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I am new to time series analysis. I am having trouble distinguishing between trend- and difference stationary time series: my application is a VARX model. Consider the following plot of gross fixed capital formation (GFCF) in the US from 1960 through 2019 (data below).

enter image description here

If I run the ADF and KPSS tests with a linear trend on this series I get the following test statistics and $p$-values:

Results of Dickey-Fuller Test:
Test Statistic                 -2.573359
p-value                         0.292306
#Lags Used                      2.000000
Number of Observations Used    57.000000
Critical Value (1%)            -4.127070
Critical Value (5%)            -3.490541
Critical Value (10%)           -3.173740

Results of KPSS Test:
Test Statistic           0.172378
p-value                  0.028019
Lags Used                4.000000
Critical Value (10%)     0.119000
Critical Value (5%)      0.146000
Critical Value (2.5%)    0.176000
Critical Value (1%)      0.216000

Clearly, the series is not linear-trend stationary. So I use statsmodel's detrend function to investigate further; the plots of a linear, quadratic, and cubic detrending look like

enter image description here

I see that there is not much difference between the quadratic and cubic cases, so a quadratic detrending should suffice. Indeed, when I run the ADF and KPSS tests (now with no trend) on the quadratically-detrended series I obtain

Results of Dickey-Fuller Test:
Test Statistic                 -4.743919
p-value                         0.000070
#Lags Used                      1.000000
Number of Observations Used    58.000000
Critical Value (1%)            -3.548494
Critical Value (5%)            -2.912837
Critical Value (10%)           -2.594129

Results of KPSS Test:
Test Statistic           0.063162
p-value                  0.100000
Lags Used                3.000000
Critical Value (10%)     0.347000
Critical Value (5%)      0.463000
Critical Value (2.5%)    0.574000
Critical Value (1%)      0.739000
InterpolationWarning: The test statistic is outside of the range of p-values available in the look-up table. The actual p-value is greater than the p-value returned.

I conclude that I can run a VAR model on the quadratically-detrended series (or alternatively, include these trend terms in the regression). I like this conclusion, since it accords with my economic intuition that a time series like GFCF should be mean reverting.

However, I could just as well have differenced my data to make it stationary. Indeed, a single difference passes both statistical tests, but since I see visually that there is a slight trend in the first differenced series, I take two differences just to be on the safe side.

enter image description here

Results of Dickey-Fuller Test:
Results of Dickey-Fuller Test:
Test Statistic                -6.295754e+00
p-value                        3.509375e-08
#Lags Used                     5.000000e+00
Number of Observations Used    5.200000e+01
Critical Value (1%)           -3.562879e+00
Critical Value (5%)           -2.918973e+00
Critical Value (10%)          -2.597393e+00

Results of KPSS Test:
Test Statistic            0.272058
p-value                   0.100000
Lags Used                35.000000
Critical Value (10%)      0.347000
Critical Value (5%)       0.463000
Critical Value (2.5%)     0.574000
Critical Value (1%)       0.739000
    
InterpolationWarning: The test statistic is outside of the range of p-values available in the look-up table. The actual p-value is greater than the p-value returned.

Needless to say, my conclusion regarding the type of stationarity in my series makes a big difference for my VARX model. Hence my questions:

  1. Is there any (statisatical) reason to go with one type of stationarity?
  2. Am I missing anything in my analysis, e.g. could the series be both trend and difference stationary?
  3. More generally, is the final decision ultimately a matter of interpretation and judgement? I am inclined to believe this: according to this webpage, "Unfortunately, for any finite amount of data there is a deterministic and stochastic trend that fits the data equally well."

Your thoughts and suggestions are appreciated.

Data. (GFCF, Chained index, divided by first observation.)

year
1960-01-01    1.000000
1961-01-01    0.997483
1962-01-01    1.080027
1963-01-01    1.153613
1964-01-01    1.326325
1965-01-01    1.607492
1966-01-01    1.922342
1967-01-01    1.914125
1968-01-01    1.960172
1969-01-01    2.072179
1970-01-01    2.039088
1971-01-01    1.940480
1972-01-01    2.109713
1973-01-01    2.285238
1974-01-01    2.600903
1975-01-01    2.405982
1976-01-01    2.457062
1977-01-01    2.624371
1978-01-01    2.893248
1979-01-01    3.034054
1980-01-01    3.153835
1981-01-01    3.338244
1982-01-01    3.038570
1983-01-01    2.853642
1984-01-01    3.334765
1985-01-01    3.623779
1986-01-01    3.411756
1987-01-01    3.391694
1988-01-01    3.449882
1989-01-01    3.859639
1990-01-01    4.001555
1991-01-01    4.005552
1992-01-01    4.092464
1993-01-01    4.084617
1994-01-01    4.473127
1995-01-01    5.054930
1996-01-01    5.607418
1997-01-01    5.862822
1998-01-01    6.159239
1999-01-01    6.165458
2000-01-01    6.471869
2001-01-01    6.440184
2002-01-01    5.788348
2003-01-01    5.558410
2004-01-01    5.685298
2005-01-01    6.158647
2006-01-01    6.472757
2007-01-01    7.255922
2008-01-01    7.437444
2009-01-01    6.385549
2010-01-01    6.459950
2011-01-01    7.015842
2012-01-01    7.403020
2013-01-01    7.737785
2014-01-01    7.791457
2015-01-01    7.904501
2016-01-01    7.981789
2017-01-01    8.050711
2018-01-01    8.336393
2019-01-01    8.656204
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  • $\begingroup$ Recall that differencing a time series that is not integrated causes the problem of overdifferencing with all of its nasty features (a unit-root MA component, increased error variance, ...). $\endgroup$ Aug 17 '21 at 10:16
  • $\begingroup$ @RichardHardy Thanks for the search term! If anything it makes me even more wary of differencing (and consequently biases me towards detrending). $\endgroup$
    – Anthony
    Aug 17 '21 at 10:43
  • 1
    $\begingroup$ This may be of interest: ideas.repec.org/p/fip/fedgif/447.html $\endgroup$ Oct 27 '21 at 13:53

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