Lesson 42

Nonlinear Models, Regimes, Market Microstructure, and Ordered Outcomes

Big question

What if the same lag has a different effect in calm and stressed regimes, or the observed price change is an ordered category?

Lesson progress

Complete checkpoints as you learn

0% complete0 checkpoint streak
Progress tracking available after sign-in. Sign in to save your work.
Big question
Concept
Activity
Quiz

Learning objectives

  • Simulate and interpret threshold autoregression.
  • Explain Markov-switching regimes and expected duration.
  • Describe market-microstructure effects.
  • Estimate and interpret ordered probit probabilities.
  • Prerequisites: Chapters 18, 19, and 22.
  • Key terms: TAR, regime, Markov switching, bid-ask bounce, ordered probit, threshold.

Simple explanation

A threshold autoregression applies different linear equations depending on a lagged state variable. Each regime may be simple even when the combined process is nonlinear and asymmetric. Regime-specific coefficients can exceed one in magnitude without automatically making the complete process explosive, because switching rules constrain the path. Simulation and stability analysis are essential.

Key terms

Simulate and interpret threshold autoregression
A core idea in Chapter 42 that students apply carefully in economic analysis.
Markov-switching regimes and expected duration
A core idea in Chapter 42 that students apply carefully in economic analysis.
market-microstructure effects
A core idea in Chapter 42 that students apply carefully in economic analysis.
Estimate and interpret ordered probit probabilities
A core idea in Chapter 42 that students apply carefully in economic analysis.
Prerequisites: Chapters 18, 19, and 22
A core idea in Chapter 42 that students apply carefully in economic analysis.
Key terms: TAR, regime, Markov switching, bid-ask bounce, ordered probit, threshold
A core idea in Chapter 42 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 unconditional mean is positive even though neither regime includes an intercept, illustrating nonlinear asymmetry.

Prerequisites

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

Full theory and examples

42.2

Threshold models make dynamics piecewise

A threshold autoregression applies different linear equations depending on a lagged state variable. Each regime may be simple even when the combined process is nonlinear and asymmetric. Regime-specific coefficients can exceed one in magnitude without automatically making the complete process explosive, because switching rules constrain the path. Simulation and stability analysis are essential.

42.3

Markov switching treats the regime as latent

A Markov-switching model assumes an unobserved state that follows transition probabilities. The expected duration of a state is related to the probability of remaining in it. Regime labels are arbitrary and should be interpreted through estimated means, variances, and posterior probabilities. Apparent regimes can also reflect structural breaks or omitted variables rather than a persistent hidden process.

42.4

Market data are generated by trading mechanisms

Transaction prices are discrete, trades arrive irregularly, and bid-ask bounce can create negative short-lag dependence. Ordered probit models represent observed categories as intervals of a latent continuous variable. Thresholds and covariates determine category probabilities. The model describes ordering, not equal distance between categories, and a standard implementation assumes a common latent error scale unless extended.

42.5

Core equations

Ordered response

Observed categories arise when a latent index crosses ordered thresholds.

42.6

Python demonstrations

42.6.1

Demonstration 23.1: Simulate a two-regime TAR process

Verified output

Interpretation. The unconditional mean is positive even though neither regime includes an intercept, illustrating nonlinear asymmetry.

42.6.2

Demonstration 23.2: Fit an ordered probit

Verified output

Interpretation. At x=0, the central category is most probable. The probabilities sum to one and depend on both slope and thresholds.

42.7

Visual evidence

42.8

Reference table

Why This Matters

Nonlinear and microstructure models prevent a single linear equation from flattening state-dependent behaviour.

Common Mistake

Interpreting estimated regimes as known historical labels without examining posterior probabilities and uncertainty.

Ceteris LAB Tip

For ordered models, graph predicted category probabilities over a meaningful covariate range instead of reading thresholds in isolation.

R-to-Python / Source Bridge

The source nonlinear lecture and separate ordered-probit handout are integrated here. Python replaces custom R indicators and polr() with transparent feature construction and statsmodels OrderedModel (Tsay 2013).

Table 23. Chapter reference.
ModelState observed?Key interpretation
TARYes, threshold variablepiecewise coefficients
Markov switchingNotransition probabilities and filtered states
ordered probitObserved category onlylatent index and thresholds
microstructure modelTrades/quotes observedinstitutional price formation

Visual evidence

Figure 31. A simulated two-regime threshold autoregressive process.
Figure 31. A simulated two-regime threshold autoregressive process.
Figure 32. A two-regime Markov model separates periods with different mean and variance patterns.
Figure 32. A two-regime Markov model separates periods with different mean and variance patterns.
Figure 33. Ordered models map a latent continuous score into ranked observed categories.
Figure 33. Ordered models map a latent continuous score into ranked observed categories.
Figure 34. Predicted category probabilities from a fitted ordered-probit model.
Figure 34. Predicted category probabilities from a fitted ordered-probit model.

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 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 7Create or update a Python object used in the analysis.
  7. Line 8Create or update a Python object used in the analysis.
  8. Line 9Create or update a Python object used in the analysis.
  9. Line 10Display a result so students can inspect the output.
  10. Line 11Display a result so students can inspect the output.

Verified source output

0.726 0.263
0.805 [0.282 0.518 0.2 ]
0.805
[0.282 0.518 0.2  ]

Interpretation. The unconditional mean is positive even though neither regime includes an intercept, illustrating nonlinear asymmetry.

Interpretation. At x=0, the central category is most probable. The probabilities sum to one and depend on both slope and thresholds.

Guided practice

  1. 1Re-run Demonstration 23.1 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 a TAR process under two threshold rules.
  2. 2Fit a Markov-switching mean model.
  3. 3Explain how bid-ask bounce affects short-lag returns.
  4. 4Plot ordered-probit category probabilities.

Source and downloads

Chapter 42 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

Nonlinear Models, Regimes, Market Microstructure, and Ordered Outcomes: live Python

Nonlinear Models, Regimes, Market Microstructure, and Ordered Outcomes: 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 3Create or update a Python object used in the analysis.
  3. Line 4Create or update a Python object used in the analysis.
  4. Line 5Run this Python instruction as part of the lesson workflow.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Create or update a Python object used in the analysis.
  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`rng = np.random.default_rng(23)`: Creates or updates a named object used by later steps.
  3. 3`x = np.zeros(300)`: Creates or updates a named object used by later steps.
  4. 4`for t in range(1, len(x)):`: Repeats the indented calculation across observations or simulation draws.
  5. 5`phi = -1.2 if x[t-1] < 0 else 0.55`: Creates or updates a named object used by later steps.
  6. 6`x[t] = phi*x[t-1] + rng.normal()`: Creates or updates a named object used by later steps.
  7. 7`print(round(x.mean(), 3), round((x < 0).mean(), 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, statsmodels, patsy

Expected output

A regression or inference table with coefficients, uncertainty, and short interpretation notes.

Learning goals

  • Load and inspect Ceteris Lab teaching sample.
  • Run the Python cells connected to Nonlinear Models, Regimes, Market Microstructure, and Ordered Outcomes.
  • Interpret the output using Simulate and interpret threshold autoregression and Markov-switching regimes and expected duration.

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 42 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 42 opening question: What if the same lag has a different effect in calm and stressed regimes, or the observed price change is an ordered category?

Quick quiz

Which practice should be avoided when applying Nonlinear Models, Regimes, Market Microstructure, and Ordered Outcomes?

Quick quiz

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

Quick quiz

Why does Chapter 42 matter in an applied econometrics workflow?

Key takeaway

Threshold models use observed regime rules; Markov models use latent states. Trading mechanisms can create dependence in transaction data. Ordered probit maps a latent index into ordered categories.

Download dataDownload resultsDownload interpretation