# trade off between stationary and seassonal time series

I have the annual change in natural logs of house prices as my monthly series. When I undertake the augmented Dickey-Fuller test I observe that is not stationary, and when I plot the auto-correlation function I observe that any seasonality has been removed. Given that is not stationary I take the first differences of the series (after noting that the Hodrik-Prescott filter does get rid of the non-stationary problem) but this process leads to my series being yearly seasonal. It seems that I am facing a trade-off between tackling stationarity and seasonality of the series. Here is the trouble maker:

House Prices =                [1]  0.060761055  0.069950884  0.092563502  0.096516298  0.081337114  0.083240450  0.054460548
[8]  0.080128787  0.068890937  0.083673567  0.064829087  0.041281629  0.061894862  0.091175931
[15]  0.080636150  0.087343516  0.111036660  0.108404162  0.257142914  0.102603577  0.116511896
[22]  0.114216916  0.127491093  0.137534698  0.129671913  0.111374618  0.108207369  0.131192210
[29]  0.132858904  0.131548705  0.000000000  0.114723242  0.111291747  0.112244978  0.102148870
[36]  0.092714275  0.054940020  0.085817967  0.074571996  0.055771684  0.042966671  0.044293063
[43]  0.032258984 -0.013020362 -0.030328532 -0.027837669 -0.035178644 -0.030125696 -0.054178897
[50] -0.058369321 -0.086977111 -0.082298625 -0.055672332 -0.070839196 -0.063850213 -0.025592358
[57] -0.004794181  0.003381210  0.024675240  0.011343808  0.042312953  0.059378202  0.056705519
[64]  0.076140995  0.039093124  0.040937556  0.048601014  0.083007151  0.033627306  0.021100200
[71]  0.008240520  0.021195008  0.005731057 -0.019777949 -0.013459304 -0.000481840  0.019458685
[78] -0.006134143  0.014640055 -0.018973359 -0.000635282  0.001130525  0.010367129 -0.031045975
[85] -0.000818617  0.016362232  0.028955592 -0.010045934 -0.015040926  0.022725145 -0.020442085
[92] -0.037055871 -0.013855981 -0.030070854 -0.042353414  0.023625106 -0.007840661 -0.001485553
[99] -0.001935076 -0.007745723  0.009077794 -0.012910669  0.013267345  0.020921684  0.012565305
[106]  0.025839041  0.057484514  0.011076450  0.033411114  0.041928308  0.027035973  0.046183315
[113]  0.054138355  0.068260861  0.032538699  0.050712983  0.072123289  0.033179492  0.018661604
[120]  0.051411799  0.073918352  0.070047567  0.200971642  0.143036668 -0.031191576 -0.014253957
[127] -0.003606198  0.012085764 -0.021088330  0.017174875  0.020631125  0.007208228


Any reasons why this might be happening?

Kind regards!

It would be a bit strange to find a unit root in growth rates. I ran the ADF test on the data series supplied above and rejected a unit root. (I used function ur.df from "urca" package in R. I varied lag order selection between AIC and BIC. I also tried different specifications: "none", "drift" and "trend", although only "none" is sensible for monthly growth rates in my opinion.) The graph of the data does not seem to suggest an obvious unit root either: