Last modified: January 24, 2026

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Randomness and Trend Tests

When a series looks noisy, it is still useful to check whether the noise is random or whether weak structure (trend or dependence) is present. The tests below are lightweight diagnostics for an IID or weak-dependence null.

Ljung-Box Q Test (Independence)

The Ljung-Box test checks whether a collection of autocorrelations is jointly zero:

$$ Q = n(n + 2) \sum_{k=1}^{m} \frac{\hat{r}_k^2}{n - k} $$

where $\hat{r}k$ is the sample autocorrelation at lag $k$, and $m$ is the maximum lag. Under the IID null, $Q$ is approximately $\chi^2_m$ (or $\chi^2{m - p - q}$ when testing model residuals).

McLeod-Li Test (Squared Series)

The McLeod-Li test applies the Ljung-Box statistic to squared data (or squared residuals) to detect conditional heteroskedasticity:

$$ Q_{\text{ML}} = n(n + 2) \sum_{k=1}^{m} \frac{\hat{r}_{k,\, \text{sq}}^2}{n - k} $$

Significant autocorrelation in the squared series indicates volatility clustering and suggests ARCH/GARCH effects.

Turning Point Test (IID vs. Not IID)

A turning point occurs at time $t$ if the series changes direction:

$$ (x_{t-1} < x_t > x_{t+1}) \quad \text{or} \quad (x_{t-1} > x_t < x_{t+1}) $$

Let $T$ be the number of turning points in a series of length $n$. For a continuous IID sequence:

$$ E[T] = \frac{2(n - 2)}{3}, \quad \text{Var}(T) = \frac{16n - 29}{90} $$

A standardized score $(T - E[T]) / \sqrt{\text{Var}(T)}$ can be compared to a normal reference.

Turning points marked on a synthetic series:

turning points example

Difference-Sign Test (Randomness)

This test looks at the signs of successive differences:

$$ d_t = x_t - x_{t-1}, \quad t = 2, \ldots, n $$

Let $s_t = \text{sign}(d_t) \in \{-1, +1\}$. Under a randomness assumption with a continuous distribution, the signs are approximately independent and equally likely.

One common version counts sign changes between adjacent differences:

$$ C = \sum_{t=3}^{n} I(s_t \neq s_{t-1}) $$

For $m = n - 1$ signs, there are $m - 1$ possible changes. Under the null:

Large deviations from the expected number of sign changes suggest non-random structure (e.g., trend or negative dependence).

Rank Test (Detecting Trend)

The rank test checks for a monotonic trend without assuming normality.

  1. Replace the series values with their ranks $r_t$.
  2. Compute Spearman's rank correlation between time $t$ and rank $r_t$:

$$ \rho_s = 1 - \frac{6 \sum_{t=1}^{n} (r_t - t)^2}{n(n^2 - 1)} $$

If $\rho_s$ is far from zero, the series likely has a monotone trend. This test is robust to outliers and does not require a parametric model.

Because multiple tests are often applied together, the probability of at least one false positive increases. Treat these as screening tools and follow up with model-based diagnostics (ACF/PACF, unit root tests, or regression diagnostics).