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A linear regression between "Number of Australian Air Passengers" and "Rice Production in Guinea" reveals a "strong" but probably spurious relationship between the two time series.

library(forecast)
library(fpp)

?ausair
?guinearice

tseries::adf.test(ausair) # Non-stationary
tseries::adf.test(guinearice) # Non-stationary

## Is the number of Air Transport Passengers in Australia related to
## rice production in the country of Guinea (in Africa)?

spurious_lm <- tslm(ausair ~ guinearice)
summary(spurious_lm)

A Phillips-Ouliaris test reveals that the regression is spurious and should be thrown out.

tseries::po.test(spurious_lm$model)

I want alternate viewpoints, so I test the residuals using the ADF, KPSS, and PP tests from the tseries package.

## ADF test (null = random walk)

tseries::adf.test(spurious_lm$residuals) # spurious

## KPSS test (null = NOT random walk)

tseries::kpss.test(spurious_lm$residuals, null="Trend") # spurious

## PP Test (null = random walk)

tseries::pp.test(spurious_lm$residuals) # spurious

All the tests agree. The relationship is spurious. Throw the model out.

However, I have questions about the default arguments in the tests. All the default arguments in the 3 tests include "drift" and "trend". When I remove the drift and trend (via the urca package), I get the exact opposite results.

urca::summary(urca::ur.df(spurious_lm$residuals, type="none"))  ### looks wrong
urca::summary(urca::ur.kpss(spurious_lm$residuals, type="mu")) ### looks wrong
urca::summary(urca::ur.pp(spurious_lm$residuals, model="constant")) ### looks wrong

Questions:

1. Why is the drift and trend terms so important for testing the stationarity of regression residuals?

2. Should I always include drift and trend when testing for unit roots in the residuals? Or in any time series?

Can I technically use following phrases interchangeably?

  1. "Testing for stationary residuals"
  2. "Testing for units roots in the residuals"
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