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(2.8554765738231,2.86031919829889,2.86031919829889,2.86273174470428,2.86513848473484,2.86753944627238,2.86993465699834,2.87232414439572,2.87708605815596,2.87945853850959,2.88418667970554,2.886542393398,2.8912372377022,2.89823843316119,2.90749775857399,2.91438641956917,2.91895263754875,2.92576310655127,2.92576310655127,2.94370080723794,2.94813540430581,2.95694603398797,2.96350343453412,2.97218027855981,2.98292152039122,2.99354861296551,3.00406395696504,3.01032049457934,3.02066184837407,3.02885861557825,3.03496239851627,3.0470593202251,3.06098989376104,3.06886395019195,3.07862012513732,3.08635710763947,3.0940346885385,3.09975443721129,3.10354950817984,3.10733023101975,3.10921524571552,3.11671997136958,3.1260223640319,3.12971922591323,3.13707220021848,3.14255166598311,3.14981122599591,3.15522151872416,3.15881218685489,3.16060269762868,3.16239000820277,3.16417412999628,3.16950747597164,3.17481252820133,3.18008958530218,3.18533894118832,3.19056088516947,3.19575570204658,3.19920398132549,3.20435420730181,3.20947804430168,3.2196476234713,3.22469389153893,3.23138287968973,3.2347106697824,3.23968168250442,3.24462810643975,3.24955018364518,3.25281816040979,3.2577001880071,3.26417270251271,3.26899974326103,3.2753997651065,3.28017304385916,3.28650217891081,3.29435767885111,3.30370354126935,3.31142558736326,3.31908846010883,3.32517675897609,3.33424026512943,3.34322236144526,3.34916591534611,3.35507435203228,3.3638720627358,3.37403918009125,3.38410396670739,3.39406846196682,3.40533618681316,3.4164783633664,3.42749775861602,3.43704111666292,3.44649425930667,3.45718954842342,3.46777165775396,3.47954419397922,3.49374714415722,3.50648616993465,3.52031417154194,3.53025214153207,3.54009231933358,3.54983661080826,3.55104799799456,3.55828566529479,3.56666393094079,3.57615391912563,3.58672360061532,3.59602599327764,3.60524264838256,3.61437513194583,3.62117017507866,3.62679765194439,3.63350908653237,3.6423881681735,3.6533772897491,3.66099844803017,3.67071250175065,3.67392966270242,3.6782031734798,3.68139636405896,3.68457939061908,3.68775231765966,3.68775231765966,3.69091520906816,3.70034429935702,3.71174929597101,3.71689069547143,3.71893987651636,3.71893987651636,3.72302568403894,3.7220057961382,3.71893987651636,3.72098486700409,3.7220057961382,3.72302568403894,3.73013592222145,3.73417633175845,3.73619043113015,3.74020650078504,3.7432080037888,3.74620052472525,3.75017667310489,3.75314844349405,3.75611140862471,3.76299104263824,3.76787625020429,3.77079596030763,3.77467569359322,3.77660993017305,3.77854043270311,3.78239029308855,3.78526799291616,3.78813743534412,3.79195059535053,3.79385173660755,3.79574927038375,3.79764321034382,3.80330360281553,3.80799624399372,3.80986715192953,3.81173456610433,3.81452916503534,3.81638790162997,3.81824318973171,3.82009504211279,3.82378849044814,3.82838619969677,3.83296286672418,3.8366091788772,3.83478768474881,3.82930321030665,3.82563011159479,3.82838619969677,3.83204920745551,3.83296286672418,3.83387569198154,3.83751868326004,3.83933521418644,3.84114845131062,3.84476509178081,3.85016564496081,3.85374987818862,3.85732131055622,3.86176773851335,3.86442613617996,3.86884115438908,3.87148083915985,3.87586488826839,3.87935834160295,3.88197044583087,3.88544267154162,3.8871742737059,3.89062850857399,3.89235116188544,3.89492959047338,3.90092019461054,3.90347665105184,3.90772294193329,3.91195127804281,3.91616181057915,3.92035468883919,3.92369637950109,3.92702694044826,3.93034644557111,3.93448039110284,3.93777528699969,3.94269736420512,3.95003535065803,3.95489759686607,3.95812600967819,3.96134403337239,3.96134403337239,3.96375077340295,3.96854694566645,3.97252625851793,3.97569834682407,3.98515468206611,3.98906858138724,3.99374512927114,3.99607522741277,3.99762561532022,4.00380324114418,4.00841154423037,4.01680533635884,4.0236209628939,4.03039045094915,4.03263680157552,4.0363695420052,4.04008840088394,4.04083051594583,4.04083051594583,4.04305356243378,4.04674769297392,4.04969320320368,4.05116271118304,4.05409526460431,4.05701924329575,4.05847803275941,4.06284167604718,4.06574022880119,4.06646355613834,4.06863040422343,4.07223141639702,4.07438581579405,4.07653558373054,4.07939477081059,4.08224580624142,4.08437876119157,4.08650717632639,4.09075046416426,4.09356935257068,4.09497582256343,4.09848336737311,4.10058200403003,4.10197865230175,4.10546176275457,4.10823954231847,4.10893278324745,4.11031782502011,4.11239179741924,4.1137720598882,4.11790145323021,4.12064493917596,4.1226976254937,4.1226976254937,4.12542800261656,4.12815094507502,4.1288305237624,4.1308664932475,4.13357468718438,4.1369496482009,4.14098462341308,4.14299602201277,4.14366559016634,4.14633938855074,4.14834005610704,4.15100140165655,4.15365568324143,4.15564178121306,4.15960218242915,4.16157651673287,4.16354696072017,4.16485843631798,4.16682243040264,4.16812962013126,4.17073888449488,4.17204096801217,4.17334135831246,4.17852607504665,4.18046343438687,4.18368404908691,4.18753514498901,4.18945514557884,4.1913714668296,4.19328412281593,4.19455719746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