Package {FCUSUM}

Fourier CUSUM Test for the Null of Cointegration

Description

Tests the null hypothesis of cointegration against the alternative of no cointegration, allowing for smooth structural change in the intercept approximated by a Fourier pair at a fractional (default) or integer frequency. The procedure extends the residual-based CUSUM test of Xiao and Phillips (2002) with the flexible Fourier form, in three steps:

1. estimate the cointegrating regression by dynamic OLS and search the Fourier frequency that minimises the residual sum of squares;

2. pre-test the Fourier terms with a sup-F statistic;

3. if the Fourier terms are significant, compute the Fourier CUSUM statistic with bootstrap critical values that replicate the frequency search; otherwise compute the Xiao-Phillips CUSUM statistic without Fourier terms.

Usage

fcum(
  y,
  x,
  kstar = 3,
  freq = c("fractional", "integer"),
  fourier = c("auto", "always", "never"),
  pretest_level = 0.05,
  cv = c("bootstrap", "table"),
  boot = c("iid", "sieve"),
  nboot = 499,
  q = NULL,
  qmax = NULL,
  bw = c("nw", "kurozumi", "andrews"),
  step = 0.01,
  seed = NULL
)

Arguments

Details

The Fourier cointegrating regression is

y_t = \alpha_0 + \alpha_1 \sin(2\pi k t/T) + \alpha_2 \cos(2\pi k t/T) + \beta' x_t + \sum_{j=-q}^{q} \gamma_j' \Delta x_{t-j} + u_t,

estimated by dynamic OLS so that the limit distribution of the statistic is free of nuisance parameters arising from endogeneity and serial correlation (Xiao and Phillips 2002). The CUSUM statistic is

CS = \max_{t} \frac{1}{\hat{\omega}\sqrt{T}} \left| \sum_{s=1}^{t} \hat{u}_s \right|,

with \hat\omega^2 a Bartlett long-run variance estimate. Large values reject cointegration.

Why a pre-test and a bootstrap. With a fractional frequency the null distribution of the Fourier CUSUM depends on the true frequency, and the search over k shrinks the statistic. When there is no break the selected frequency is spurious; when there is one, it is estimated with error. The sup-F pre-test (Becker, Enders and Lee 2006; Tsong et al. 2016) sends samples without a break to the standard Xiao-Phillips test, and the parametric bootstrap reproduces the frequency search around the estimated break so that the Fourier CUSUM has correct size. In Monte Carlo experiments (see data-raw/size_power.R) this keeps size close to the nominal level both with and without a break, whereas critical values conditional on the selected frequency alone are very conservative and have little power.

Value

An object of class fcum, a list with

test

"Fourier CUSUM" or "Xiao-Phillips CUSUM": the statistic on which the decision is based.

statistic, p.value, critical_values, decision, significance

For the reported test.

pretest

List with the sup-F statistic, its critical values, p-value and whether the Fourier terms were retained.

stat_fourier, stat_plain

Both CUSUM statistics.

best_frequency

Selected Fourier frequency.

coefficients, residuals, lrv, bw_lags, nobs

Of the model behind the reported test.

best_model

That model as an lm object (its default standard errors are not valid for DOLS inference).

settings

kstar, freq, q, bw, cv, boot, nboot, p.

call

The matched call.

Changes from version 1.0.0

The 1.0.0 implementation (i) used critical values that were lower-tail quantiles, so a true null was rejected in about 90% of samples at the nominal 5% level; (ii) used OLS residuals scaled by their short-run standard deviation, so the test was not robust to serial correlation or endogeneity; (iii) used critical values indexed by kstar that ignore the dependence on the frequency; and (iv) flattened a multivariate x into one column. All are corrected here.

References

Xiao, Z., & Phillips, P. C. B. (2002). A CUSUM test for cointegration using regression residuals. Journal of Econometrics, 108(1), 43-61.

Tsong, C.-C., Lee, C.-F., Tsai, L.-J., & Hu, T.-C. (2016). The Fourier approximation and testing for the null of cointegration. Empirical Economics, 51(3), 1085-1113.

Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

Stock, J. H., & Watson, M. W. (1993). A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica, 61(4), 783-820.

Kurozumi, E. (2002). Testing for stationarity with a break. Journal of Econometrics, 108(1), 63-99.

Examples

set.seed(123)
n <- 150
x <- cumsum(rnorm(n))
brk <- 2 * sin(2 * pi * 1.5 * seq_len(n) / n)
y <- 2 + brk + 1.5 * x + rnorm(n)

res <- fcum(y, x, kstar = 3, nboot = 99, seed = 1)
res
summary(res)

# Two regressors, integer frequencies, fast table critical values
x2 <- cbind(x1 = x, x2 = cumsum(rnorm(n)))
y2 <- 1 + brk + x2 %*% c(1, -0.5) + rnorm(n)
fcum(y2, x2, kstar = 2, freq = "integer", cv = "table")

Simulate Null Distributions for the Fourier CUSUM Test

Description

Simulates, under the null of cointegration without structural change, the distribution of one of the statistics used by fcum, with exactly the same estimation steps (DOLS, Fourier terms, Bartlett long-run variance). It reproduces the built-in tables and gives finite-sample critical values for any configuration.

Usage

fcum_cv_sim(
  n,
  p = 1,
  type = c("plain", "supF", "fixed"),
  k = 1,
  kstar = 3,
  freq = c("fractional", "integer"),
  step = 0.01,
  q = 1,
  bw = c("nw", "kurozumi", "andrews"),
  nsim = 2000,
  probs = c(0.5, 0.75, 0.8, 0.85, 0.9, 0.925, 0.95, 0.975, 0.99),
  seed = NULL
)

Arguments

Details

Under the null, y_t = u_t with u_t iid N(0, 1) and x_t a p-dimensional Gaussian random walk independent of u_t. The statistics are invariant to the cointegrating vector, the intercept and the scale of u_t.

Value

A list with the simulated quantiles and all simulated statistics (draws).

Examples

fcum_cv_sim(n = 100, p = 1, type = "plain", nsim = 200, seed = 1)$quantiles
fcum_cv_sim(n = 100, p = 1, type = "supF", kstar = 2, step = 0.05,
            nsim = 100, seed = 1)$quantiles

Plot Method for fcum Objects

Description

Plots the scaled cumulative sum of residuals of the reported model with the critical band.

Usage

## S3 method for class 'fcum'
plot(x, level = c("5%", "10%", "1%"), ...)

Arguments

Value

Invisibly returns the input object.

Print Method for fcum Objects

Description

Print Method for fcum Objects

Usage

## S3 method for class 'fcum'
print(x, digits = 4, ...)

Arguments

Value

Invisibly returns the input object.

Summary Method for fcum Objects

Description

Summary Method for fcum Objects

Usage

## S3 method for class 'fcum'
summary(object, ...)

## S3 method for class 'fcum'
coef(object, ...)

## S3 method for class 'fcum'
residuals(object, ...)

Arguments

Value

Invisibly returns the input object.