*! version 1.0.0 18may2026
help multicoint (Stata)

Title

multicoint -- Estimation and testing of multicointegrated
time-series in the sense of Granger-Lee (1989, 1990)

Package contents

The multicoint library provides every estimator and every test that has
been proposed for the analysis of multicointegrated time-series. Components:

multicoint - main: multicointegration estimation and testing.

multicoint_sim - simulate a multicointegrated DGP for Monte-Carlo work.

multicoint_graph - diagnostic graphs after multicoint.

multicoint_cv - Engsted-Gonzalo-Haldrup (1997) critical-value tables.

Syntax

multicoint depvar indepvars [if] [in] [, options]

Main
estimator(name) choice of estimator (see Estimators)

test(name) choice of multicoint test (see Tests)

trend(spec) deterministics: none, c, ct, ctt

noconstant suppress constant term

Test/lag options
lags(#) fixed ADF lag order (default 0)

autolag(crit) IC for lag selection: aic, bic, hqic, fixed

maxlag(#) maximum lag for auto-selection (default 8)

DOLS-only options
leads(#) leads of Δx (default 2)

dlags(#) lags of Δx (default 2)

FM-OLS / CCR options
kernel(name) bartlett (default), parzen, qs

bwidth(#) fixed bandwidth (default 0 = Andrews auto)

TAOLS options
k(#) number of orthonormal basis functions (default 12)

Reporting
level(#) confidence level; default 95

graph call multicoint_graph after estimation

grsave(filename) export the diagnostic graph to a file

notable suppress result tables

depvar is the flow series y_t (I(1) under the Granger-Lee /
Engsted-Haldrup set-up).
indepvars is one or more flow regressors x_t (I(1)).
multicoint internally constructs the cumulated series
Y_t = Σy_s, X_t = Σx_s, and estimates the
multicointegration regression

Y_t = α + δ_1 t + δ_2 t² + β'·X_t + γ'·x_t + u_t

Coefficients on the cumulated I(2) regressors are reported with suffix
_cum; the original flow I(1) regressors are reported by their original
name. Residuals are saved as _mc_uhat; cumulated series as
_mc_Ycum, _mc_Xcum1, etc.

Description

Two I(1) flow series are said to be multicointegrated when their
cumulated sums - which are I(2) by construction - cointegrate together
and with the original flows. In that case there is a "second layer"
of long-run equilibrium relating the stock variables to the underlying
flow variables, on top of the standard I(1) cointegration
y - β x ~ I(0).

The classical example (Granger & Lee, 1989) is the production-sales-inventory
identity: y_t (production) and x_t (sales) are I(1) and cointegrated; the
level of inventories Q_t = Σ(y - x)_s would normally inherit one unit root,
but in the presence of multicointegration Q_t (or a linear combination of
Q_t and the flows) is stationary. Stock-flow data of housing starts /
completions, income / consumption / wealth, and many other systems exhibit
this feature (Engsted & Haldrup, 1999).

multicoint bundles all the estimators and tests that have been
proposed for this set-up.

Estimators

est(ols) -- ordinary least squares

Plain OLS on the multicoint regression. Super-consistent for β (rate T²)
and γ (rate T) under multicoint (Haldrup, 1994). The reported standard
errors are non-standard; use TAOLS for valid inference.

est(fmols) -- fully-modified OLS

Phillips-Hansen (1990). Corrects OLS for long-run endogeneity and serial
correlation using a kernel HAC long-run variance. Standard t/F inference
applies asymptotically. Requires the cointreg package
(Wang, 2011).

est(dols) -- dynamic OLS

Saikkonen (1991), Stock & Watson (1993). Augments the regression with
leads and lags of Δx_t to neutralise long-run endogeneity. Standard
inference applies. Requires cointreg.

est(ccr) -- canonical cointegrating regression

Park (1992). Pre-transforms y_t and x_t using a long-run covariance
estimate so OLS on the transformed series yields asymptotically efficient
coefficients with standard inference. Requires cointreg.

est(imols) -- integrated-modified OLS

Vogelsang & Wagner (2014). Partial-sum-transforms the regression and
estimates by OLS. Has standard normal/mixed-normal limiting theory under
cointegration and multicointegration without needing a kernel bandwidth.

est(taols) -- transformed and augmented OLS

Hwang & Sun (2018), Sun et al. (2025, 2026). Applies an orthonormal
Fourier (shifted sine) basis transformation to the regression augmented
with the endogeneity-correction term Δx_t. The resulting transformed
regression behaves as a classical linear normal regression, so standard
t / F inference is exact in the asymptotic limit. Recommended when the
cointegration regime is uncertain.

Tests of multicointegration

test(gl) -- Granger-Lee (1989, 1990) two-step

Stage 1: regress y_t on x_t and save residual Z_t. Stage 2: cumulate Z_t
into S_t = Σ Z_s, regress S_t on x_t and ADF-test the second-stage
residual. Note: this is the original multicoint test;
Engsted-Gonzalo-Haldrup (1997) show that its limit distribution is a
Brownian-bridge functional - we provide it for completeness only.

test(egh) -- Engsted-Gonzalo-Haldrup (1997) one-step

ADF t-test applied to the residual u_t of the single multicoint
regression Y_t = α + δ_1 t + δ_2 t² + β'·X_t + γ'·x_t + u_t. Rejection
of the unit-root null supports multicointegration. Critical values from
EGH (1997, Tables 1 - 2), interpolated for the user's sample size T.
See multicoint_cv to inspect the full c.v. table.

test(taols) -- Sun et al. (2026) adaptive F-test

Constructs two Wald statistics for γ = 0 - one under the multicoint
regression and one under the conventional cointegration regression - and
combines them with a data-driven weight w that converges to 1 under
multicoint and to 0 under plain coint. The adaptive Wald statistic is
asymptotically F-distributed with (m_2, K - 3·d_x - 1) degrees of
freedom under either regime.

test(all)

Run all three tests and report them side-by-side.

Stored results

Scalars

e(N) number of observations

e(K) number of flow regressors

e(rss) residual sum of squares

e(r2) R²

e(lrse) long-run S.E. (FM-OLS / CCR)

e(gl_stat) Granger-Lee statistic

e(gl_cv05) GL 5% c.v.

e(egh_stat) EGH t-statistic

e(egh_cv01/cv025/cv05/cv10) EGH c.v. at 1, 2.5, 5, 10%

e(egh_lags) lag order used by ADF

e(taols_Fm/Fc/Fa) TAOLS Wald under multicoint / coint / adaptive

e(taols_w) adaptive weight

Macros

e(cmd) multicoint

e(estimator) selected estimator

e(test) selected test(s)

e(trend) deterministics

e(depvar) y flow variable

e(indepvars) x flow variables

e(resvar) name of residual variable (_mc_uhat)

e(Ycumvar) cumulated y variable

e(Xcumvars) cumulated x variables

Matrices

e(b) coefficient vector

e(V) covariance matrix

Examples

1. Simulate a multicointegrated DGP and run all tests/estimators

multicoint_sim, n(300) beta(1) gamma(0.95) regime(multicoint) clear

multicoint y x, est(taols) test(all) graph

2. Estimate via FM-OLS, then DOLS (delegated to cointreg)

multicoint y x, est(fmols) test(egh) trend(ct)

multicoint y x, est(dols) leads(3) dlags(3)

3. Granger-Lee two-step test under OLS estimation

multicoint y x, est(ols) test(gl)

4. Inspect EGH critical values

multicoint_cv, trend(ct) m1(2) m2(1) tsize(200)

5. Six-panel diagnostic graph after estimation

multicoint y x, est(taols) test(taols)

multicoint_graph, sixpanel save(mc_diag.png)

Dependencies

The FM-OLS, DOLS and CCR estimators delegate to Qunyong Wang's
cointreg package, which can be installed with:

. ssc install cointreg

All other estimators (ols, imols, taols) and all three tests
are implemented natively in this package.

References

Engsted, T., Gonzalo, J. & Haldrup, N. (1997). Testing for
multicointegration. Economics Letters 56, 259-266.

Engsted, T. & Haldrup, N. (1999). Multicointegration in stock-flow
models. Oxford Bulletin of Economics and Statistics 61(2), 237-254.

Engsted, T. & Johansen, S. (1997). Granger's representation theorem and
multicointegration. EUI Working Paper ECO 97/15.

Granger, C.W.J. & Lee, T.-H. (1989). Investigation of production, sales
and inventory relationships using multicointegration and non-symmetric
error correction models. J. Applied Econometrics 4, S145-S159.

Granger, C.W.J. & Lee, T.-H. (1990). Multicointegration. In G.F. Rhodes &
T.B. Fomby (eds.), Advances in Econometrics 8, 71-84.

Haldrup, N. (1994). The asymptotics of single-equation cointegration
regressions with I(1) and I(2) variables. J. Econometrics 63, 153-181.

Hwang, J. & Sun, Y. (2018). Should we go one step further? An accurate
comparison of size and power for cointegrating regressions.
J. Econometrics.

Park, J. (1992). Canonical cointegrating regressions. Econometrica
60, 119-143.

Phillips, P.C.B. & Hansen, B.E. (1990). Statistical inference in
instrumental variables regression with I(1) processes.
Review of Economic Studies 57, 99-125.

Saikkonen, P. (1991). Asymptotically efficient estimation of cointegrating
regressions. Econometric Theory 7, 1-21.

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

Sun, Y. et al. (2025, 2026). TAOLS: Adaptive F and t tests for
cointegration and multicointegration. Working paper.

Vogelsang, T.J. & Wagner, M. (2014). Integrated modified OLS estimation
and fixed-b inference for cointegrating regressions.
J. Econometrics.

Wang, Q. (2011). cointreg: Stata module for cointegration regression
(FM-OLS, CCR, DOLS). Available from SSC.

Author

Dr Merwan Roudane

Department of Economics

Email: merwanroudane920@gmail.com

multicoint v1.0.0 - 18 May 2026. Bug reports and questions welcome.

Also see

Online: multicoint_sim, multicoint_graph,
multicoint_cv, cointreg