Randomness tests look for specific kinds of structure that should not appear under a stated null model. In time-series work, they are most useful as diagnostics: one test may target linear autocorrelation, another unusual turning-point behavior, another monotone trend, and another dependence in squared residuals.
No finite set of tests can prove that a series is truly random. Their role is to identify particular departures from an assumed noise process and to be interpreted alongside plots, model assumptions, residual behavior, and out-of-sample forecasting evidence.
Worked calculation: counting turning points
For the short sequence
$$ 1,\ 3,\ 2,\ 4,\ 3, $$
the interior observations at positions 2, 3, and 4 are turning points: the sequence goes up then down at 3, down then up at 2, and up then down at 4. Thus the count is 3. A turning-point test compares the observed count with the distribution expected under a continuous IID null; it is not a general test of every possible dependence pattern.
For a fitted model, the more important question is whether residuals still contain predictable structure. A residual ACF, a Ljung-Box test, and a plot of squared residuals address different kinds of remaining dependence and should be interpreted together.
The figure marks local peaks and troughs directly. A sequence with unusually few turning points tends to move persistently in one direction, while unusually many turning points indicate excessive alternation.
When a series looks noisy, diagnostics can still test whether particular forms of structure remain. No single procedure proves that a series is "random"; each test defines a specific null hypothesis and is sensitive to particular alternatives.
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 included in the test.
For a raw IID series, the reference distribution is commonly approximated by $\chi^2_m$. When the test is applied to residuals from a fitted ARMA model, the degrees of freedom are often reduced to reflect estimated AR and MA parameters, although the exact adjustment depends on the setting and implementation.
A rejection means that at least some of the tested linear autocorrelations are inconsistent with the null. It does not identify which new model should be fitted.
McLeod-Li Test (Squared Series)
The McLeod-Li test applies a Ljung-Box-type statistic to squared observations or squared residuals:
$$ Q_{\mathrm{ML}}=n(n+2)\sum_{k=1}^{m}\frac{\hat r_{k,\mathrm{sq}}^2}{n-k}. $$
Autocorrelation in the squared series indicates dependence in the magnitude of fluctuations even when the original residuals have little linear autocorrelation. This pattern is consistent with conditional heteroskedasticity and motivates checking models such as ARCH or GARCH, but the test alone does not establish a specific variance model.
The figure places mean-dependence and variance-dependence diagnostics side by side. Residual autocorrelation points to structure left in the conditional mean, while autocorrelation in squared residuals points to time-varying conditional variance.
Turning Point Test (IID vs. Not IID)
A turning point occurs at an interior time $t$ when the series changes direction:
$$
(x_{t-1}