# How to interpret the the results of augmented Dickey-Fuller test to make conclusions about the order of integration

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:

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:

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)


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 with model and here with the following statistics

The model residuals suggest sufficiency with an ACF 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 .

• If you are happy with my response please accept it to close the question Aug 15, 2019 at 8:21
• 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? Aug 19, 2019 at 18:49
• 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. Aug 19, 2019 at 19:35
• 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. Aug 19, 2019 at 22:08
• 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 . . Aug 20, 2019 at 2:29