Testing Cointegration with Regression Residuals

Summary

The document considers whether an integrated dependent series and several integrated explanatory series are cointegrated when a principal component regression produces a stationary residual. Under the definition given, a stationary linear combination of integrated variables establishes cointegration, so the residual can serve as evidence against a spurious regression.

The key caution concerns inference: regression residuals are estimated from the same data and can appear more stationary by construction. Therefore, a unit-root test on the residual should use the appropriate residual-based null distribution and critical values, such as those associated with the Engle–Granger procedure, rather than ordinary augmented Dickey–Fuller critical values. The discussion is brief and does not specify model details, deterministic terms, lag selection, or how principal component estimation affects the test procedure.

Key ideas

• A stationary linear combination of integrated variables is the defining condition for cointegration.
• A stationary regression residual can therefore indicate cointegration among the modeled series.
• Residual-based unit-root tests require critical values adjusted for the fact that the residual was estimated.
• Ordinary augmented Dickey–Fuller critical values are not appropriate for this residual test.
• The explanation does not address specification choices or principal component estimation effects.

Tags

# How do I test if my betas form a co-integrated vector?


# How do I test if my betas form a co-integrated vector?












I have identified a model using principal component regression where $Y_t$ is explained by 4 factors such as:

$$Y_t = \beta_1 X_{1t} + \beta_2 X_{2t} + \beta_3 X_{3t} + \beta_4 X_{4t} + \epsilon_t$$

Where: $Y, X_1, X_2, X_3, X_4$ are $\text{I(1)}$ variables.

If I check that $\epsilon_t$ is stationary, can I assume that the variables are co-integrated and hence there was no spurious regression in my principal component regression?

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/29894

If a regression of an integrated variable on one or more integrated variables yields a stationary residual, the variables are cointegrated. This is a special case of the definition of cointegration.

Make sure to use appropriate null distribution and corresponding critical values when testing for absence of a unit root in $\epsilon_t$. Since $\epsilon_t$ is a residual rather than raw data, it is more likely to appear stationary by construction; that is why you need critical values as in Engle & Granger's procedure rather than the regular critical values of an augmented Dickey-Fuller (ADF) test.

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