## TESTING THE MARTINGALE HYPOTHESIS J

This paper shows how regularity conditions on the behavior of theprice process, together with absence of arbitrage, imply thateconometric tests of the martingale hypothesis on asset priceswill tend to accept it when the data are at small timeintervals.

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov-Smirnov and Cramér-von Mises tests. The tests are distribution free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The paper develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, non-martingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroskedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research. JEL Classification: C12

## Testing the Martingale Difference Hypothesis (MDH) …

The martingale hypothesis defines that the level of any variable in is equal to the price of the same variable in t using all the past information set.

Analytically, the martingale is a stochastic process of if the conditions , and hold. If I_{t} represents the information set available at time t built on the past history of the variable, we obtain , or equivalently: (using the iterated expectation law).

Note that the stochastic process is also defined as fair game because the expected value of the variable in a defined interval, given the information set available, equals zero. Using an alternative definition, the expected variation, influenced by the past history of the variable, is zero. It follows that the probability of positive/negative variation of the variable is the same.

Using the asset price as an example, we can say that the best forecast of tomorrow’s price is today’s price, where the "best" is understood as "with lower average square root".

The martingale hypothesis is generally associated with the efficient market theory. We can use the martingale to define the weak form of : if the market is efficient, in a weak form, it is not possible to systematically generate return trading on the information of past prices. Hence, the expectation on a future variation of price influenced by the price history set must be equal to zero. Consequently, the more efficient the market, the more random the variation of price. The most efficient market is the one in which changes in prices are completely random and unpredictable.

The martingale hypothesis is not related in any way to the hypothesis. The trade-off between expected return and risk is a fundamental pillar of the modern finance theory. Even with this limit, the martingale hypothesis is widely used in the pricing theory after checking for the risk correction model.