Lesson 37

Dependence, Stationarity, and White Noise

Big question

How can we tell whether a time series contains predictable linear structure rather than random fluctuation?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Define weak stationarity and white noise.
  • Compute and interpret ACF and PACF.
  • Apply Ljung-Box and unit-root diagnostics.
  • Recognize volatility clustering and nonlinear dependence.
  • Prerequisites: Chapter 17 and basic probability.
  • Key terms: stationarity, white noise, autocorrelation, partial autocorrelation, Ljung-Box, volatility clustering.

Simple explanation

A weakly stationary series has a constant mean, constant finite variance, and autocovariances that depend on lag rather than calendar date. Stationarity does not mean that every segment looks identical or that shocks are small. It supplies a stable reference for estimating dependence and long-run variance. Prices often appear nonstationary, while returns are closer to stationary, although their conditional variance may still change over time.

Key terms

Define weak stationarity and white noise
A core idea in Chapter 37 that students apply carefully in economic analysis.
Compute and interpret ACF and PACF
A core idea in Chapter 37 that students apply carefully in economic analysis.
Apply Ljung-Box and unit-root diagnostics
A core idea in Chapter 37 that students apply carefully in economic analysis.
Recognize volatility clustering and nonlinear dependence
A core idea in Chapter 37 that students apply carefully in economic analysis.
Prerequisites: Chapter 17 and basic probability
A core idea in Chapter 37 that students apply carefully in economic analysis.
Key terms: stationarity, white noise, autocorrelation, partial autocorrelation, Ljung-Box, volatility clustering
A core idea in Chapter 37 that students apply carefully in economic analysis.

Analytical workflow

Question+data+assumptions+transparentPython−>evidenceQuestion + data + assumptions + transparent Python -> evidence

Interpret the expression in words and units before using it in a claim.

Example

Interpretation. The simulated white-noise series has small sample autocorrelations and does not reject joint independence at lag 10 in this run.

Prerequisites

  • Complete the preceding course chapters or review their summaries as needed.

Full theory and examples

37.2

Stationarity stabilizes the probabilistic frame

A weakly stationary series has a constant mean, constant finite variance, and autocovariances that depend on lag rather than calendar date. Stationarity does not mean that every segment looks identical or that shocks are small. It supplies a stable reference for estimating dependence and long-run variance. Prices often appear nonstationary, while returns are closer to stationary, although their conditional variance may still change over time.

37.3

ACF and PACF are diagnostic fingerprints

The autocorrelation function measures linear association between observations separated by each lag. The partial autocorrelation removes the linear contribution of shorter lags. Sample values fluctuate even under white noise, so isolated spikes should be interpreted with approximate uncertainty bands and domain context. ACF patterns guide model specification but should not replace estimation and residual checks.

37.4

Uncorrelated does not mean independent

Financial returns may show little autocorrelation while squared or absolute returns show strong persistence. The mean can therefore be difficult to predict even when volatility is predictable. Ljung-Box tests summarize groups of autocorrelations, but p-values depend on lag choice and model estimation. Diagnostics should be applied to raw series, residuals, and transformed residuals according to the model question.

37.5

Core equations

Autocorrelation

For a stationary series, the denominator is constant and dependence is indexed by lag k.

37.6

Python demonstrations

37.6.1

Demonstration 18.1: ACF and Ljung-Box

Verified output

Interpretation. The simulated white-noise series has small sample autocorrelations and does not reject joint independence at lag 10 in this run.

37.6.2

Demonstration 18.2: Levels versus differences

Verified output

Interpretation. The random-walk level is consistent with a unit root, while its first difference is strongly stationary in the ADF diagnostic.

37.7

Visual evidence

37.8

Reference table

Why This Matters

Dependence diagnostics tell us whether a model has left systematic temporal structure unexplained.

Common Mistake

Calling a series white noise because its mean is near zero. White noise concerns dependence and variance, not only location.

Ceteris LAB Tip

Inspect the ACF of both returns and squared returns before choosing a conditional-mean or volatility model.

R-to-Python / Source Bridge

These concepts are central to the first and second source lectures. The Python version retains the diagnostic sequence while adding explicit distinction between linear and nonlinear dependence (Tsay 2013).

Table 18. Chapter reference.
DiagnosticNull ideaUse
ACFindividual lag correlation near zeropattern discovery
PACFdirect lag contribution near zeroAR order guidance
Ljung-Boxjoint zero autocorrelation through lag mresidual adequacy
ADFunit-root nullnonstationarity diagnostic
ACF of squared returnsno linear volatility dependenceARCH screening

Visual evidence

Figure 20. Autocorrelation and partial autocorrelation summarize lag dependence.
Figure 20. Autocorrelation and partial autocorrelation summarize lag dependence.
Figure 21. A stationary process fluctuates around a stable distribution; a random walk carries shocks forward.
Figure 21. A stationary process fluctuates around a stable distribution; a random walk carries shocks forward.

Additional Python demonstrations

Live Python

Source demonstration 2

Source demonstration 2

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 4Create or update a Python object used in the analysis.
  4. Line 5Create or update a Python object used in the analysis.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Display a result so students can inspect the output.
  7. Line 8Display a result so students can inspect the output.

Verified source output

[1. 0.032 0.046 0.033] 0.844
0.8381 0.00e+00
[1.    0.032 0.046 0.033]
0.844
0.8381
0.00e+00

Interpretation. The simulated white-noise series has small sample autocorrelations and does not reject joint independence at lag 10 in this run.

Interpretation. The random-walk level is consistent with a unit root, while its first difference is strongly stationary in the ADF diagnostic.

Guided practice

  1. 1Re-run Demonstration 18.1 and change one input while keeping the analytical question fixed.
  2. 2Explain in two sentences how the output supports, or fails to support, the chapter opening question.
  3. 3Add one validation check that would prevent a plausible error.

Exercises

  1. 1Simulate an AR(1) process and compare its ACF with white noise.
  2. 2Apply Ljung-Box at several lags and discuss multiple testing.
  3. 3Compare ADF results for a random walk and its difference.
  4. 4Explain why zero return autocorrelation does not imply independent returns.

Source and downloads

Chapter 37 of Fundamentals of Python for Financial Econometrics by Mohammad Safavi, Ph.D.. The lesson is an original Ceteris Lab web adaptation of the supplied publication package.

Download course PDFDownload student package

Live Python

Dependence, Stationarity, and White Noise: live Python

Dependence, Stationarity, and White Noise: live Python

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 3Load a Python library needed for data work or regression.
  4. Line 5Create or update a Python object used in the analysis.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Display a result so students can inspect the output.
  7. Line 8Display a result so students can inspect the output.

Python walkthrough

  1. 1`import numpy as np`: Loads a package or function used by the analysis.
  2. 2`from statsmodels.stats.diagnostic import acorr_ljungbox`: Loads a package or function used by the analysis.
  3. 3`from statsmodels.tsa.stattools import acf`: Loads a package or function used by the analysis.
  4. 4`rng = np.random.default_rng(18)`: Creates or updates a named object used by later steps.
  5. 5`x = rng.normal(size=500)`: Creates or updates a named object used by later steps.
  6. 6`print(np.round(acf(x, nlags=3, fft=True), 3))`: Displays a result so it can be checked and interpreted.
  7. 7`print(round(acorr_ljungbox(x, lags=[10], return_df=True)["lb_pvalue"].iloc[0], 3))`: Displays a result so it can be checked and interpreted.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

Ceteris Lab teaching sample

Estimated time

25 to 40 min

Packages

pandas, numpy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect Ceteris Lab teaching sample.
  • Run the Python cells connected to Dependence, Stationarity, and White Noise.
  • Interpret the output using Define weak stationarity and white noise and Compute and interpret ACF and PACF.

Common errors

  • File not found: check that wage_sample.csv is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_csv("/data/wage_sample.csv")
df.head()

Interactive activity

Chapter 37 interactive

Assumption stress test

Evidence strength: 55%
Weak designCredible design

What should determine the strength of an econometric claim?

Immediate feedback

Choose a decision, then test how the claim changes as evidence becomes stronger or weaker.

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

Return to the lesson assumptions, units, diagnostics, and source evidence to replace this shortcut with a defensible interpretation.

Quick quiz

Which statement best answers the Chapter 37 opening question: How can we tell whether a time series contains predictable linear structure rather than random fluctuation?

Quick quiz

Which practice should be avoided when applying Dependence, Stationarity, and White Noise?

Quick quiz

What is the most defensible way to interpret the Python demonstration?

Quick quiz

Why does Chapter 37 matter in an applied econometrics workflow?

Key takeaway

Stationarity provides a stable dependence framework. ACF, PACF, and Ljung-Box diagnose linear serial structure. Volatility dependence may remain when return autocorrelation is weak.

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