Lesson 6.8: Putting It Together: The Vector Error Correction Model (VECM)

This lesson introduces the culminating model of our time series journey: the Vector Error Correction Model (VECM). The VECM elegantly solves the puzzle of modeling cointegrated systems by combining the short-run, system-wide dynamics of a VAR with the long-run equilibrium constraints from cointegration. It is the definitive framework for analyzing and forecasting interconnected, non-stationary systems.

Part 1: The Central Dilemma and the VECM Solution

We are faced with a fundamental dilemma. We have two non-stationary, $I(1)$ time series, like the prices of Coca-Cola (KO) and Pepsi (PEP).

Our **VAR model** (Lesson 6.6) requires stationary data. The only way to use it is to take the first difference of both series, $\Delta KO_t$ and $\Delta PEP_t$ , and model them as a VAR in differences. This captures the short-run dynamics but **completely ignores the long-run equilibrium relationship** between the price levels. It's like modeling the drunk and the dog's individual steps without acknowledging the leash.
Our **Cointegration test** (Lesson 6.7) is all about the long-run equilibrium. It tells us that the spread, $KO_t - \beta PEP_t$ , is stationary and mean-reverting. But it doesn't, by itself, tell us about the short-run dynamics or how the system reacts from period to period.

We need a model that does both. We need a model that describes how the short-run changes ( $\Delta Y_t$ ) are driven not only by their own past dynamics but also by how far the system was from its long-run equilibrium in the previous period.

The Core Insight: The Error Correction Mechanism

The VECM is built on a simple, intuitive idea called the **error correction mechanism**. Think back to the drunk and her dog.

If, in the last period ( $t-1$ ), the dog wandered too far to the right of the drunk, the leash ( $\text{the equilibrium error}$ ) became taut. What happens in the next period ( $t$ )?

The drunk might get pulled slightly to the right.
The dog might get pulled back slightly to the left.

The **size of the error** in the previous period directly influences the **change** in their positions in the current period. The VECM is the formal model of this adjustment process. The "error" from the cointegrating relationship is "corrected" in the subsequent period.

Part 2: From VAR in Levels to VECM

The VECM is not a new model from scratch; it is a clever algebraic rearrangement of a VAR model in the levels of the data. This insight comes from the **Granger Representation Theorem**.

Let's start with a simple VAR(1) model for two $I(1)$ variables, $y_{1,t}$ and $y_{2,t}$ :

y_{1,t} = \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + \epsilon_{1,t}

y_{2,t} = \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + \epsilon_{2,t}

With some algebra (subtracting $y_{t-1}$ from both sides and rearranging), we can transform this into the VECM form:

The Vector Error Correction Model (VECM) Specification

For a bivariate, cointegrated system, the VECM takes the following form:

\Delta y_{1,t} = \gamma_1 (y_{2,t-1} - \beta y_{1,t-1}) + \text{lags}(\Delta y_{1,t-1}, \Delta y_{2,t-1}) + \epsilon_{1,t}

\Delta y_{2,t} = \gamma_2 (y_{2,t-1} - \beta y_{1,t-1}) + \text{lags}(\Delta y_{1,t-1}, \Delta y_{2,t-1}) + \epsilon_{2,t}

Let's deconstruct this:

$\Delta y_{1,t}$ , $\Delta y_{2,t}$ : The dependent variables are the **changes** in the series. The left side of the equation is stationary.
$(y_{2,t-1} - \beta y_{1,t-1})$ : The **Error Correction Term**. This is the heart of the model. It is the lagged value of the equilibrium error from the cointegrating relationship. It's the "leash."
$\gamma_1, \gamma_2$ : The **Adjustment Coefficients** or **Speed of Adjustment**. These are the most important parameters. $\gamma_1$ tells us what percentage of the previous period's disequilibrium is corrected in $y_{1,t}$ 's value in the current period.
$\text{lags}(\Delta y_{1,t-1}, \Delta y_{2,t-1})$ : The lagged changes of all variables. This is the **VAR part** of the model that captures the short-run dynamics.

Part 3: Interpreting the VECM - The Speed of Adjustment

The key to interpreting a VECM lies in the signs and significance of the adjustment coefficients ( $\gamma$ 's).

Interpreting the Adjustment Coefficients (

\gamma

)

Consider the error term defined as $Z_{t-1} = y_{2,t-1} - \beta y_{1,t-1}$ .

Suppose last period, $y_2$ was "too high" relative to its equilibrium with $y_1$ , so $Z_{t-1} > 0$ .
For the system to return to equilibrium, we expect $y_2$ to decrease ( $\Delta y_{2,t} < 0$ ) and/or $y_1$ to increase ( $\Delta y_{1,t} > 0$ ).
This means we would expect the estimated coefficient $\hat{\gamma}_2$ to be **negative** (a positive error leads to a negative change) and the coefficient $\hat{\gamma}_1$ to be **positive**.
The magnitude of the coefficient tells you how quickly the correction happens. A $\gamma$ of -0.05 means that 5% of the disequilibrium is corrected in the next period.
At least one of the adjustment coefficients must be statistically significant for the error correction mechanism to be valid.

Part 4: Estimation - The Johansen Test

The Engle-Granger two-step method is intuitive, but it has limitations. It can only test for one cointegrating relationship, and the two-step estimation process can be inefficient.

The modern, standard approach for analyzing cointegrated systems is the **Johansen test**. It is a more powerful and comprehensive procedure that allows us to:

Test for the **number of cointegrating relationships** in a system with more than two variables.
Estimate all the parameters of the VECM in a single, efficient step using Maximum Likelihood Estimation.

The Johansen test is mathematically complex (based on matrix algebra and eigenvalues), but its interpretation is straightforward. It provides two test statistics (Trace and Max-Eigen) to help the user decide on the cointegrating rank of the system.

Part 5: Python Implementation - VECM for Macro Data

Fitting and Interpreting a VECM in Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.api import VECM
from statsmodels.tsa.vector_ar.vecm import select_order, select_coint_rank

# --- 1. Load and Prepare Data ---
# Let's use two interest rate series, which are often cointegrated.
# 3-Month Treasury Bill (short-term) and 10-Year Treasury Bond (long-term)
data = pd.read_csv('https://raw.githubusercontent.com/statsmodels/statsmodels/main/statsmodels/datasets/macrodata/macrodata.csv', index_col=0, parse_dates=True)
interest_rates = data[['tbilrate', 'unemp']].dropna()

# --- 2. Determine the Number of Cointegrating Relationships (Johansen Test) ---
# The select_coint_rank function runs the Johansen test for us.
# We test with a constant term ('c') in the cointegrating relationship.
rank_test = select_coint_rank(interest_rates, det_order=0, k_ar_diff=1)
print(rank_test)
# The output table helps you decide the rank (number of cointegrating equations).
# Let's assume the test suggests a rank of 1.
coint_rank = 1

# --- 3. Determine the VAR Lag Order ---
# We use the select_order function on the differenced data.
lag_order_result = select_order(interest_rates.diff().dropna(), maxlags=10, deterministic="n")
# print(lag_order_result) # Assume it suggests 2 lags for the VAR part.
num_lags = 2

# --- 4. Fit the VECM ---
# We have a cointegrating rank of 1 and 2 lags for the short-run dynamics.
# The VECM order is p-1 = 2-1 = 1.
model = VECM(interest_rates, k_ar_diff=1, coint_order=coint_rank, deterministic='n')
vecm_fit = model.fit()
print(vecm_fit.summary())

# --- 5. Interpreting the Summary ---
# The summary is dense. Key things to look for:
# 1. The 'gamma' coefficients (loading coefficients): These are the adjustment speeds.
#    - We'd expect the short-term rate's adjustment to be larger and more significant.
# 2. The 'beta' coefficients: These define the long-run cointegrating relationship.

# --- 6. Forecast ---
# Forecast the next 12 months
forecast = vecm_fit.predict(steps=12)
print("\n12-Month Interest Rate Forecast:")
print(forecast)

What's Next? The Capstone Project

Congratulations. You have reached the summit of our time series modeling curriculum. The VECM represents the synthesis of everything we have learned: stationarity, autoregression, and long-run equilibrium.

Theory is essential, but application is where true mastery is forged. It is now time to take all the powerful tools from this module—cointegration, the VECM framework, and the logic of mean reversion—and apply them to build a complete, end-to-end quantitative trading strategy from scratch.

In our final lesson, we will embark on our capstone project: **Building a Pairs Trading Strategy with Cointegration**.

Up Next: Capstone Project: Building a Pairs Trading Strategy