Time Series Track · Part 3 of 3

White Noise and Random Walks in Time Series Analysis

Learn how to identify White Noise and Random Walk processes in financial time series using visual, ACF/PACF, and Ljung-Box test diagnostics.

4.1 Understanding White Noise

White Noise is a fundamental concept in time series analysis representing a completely random process. It is characterized by having no systematic patterns, trends, or seasonality, making its future values completely unpredictable from past observations.

4.1.1 Mathematical Properties of White Noise:

• Constant Mean: The expected value of the series is constant over time (usually zero). \mu_t = E[Y_t] = C
• Constant Variance: The volatility (spread) of the series remains stable and does not expand or shrink. \sigma^2_t = \text{Var}(Y_t) = \sigma^2
• Zero Autocorrelation: There is no correlation between observations at any non-zero lags. Each value is completely independent of past values. \text{Corr}(Y_t, Y_{t-k}) = 0 \quad \text{for } k \neq 0

4.2 Understanding Random Walk

A Random Walk is a time series in which the current value is equal to the previous value plus a random, unpredictable error term. Because these random changes accumulate over time, the series exhibits a strong cumulative, non-stationary pattern.

4.2.1 Mathematical Formula

Y_t = Y_{t-1} + \epsilon_t

Where: * Y_t: Current value at time t * Y_{t-1}: Previous value at time t-1 * \epsilon_t: Random error term (representing White Noise with mean 0 and variance \sigma^2)

4.2.2 Properties of a Random Walk

• Non-Constant Mean & Variance: The variance increases linearly with time (Var(Y_t) = t \sigma^2), which causes the series to drift further away from its starting point over time.
• Strong Autocorrelation: The current value is highly dependent on its immediate past values, yielding an ACF that decays extremely slowly.
• Unit Root: The coefficient of Y_{t-1} is exactly 1, making the series non-stationary unless first-order differencing is applied (which yields stationary white noise: \Delta Y_t = Y_t - Y_{t-1} = \epsilon_t).

4.3 Methods for Identification of White Noise or Random Walk

There are three primary methods used to identify whether a time series is White Noise or a Random Walk:

1. Visually Checking the Series
   • Plotting the raw values over time to inspect if the series exhibits structural trends, shifting averages, or changing variance (indicating a Random Walk), or looks like completely random, stable fluctuations (indicating White Noise).
2. ACF & PACF Plots (Autocorrelation & Partial Autocorrelation)
   • White Noise: Autocorrelations at all non-zero lags are near zero and fall within confidence intervals.
   • Random Walk: The ACF decays extremely slowly (near 1.0 for initial lags), while the PACF has a single dominant spike at lag 1.
3. Ljung-Box Test for Autocorrelation
   • A formal statistical test to determine whether autocorrelation exists across multiple lags simultaneously.
   • H_0 (Null Hypothesis): The data values are independently distributed (no autocorrelation / White Noise).
   • H_1 (Alternative Hypothesis): The data values exhibit serial correlation (autocorrelated / not White Noise).

4.3.1 Implementation: 3-Step Identification (Visual, ACF/PACF, and Ljung-Box Test)

Let’s apply all three diagnostic tools side-by-side to verify the statistical behavior of: 1. The Original Close Prices (expected to behave as a Random Walk). 2. The First-Order Differenced Prices (expected to behave as White Noise).

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
from statsmodels.stats.diagnostic import acorr_ljungbox

# Prepare data
close_prices = nifty_50['Close'].dropna()
diff_prices = close_prices.diff().dropna()

# Set up a multi-panel visualization comparing Original vs. Differenced Series
fig, axes = plt.subplots(3, 2, figsize=(16, 12))

# ==========================================
# METHOD 1: VISUAL CHECK
# ==========================================
# Original Prices Plot
axes[0, 0].plot(close_prices, color='royalblue')
axes[0, 0].set_title("Method 1 (Visual): Original Close Price (Strong Trend/Walk)", fontsize=11, fontweight='bold')
axes[0, 0].grid(True)

# Differenced Prices Plot
axes[0, 1].plot(diff_prices, color='purple')
axes[0, 1].axhline(y=0, color='black', linestyle='--')
axes[0, 1].set_title("Method 1 (Visual): 1st-Order Differenced Price (Fluctuates Around 0)", fontsize=11, fontweight='bold')
axes[0, 1].grid(True)

# ==========================================
# METHOD 2: ACF & PACF PLOTS
# ==========================================
# Original Prices ACF
sm.graphics.tsa.plot_acf(close_prices, lags=30, ax=axes[1, 0], color='royalblue', vlines_kwargs={"colors": 'royalblue'})
axes[1, 0].set_title("Method 2: ACF of Original Prices (Slow Linear Decay)", fontsize=11, fontweight='bold')
axes[1, 0].grid(True)

# Differenced Prices ACF
sm.graphics.tsa.plot_acf(diff_prices, lags=30, ax=axes[1, 1], color='purple', vlines_kwargs={"colors": 'purple'})
axes[1, 1].set_title("Method 2: ACF of Differenced Prices (Uncorrelated/White Noise)", fontsize=11, fontweight='bold')
axes[1, 1].grid(True)

# Original Prices PACF
sm.graphics.tsa.plot_pacf(close_prices, lags=30, ax=axes[2, 0], color='royalblue', vlines_kwargs={"colors": 'royalblue'})
axes[2, 0].set_title("Method 2: PACF of Original Prices (Spike at Lag 1)", fontsize=11, fontweight='bold')
axes[2, 0].grid(True)

# Differenced Prices PACF
sm.graphics.tsa.plot_pacf(diff_prices, lags=30, ax=axes[2, 1], color='purple', vlines_kwargs={"colors": 'purple'})
axes[2, 1].set_title("Method 2: PACF of Differenced Prices (No Significant Spikes)", fontsize=11, fontweight='bold')
axes[2, 1].grid(True)

plt.tight_layout()
plt.show()

# ==========================================
# METHOD 3: LJUNG-BOX TEST FOR AUTOCORRELATION
# ==========================================
print("=== Method 3: Ljung-Box Test on Original Prices ===")
ljung_orig = acorr_ljungbox(close_prices, lags=[10], return_df=True)
display(ljung_orig)
p_orig = ljung_orig['lb_pvalue'].values[0]
print(f"Verdict: p-value = {p_orig:.4e}. " +
      ("Significant autocorrelation present (Reject H0 -> Random Walk / Non-Stationary)." if p_orig < 0.05 else "No autocorrelation (Fail to reject H0 -> White Noise)."))

print("\n=== Method 3: Ljung-Box Test on 1st-Order Differenced Prices ===")
ljung_diff = acorr_ljungbox(diff_prices, lags=[10], return_df=True)
display(ljung_diff)
p_diff = ljung_diff['lb_pvalue'].values[0]
print(f"Verdict: p-value = {p_diff:.4f}. " +
      ("Significant autocorrelation remains." if p_diff < 0.05 else "No significant autocorrelation (Fail to reject H0 -> White Noise / Stationary)."))

=== Method 3: Ljung-Box Test on Original Prices ===

	lb_stat	lb_pvalue
10	2084.494392	0.0

Verdict: p-value = 0.0000e+00. Significant autocorrelation present (Reject H0 -> Random Walk / Non-Stationary).

=== Method 3: Ljung-Box Test on 1st-Order Differenced Prices ===

	lb_stat	lb_pvalue
10	13.269247	0.209005

Verdict: p-value = 0.2090. No significant autocorrelation (Fail to reject H0 -> White Noise / Stationary).

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