Normality Tests

Module 3: Modelling with Randomness

The Gaussian random walk model predicts normal log returns. On this page you test that prediction graphically and with the Jarque-Bera test. On the next page, Calibrated KS Test, you add a Kolmogorov-Smirnov test with a correctly computed p-value. You continue from Loading Prices and Computing Returns.

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

from amlab.finance import load_prices, log_returns

r = log_returns(load_prices()["close"])
z = (r - r.mean()) / r.std(ddof=1)  # standardized returns
n = r.size

WARNING: data/finance/prices.csv not found. Using a SYNTHETIC GARCH(1,1) series with Student-t innovations (seed 2026). These are not market data. See data/finance/README.md.

Prediction

Exercise

If returns were normal, what fraction of days would have $|z| > 4$? Write the number down. Then count them in the data.

Solution

For a standard normal, $\Pr(|Z| > 4) = 2\,\Pr(Z > 4) \approx 6 \times 10^{-5}$, about one day in 16 000. Real returns exceed this by a large factor.

print(f"normal prediction: {2 * stats.norm.sf(4):.1e}")
print(f"observed:          {np.mean(np.abs(z) > 4):.1e}")

normal prediction: 6.3e-05
observed:          5.8e-03

Graphical checks: histogram and Q-Q plot

A histogram on a logarithmic density axis shows the tails. A quantile-quantile (Q-Q) plot shows the sorted standardized returns against the quantiles of the standard normal at probabilities $(i - 0.5)/n$. Normal data lie on the diagonal.

fig, axes = plt.subplots(1, 2, figsize=(9, 3.8))
grid = np.linspace(-8, 8, 400)
axes[0].hist(z, bins=80, range=(-8, 8), density=True, alpha=0.6)
axes[0].semilogy(grid, stats.norm.pdf(grid), "k-")
axes[0].set_ylim(1e-5, 1)
axes[0].set_xlabel("standardized return")
axes[0].set_ylabel("density")

theoretical = stats.norm.ppf((np.arange(1, n + 1) - 0.5) / n)
axes[1].plot(theoretical, np.sort(z), ".", ms=2)
axes[1].plot([-4, 4], [-4, 4], "k--")
axes[1].set_xlabel("normal quantile")
axes[1].set_ylabel("sample quantile")
plt.tight_layout()
plt.show()

Figure 1: Left: histogram of standardized log returns with the standard normal density, on a log scale. Right: normal Q-Q plot. Points above the diagonal on the right and below it on the left indicate heavier tails than the normal.

The Jarque-Bera test

With sample skewness $S$ and sample excess kurtosis $K$, the Jarque-Bera statistic is

\[ JB = \frac{n}{6} \left( S^2 + \frac{K^2}{4} \right), \]

which is asymptotically $\chi^2$ with 2 degrees of freedom under normality (jarque1980efficient?).

jb = stats.jarque_bera(r)
print(f"skewness {stats.skew(r):.2f}, excess kurtosis {stats.kurtosis(r):.2f}")
print(f"JB = {jb.statistic:.1f}, p-value = {jb.pvalue:.2e}")

skewness -0.33, excess kurtosis 11.97
JB = 29942.3, p-value = 0.00e+00

The test depends on the fourth moment. For heavy-tailed returns the sample kurtosis fluctuates wildly between samples (cont2001empirical?), so read $K$ as “far from 0”, not as a precise number.

Interpretation so far

The Q-Q plot shows how the data depart from the normal, which a p-value does not. The Jarque-Bera test gives a p-value but relies on moments that may not exist for heavy tails. Next you build a test based on the whole distribution function: Calibrated KS Test.

--- title: "Normality Tests" subtitle: "Module 3: Modelling with Randomness" format: html --- The Gaussian random walk model predicts normal log returns. On this page you test that prediction graphically and with the Jarque-Bera test. On the next page, [Calibrated KS Test](ks-bootstrap.qmd), you add a Kolmogorov-Smirnov test with a correctly computed p-value. You continue from [Loading Prices and Computing Returns](returns-data.qmd). ```{python} import matplotlib.pyplot as plt import numpy as np from scipy import stats from amlab.finance import load_prices, log_returns r = log_returns(load_prices()["close"]) z = (r - r.mean()) / r.std(ddof=1) # standardized returns n = r.size ``` ## Prediction ::: {.callout-note} ## Exercise If returns were normal, what fraction of days would have $|z| > 4$? Write the number down. Then count them in the data. ::: ::: {.callout-tip collapse="true"} ## Solution For a standard normal, $\Pr(|Z| > 4) = 2\,\Pr(Z > 4) \approx 6 \times 10^{-5}$, about one day in 16 000. Real returns exceed this by a large factor. ```{python} print(f"normal prediction: {2 * stats.norm.sf(4):.1e}") print(f"observed: {np.mean(np.abs(z) > 4):.1e}") ``` ::: ## Graphical checks: histogram and Q-Q plot A histogram on a logarithmic density axis shows the tails. A quantile-quantile (Q-Q) plot shows the sorted standardized returns against the quantiles of the standard normal at probabilities $(i - 0.5)/n$. Normal data lie on the diagonal. ```{python} #| label: fig-lab-hist-qq #| fig-cap: "Left: histogram of standardized log returns with the standard normal density, on a log scale. Right: normal Q-Q plot. Points above the diagonal on the right and below it on the left indicate heavier tails than the normal." fig, axes = plt.subplots(1, 2, figsize=(9, 3.8)) grid = np.linspace(-8, 8, 400) axes[0].hist(z, bins=80, range=(-8, 8), density=True, alpha=0.6) axes[0].semilogy(grid, stats.norm.pdf(grid), "k-") axes[0].set_ylim(1e-5, 1) axes[0].set_xlabel("standardized return") axes[0].set_ylabel("density") theoretical = stats.norm.ppf((np.arange(1, n + 1) - 0.5) / n) axes[1].plot(theoretical, np.sort(z), ".", ms=2) axes[1].plot([-4, 4], [-4, 4], "k--") axes[1].set_xlabel("normal quantile") axes[1].set_ylabel("sample quantile") plt.tight_layout() plt.show() ``` ## The Jarque-Bera test With sample skewness $S$ and sample excess kurtosis $K$, the Jarque-Bera statistic is $$ JB = \frac{n}{6} \left( S^2 + \frac{K^2}{4} \right), $$ which is asymptotically $\chi^2$ with 2 degrees of freedom under normality [@jarque1980efficient]. ```{python} jb = stats.jarque_bera(r) print(f"skewness {stats.skew(r):.2f}, excess kurtosis {stats.kurtosis(r):.2f}") print(f"JB = {jb.statistic:.1f}, p-value = {jb.pvalue:.2e}") ``` The test depends on the fourth moment. For heavy-tailed returns the sample kurtosis fluctuates wildly between samples [@cont2001empirical], so read $K$ as "far from 0", not as a precise number. ## Interpretation so far The Q-Q plot shows *how* the data depart from the normal, which a p-value does not. The Jarque-Bera test gives a p-value but relies on moments that may not exist for heavy tails. Next you build a test based on the whole distribution function: [Calibrated KS Test](ks-bootstrap.qmd).