In luisdamiano/LeastFracDiff: Grid Search for the Least Fractional Differenciation Order

knitr::opts_chunk$set(echo = TRUE, eval = TRUE, cache = TRUE)

LeastFracDiff provides minimal tools for finding an approximation to the least fractional difference order $d$ that transforms a time series into a stationary one, as suggested by de Prado (2018)^[de Prado, M. L. (2018). Advances in financial machine learning. John Wiley & Sons.]. As we typically work with log prices, we rely on quantmod to download data for our example.

library(LeastFracDiff)
suppressPackageStartupMessages(library(quantmod))

penv <- new.env()
getSymbols(
  Symbols = c("SPY", "QQQ"),
  src     = "yahoo",
  from    = "1990-01-01", 
  to      = "2018-06-30",
  env     = penv
)
p    <- penv$SPY
x    <- log(Cl(na.omit(p)))

We approximate the minimal fractional differentiation coefficient $d$ so that the p-value from the Augmented Dickey Fuller test is below an arbitrary threshold (say, $0.05$). The function min_d takes three arguments: the time series x, a function fun taking a time series and returning a p-value, and a threshold. The package includes a function adf relying on the package tseries for convenience.

print(adf)

d   <- min_d(x, adf, 0.05)
print(d)

A series can be transformed via the function diff_series, which in turns relies on the package fracdiff.

dx  <- diff_series(x, d)
plot(
  cbind(
    "Log Price" = x, 
    "Differenced Log Price" = dx
  ),
  main       = colnames(x),
  grid.col   = "white",
  legend.loc = "right"
)

Instead of keeping only the least $d$, we may explore the relationship between the order of fractional differentiation, some measures of stationarity, and a metric for loss of information with respect to the original series. The function grid_d takes the already discussed arguments x and fun to produce a grid for values of $d$ from dFrom to dTo increase by dBy each step.

g   <- grid_d(x, adf, dFrom = 0, dTo = 1, dBy = 0.1)

knitr::kable(g)

plot(g, ylab = "ADF p-value")
abline(h = 0.05)

matplot(
  g[, 1], g[, -1], type = "l", ylab = "", xlab = "d",
  col = 1:3, lty = 1, lwd = 1
)
legend(
  x      = "topright",
  legend = c("p-value", "Pearson", "Spearman"),
  bty    = "n",
  cex    = 0.7,
  lty    = 1,
  col    = 1:3
)

We can also compute min $d$. The function roll_min_d takes three mandatory arguments: x a time series, width the number of observations per rolling window, and by the number of time steps between each window. Additionally, the user may pass other arguments for min_d or zoo::rollapply.

For example, the following snippet minimizes $d$ so that the p-value of the ADF statistic is below a threshold of $0.05$. The minimization is run on a rolling window with approximately five years of daily close prices. Each run is separated by a month approximately.

dRolling <- roll_min_d(x, fun = adf, threshold = 0.05, width = 5 * 252, by = 21)

plot(
  na.omit(dRolling),
  main       = sprintf(
    "Auto d for %s (5ys daily obs per window - one window per month)", 
    colnames(x)
  ),
  ylab       = "min d s.t. ADF p-value = 0.05",
  grid.col   = "white",
  legend.loc = "topleft"
)

Compare the fractional differentiation order found using the entire time series $d^ = r print(d)$ versus the time-varying quantity observed for each window. We observe that min $d$ is approximately zero for the rolling window ending on March 2015, i.e. there was not enough evidence to reject the null hypothesis that log prices had a non-stationary distribution according to ADF.

subx   <- last(x["/2015-04-01"], 5 * 252)
plot(
  subx,
  main       = colnames(x),
  grid.col   = "white",
  legend.loc = "right"
)

print(tseries::adf.test(subx))

The code to expand this analysis to other assets is trivial:

dMat <- do.call(
  cbind,
  lapply(penv, function(p) {
    x    <- log(Cl(na.omit(p)))
    as.numeric(roll_min_d(x, fun = adf, threshold = 0.05, width = 5 * 252, by = 21))
  })
)

matplot(
  na.omit(dMat), type = "l", ylab = "d", xlab = "Time",
  col = 1:2, lty = 1, lwd = 1
)
legend(
  x      = "topright",
  legend = colnames(dMat),
  bty    = "n",
  cex    = 0.7,
  lty    = 1,
  col    = 1:2
)

Final remarks

• A more granular value for $d$ may be find at the cost of computational complexity by setting a smaller value for dBy.
• P-values from other tests, or completely different metrics, may be used by changing the argument fun. For example, set fun = kpss to run the KPSS test.

luisdamiano/LeastFracDiff documentation built on May 10, 2019, 3:18 a.m.