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I am following Pfaff 2011 chapter 3 and 5.1 to find the order of integration of a time series $y_t$ by augmented Dickey-Fuller (ADF). Basically what we do here is testing whether $y$ has a unit root (0-hypothesis) against the alternative hypothesis $|$root(s)$|$ $<$ 1. Right?

I am using the ur.df function from ucra package. This is how my data looks like:

enter image description here

I am performing the regressions for three different cases: $$ \Delta y_t = \beta_1+\beta_2t+\pi y_{t-1} \sum^{k=2}_{j=1} \gamma_j\Delta y_{t-i} + \epsilon_t \text{ (trend)} $$

$$ \Delta y_t = \beta_1+\pi y_{t-1} \sum^{2}_{j=1} \gamma_j\Delta y_{t-i} + \epsilon_t \text{ (drift)} $$

$$ \Delta y_t = \pi y_{t-1} \sum^{2}_{j=1} \gamma_j\Delta y_{t-i} + \epsilon_t \text{ (none)} $$ The test-statistics values for the three different regressions are presented here:

enter image description here

The critical values are the 5% ones and are stated in the R output from ur.df.

MY QUESTION(s): How can we use the test statistics (output) to make conclusions about the order of integration? In my case, I need to reject the hypothesis about unit-root (with 95% confidence) in all three cases but where does that leave me in determination of order of integration? The alternativ hypothesis is that $y$ is stationary but by looking at the data it is nowhere near stationary.

As far as I know the conclusion has to be made in two steps: does $y$ unit root? If yes, can we remove the unit root by differencing once, then $y$ is I(1). Am I correct?

It basically comes down to the fact that I don't know what those output statistics statistics (tau and phi) actually are... The book explains it a bit in chapter 3 but not to much.

My code:

y.diff = diff(y)
par(mfrow = c(1,2))
plot(y, type  = "l"); plot(y.diff, type = "l")


ur.trend = ur.df(y, lags = 2, type = 'trend')
summary(ur.trend)

ur.drift = ur.df(y, lags = 2, type = 'drift')
summary(ur.drift)

ur.none = ur.df(y, lags = 2, type = 'none')
summary(ur.none)

My data

y = c(2.8554765738231,2.86031919829889,2.86031919829889,2.86273174470428,2.86513848473484,2.86753944627238,2.86993465699834,2.87232414439572,2.87708605815596,2.87945853850959,2.88418667970554,2.886542393398,2.8912372377022,2.89823843316119,2.90749775857399,2.91438641956917,2.91895263754875,2.92576310655127,2.92576310655127,2.94370080723794,2.94813540430581,2.95694603398797,2.96350343453412,2.97218027855981,2.98292152039122,2.99354861296551,3.00406395696504,3.01032049457934,3.02066184837407,3.02885861557825,3.03496239851627,3.0470593202251,3.06098989376104,3.06886395019195,3.07862012513732,3.08635710763947,3.0940346885385,3.09975443721129,3.10354950817984,3.10733023101975,3.10921524571552,3.11671997136958,3.1260223640319,3.12971922591323,3.13707220021848,3.14255166598311,3.14981122599591,3.15522151872416,3.15881218685489,3.16060269762868,3.16239000820277,3.16417412999628,3.16950747597164,3.17481252820133,3.18008958530218,3.18533894118832,3.19056088516947,3.19575570204658,3.19920398132549,3.20435420730181,3.20947804430168,3.2196476234713,3.22469389153893,3.23138287968973,3.2347106697824,3.23968168250442,3.24462810643975,3.24955018364518,3.25281816040979,3.2577001880071,3.26417270251271,3.26899974326103,3.2753997651065,3.28017304385916,3.28650217891081,3.29435767885111,3.30370354126935,3.31142558736326,3.31908846010883,3.32517675897609,3.33424026512943,3.34322236144526,3.34916591534611,3.35507435203228,3.3638720627358,3.37403918009125,3.38410396670739,3.39406846196682,3.40533618681316,3.4164783633664,3.42749775861602,3.43704111666292,3.44649425930667,3.45718954842342,3.46777165775396,3.47954419397922,3.49374714415722,3.50648616993465,3.52031417154194,3.53025214153207,3.54009231933358,3.54983661080826,3.55104799799456,3.55828566529479,3.56666393094079,3.57615391912563,3.58672360061532,3.59602599327764,3.60524264838256,3.61437513194583,3.62117017507866,3.62679765194439,3.63350908653237,3.6423881681735,3.6533772897491,3.66099844803017,3.67071250175065,3.67392966270242,3.6782031734798,3.68139636405896,3.68457939061908,3.68775231765966,3.68775231765966,3.69091520906816,3.70034429935702,3.71174929597101,3.71689069547143,3.71893987651636,3.71893987651636,3.72302568403894,3.7220057961382,3.71893987651636,3.72098486700409,3.7220057961382,3.72302568403894,3.73013592222145,3.73417633175845,3.73619043113015,3.74020650078504,3.7432080037888,3.74620052472525,3.75017667310489,3.75314844349405,3.75611140862471,3.76299104263824,3.76787625020429,3.77079596030763,3.77467569359322,3.77660993017305,3.77854043270311,3.78239029308855,3.78526799291616,3.78813743534412,3.79195059535053,3.79385173660755,3.79574927038375,3.79764321034382,3.80330360281553,3.80799624399372,3.80986715192953,3.81173456610433,3.81452916503534,3.81638790162997,3.81824318973171,3.82009504211279,3.82378849044814,3.82838619969677,3.83296286672418,3.8366091788772,3.83478768474881,3.82930321030665,3.82563011159479,3.82838619969677,3.83204920745551,3.83296286672418,3.83387569198154,3.83751868326004,3.83933521418644,3.84114845131062,3.84476509178081,3.85016564496081,3.85374987818862,3.85732131055622,3.86176773851335,3.86442613617996,3.86884115438908,3.87148083915985,3.87586488826839,3.87935834160295,3.88197044583087,3.88544267154162,3.8871742737059,3.89062850857399,3.89235116188544,3.89492959047338,3.90092019461054,3.90347665105184,3.90772294193329,3.91195127804281,3.91616181057915,3.92035468883919,3.92369637950109,3.92702694044826,3.93034644557111,3.93448039110284,3.93777528699969,3.94269736420512,3.95003535065803,3.95489759686607,3.95812600967819,3.96134403337239,3.96134403337239,3.96375077340295,3.96854694566645,3.97252625851793,3.97569834682407,3.98515468206611,3.98906858138724,3.99374512927114,3.99607522741277,3.99762561532022,4.00380324114418,4.00841154423037,4.01680533635884,4.0236209628939,4.03039045094915,4.03263680157552,4.0363695420052,4.04008840088394,4.04083051594583,4.04083051594583,4.04305356243378,4.04674769297392,4.04969320320368,4.05116271118304,4.05409526460431,4.05701924329575,4.05847803275941,4.06284167604718,4.06574022880119,4.06646355613834,4.06863040422343,4.07223141639702,4.07438581579405,4.07653558373054,4.07939477081059,4.08224580624142,4.08437876119157,4.08650717632639,4.09075046416426,4.09356935257068,4.09497582256343,4.09848336737311,4.10058200403003,4.10197865230175,4.10546176275457,4.10823954231847,4.10893278324745,4.11031782502011,4.11239179741924,4.1137720598882,4.11790145323021,4.12064493917596,4.1226976254937,4.1226976254937,4.12542800261656,4.12815094507502,4.1288305237624,4.1308664932475,4.13357468718438,4.1369496482009,4.14098462341308,4.14299602201277,4.14366559016634,4.14633938855074,4.14834005610704,4.15100140165655,4.15365568324143,4.15564178121306,4.15960218242915,4.16157651673287,4.16354696072017,4.16485843631798,4.16682243040264,4.16812962013126,4.17073888449488,4.17204096801217,4.17334135831246,4.17852607504665,4.18046343438687,4.18368404908691,4.18753514498901,4.18945514557884,4.1913714668296,4.19328412281593,4.19455719746273,4.19773281143662,4.20089837280118,4.20405394499999,4.20657125281137,4.20845508382374,4.21033537268859,4.2109613507998,4.21158693730753,4.21158693730753,4.21346135210188,4.21470901290004,4.21719967421249,4.21968414754015,4.22154346013034,4.22278108405069,4.22339932209202,4.22463465270001,4.22463465270001,4.22463465270001,4.22586845914894,4.22833151458133,4.22956077103606,4.23201475999763,4.23323949989353,4.23385130780493,4.23629480321741,4.23751431556367,4.2393408000897,4.24116395465122,4.24116395465122,4.241770934943,4.24842351466402,4.24902610582417,4.24902610582417,4.25323410722437,4.25563075306897,4.25981111150073,4.26159735788219,4.26338041927585,4.26575289962948,4.26871060660671,4.27283675451789,4.27870187397029,4.27811690758917,4.27987078117527,4.28569490946126,4.28859430262026,4.28859430262026,4.29379211969387,4.29551873882786,4.29724238188603,4.29953596087313,4.30524699866864,4.30752231250578,4.30809033296979,4.30979246104032,4.31488153054779,4.31713505262823,4.3154453873431,4.3154453873431,4.31938350776324,4.31657214802569,4.31600892638309,4.3154453873431,4.31713505262823,4.31882186775971,4.32162691868732,4.32609869807871,4.32721352539596,4.32777047333572,4.32999516835784,4.33276909524056,4.3344297654257,4.3366397110765,4.33829397117253,4.339945499211,4.34433628562849,4.34979779565401,4.3514304490779,4.34761676970765,4.34597787286444,4.34707076914809,4.3503423096887,4.35468778094821,4.35793453704708,4.3568534558607,4.35739414254595,4.36009319951511,4.36439659507517,4.3665413680153,4.36868155074767,4.37028368727195,4.3745435446189,4.37825609197509,4.37931429313204,4.3798429740854,4.38300920364345,4.38826407648181,4.39296996751922,4.39296996751922,4.39244818300438,4.39661485569923,4.40024650690209,4.40334888788057,4.4028324923101,4.40334888788057,4.40952492398114,4.41566305066993,4.4293376116901,4.43084552853778,4.42581027117213,4.42581027117213,4.43184954468637,4.43235117499536,4.43385455788939,4.43884957326992,4.44183465011337,4.44431542538713,4.44975156429939,4.45417743825625,4.44925858918465,4.44481084278642,4.44530601486883,4.45073678605396,4.4523946921102,4.45626554124374,4.46145218873693,4.46444835821893,4.46857283487586,4.47088690705532,4.472665919522,4.47297415272212,4.47720275296755,4.48028124736865,4.48810941145153,4.49100323485448,4.49444501022165,4.4968599188591,4.50043156075185,4.50274326771736,4.50864331978548,4.51906703858265,4.52618310397615,4.52469351936786,4.52554824566753,4.51691262926797,4.49904853517515,4.49078093012107,4.49330850445334,4.49694454668478,4.49595677614351,4.49696335189571,4.49843376422835,4.5066991240838,4.50640111422864,4.50974397401952,4.51167299948636,4.51467043448494,4.51801343102987,4.51853347218467,4.51918199406052,4.51822976418377,4.51856107742967,4.51879109150754,4.51827118434297,4.51785230146524,4.5197198101266,4.52118010685384,4.5227940531645,4.52626985186629,4.52880048872552,4.53280902016437,4.53604681504518,4.5392561342966,4.54441634568226,4.54909946173132,4.55227612552244,4.55227612552244,4.5548927360411,4.55804223333001,4.5602114267859,4.56088640646287,4.56273255134337,4.56297023158533,4.5656907237426,4.56782588877946,4.56991717091086,4.57157658167694,4.56950625990665,4.56867955510908,4.56896832333958,4.5747610410827,4.57952096094616,4.58221412570056,4.58053336952648,4.58041228059084,4.58239111035732,4.58780634820627,4.58499046823232,4.58290030609798,4.58331437584877,4.58569195537072,4.58764749435792,4.59003193998929,4.59040881365542,4.59094390151891,4.59278669196449,4.59542737145786,4.59784861317543,4.59894878638313,4.60098875952338,4.60285120969094,4.60475240717268,4.60607266745148,4.60719751985423,4.60703750570527,4.60710909414755,4.60691116065678,4.60502672607223,4.60193734941411,4.59542311103526,4.59762758300391,4.60089130870948,4.60153092106691,4.60498874794341,4.60798037783075,4.6095316554232,4.60942663127805,4.60723120378111,4.60838839945735,4.60937621577023,4.60853978781504,4.60860706417373,4.60707540610723,4.60946864225973,4.6127777579213,4.61544140986804,4.61837283216318,4.61830204557846,4.62018243352561,4.62204684682755,4.6246983682405,4.62609587115545,4.6291610487509,4.63330449669531,4.6340466939201,4.63317732477307,4.63441553731339,4.63405079292753,4.6349521663538,4.63580362739768,4.63950803873474,4.64406759840688,4.64475318985775,4.64776988111679,4.65006779038527,4.65402524193043,4.65597204467192,4.656489216595,4.65832321984005,4.66107990254192,4.66302497868627,4.66486691853443,4.66598713093577,4.6665070867314,4.66961324528145,4.66947873937287,4.6693323447362,4.6691344800908)
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The suggested order of integration is 1 as the non-stationarity should not be remedied by differencing and time trends.

There is more that one remedy to treat a non-stationary series such as yours besides simply differencing. It is possible to combine both differencing (stochastic structure) with deterministic structure ( counting numbers) suggesting possible trends and trend breakpoints . In your case the latter is the appropriate remedy/solution/model along with two pulses with trend breakpoints at or about periods 118 and 230 . This is readily confirmed by eye .

Here is the Actual/Fit and Forecast graph enter image description here with model enter image description here and here enter image description here with the following statistics enter image description here

The model residuals suggest sufficiency enter image description here with an ACF here enter image description here with a somewhat curious spike at lag 11 . Is the data monthly perhaps ?

In summary one level of differencing , three trends and two anomalies (pulses) thus a simple (1,1,0) arima structure with 3 exogenous variables.

The methods used here to identify the latent deterministic structure (the three trend indicators) are EXTENSIONS of the pionering work done by TSAY . Overlooked by many but not by everyone http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html.

The 3 regressions you ran are just a subset of the class of regressions available as suggested herein. Good analytics actually detects/pushes out the preferred regression.

Finally the forecasts with 95% confidence limits for the next 56 periods allowing for possible future pulses/anomalies .enter image description here

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  • $\begingroup$ If you are happy with my response please accept it to close the question $\endgroup$ – IrishStat Aug 15 at 8:21
  • $\begingroup$ thanks for the very! thorough answer.. Just out of curiosity: When I use ADF the way I do, have I then ASSUMED that the process is not explosive? Without verification before/after? $\endgroup$ – k.dkhk Aug 19 at 18:49
  • $\begingroup$ if you are asking is the null hypothesis = no growth yes ...... this is the assumption that you are testing and wishing to counter with a more realistic hypothesis . The most correct alternative to this hypothesis is what NEEDS to be found NOT assumed ala DF. $\endgroup$ – IrishStat Aug 19 at 19:35
  • $\begingroup$ Thank you very much for your help. I have not accepted the answer. While you do help me get around the problem in a different way, my main question: "How can we use the test statistics (output) to make conclusions about the order of integration?" have not been answered. $\endgroup$ – k.dkhk Aug 19 at 22:08
  • $\begingroup$ The order of integration is 1 in conjunction with 3 trends. Your test statistic does not per se tell anything about the order other than it is not a 0 , By explicitly restricting the incorporation of time trends , I obtained that it is a 1 wiith a strong ar(1) coefficient of .6 and some pulses with a null acf . . $\endgroup$ – IrishStat Aug 20 at 2:29

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