# Stock market prediction equations

Stock market data have thwarted decades of effort by mathematicians and statisticians to discover their hidden pattern. Simple time series analyses including AR, MA, ARMA, and ARIMA were eventually replaced with more sophisticated instruments of torture such as spectral analysis. But the data refused to confess.

The failure to discover the structure in price movements convinced many researchers that the movements were random. The so called random walk hypothesis (RWH) of Osborne and others was developed into the efficient market hypothesis (EMH) by Eugene Fama. The `weak form of the EMH says that movements in stock returns are independent random events independent of historical values.

The rationale is that if patterns did exist, arbitrageurs would take advantage and thereby quickly eliminate them. Both the RWH and the EMH came under immediate attack from market analysts and this attack continues to this day partly because statistics used in tests of the EMH are controversial. The null hypothesis states that the market is efficient. The test then consists of presenting convincing evidence that it is not. The tests usually fail.

Many argue that the failure of these tests represent a Type II error, that is, a failure to detect a real effect because of low power of the statistical test employed.

Besides, the methods of analysis assume a normal and linear world that is difficult to defend. All residuals are assumed to be independent and normally disrtributed, all relationships are assumed to be linear, and all effects are assumed to be linearly additive with no interactions. At each point in time the data are assumed to be taken from identically distributed independent populations of numbers the other members of which are unobservable. Econometric models such as ARIMA assume that all dependencies in time are linear. It is therefore logical to conjecture that the reason for the failure of statistics to reject the EMH is due not to the strength of the theory but to the weakness of the statistics. Many hold that a different and more powerful mathematical device that allowed for non-linearities to exist might be more successful in discovering the hidden structure of stock prices.

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#### Popular Q&A

##### Is there a equation to predict stock market exchange or is it roots in chaos and non-linear theory ?

Is there a differential equation to predict the dynamics of stock market?

Some texts list stochastic differential equations, generally geometric/arithemetic Brownian motion as possible models.