fully reflect all available relevant information:The only systematic/predictable gain (change in price) is the required rate of return on the asset. Other gains/losses are attributable to unpredictable events: news ⇒Investors cannot make abnormal profits systematically from buying and selling assets: risk adjusted returns are a fair gameThe key empirical prediction of the MM/EMH is that future returns are linearlyindependent from information available in the current or previous periods. ()()011=Ω=−Ω++ttttErEεμInvestors form rational expectations:i) They know and the true model for returns; and ii) They use this information to predictfuture returnstΩRE implies forecast errors are unpredictable given tΩWarwick Business School 8
Empirical testing of EMH
EMH comes in 3 flavours depending on what constitutes the relevant information set:
1.Weak form efficiency –current price incorporates all the information on past prices.
2.Semi strong e