Simulate a Phillips-Perron Test to Assess Unit Root in a Time Series
spp_test.Rdspp_test() provides a simulation approach to assessing
unit root in a time series by way of the Phillips-Perron test. It takes a
vector and performs three Phillips-Perron tests (no drift, no trend; drift, no
trend; drift and trend) and calculates both rho and tau statistics as one
normally would. Rather than interpolate or approximate a p-value, it
simulates some user-specified number of Phillips-Perron tests of either a
known, non-stationary time series or a known, white-noise time series
matching the length of the time series the user provides. This allows the
user to make assessments of non-stationarity or stationarity by way of
simulation rather than approximation from received critical values by way of
various books/tables.
Arguments
- x
a vector
- lag_short
logical, defaults to
TRUE. IfTRUE, the "short-term" lag is used for the Phillips-Perron test. IfFALSE, the "long-term" lag is used.- n_sims
the number of simulations for calculating an interval or distribution of test statistics for assessing stationarity or non-stationarity. Defaults to 1,000.
- sim_hyp
can be either "stationary" or "nonstationary". If "stationary", the function runs Phillips-Perron tests on simulated stationary (pure white noise) data. This allows the user to assess compatibility/plausibility of the test statistic against a distribution of test statistics that are known to be pure white noise (in expectation). If "nonstationary" (default), the function generates three different data sets of a pure random walk, a random walk with a drift, and a random walk with a drift and trend. It then runs Phillips-Perron tests on all those. This allows the user to assess the compatibility/plausibility of their test statistics with data that are known to be nonstationary in some form.
Value
spp_test() returns a list of length 3. The first element
in the list is a matrix of rho statistics and tau statistics calculated by
the Phillips-Perron test. The second element is a data frame of the simulated
rho and tau statistics of either a known white-noise time series or three
different non-stationary time series (pure random walk, random walk with
drift, random walk with drift and trend). The third element is some
attributes about the procedure for post-processing.
Details
Some knowledge of Augmented Dickey-Fuller and the Phillips-Perron procedure is assumed here. Generally, the Phillips-Perron test purports to build on the Augmented Dickey-Fuller procedure through two primary means. The first is relaxing the need to specify or assume lag structures ad hoc or ex ante. Only a short-term lag or long-term lag are necessary. The second is that its robust to various forms of heteroskedasticity in the error term.
The short-term and long-term lags follow the convention introduced in the
Phillips-Perron test. The short-term lag uses the default number of
Newey-West lags, defined as the floor of 4*(n/100)^.25 where n is the length
of the time series. The long-term lag substitutes 4 for 12 in this equation.
We have Schwert (1989) to thank for these defaults.
This function specifies three different types of tests: 1) no drift, no trend,
2) drift, no trend, and 3) drift and trend. In the language of the lm()
function, the first is lm(y ~ ly - 1) where y is the value of y and
ly is its first-order lag. The second test is lm(y ~ ly), intuitively
suggesting the y-intercept in this equation is the "drift". The third would
be lm(y ~ ly + t) with t being a simple integer that increases by 1 for
each observation (i.e. a time-trend).
There are two types of statistics in the Phillips-Perron test: rho and tau. Of the two, tau is the more intuitive statistic and compares favorably to its corollary statistic in the Augmented Dickey-Fuller test. It's why you'll typically see tau reported as the statistic of interest in other implementations. rho has its utility for more advanced diagnostics, though. Both are calculated in this function, though tau is the default statistic.
None of this is meant to discourage the use of Fuller (1976) or its various reproductions for the sake of diagnosing stationarity or non-stationary, and I will confess their expertise on these matters outpaces mine. Consider the justification for this function to be largely philosophical and/or experimental. Why not simulate it? It's not like time or computing power are huge issues anymore.
This is always awkwardly stated, but it's a good reminder that the classic Dickey-Fuller statistics are mostly intended to come back negative. That's not always the case, to be clear, but it is the intended case. You assess the statistic by "how negative" it is. Stationary time series will produce test statistics more negative ("smaller") than those produced by non-stationary time series. In a way, this makes the hypotheses implicitly one-tailed (to use that language).
This function removes missing values from the vector before calculating test statistics.
Examples
a <- rnorm(25) # white noise
b <- cumsum(a) # random walk
spp_test(a, n_sims = 25)
#> $stats
#> [,1] [,2]
#> [1,] -23.48328 -4.605517
#> [2,] -23.93430 -4.742632
#> [3,] -24.00217 -4.645973
#>
#> $sims
#> z_rho z_tau sim cat
#> 1 1.1294248 0.6367477 1 No Drift, No Trend
#> 2 -0.9068868 -0.3079832 1 Drift, No Trend
#> 3 -2.8966041 -1.1404323 1 Drift and Trend
#> 4 -0.7370774 -0.5985780 2 No Drift, No Trend
#> 5 -4.3622620 -1.4757252 2 Drift, No Trend
#> 6 -13.1058554 -2.7077509 2 Drift and Trend
#> 7 -10.8416351 -2.5858515 3 No Drift, No Trend
#> 8 -11.9029908 -2.6907983 3 Drift, No Trend
#> 9 -14.9289381 -2.2962002 3 Drift and Trend
#> 10 -13.4938711 -2.9331418 4 No Drift, No Trend
#> 11 -13.5393488 -2.8677585 4 Drift, No Trend
#> 12 -7.0149008 -1.6514995 4 Drift and Trend
#> 13 -1.6569621 -0.7541765 5 No Drift, No Trend
#> 14 -1.3397403 -0.4127787 5 Drift, No Trend
#> 15 -9.1850058 -2.1199585 5 Drift and Trend
#> 16 -0.2128096 -0.1053604 6 No Drift, No Trend
#> 17 -1.6067642 -0.7113448 6 Drift, No Trend
#> 18 -14.3960703 -2.9488911 6 Drift and Trend
#> 19 -0.8852778 -0.4728312 7 No Drift, No Trend
#> 20 -10.2493271 -2.4628553 7 Drift, No Trend
#> 21 -6.6265427 -2.4029418 7 Drift and Trend
#> 22 -2.2075688 -1.0235998 8 No Drift, No Trend
#> 23 -4.5009687 -1.6618158 8 Drift, No Trend
#> 24 -11.2699230 -2.6610279 8 Drift and Trend
#> 25 -1.7287100 -0.7520601 9 No Drift, No Trend
#> 26 -3.9257714 -1.3600884 9 Drift, No Trend
#> 27 -9.1225129 -2.1384250 9 Drift and Trend
#> 28 -1.7120964 -0.9058208 10 No Drift, No Trend
#> 29 -3.9958306 -1.8616200 10 Drift, No Trend
#> 30 -14.1509870 -2.7001421 10 Drift and Trend
#> 31 0.3350596 0.3517101 11 No Drift, No Trend
#> 32 -8.3573022 -4.1165346 11 Drift, No Trend
#> 33 -7.1841876 -1.8738945 11 Drift and Trend
#> 34 -0.9725538 -0.6543340 12 No Drift, No Trend
#> 35 -3.3052661 -1.5300153 12 Drift, No Trend
#> 36 -9.5129526 -2.0997490 12 Drift and Trend
#> 37 -2.4244423 -0.8173012 13 No Drift, No Trend
#> 38 -1.4792524 -0.4398537 13 Drift, No Trend
#> 39 -10.5034794 -3.0627737 13 Drift and Trend
#> 40 1.0489293 1.0831857 14 No Drift, No Trend
#> 41 -0.5863368 -0.4401832 14 Drift, No Trend
#> 42 -1.4004534 -0.4861305 14 Drift and Trend
#> 43 -5.5971047 -1.6821989 15 No Drift, No Trend
#> 44 -5.3897665 -1.6632667 15 Drift, No Trend
#> 45 -9.9472249 -2.3416121 15 Drift and Trend
#> 46 -2.3468982 -0.6816219 16 No Drift, No Trend
#> 47 -3.4135118 -0.9665459 16 Drift, No Trend
#> 48 -10.1314544 -2.8415270 16 Drift and Trend
#> 49 0.3331753 0.2755908 17 No Drift, No Trend
#> 50 -7.0605106 -2.0357348 17 Drift, No Trend
#> 51 -10.9990568 -2.1880948 17 Drift and Trend
#> 52 -4.1434352 -1.4798280 18 No Drift, No Trend
#> 53 -4.7661190 -1.5652520 18 Drift, No Trend
#> 54 -14.5727902 -3.2601228 18 Drift and Trend
#> 55 -0.6899220 -0.2818260 19 No Drift, No Trend
#> 56 -5.1785999 -1.3979981 19 Drift, No Trend
#> 57 -11.1952575 -2.4972148 19 Drift and Trend
#> 58 -4.0861377 -1.4702544 20 No Drift, No Trend
#> 59 -9.0703605 -2.9483743 20 Drift, No Trend
#> 60 -5.5077254 -1.5515693 20 Drift and Trend
#> 61 1.1451524 1.4419096 21 No Drift, No Trend
#> 62 -1.5161716 -1.0733236 21 Drift, No Trend
#> 63 -9.4416130 -2.0419656 21 Drift and Trend
#> 64 -1.3146850 -0.8281979 22 No Drift, No Trend
#> 65 -5.3907277 -1.4219443 22 Drift, No Trend
#> 66 -3.6900973 -1.3826374 22 Drift and Trend
#> 67 -1.6277517 -0.7807940 23 No Drift, No Trend
#> 68 -0.5909836 -0.1782738 23 Drift, No Trend
#> 69 -3.7388786 -0.9820553 23 Drift and Trend
#> 70 -5.3306989 -1.5107292 24 No Drift, No Trend
#> 71 -7.4218936 -1.4706567 24 Drift, No Trend
#> 72 -1.0078166 -0.4979061 24 Drift and Trend
#> 73 0.9115373 0.8317744 25 No Drift, No Trend
#> 74 -4.5454840 -1.8091737 25 Drift, No Trend
#> 75 -10.3592143 -2.8115282 25 Drift and Trend
#>
#> $attributes
#> lags sim_hyp n_sims n test
#> 1 2 nonstationary 25 25 pp
#>
#> attr(,"class")
#> [1] "spp_test"
spp_test(b, n_sims = 25)
#> $stats
#> [,1] [,2]
#> [1,] 0.3242636 0.2688907
#> [2,] -4.7454538 -1.8513845
#> [3,] -9.1278106 -2.3117382
#>
#> $sims
#> z_rho z_tau sim cat
#> 1 -1.665619e-01 -0.146684873 1 No Drift, No Trend
#> 2 -3.673281e+00 -1.801378886 1 Drift, No Trend
#> 3 -3.628388e+00 -0.968229690 1 Drift and Trend
#> 4 -8.205472e+00 -2.070028595 2 No Drift, No Trend
#> 5 -8.356446e+00 -1.963371279 2 Drift, No Trend
#> 6 -1.003390e+01 -2.226467957 2 Drift and Trend
#> 7 -2.851732e+00 -1.149330596 3 No Drift, No Trend
#> 8 -2.505238e+00 -1.046419455 3 Drift, No Trend
#> 9 -1.321583e+01 -2.762012427 3 Drift and Trend
#> 10 -7.132989e+00 -1.995266950 4 No Drift, No Trend
#> 11 -8.032439e+00 -2.007359026 4 Drift, No Trend
#> 12 -8.263651e+00 -1.907754237 4 Drift and Trend
#> 13 -3.475496e-02 -0.035872616 5 No Drift, No Trend
#> 14 -2.922238e+00 -1.715247990 5 Drift, No Trend
#> 15 -1.038536e+01 -2.192730337 5 Drift and Trend
#> 16 2.333278e+00 1.487522698 6 No Drift, No Trend
#> 17 4.985550e-01 0.224519350 6 Drift, No Trend
#> 18 -5.879713e+00 -1.974101549 6 Drift and Trend
#> 19 1.130417e+00 0.686076168 7 No Drift, No Trend
#> 20 -2.847960e+00 -0.878231831 7 Drift, No Trend
#> 21 -7.903666e+00 -1.770855762 7 Drift and Trend
#> 22 -8.617611e-01 -0.518689155 8 No Drift, No Trend
#> 23 -5.741963e+00 -1.913131888 8 Drift, No Trend
#> 24 -1.697466e+01 -3.930401854 8 Drift and Trend
#> 25 -3.634477e+00 -1.353857074 9 No Drift, No Trend
#> 26 -5.972684e+00 -1.754341983 9 Drift, No Trend
#> 27 -6.218477e+00 -1.712368319 9 Drift and Trend
#> 28 -3.911904e+00 -1.341573914 10 No Drift, No Trend
#> 29 -4.124724e+00 -1.281217200 10 Drift, No Trend
#> 30 -1.085643e+01 -2.247249365 10 Drift and Trend
#> 31 1.014780e+00 1.707974563 11 No Drift, No Trend
#> 32 -1.263663e+00 -0.854009961 11 Drift, No Trend
#> 33 -1.164529e+00 -0.224714447 11 Drift and Trend
#> 34 -1.561863e+00 -0.807364973 12 No Drift, No Trend
#> 35 -3.000557e+00 -1.311735953 12 Drift, No Trend
#> 36 -1.680064e+01 -3.354111709 12 Drift and Trend
#> 37 1.667810e+00 1.639584873 13 No Drift, No Trend
#> 38 -3.047593e-01 -0.194022367 13 Drift, No Trend
#> 39 -1.587200e+00 -0.670641555 13 Drift and Trend
#> 40 4.024234e-02 0.030915219 14 No Drift, No Trend
#> 41 -1.613374e+00 -1.036147183 14 Drift, No Trend
#> 42 -1.847518e+01 -3.740452738 14 Drift and Trend
#> 43 3.330698e-01 0.275905325 15 No Drift, No Trend
#> 44 -2.439247e+00 -1.197556256 15 Drift, No Trend
#> 45 -1.247711e+01 -2.515982393 15 Drift and Trend
#> 46 1.009456e+00 0.704712093 16 No Drift, No Trend
#> 47 -2.960559e+00 -0.926808108 16 Drift, No Trend
#> 48 -4.600844e+00 -1.407315036 16 Drift and Trend
#> 49 5.731102e-02 0.051154589 17 No Drift, No Trend
#> 50 -2.884286e+00 -1.470308083 17 Drift, No Trend
#> 51 -1.010458e+01 -2.356348984 17 Drift and Trend
#> 52 1.480332e+00 1.850215004 18 No Drift, No Trend
#> 53 -6.276449e-04 0.006203227 18 Drift, No Trend
#> 54 -5.099533e+00 -1.409227882 18 Drift and Trend
#> 55 -6.306361e-01 -0.323474973 19 No Drift, No Trend
#> 56 -3.820332e+00 -1.313354679 19 Drift, No Trend
#> 57 -5.704391e+00 -1.905547132 19 Drift and Trend
#> 58 1.598719e-01 0.183638205 20 No Drift, No Trend
#> 59 -1.240748e+01 -4.370252774 20 Drift, No Trend
#> 60 -6.028731e+00 -2.103354734 20 Drift and Trend
#> 61 5.273312e-02 0.037127424 21 No Drift, No Trend
#> 62 -2.160675e+00 -1.255057063 21 Drift, No Trend
#> 63 -3.296475e+00 -1.225400068 21 Drift and Trend
#> 64 1.854018e+00 1.505443416 22 No Drift, No Trend
#> 65 5.742201e-01 0.463514798 22 Drift, No Trend
#> 66 -1.343317e+01 -2.754901321 22 Drift and Trend
#> 67 -9.768731e-01 -0.518062974 23 No Drift, No Trend
#> 68 -1.119991e+01 -2.726869427 23 Drift, No Trend
#> 69 -9.227291e+00 -2.093362415 23 Drift and Trend
#> 70 -6.078577e-01 -0.240509603 24 No Drift, No Trend
#> 71 -1.934166e+00 -0.674362003 24 Drift, No Trend
#> 72 -1.088283e+01 -2.847411988 24 Drift and Trend
#> 73 2.223602e+00 2.957131020 25 No Drift, No Trend
#> 74 1.137741e+00 0.867337028 25 Drift, No Trend
#> 75 -1.566082e+01 -3.088469755 25 Drift and Trend
#>
#> $attributes
#> lags sim_hyp n_sims n test
#> 1 2 nonstationary 25 25 pp
#>
#> attr(,"class")
#> [1] "spp_test"