Choosing Deterministic Trends in Johansen and Engle–Granger Cointegration Tests

Summary

The document explains how deterministic constants and trends can be specified in Johansen cointegration models. It lists five model forms, ranging from unrestricted constants and trends to no constant or trend, and says nested specifications can be compared sequentially with likelihood ratio tests. The choice affects the model being estimated, so testing trend significance in a single VEC specification may not capture the full set of alternatives.

For residual based cointegration tests, the constant or trend can instead be included when estimating the first-stage OLS relationship; the residual test regression then need not include them again. The suggested choice should be guided by economic theory. The answer cautions that the deterministic specifications can look similar in typical samples and that tests may have low power, limiting the reliability of a purely statistical choice. It points to Johansen's treatment for further implications, but gives no empirical example or detailed testing procedure.

Key ideas

• Johansen cointegration models allow several specifications for constants and deterministic trends.
• Nested Johansen specifications can be compared sequentially with likelihood ratio tests.
• Residual based tests can place deterministic terms in the first-stage OLS regression.
• Economic reasoning matters because the specifications may be difficult to distinguish and tests may have low power.

Tags

# Cointegration tests: how do you accurately test the necessity of time trends in the Johansen and Engle-Granger Test?


# Cointegration tests: how do you accurately test the necessity of time trends in the Johansen and Engle-Granger Test?












Is there a correct and up to date procedure? I just run the equation in VEC form and test the significance of the time trends? What are the possible problems that I should be aware of?

## Answer by user1483 (score 1)

https://quant.stackexchange.com/a/17913

In the Johansen methodology there are five models

- unrestricted constant and unrestricted trend

- unrestricted constant and restricted trend

- unrestricted constant and no trend

- restricted constant and no trend

- no constant and no trend.

As these models are nested they can be tested sequentially using likelihood ratio tests.

For the usual sample sizes these processes can look very similar and the tests probably have low power and it is better to use your economic understanding of the variables involved. Chapter 5 of Johansen (1955), Likelihood -based inference in cointegrated vector autoregressive models contains more details of these models and their implication on the models being estimated.

In the residual based cointegration test the trend and constant can be included in the OLS estimation of the residuals and there is then no need to include a constant or trend in the cointegrating test regression. The decision to include a constant or trend in the first OLS estimation should be based on economic theory.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.