spp_test: Simulate a Phillips-Perron Test to Assess Unit Root in a Time...

In sTSD: Simulate Time Series Diagnostics

Simulate a Phillips-Perron Test to Assess Unit Root in a Time Series

Description

spp_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.

Usage

spp_test(x, lag_short = TRUE, n_sims = 1000, sim_hyp = "nonstationary")

Arguments

x	a vector
lag_short	logical, defaults to TRUE. If TRUE, the "short-term" lag is used for the Phillips-Perron test. If FALSE, 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.

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 (ADF) procedure through two primary means. The first is relaxing the need to be as deliberate about lag structures as ADF is, typically asking for some kind of long- or short-term lag for the bandwidth/kernel. The second, related to this, is that its nonparametric procedure makes it 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. Future iterations of this function may permit manual lag selection, but this is fine for now.

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.

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.

Author(s)

Steven V. Miller

Examples

a <- rnorm(25) # white noise
b <- cumsum(a) # random walk

spp_test(a, n_sims = 25)
spp_test(b, n_sims = 25)

sTSD documentation built on April 3, 2025, 7:37 p.m.