Null Hypothesis

The null hypothesis is the default assumption that a statistical test sets out to challenge. It usually states that there is no effect, no difference or no relationship: a drug does nothing, two groups have the same average, a trading strategy has no edge. It is written $H_0$ and is paired with an alternative hypothesis, $H_1$, which states that an effect exists. The test asks whether the observed data would be surprising if the null were true.

The concept is the centre of hypothesis testing. The term was introduced by the statistician Ronald Fisher in his 1935 book The Design of Experiments. In finance the null hypothesis is the sceptic's position, and the burden of proof falls on anyone claiming to have found a pattern in markets.

The procedure is a form of argument by contradiction. The researcher assumes the null is true and works out what results would then be likely. If the actual result falls far outside that range, the assumption is judged implausible and the null is rejected in favour of the alternative. If the result is within the range that chance could easily produce, the null is not rejected.

The wording matters. A test never proves the null. It can only fail to find enough evidence against it. A court offers a close parallel: the defendant is presumed innocent, the prosecution must show guilt beyond reasonable doubt, and an acquittal means the case was not proved, not that innocence was established. The null is chosen as the claim that will stand unless the data clearly contradict it, which is why it is normally the hypothesis of no effect.

Suppose a strategy has been tested over 900 trading days and produced a mean daily return of 0.08 percent with a standard deviation of 1.2 percent. The hypotheses are:

Here $\mu$ is the strategy's true mean daily return. The test statistic is:

• $\bar{x}$ is the sample mean, 0.08.
• $s$ is the sample standard deviation, 1.2.
• $n$ is the number of observations, 900.

The standard error is $1.2 / \sqrt{900} = 0.04$, so $t = 0.08 / 0.04 = 2.0$. The critical value for a two-sided test at the 5 percent significance level is 1.96. Since 2.0 exceeds it, the null of zero mean return is rejected, though only narrowly. The corresponding p-value is about 0.046.

• No skill. In performance analysis the null is that a fund's alpha is zero, meaning its returns are explained by its exposure to market factors.
• Unit root. In the augmented Dickey-Fuller test the null is that a series is non-stationary. Here rejecting the null is the hoped-for result for a trader looking for mean reversion.
• No cointegration. The Johansen test starts from the null that a set of price series share no long-run relationship.
• No predictive power. In a test of Granger causality the null is that past values of one series do not help forecast another.
• Random walk. Tests of market efficiency take as the null that price changes are unpredictable, as a random walk implies.
• Normality. Tests of distribution take as the null that returns are normally distributed, which daily returns usually reject.

As the examples show, the null is not always the outcome the researcher expects or wants. It is the hypothesis whose consequences can be calculated exactly, which is what makes a test possible.

Rejecting a null that is in fact true is a Type I error, a false positive. Failing to reject a null that is false is a Type II error. The significance level fixes the probability of the first. In trading research, a Type I error means concluding that a worthless strategy has an edge and committing capital to it.

The risk multiplies when many hypotheses are tested. A researcher who examines 200 variations of a rule should expect about ten to reject the null at the 5 percent level by chance alone. Selecting the best of them and presenting it as a single test misstates the evidence, and is a principal cause of overfitting in backtesting. Stricter thresholds and adjustments for the number of trials are the standard responses.

Statisticians have long pointed out weaknesses in how the null is used. A point null, such as a mean of exactly zero, is almost never literally true, so with enough data it will always be rejected, even when the effect is too small to matter. Rejection says nothing about the size of an effect, and failure to reject is often wrongly read as evidence of no effect. Reporting an estimate with a confidence interval conveys more than the verdict of a test alone.