Stationary vs. Trend-Stationary Time Series: Auto.Arima difference parameter

Question

I have the following time series training_ts that looks like this:

It appears to be stationary or "trend-stationary". When I analyze the stationarity using adf and kpss as well as nsdiffs, I receive the following output:

> adf.test(training_ts)

Augmented Dickey-Fuller Test

data:  training_ts
Dickey-Fuller = -5.5779, Lag order = 11, p-value = 0.01
alternative hypothesis: stationary

Warning message:
In adf.test(training_ts) : p-value smaller than printed p-value
> kpss.test(training_ts, null = c("Level"))

KPSS Test for Level Stationarity

data:  training_ts
KPSS Level = 10.369, Truncation lag parameter = 9, p-value = 0.01

Warning message:
In kpss.test(training_ts, null = c("Level")) :
  p-value smaller than printed p-value
> kpss.test(training_ts, null = c("Trend"))

KPSS Test for Trend Stationarity

data:  training_ts
KPSS Trend = 0.52976, Truncation lag parameter = 9, p-value = 0.01

Warning message:
In kpss.test(training_ts, null = c("Trend")) :
  p-value smaller than printed p-value
> nsdiffs(training_ts)
[1] 0

All of which seems to suggest that training_ts is indeed stationary. My next step is then to use the auto.arima package to do some forecasting with external regressors (training_xregs). However, the model that auto.arima returns is a d = 1 model, which is counterintuitive given that the nsdiff, adf.test, kpss.test output suggests training_ts is already stationary. Here is the auto.arima output:

> arima_model = auto.arima(y = training_ts, trace = TRUE, approximation = 
TRUE, xreg = training_xregs[, 2:ncol(training_xregs)])

 Fitting models using approximations to speed things up...

 Regression with ARIMA(2,1,2)(1,0,1)[7] errors : 9939.105
 Regression with ARIMA(0,1,0)           errors : 10624.08
 Regression with ARIMA(1,1,0)(1,0,0)[7] errors : 10289.05
 Regression with ARIMA(0,1,1)(0,0,1)[7] errors : 9949.029
 ARIMA(0,1,0)                              : 10622.07
 Regression with ARIMA(2,1,2)(0,0,1)[7] errors : 9936.425
 Regression with ARIMA(2,1,2)           errors : 9944.379
 Regression with ARIMA(2,1,2)(0,0,2)[7] errors : 9935.7
 Regression with ARIMA(1,1,2)(0,0,2)[7] errors : 9933.618
 Regression with ARIMA(1,1,1)(0,0,2)[7] errors : 9932.408
 Regression with ARIMA(0,1,0)(0,0,2)[7] errors : 10623.19
 ARIMA(1,1,1)(0,0,2)[7]                    : 9930.753
 ARIMA(1,1,1)(1,0,2)[7]                    : 9934.871
 ARIMA(1,1,1)(0,0,1)[7]                    : 9931.152
 ARIMA(0,1,1)(0,0,2)[7]                    : 9946.309
 ARIMA(2,1,1)(0,0,2)[7]                    : 9932.426
 ARIMA(1,1,0)(0,0,2)[7]                    : 10279.04
 ARIMA(1,1,2)(0,0,2)[7]                    : 9931.963
 ARIMA(0,1,0)(0,0,2)[7]                    : 10621.17
 ARIMA(2,1,2)(0,0,2)[7]                    : 9934.051

 Now re-fitting the best model(s) without approximations...

 ARIMA(1,1,1)(0,0,2)[7]                    : 9934.814

 Best model: Regression with ARIMA(1,1,1)(0,0,2)[7] errors 

So auto.arima believes that training_ts is not a stationary time series. Which is confusing given the above.

I then used auto.arima to specify a d=0 difference order with the following results:

> arima_model = auto.arima(y = training_ts, d = 0, trace = TRUE, approximation = TRUE, xreg = training_xregs[, 2:ncol(training_xregs)])

 Fitting models using approximations to speed things up...

 ARIMA(2,0,2)(1,0,1)[7] with non-zero mean : 9937.261
 ARIMA(0,0,0)           with non-zero mean : 11318.17
 ARIMA(1,0,0)(1,0,0)[7] with non-zero mean : 10311.81
 ARIMA(0,0,1)(0,0,1)[7] with non-zero mean : 10593.26
 ARIMA(0,0,0)           with zero mean     : 11343.59
 ARIMA(2,0,2)(0,0,1)[7] with non-zero mean : Inf
 ARIMA(2,0,2)(2,0,1)[7] with non-zero mean : 9945.74
 ARIMA(2,0,2)(1,0,0)[7] with non-zero mean : Inf
 ARIMA(2,0,2)(1,0,2)[7] with non-zero mean : Inf
 ARIMA(2,0,2)           with non-zero mean : Inf
 ARIMA(2,0,2)(2,0,2)[7] with non-zero mean : Inf
 ARIMA(1,0,2)(1,0,1)[7] with non-zero mean : 9937.014
 ARIMA(1,0,1)(1,0,1)[7] with non-zero mean : Inf
 ARIMA(1,0,3)(1,0,1)[7] with non-zero mean : 9937.009
 ARIMA(0,0,2)(1,0,1)[7] with non-zero mean : Inf
 ARIMA(2,0,4)(1,0,1)[7] with non-zero mean : 9941.742
 ARIMA(1,0,3)(1,0,1)[7] with zero mean     : 9937.686
 ARIMA(1,0,3)(0,0,1)[7] with non-zero mean : Inf
 ARIMA(1,0,3)(2,0,1)[7] with non-zero mean : 9945.782
 ARIMA(1,0,3)(1,0,0)[7] with non-zero mean : Inf
 ARIMA(1,0,3)(1,0,2)[7] with non-zero mean : 9939.034
 ARIMA(1,0,3)           with non-zero mean : Inf
 ARIMA(1,0,3)(2,0,2)[7] with non-zero mean : 9944.402
 ARIMA(0,0,3)(1,0,1)[7] with non-zero mean : Inf
 ARIMA(2,0,3)(1,0,1)[7] with non-zero mean : 9939.694
 ARIMA(1,0,4)(1,0,1)[7] with non-zero mean : 9938.988

 Now re-fitting the best model(s) without approximations...

 ARIMA(1,0,3)(1,0,1)[7] with non-zero mean : 9934.032

 Best model: Regression with ARIMA(1,0,3)(1,0,1)[7] errors 

So the error for the d = 0 model is less than the d = 1 error. A few questions:

1. Is training_ts truly stationary or is it "trend stationary"? And if it is "trend stationary", I'll assume that means that when the trend is removed, the variance is stable vs. a truly stationary time series that has a stable variance without adjusting for the trend.
2. If training_ts is truly stationary, as adf.test, kpss.test, nsdiffs seem to suggest, why would auto.arima return a d = 1 model with a higher error than the specified d = 0 model?
3. If 'training_ts' is "trend-stationary", should I be using the 'd = 1' model? I assumed that auto.arima would handle trend-stationary time series by removing the trend without a need to apply any differencing?

Thank you!

Answer 1

For KPSS test, the null hypothesis is stationary. Since value is 0.01, it shows that series is non statoinary. Since Auto arima use KPSS test to test for stationary, it is differe