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Using Regression Evidence to Assess a Time Series’ Integration Level

Article Quant Q&A · Author: macgivera
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Summary

The document presents four regressions for a series and its first and second differences, then asks which integration level is supported. The examples include lagged-level coefficients, standard errors, and Breusch–Godfrey p-values for residual autocorrelation. They illustrate evidence that may be considered when distinguishing a stationary series from one that becomes stationary after differencing.

No conclusion or interpretation is supplied, so the results do not establish a definitive integration order on their own. In particular, the listed regressions and autocorrelation checks are not a complete unit-root testing procedure. The note is useful as a prompt to examine regression specification and diagnostics, but readers need additional testing assumptions and inference before classifying the series.

Key ideas

  • Integration order concerns how many differences are needed to obtain stationarity.
  • The examples compare regressions in levels, first differences, and second differences.
  • Lagged-level estimates and their standard errors provide clues but do not alone determine integration order.
  • Breusch–Godfrey p-values assess residual autocorrelation, not whether a series has a unit root.

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Full text
# Specifying integration level of time series


# Specifying integration level of time series












Following model was estimated on 200 observations. How to specify the level of integration of $X_t?$ In brackets there are standard errors and p-value of Breusch-Godfrey test is also shown.

$X_t=0,02+\underset{(0,06)}{0,94}X_{t-1}+\varepsilon_t\qquad pv(BG)=0,765$

$\Delta X_t=0,03-\underset{(0,71)}{0,52}X_{t-1}+\varepsilon_t\qquad pv(BG)=0,878$

$\Delta X_t=0,03-\underset{(0,05)}{0,51}X_{t-1}-\underset{(0,09)}{0,02}\Delta X_{t-1}+\varepsilon_t\qquad pv(BG)=0,775$

$\Delta \Delta X_t=0,02-\underset{(0,05)}{0,62} \Delta X_{t-1}+\varepsilon_t\qquad pv(BG)=0,852$

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