Stylized Facts of Financial Returns¶
Having computed log return series and estimated their moments, we now document the empirical regularities that appear robustly across assets, markets, and time periods. These regularities — called stylized facts — are the empirical constraints that any credible model must respect. They are also the direct motivation for stochastic differential equations: each fact exposes a failure mode of simpler models and points toward the mathematical structure needed to fix it.
Concept Definition¶
A stylized fact is an empirical regularity of financial returns that:
- appears consistently across many assets, markets, and sample periods,
- is robust to the precise measurement method used,
- cannot be explained by an i.i.d. Gaussian model, and
- constrains the class of plausible mathematical models.
The term was popularised in finance by Rama Cont (2001) in a systematic survey of equity and FX return data. The five facts documented below are the ones most directly relevant to the SDE models developed in this book.
Stylized Fact 1: Heavy Tails (Leptokurtosis)¶
Observation¶
The marginal distribution of daily log returns has fatter tails than the Gaussian distribution with the same mean and variance. This is measured by excess kurtosis:
For a Gaussian distribution \(\kappa = 0\) by definition. Empirical estimates for major equity indices at daily frequency:
| Index | Sample period | Excess kurtosis |
|---|---|---|
| S&P 500 | 1928–2023 | ≈ 20 (includes 1929, 1987 crashes) |
| S&P 500 | 2000–2023 | ≈ 9 |
| KOSPI | 2000–2023 | ≈ 7 (approximate; varies by sample) |
| EUR/USD | 2000–2023 | ≈ 5 |
As a consequence, extreme events occur far more frequently than Gaussian models predict:
| Event | Gaussian probability | Empirical frequency (S&P 500, 2000–2023) |
|---|---|---|
| \(\|r\| > 3\sigma\) | 1 in 370 days | 1 in ~50 days |
| \(\|r\| > 4\sigma\) | 1 in 31 600 days | 1 in ~200 days |
| \(\|r\| > 5\sigma\) | 1 in 3.5 million days | 1 in ~1 000 days |
Mathematical Characterisation¶
Tail behaviour is well approximated by a power law:
with tail index \(\alpha \approx 3\)–\(5\) for daily equity returns. A Gaussian has exponentially thin tails; any finite \(\alpha\) implies dramatically heavier tails.
A tractable parametric family capturing this is the Student-\(t\) distribution with \(\nu\) degrees of freedom:
Empirical fits for equity indices typically give \(\nu \approx 4\)–\(6\).
The boundary case \(\\nu = 4\)
For \(\nu \leq 4\) the fourth moment of the Student-\(t\) does not exist, so the kurtosis is formally infinite. Empirical kurtosis estimates are always finite (the sample is finite), so \(\nu = 4\) marks a boundary: fits with \(\nu\) slightly above 4 produce very large but finite kurtosis consistent with the table above. Use \(\nu > 4\) strictly when finite kurtosis is required.
Implication for Modelling¶
Basic GBM assumes Gaussian log-returns (\(\kappa = 0\)). Capturing heavy tails requires either jump-diffusion models (compound Poisson jumps added to the SDE) or stochastic volatility models (Heston, SABR), both of which are extensions of the Itô SDE framework developed in this book.
Stylized Fact 2: Volatility Clustering¶
Observation¶
Mandelbrot (1963) noted: large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.
Quantitatively:
- Log returns \(r_t\) show little or no autocorrelation: \(\operatorname{Corr}(r_t, r_{t-k}) \approx 0\) for \(k \geq 1\).
- Squared returns \(r_t^2\) (a proxy for volatility) show strong positive autocorrelation that decays slowly: \(\operatorname{Corr}(r_t^2, r_{t-k}^2) > 0\) for lags \(k\) up to 100 days or more.
- Absolute returns \(|r_t|\) show similarly persistent autocorrelation.
The combination — unpredictable direction, predictable magnitude — is the defining signature of volatility clustering.
Why can magnitude be predictable when direction is not?
This is not a contradiction. Returns being unpredictable means \(\mathbb{E}[r_t \mid \mathcal{F}_{t-1}] \approx 0\): the conditional mean is zero. Volatility clustering means \(\operatorname{Var}(r_t \mid \mathcal{F}_{t-1})\) is time-varying and autocorrelated. A process can have zero conditional mean and non-constant conditional variance simultaneously — this is precisely the GARCH family of models.
Testing¶
Engle's ARCH test regresses \(r_t^2\) on its own lags and tests whether all slope coefficients are zero. For virtually every equity index at daily frequency, the null is rejected at \(p \ll 0.001\).
```python from statsmodels.stats.diagnostic import het_arch
def test_arch(returns, nlags=10): stat, pval, _, _ = het_arch(returns, nlags=nlags) print(f"ARCH({nlags}): stat = {stat:.2f}, p-value = {pval:.2e}") conclusion = "ARCH effects present" if pval < 0.05 else "No ARCH effects" print(f"Conclusion: {conclusion}") ```
Implication for Modelling¶
Basic GBM has constant volatility \(\sigma\). Capturing clustering requires either discrete-time GARCH processes or their continuous-time analogues — stochastic volatility SDEs such as the Heston model, where \(\sigma_t\) itself satisfies a mean-reverting SDE.
Stylized Fact 3: The Leverage Effect¶
Observation¶
There is a negative correlation between today's signed return and tomorrow's variance — specifically, \(\operatorname{Corr}(r_t, \sigma_{t+1}^2)\) — whose magnitude for large-cap equities is typically in the range \(-0.5\) to \(-0.8\). The relationship is asymmetric:
- A large negative return today substantially increases tomorrow's volatility.
- A large positive return today has a smaller, sometimes negligible, effect.
The effect was first documented by Black (1976), who attributed it to the increase in financial leverage following a price decline (equity falls, debt/equity ratio rises, making the firm riskier). An alternative explanation is the volatility feedback hypothesis: anticipated higher future volatility raises the risk premium, requiring a price fall today.
Testing¶
The standard non-parametric diagnostic computes the cross-correlation between the signed return \(r_t\) and the future squared return \(r_{t+k}^2\) at lags \(k = 1, 2, \ldots\). Negative values confirm the direction of the leverage effect.
```python import numpy as np
def leverage_test(returns, max_lag=20): """ Compute Corr(r_t, r_{t+k}^2) for k = 1, ..., max_lag.
Negative values at positive lags indicate the leverage effect:
a negative return today predicts higher variance tomorrow.
Parameters
----------
returns : pd.Series of signed log returns
max_lag : int, number of forward lags to compute
Returns
-------
list of float, one correlation per lag
"""
corrs = []
for k in range(1, max_lag + 1):
r_signed = returns.iloc[:-k].values # r_t (signed)
r_sq_lead = (returns.iloc[k:].values) ** 2 # r_{t+k}^2 (variance proxy)
corrs.append(float(np.corrcoef(r_signed, r_sq_lead)[0, 1]))
return corrs
```
Why the diagnostic values are smaller than the true correlation
This test correlates \(r_t\) with \(r_{t+k}^2\), which is a noisy proxy for the true variance \(\sigma_{t+k}^2\). Measurement noise in the proxy attenuates the correlation toward zero. Typical values from this test are \(-0.1\) to \(-0.3\) at short lags, whereas the underlying \(\operatorname{Corr}(r_t, \sigma_{t+1}^2)\) is closer to \(-0.5\) to \(-0.8\). Both numbers are consistent: the attenuation is a known property of noisy-proxy regressions, not a sign that the effect is weaker than claimed.
Implication for Modelling¶
The leverage effect cannot be captured by any model in which volatility is independent of the price process. It requires a negative correlation between the Brownian motions driving price and volatility — precisely the parameter \(\rho < 0\) in the Heston model:
where \(dW_t^S\) and \(dW_t^V\) are standard Brownian increments each with variance \(dt\), so \(\rho\) is their correlation coefficient. In the Heston model, \(dW_t^V\) drives the variance process \(V_t = \sigma_t^2\), so \(\rho < 0\) is the continuous-time counterpart of the empirical \(\operatorname{Corr}(r_t, \sigma_{t+1}^2) < 0\) observed above.
Stylized Fact 4: Absence of Return Autocorrelation¶
Observation¶
For daily and lower frequencies, log returns show no statistically significant autocorrelation:
This is the empirical content of the Efficient Market Hypothesis in its weak form: past prices contain no exploitable information about future returns.
Minor exceptions exist at very short horizons (intraday) due to bid-ask bounce and market microstructure, but these disappear at daily and lower frequencies.
The Ljung–Box test provides a formal check:
```python try: # statsmodels >= 0.13 from statsmodels.stats.diagnostic import acorr_ljungbox except ImportError: from statsmodels.tsa.stattools import acorr_ljungbox
def test_return_autocorrelation(returns, lags=20): result = acorr_ljungbox(returns, lags=lags, return_df=True) n_sig = (result["lb_pvalue"] < 0.05).sum() print(f"Ljung-Box: {n_sig}/{lags} lags significant at 5 %") ```
For most equity indices fewer than 1–2 lags out of 20 will be significant, consistent with no systematic autocorrelation.
Implication for Modelling¶
The absence of return autocorrelation motivates martingale models for discounted price processes — a property satisfied by GBM under the risk-neutral measure and by all arbitrage-free pricing models.
Stylized Fact 5: Aggregational Gaussianity¶
Observation¶
Daily log returns are non-Gaussian (heavy tails, negative skewness). As we aggregate over longer horizons, the distribution becomes progressively more Gaussian. The kurtosis table in Fact 1 shows that the full S&P 500 history reaches excess kurtosis ≈ 20; at shorter post-2000 samples the daily figure is ≈ 9, and it decreases steadily with horizon:
| Horizon | Typical excess kurtosis |
|---|---|
| Daily | 7–10 (post-2000; see Fact 1 for century-long estimates) |
| Weekly (5 days) | 3–8 |
| Monthly (21 days) | 1–3 |
| Quarterly (63 days) | 0.5–1.5 |
This convergence is predicted by the Central Limit Theorem: the sum of \(n\) weakly-dependent random variables with finite variance converges to a Gaussian as \(n \to \infty\).
Convergence is slower than i.i.d. CLT predicts
Because volatility clusters (Fact 2), daily returns are not independent. The effective sample size for the CLT is smaller than the nominal \(n\), so convergence to Gaussianity is slower than for i.i.d. data. At monthly horizons non-Gaussianity is still detectable but much reduced. The mathematical content of this fact is that skewness decreases as \(\sim n^{-1/2}\) and excess kurtosis as \(\sim n^{-1}\) with horizon under i.i.d. assumptions — the direction is correct even if the rate is slower in practice due to clustering.
Implication for Modelling¶
GBM's Gaussian log-return distribution is a reasonable approximation at monthly and quarterly horizons, but not at daily horizons. Models that need to be accurate at short horizons require non-Gaussian innovations, jump components, or stochastic volatility.
Summary of Stylized Facts and Model Requirements¶
| Stylized fact | Basic GBM | Required extension |
|---|---|---|
| Heavy tails | ✗ Gaussian | Jump diffusion or stochastic volatility |
| Volatility clustering | ✗ Constant \(\sigma\) | Stochastic volatility SDE |
| Leverage effect | ✗ Independent noise | Correlated Brownian motions (\(\rho < 0\)) |
| No return autocorrelation | ✓ Martingale structure | Already satisfied |
| Aggregational Gaussianity | ✓ At long horizons | Already approximately satisfied |
Basic GBM satisfies two of the five facts. The remaining three — heavy tails, clustering, and the leverage effect — require extensions that all remain within the Itô SDE framework. This is why developing that framework rigorously is worth the mathematical investment.
Conclusion¶
The stylized facts reveal that financial returns are:
- non-Gaussian (heavy tails, negative skewness),
- conditionally heteroskedastic (volatility clusters),
- asymmetrically responsive to shocks (leverage effect),
- unpredictable in direction (no return autocorrelation), and
- approximately Gaussian at long horizons (aggregational Gaussianity).
A model that ignores these facts will produce incorrect risk measures, mispriced options, and unreliable simulations. The next section shows rigorously why deterministic ODEs fail to reproduce even the most basic of these properties, and motivates the transition to stochastic differential equations.
Next: Why Deterministic Models Fail. \(\square\)
Exercises¶
Exercise 1. The excess kurtosis of a Student-\(t\) distribution with \(\nu > 4\) degrees of freedom is \(6/(\nu - 4)\). If an empirical estimate of excess kurtosis for daily equity returns is \(\hat{\kappa} = 6\), what value of \(\nu\) does this imply? For this \(\nu\), does the fourth moment of the Student-\(t\) distribution exist? What happens to the theoretical kurtosis as \(\nu \to 4^+\)?
Solution to Exercise 1
The excess kurtosis of a Student-\(t\) distribution with \(\nu > 4\) degrees of freedom is \(6/(\nu - 4)\). Setting this equal to the empirical estimate:
For \(\nu = 5\), the fourth moment exists because the \(k\)-th moment of the Student-\(t\) exists if and only if \(\nu > k\), and here \(\nu = 5 > 4\). Therefore the fourth moment (and hence the kurtosis) is finite.
As \(\nu \to 4^+\), we have \(\nu - 4 \to 0^+\), so
The theoretical kurtosis diverges to infinity. At \(\nu = 4\) exactly, the fourth moment ceases to exist, and for \(\nu < 4\) the kurtosis is undefined. This boundary behaviour reflects the fact that heavier tails (smaller \(\nu\)) produce more extreme observations, inflating the fourth moment relative to the squared variance.
Exercise 2. Suppose daily log returns \(\{r_t\}\) satisfy \(\operatorname{Corr}(r_t, r_{t-k}) = 0\) for all \(k \geq 1\), but \(\operatorname{Corr}(r_t^2, r_{t-1}^2) = 0.25\). Explain why these two conditions are not contradictory. Write down a simple GARCH(1,1) model
and verify that \(\mathbb{E}[r_t \mid \mathcal{F}_{t-1}] = 0\) while \(\operatorname{Var}(r_t \mid \mathcal{F}_{t-1}) = \sigma_t^2\) varies over time.
Solution to Exercise 2
The two conditions are not contradictory because they concern different aspects of the return process.
The condition \(\operatorname{Corr}(r_t, r_{t-k}) = 0\) means that the signed direction of returns is unpredictable — the conditional mean is (approximately) zero. The condition \(\operatorname{Corr}(r_t^2, r_{t-1}^2) = 0.25 > 0\) means that the magnitude of returns is predictable — the conditional variance varies over time and is autocorrelated.
A process can have zero conditional mean and non-constant conditional variance simultaneously. The GARCH(1,1) model provides exactly this structure:
Verification that \(\mathbb{E}[r_t \mid \mathcal{F}_{t-1}] = 0\): Since \(\sigma_t^2\) depends only on \(r_{t-1}^2\) and \(\sigma_{t-1}^2\), \(\sigma_t\) is \(\mathcal{F}_{t-1}\)-measurable. Also \(Z_t\) is independent of \(\mathcal{F}_{t-1}\) by construction. Therefore:
Verification that \(\operatorname{Var}(r_t \mid \mathcal{F}_{t-1}) = \sigma_t^2\) varies over time: Since the conditional mean is zero:
Because \(\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2\) depends on the past squared return, the conditional variance changes over time. After a large \(|r_{t-1}|\), \(\sigma_t^2\) increases, making a large \(|r_t|\) more likely (regardless of sign). This is precisely volatility clustering with unpredictable returns.
Exercise 3. The tail of a distribution follows a power law \(P(|r| > x) \sim C x^{-\alpha}\) with \(\alpha = 4\). A Gaussian distribution with the same variance \(\sigma^2\) has \(P(|r| > x) \approx \frac{\sigma}{x\sqrt{2\pi}} e^{-x^2/(2\sigma^2)}\) for large \(x\). Taking \(\sigma = 0.01\) (daily), compute both probabilities at \(x = 5\sigma = 0.05\) (choose \(C\) so that the power-law probability at \(x = 3\sigma\) matches the Gaussian probability at \(x = 3\sigma\)). How many times more likely is a \(5\sigma\) event under the power-law tail than under the Gaussian?
Solution to Exercise 3
Step 1: Calibrate \(C\) so that the power-law and Gaussian probabilities match at \(x = 3\sigma\).
The Gaussian probability at \(x = 3\sigma = 0.03\):
More precisely, \(P(|Z| > 3) = 2\Phi(-3) \approx 2 \times 0.001350 = 0.002700\).
The power-law probability at \(x = 3\sigma = 0.03\):
Setting equal: \(C / (0.03)^4 = 0.002700\), so
Step 2: Compute both probabilities at \(x = 5\sigma = 0.05\).
Gaussian:
Power law:
Step 3: Ratio.
A \(5\sigma\) event is roughly 600 times more likely under the power-law tail than under the Gaussian. This demonstrates quantitatively why Gaussian risk models severely underestimate the frequency of extreme events.
Exercise 4. The leverage effect is characterised by \(\operatorname{Corr}(r_t, \sigma_{t+1}^2) < 0\). In the Heston model the correlation between the Brownian motions driving the price and variance processes is \(\rho\). Explain qualitatively why \(\rho < 0\) generates the leverage effect. Specifically, if \(dS_t = \mu S_t\,dt + \sqrt{V_t}\,S_t\,dW_t^S\) and \(dV_t = \kappa(\theta - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^V\) with \(\operatorname{Corr}(dW_t^S, dW_t^V) = \rho\,dt\), trace through what happens to \(V_t\) when \(S_t\) experiences a large negative shock (i.e., \(dW_t^S \ll 0\)) and \(\rho = -0.7\).
Solution to Exercise 4
In the Heston model:
with \(\operatorname{Corr}(dW_t^S, dW_t^V) = \rho\,dt\).
When \(\rho < 0\), the Brownian motions \(W_t^S\) and \(W_t^V\) are negatively correlated. This means that when \(dW_t^S\) takes a large negative value (a negative price shock), \(dW_t^V\) tends to take a large positive value.
Tracing through the mechanism with \(\rho = -0.7\) and \(dW_t^S \ll 0\):
-
Price falls: A large negative \(dW_t^S\) causes \(dS_t\) to be strongly negative (the diffusion term \(\sqrt{V_t} S_t\,dW_t^S\) dominates), so the stock price drops sharply.
-
Variance increases: Because \(\rho = -0.7\), when \(dW_t^S\) is very negative, \(dW_t^V\) tends to be positive (with correlation \(-0.7\)). The positive \(dW_t^V\) feeds into the variance dynamics through the term \(\xi\sqrt{V_t}\,dW_t^V > 0\), pushing \(dV_t\) upward beyond what the mean-reversion term \(\kappa(\theta - V_t)\,dt\) alone would produce.
-
Result: A negative return at time \(t\) is associated with an increase in the variance process \(V_t\). Since \(\sigma_t = \sqrt{V_t}\), this means volatility rises after a price decline.
This is exactly the leverage effect: \(\operatorname{Corr}(r_t, \sigma_{t+1}^2) < 0\). The negative correlation \(\rho\) between the two driving Brownian motions is the continuous-time mechanism that generates the asymmetric volatility response observed empirically. A positive return (large positive \(dW_t^S\)) would tend to produce negative \(dW_t^V\), decreasing variance — but the effect is weaker in magnitude for positive returns due to the asymmetric nature of the observed leverage effect, which additional model features (such as non-linear drift in \(V_t\)) can capture.
Exercise 5. Aggregational Gaussianity states that excess kurtosis decreases with the return horizon. Under i.i.d. assumptions, if a daily return has excess kurtosis \(\kappa_1\), show that the \(n\)-day return (sum of \(n\) daily returns) has excess kurtosis
Using this formula, if \(\kappa_1 = 8\) at daily frequency, compute the excess kurtosis at weekly (\(n = 5\)), monthly (\(n = 21\)), and quarterly (\(n = 63\)) horizons. At what horizon does the excess kurtosis drop below 0.5?
Solution to Exercise 5
If \(r_1, r_2, \ldots, r_n\) are i.i.d. daily returns each with excess kurtosis \(\kappa_1\) and variance \(\sigma^2\), the \(n\)-day return is \(R_n = \sum_{i=1}^n r_i\).
By the additivity of cumulants for independent random variables, the fourth cumulant of a sum of independent variables equals the sum of the fourth cumulants. The fourth cumulant is \(\kappa_4 = \kappa_1 \sigma^4\) for each daily return, and \(\operatorname{Var}(R_n) = n\sigma^2\). Therefore:
With \(\kappa_1 = 8\):
| Horizon | \(n\) | \(\kappa_n = 8/n\) |
|---|---|---|
| Weekly | 5 | \(8/5 = 1.60\) |
| Monthly | 21 | \(8/21 \approx 0.381\) |
| Quarterly | 63 | \(8/63 \approx 0.127\) |
To find when \(\kappa_n < 0.5\):
The excess kurtosis drops below 0.5 at horizon \(n = 17\) days (roughly 3.5 weeks). This confirms aggregational Gaussianity: as the horizon increases, the return distribution becomes progressively closer to Gaussian.
Exercise 6. You are given a time series of 2520 daily log returns (approximately 10 years). Describe step-by-step how you would test for each of the five stylized facts documented in this section. For each fact, state: (a) the specific quantity you would compute or the test you would run, (b) the null hypothesis, and (c) what outcome would confirm the presence of the stylized fact. You do not need to perform the computations; outline the methodology only.
Solution to Exercise 6
Fact 1: Heavy tails (leptokurtosis).
(a) Compute the sample excess kurtosis \(\hat{\kappa} = \frac{1}{T}\sum_{t=1}^T \left(\frac{r_t - \hat{\mu}}{\hat{\sigma}}\right)^4 - 3\). Alternatively, perform a Jarque–Bera test, which combines skewness and kurtosis into a single test statistic \(\text{JB} = \frac{T}{6}\left(\hat{\gamma}_1^2 + \frac{(\hat{\kappa})^2}{4}\right)\).
(b) \(H_0\): Returns are drawn from a Gaussian distribution, i.e., \(\kappa = 0\).
(c) Reject if \(\hat{\kappa}\) is significantly positive (excess kurtosis \(\gg 0\)). For \(T = 2520\), the standard error is \(\sqrt{24/2520} \approx 0.098\). Any \(\hat{\kappa} > 0.19\) would be significant at 5%.
Fact 2: Volatility clustering.
(a) Compute the autocorrelation function (ACF) of squared returns \(r_t^2\) for lags \(k = 1, \ldots, 50\). Run Engle's ARCH test: regress \(r_t^2\) on \(r_{t-1}^2, \ldots, r_{t-p}^2\) and test whether all slope coefficients are jointly zero using an \(F\)-test or \(TR^2 \sim \chi^2_p\).
(b) \(H_0\): No ARCH effects, i.e., \(\operatorname{Corr}(r_t^2, r_{t-k}^2) = 0\) for all \(k \geq 1\) (constant conditional variance).
(c) Reject if the ARCH test \(p\)-value is below 0.05, or equivalently if the ACF of \(r_t^2\) shows significant positive values at multiple lags.
Fact 3: Leverage effect.
(a) Compute the cross-correlation \(\operatorname{Corr}(r_t, r_{t+k}^2)\) for forward lags \(k = 1, 2, \ldots, 20\).
(b) \(H_0\): No leverage effect, i.e., \(\operatorname{Corr}(r_t, r_{t+k}^2) = 0\) for all \(k \geq 1\).
(c) Confirm the leverage effect if these cross-correlations are significantly negative at short positive lags (typically \(k = 1, \ldots, 5\)), indicating that negative returns precede higher volatility.
Fact 4: Absence of return autocorrelation.
(a) Compute the ACF of signed returns \(r_t\) for lags \(k = 1, \ldots, 20\). Run the Ljung–Box test: \(Q = T(T+2)\sum_{k=1}^{m}\frac{\hat{\rho}_k^2}{T-k}\), where \(\hat{\rho}_k\) is the sample autocorrelation at lag \(k\).
(b) \(H_0\): Returns are serially uncorrelated, i.e., \(\rho_k = 0\) for all tested lags.
(c) Fail to reject \(H_0\) — the stylized fact is confirmed by the absence of significant autocorrelation. Expect fewer than 1–2 lags out of 20 to be significant (consistent with the 5% false positive rate).
Fact 5: Aggregational Gaussianity.
(a) Aggregate daily returns into weekly (\(n = 5\)), monthly (\(n = 21\)), and quarterly (\(n = 63\)) returns by summing log returns. Compute excess kurtosis at each horizon. Optionally, apply the Jarque–Bera test at each horizon.
(b) \(H_0\): Returns at the given horizon are Gaussian (\(\kappa = 0\), \(\gamma_1 = 0\)).
(c) Confirm aggregational Gaussianity if the excess kurtosis decreases monotonically with horizon, and the Jarque–Bera test becomes non-significant (fail to reject normality) at longer horizons such as monthly or quarterly.