Package {hatemicoint}

Type:	Package
Title:	Hatemi-J Cointegration Test with Two Unknown Regime Shifts
Version:	1.1.0
Description:	Implements the Hatemi-J (2008) cointegration test which allows for two unknown structural breaks (regime shifts) in the cointegrating relationship. The test provides three test statistics: ADF* (Augmented Dickey-Fuller), Zt* (Phillips-Perron Z_t), and Za* (Phillips-Perron Z_alpha), along with endogenously determined break dates. Critical values are based on simulations from Hatemi-J (2008) <doi:10.1007/s00181-007-0175-9>. The long-run variance in the Phillips statistics is estimated by default with a prewhitened quadratic spectral kernel and the automatic bandwidth of Andrews (1991) <doi:10.2307/2938229>, following Andrews and Monahan (1992) <doi:10.2307/2951574>.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/hatemicoint
BugReports:	https://github.com/muhammedalkhalaf/hatemicoint/issues
Date:	2026-09-30
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	stats
Suggests:	testthat (≥ 3.0.0)
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-09-30 22:53:04 UTC; root
Author:	Muhammad Alkhalaf [aut, cre, cph]
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-10-01 15:50:07 UTC

hatemicoint: Hatemi-J Cointegration Test with Two Unknown Regime Shifts

Description

Implements the Hatemi-J (2008) cointegration test which allows for two unknown structural breaks (regime shifts) in the cointegrating relationship. The test provides three test statistics: ADF* (Augmented Dickey-Fuller), Zt* (Phillips-Perron Z_t), and Za* (Phillips-Perron Z_alpha), along with endogenously determined break dates.

Details

The main function in this package is hatemicoint, which performs the cointegration test with two structural breaks.

The test is particularly useful when:

• Standard cointegration tests fail to reject the null of no cointegration

• There is reason to believe the relationship has changed over time

• Two structural breaks are suspected in the data

Test Statistics

ADF*

Augmented Dickey-Fuller test on residuals with optimal lag selection

Zt*

Phillips-Perron Z_t test with kernel-based long-run variance

Za*

Phillips-Perron Z_alpha test with kernel-based long-run variance

Critical Values

Critical values depend on the number of regressors (k = 1, 2, 3, or 4) and are taken from Table 1 of Hatemi-J (2008). The null hypothesis of no cointegration is rejected when the test statistic is smaller (more negative) than the critical value.

Author(s)

Maintainer: Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

Authors:

• Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

References

Hatemi-J, A. (2008). Tests for cointegration with two unknown regime shifts with an application to financial market integration. Empirical Economics, 35, 497-505. doi:10.1007/s00181-007-0175-9

Gregory, A.W. and Hansen, B.E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics, 70(1), 99-126. doi:10.1016/0304-4076(69)41685-7

See Also

Useful links:

• https://github.com/muhammedalkhalaf/hatemicoint

• Report bugs at https://github.com/muhammedalkhalaf/hatemicoint/issues

Hatemi-J Cointegration Test with Two Unknown Regime Shifts

Description

Performs the Hatemi-J (2008) cointegration test which allows for two unknown structural breaks (regime shifts) in the cointegrating relationship. The test searches over all admissible break date pairs and returns the minimum test statistics along with the endogenously determined break dates.

Usage

hatemicoint(
  y,
  x,
  maxlags = NULL,
  lag_selection = c("tstat", "aic", "sic", "fixed"),
  kernel = c("qs", "bartlett", "iid"),
  bwl = NULL,
  prewhite = TRUE,
  trimming = 0.15
)

Arguments

Details

The Hatemi-J (2008) test extends the Gregory and Hansen (1996) cointegration test by allowing for two structural breaks instead of one. The test is based on the residuals from the cointegrating regression with regime shift dummies:

y_t = \alpha_0 + \alpha_1 D_{1t} + \alpha_2 D_{2t} + \beta_0' x_t + \beta_1' D_{1t} x_t + \beta_2' D_{2t} x_t + u_t

where D_{1t} = 1 if t > tb_1 and D_{2t} = 1 if t > tb_2.

Three test statistics are computed for every break pair and the smallest value over the grid is reported (equations 7 to 9 of the paper):

• ADF*: t-ratio of \delta in \Delta \hat u_t = \delta \hat u_{t-1} + \sum_{i=1}^p \phi_i \Delta \hat u_{t-i} + e_t (no intercept).

• Zt*: Phillips Z_t statistic, equation (6) (see below for the square root).

• Za*: Phillips Z_\alpha statistic, equation (5).

Phillips statistics (equations 3 to 6). With \hat\rho the OLS coefficient of \hat u_{t-1} on \hat u_t (no intercept), v_t = \hat u_t - \hat\rho \hat u_{t-1}, autocovariances \hat\gamma(j) = n^{-1} \sum_t v_t v_{t-j} (equation 4), and S = \sum_{t=1}^{n-1} \hat u_t^2, the long-run variance is \hat\sigma^2 = \hat\gamma(0) + 2 \sum_{j \ge 1} w(j/B) \hat\gamma(j), the one-sided correction is \hat\lambda = (\hat\sigma^2 - \hat\gamma(0))/2, \hat\rho^* = (\sum_{t=1}^{n-1} \hat u_t \hat u_{t+1} - n \hat\lambda)/S, Z_\alpha = n(\hat\rho^* - 1) and Z_t = (\hat\rho^* - 1)/\sqrt{\hat\sigma^2 / S}. The factor n in front of \hat\lambda compensates the 1/n in equation (4), as in Phillips (1987); equation (3) of the paper prints the difference without it. The Z_t formula is the one of Gregory and Hansen (1996, p. 105) and Phillips (1987); equation (6) of the paper as printed omits the square root in the denominator (that form would grow with n). All sums use the normalisation of the paper (n, not n - 1).

Long-run variance. The paper (footnote 3) estimates \hat\sigma^2 with a prewhitened quadratic spectral (QS) kernel with first-order autoregressive prewhitening and the automatic bandwidth of Andrews (1991), following Andrews and Monahan (1992). This is the default since version 1.1.0 (kernel = "qs", prewhite = TRUE, bwl = NULL) and is implemented as follows for every break pair: an AR(1) without intercept is fitted to v_t, its coefficient a_1 is capped at 0.97 in absolute value, the AR(1) residuals e_t are used to compute the Andrews bandwidth B = 1.3221 (\hat\alpha(2) T)^{1/5} with \hat\alpha(2) = 4 \hat\phi^2 / (1 - \hat\phi)^4, where \hat\phi is the AR(1) coefficient of e_t, also capped at 0.97 in absolute value (a package choice, not part of Andrews 1991), and T is the length of e_t; if the bandwidth is zero (\hat\phi = 0) the kernel estimate reduces to \hat\gamma(0) of e_t. The QS kernel k(x) = 3/z^2 (\sin z / z - \cos z), z = 6\pi x/5, is applied to all lags j = 1, \ldots, T - 1 of e_t (the QS kernel is not truncated), and the result is recoloured by dividing by (1 - a_1)^2. With kernel = "bartlett" the weights are 1 - j/B for j < B and zero beyond, that is the kernel k(x) = 1 - |x| evaluated at x = j/B as in the paper's w(j/B) notation; versions before 1.1.0 used the Newey-West form 1 - j/(B + 1). This is the package's choice, not a correction from the paper. With integer B only lags 1, \ldots, B - 1 enter, so bwl = 1 gives no correction. If bwl is NULL the Bartlett bandwidth is round(4 (n/100)^(2/9)) (the rule used by earlier versions). With kernel = "iid" no serial-correlation correction is applied and Z_t equals the Dickey-Fuller t-ratio up to the residual-variance divisor. Versions before 1.1.0 used kernel = "iid" by default.

ADF lag selection. The paper does not state a lag rule. The package compares the candidate lags 0, \ldots, maxlags on a common sample (the first maxlags usable observations are dropped for every candidate) and computes the final statistic for the chosen lag on the largest sample available for that lag. "tstat" is the general-to-specific rule (largest lag whose last coefficient has |t| > 1.645), "aic" and "sic" minimise \log(RSS/T) + c (p + 1)/T with c = 2 and c = \log T, and "fixed" uses maxlags lags. The lag is re-selected for every break pair, as in Gregory and Hansen (1996) style infimum tests. The default maxlags = floor(4 (n/100)^(1/4)) is the Schwert rule and is the package's choice, not the paper's.

Break grid. With trimming \tau (default 0.15), tb_1 runs from \lceil \tau n \rceil to \lfloor (1 - 2\tau) n \rfloor and tb_2 from tb_1 + \lceil \tau n \rceil to \lfloor (1 - \tau) n \rfloor, so that tb_1 \ge \tau n, tb_2 \le (1 - \tau) n and tb_2 - tb_1 \ge \tau n for every n. Break pairs for which a statistic cannot be computed (singular regressor matrix or non-positive long-run variance) are skipped with a warning and never replaced by a substitute value.

The null hypothesis is no cointegration. Rejection occurs when the test statistic is smaller (more negative) than the critical value. The critical values of Table 1 are asymptotic (response-surface intercepts). In a Monte Carlo with the data generating process of Section 4 of the paper (n = 100, m = 1, 400 replications, package defaults) the empirical 5 percent null quantiles were about -6.6 for ADF*, -6.5 for Zt* and -62 for Za*, against the asymptotic values -6.015, -6.015 and -76.003, so at n = 100 ADF* and Zt* reject more often than the nominal level and Za* much less often. With the asymptotic Table 1 critical values the package therefore does not reproduce Table 2 of the paper at n = 100: a 200 replication run with the defaults gave rejection frequencies of 0.175 (ADF*), 0.140 (Zt*) and 0.000 (Za*) under the null against 0.032, 0.069 and 0.069 in Table 2, and the ADF* size depends strongly on the lag rule (0.005 with a fixed lag of 4, 0.105 with lag 0, 0.175 with the default "tstat" rule). The package does not adjust for this.

Value

An object of class "hatemicoint" containing:

adf_min

Minimum ADF* test statistic

tb1_adf

First break location (observation number) for ADF*

tb2_adf

Second break location (observation number) for ADF*

lag_adf

ADF lag used at the ADF* break pair

zt_min

Minimum Zt* test statistic

tb1_zt

First break location for Zt*

tb2_zt

Second break location for Zt*

bwl_zt

Bandwidth used at the Zt* break pair (NA for "iid")

za_min

Minimum Za* test statistic

tb1_za

First break location for Za*

tb2_za

Second break location for Za*

bwl_za

Bandwidth used at the Za* break pair (NA for "iid")

cv_adfzt

Critical values for ADF* and Zt* tests (1%, 5%, 10%)

cv_za

Critical values for Za* test (1%, 5%, 10%)

nobs

Number of observations

k

Number of regressors

maxlags

Maximum lags used

lag_selection

Lag selection method used

kernel

Kernel used

bwl

Bandwidth argument (NA when chosen automatically)

prewhite

Whether prewhitening was used

trimming

Trimming parameter

n_skipped

Number of break pairs skipped because a statistic could not be computed

References

Hatemi-J, A. (2008). Tests for cointegration with two unknown regime shifts with an application to financial market integration. Empirical Economics, 35, 497-505. doi:10.1007/s00181-007-0175-9

Gregory, A.W. and Hansen, B.E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics, 70(1), 99-126. doi:10.1016/0304-4076(69)41685-7

Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-858. doi:10.2307/2938229

Andrews, D.W.K. and Monahan, J.C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60(4), 953-966. doi:10.2307/2951574

Phillips, P.C.B. (1987). Time series regression with a unit root. Econometrica, 55(2), 277-301. doi:10.2307/1913237

Examples

# Generate example data with structural breaks
set.seed(123)
n <- 200
x <- cumsum(rnorm(n))

# Create cointegrated series with two breaks
y <- numeric(n)
y[1:70] <- 1 + 0.8 * x[1:70] + rnorm(70, sd = 0.5)
y[71:140] <- 3 + 1.2 * x[71:140] + rnorm(70, sd = 0.5)
y[141:200] <- 2 + 0.6 * x[141:200] + rnorm(60, sd = 0.5)

# Run the test
result <- hatemicoint(y, x)
print(result)
summary(result)

Print Method for hatemicoint Objects

Description

Print Method for hatemicoint Objects

Usage

## S3 method for class 'hatemicoint'
print(x, ...)

Arguments

Value

Invisibly returns the input object.

Summary Method for hatemicoint Objects

Description

Summary Method for hatemicoint Objects

Usage

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

Arguments

Value

Invisibly returns a summary list with inference at various significance levels.