Deterministic Trends and Unit-Root Tests in Cointegration

Summary

The document considers a long-run relation between exchange rates and purchasing-power parity whose residual appears stationary only when a deterministic trend is included. It asks whether that trend is acceptable and how to justify its inclusion. The response explains that unit-root test critical values depend on the deterministic terms in the test specification. In particular, including a trend changes the relevant Dickey–Fuller distribution and its critical values.

Those critical values are generated for different deterministic specifications, so the test must match the model being assessed. This addresses the statistical mechanics, but does not show that a trend is substantively appropriate for a particular economic relationship. The document gives no data, model selection procedure, or empirical evidence; researchers still need economic justification and careful specification to avoid treating trend inclusion as a way to force stationarity.

Key ideas

• Unit-root test critical values depend on whether deterministic terms such as a trend are included.
• Dickey–Fuller critical values differ between specifications with and without a trend.
• A unit-root test should use critical values calibrated to its deterministic specification.
• The response explains test calibration but does not establish that a trend fits the economic relationship.

Tags

# Trend in Cointegration relationship


# Trend in Cointegration relationship












I was estimating a long-run relationship of exchange rate and purchasing power parity. The residual of the long-run relation which should be $I(0)$, but it is only $I(0)$ when I introduce trend in the long-run relationship. Can someone provide the logic or study material that shows introducing trend is not a problem and how to justify it?

## Answer by user21240 (score 1)

https://quant.stackexchange.com/a/8634

The critical values of the unit root test you are using depends if there is a trend or not. For example, the quantiles of the Dickey-Fuller distribution is different when a trend is included from when a trend is not included, hence the critical values for your unit root test are different. The critical values of unit root tests are generated by simulation, with different kinds of deterministic terms to match the series you are considering.

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