Chapter 8: Modelling Long-Run Relationships in Finance#

This chapter demonstrates cointegration analysis following Chapter 8 of the textbook. Cointegration captures long-run equilibrium relationships between non-stationary time series - a key concept for analysing asset prices, interest rates, and other financial variables.

Note

This chapter requires the TSMT (Time Series MT) module in addition to base GAUSS.

Example 1: Cointegration of Interest Rates#

This example tests for cointegration between short-term and long-term interest rates (the term structure). We demonstrate:

  • Unit root testing of individual series

  • Engle-Granger two-step cointegration test

  • Johansen cointegration test

  • VAR model estimation with cointegration diagnostics

Getting Started#

To run this example you will need:

  • The TSMT module installed

  • The BrooksEconFinLib package with the example data

Step One: Load and examine the data#

We use U.S. Treasury yields: the 3-month T-Bill rate and 10-year Treasury bond rate.

// Load the TSMT library
library tsmt;

// Create file name with full path
data_file = getGAUSSHome() $+ "pkgs/BrooksEcoFinLib/examples/macro.dta";

// Load the data
macro = loadd(data_file);

// Extract interest rate series
ustb3m = packr(macro[., "USTB3M"]);
ustb10y = packr(macro[., "USTB10Y"]);

print "Interest Rate Data";
print "==================";
print "3-Month T-Bill mean:  " meanc(ustb3m);
print "10-Year T-Bond mean:  " meanc(ustb10y);
print "Observations:         " rows(ustb3m);

Interest Rate Data
==================
3-Month T-Bill mean:      3.2969
10-Year T-Bond mean:      5.0754
Observations:               385

Step Two: Test for unit roots#

Before testing for cointegration, we must verify that each series is I(1) - integrated of order one. We use the ADF test.

// DF-GLS test on 3-month rate
print "DF-GLS Test for 3-Month T-Bill";
print "==============================";
{ tstat3m, crit3m } = dfgls(ustb3m, 12, 0);

// DF-GLS test on 10-year rate
print "";
print "DF-GLS Test for 10-Year T-Bond";
print "==============================";
{ tstat10y, crit10y } = dfgls(ustb10y, 12, 0);

Both series should fail to reject the unit root null, indicating they are non-stationary in levels. However, their first differences should be stationary.

Step Three: Engle-Granger cointegration test#

The Engle-Granger two-step method:

  1. Estimate the cointegrating regression: \(y_t = \alpha + \beta x_t + u_t\)

  2. Test the residuals for stationarity using ADF

If the residuals are stationary, the series are cointegrated.

// Engle-Granger cointegration test
// vmcadfmt(y, x, p, l)
// y = dependent variable (10-year rate)
// x = explanatory variable (3-month rate)
// p = 0 (include constant)
// l = 4 (number of lagged differences)

print "Engle-Granger Cointegration Test";
print "=================================";

{ alpha, tstat, crit } = vmcadfmt(ustb10y, ustb3m, 0, 4);

print "";
print "ADF test on cointegrating residuals";
print "Alpha (autoregressive parameter): " alpha;
print "t-statistic:                      " tstat;
print "";
print "Critical values (1%, 5%, 10%):";
print crit[1:3]';

Engle-Granger Cointegration Test
=================================

ADF test on cointegrating residuals
Alpha (autoregressive parameter):    0.9638
t-statistic:                        -3.0084

Critical values (1%, 5%, 10%):
      -3.9024       -3.3271       -3.0372

The t-statistic (-3.01) is compared to special critical values for cointegrating regressions (not standard ADF critical values). Here, the test is marginally significant at the 10% level.

Step Four: Johansen cointegration test#

The Johansen procedure uses a VAR framework to test for multiple cointegrating relationships simultaneously.

// Combine series into matrix
y = ustb3m ~ ustb10y;

// Johansen test
// vmsjmt(x, p, k, nodet)
// x = data matrix
// p = 0 (constant in cointegrating equation)
// k = 2 (number of lags)
// nodet = 0 (include constant)

print "Johansen Cointegration Test";
print "===========================";

{ ev, evec, lr1, lr2 } = vmsjmt(y, 0, 2, 0);

print "";
print "Eigenvalues:";
print ev';
print "";
print "Trace Statistics:";
print "H0: r=0  " lr1[1];
print "H0: r<=1 " lr1[2];
print "";
print "Maximum Eigenvalue Statistics:";
print "H0: r=0  " lr2[1];
print "H0: r<=1 " lr2[2];

Johansen Cointegration Test
===========================

Eigenvalues:
     0.02429        0.00476

Trace Statistics:
H0: r=0     11.2150
H0: r<=1     1.8221

Maximum Eigenvalue Statistics:
H0: r=0      9.3929
H0: r<=1     1.8221

The trace statistic tests the null hypothesis of at most r cointegrating vectors. Compare to critical values to determine the cointegration rank.

Step Five: VAR model with cointegration diagnostics#

The varmaFit function provides comprehensive cointegration testing as part of VAR estimation:

// Estimate VAR(2) model
struct varmamtOut vmo;
vmo = varmaFit(y, 2);

This produces extensive output including:

  • VAR coefficient estimates

  • ADF and Phillips-Perron unit root tests

  • ADF cointegration test

  • Johansen trace and maximum eigenvalue statistics

  • Characteristic roots for stationarity checking

  • Multivariate ACF and goodness-of-fit tests

================================================================================
Model:                VARIMA(2,0,0)            Number of Eqs.:                 2
Time Span:                  Unknown            Valid cases:                  385
Log Likelihood:            -162.921            AIC:                     -347.842
                                               SBC:                     -260.356
================================================================================

Johansen's Trace and Maximum Eigenvalue Statistics
                               r    Trace    Max. Eig
   Intercept                   0   11.2150    9.3929
                               1    1.8221    1.8221

Interpreting Results#

Cointegration evidence:

The Engle-Granger and Johansen tests provide marginal evidence for cointegration between the two interest rates. This is consistent with economic theory - the term structure suggests long-term and short-term rates should move together in the long run, with deviations representing the term premium.

Economic interpretation:

If rates are cointegrated, short-run deviations from the long-run relationship are temporary and the spread between rates is mean-reverting. This has implications for:

  • Fixed income portfolio management

  • Interest rate forecasting

  • Monetary policy transmission

Example 2: Error Correction Model#

When series are cointegrated, we can estimate an Error Correction Model (ECM) that captures both short-run dynamics and long-run equilibrium adjustment.

The ECM specification:

\(\Delta y_t = \alpha(y_{t-1} - \beta x_{t-1}) + \gamma \Delta x_t + \epsilon_t\)

where \((y_{t-1} - \beta x_{t-1})\) is the error correction term representing deviation from long-run equilibrium.

// VAR in differences with error correction
// First, get the cointegrating residuals
b = ustb3m / ustb10y;  // OLS coefficient
ecm_term = ustb10y - b .* ustb3m;  // Error correction term

// The error correction term should be stationary
print "Testing stationarity of error correction term:";
{ tstat_ecm, crit_ecm } = dfgls(ecm_term, 12, 0);

Summary of Cointegration Functions#

TSMT provides the following cointegration-related functions:

Function

Description

vmsjmt()

Johansen trace and maximum eigenvalue tests

vmcadfmt()

Engle-Granger ADF test on cointegrating residuals

vmppmt()

Phillips-Perron unit root test

varmaFit()

VAR/VARMA estimation with cointegration diagnostics

vmc_sjtmt()

Johansen trace statistic critical values

vmc_sjamt()

Johansen max eigenvalue critical values

Function reference: vmsjmt(), vmcadfmt(), varmaFit(), dfgls()

Further reading:

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