*! version 1.0.0 20may2026 Dr Merwan Roudane (merwanroudane920@gmail.com)

Title

mixi01_acl -- Augmented Cointegrating Linear (ACL) regression
with strongly correlated I(1) and I(0) regressors

Syntax

mixi01_acl depvar indepvars [if] [in]
[, options]

Integration classification
i1(varlist) list of I(1) (nonstationary) regressors

i0(varlist) list of I(0) (stationary) regressors

i1vars(varlist) synonym for i1()

i0vars(varlist) synonym for i0()

auto auto-classify regressors with Augmented Dickey-Fuller pre-test

Deterministic terms
noconstant suppress the intercept

trend(#) include a polynomial trend of degree # (0, 1 or 2)

Specification check (Peng-Dong, 2021, Section 5)
cointtest after estimation, run an Augmented Dickey-Fuller
test on the residuals to confirm the ACL specification (residuals should be I(0))

adflags(#) number of lags in the residual ADF; default adflags(4)

Reporting
level(#) confidence level; default is level(95)

Description

mixi01_acl estimates the Augmented Cointegrating Linear (ACL) regression
model of Peng and Dong (2021),

y_t = β x_t + α z_t + e_t,
t = 1, ..., n,

where x_t is a d_1-vector of I(1) (unit-root) processes and
z_t is a d_2-vector of I(0) (stationary) processes. Crucially,
the two regressor blocks may be strongly correlated in the sense that they
share the same innovation sequence (Peng-Dong Assumption 2.2(b)). This goes
beyond the weak-correlation setting of Dong and Linton (2018) and most prior
work on mixed-integration regression.

The estimator is plain OLS — no kernel-based long-run-covariance correction is
needed. Peng and Dong's Theorem 1 shows that the OLS coefficients converge:

sqrt(n)(α̂ - α) →_D N(0, σ²Σ^{-1}),
where Σ = E[z_1 z_1], at the standard rate;

n(β̂ - β) →_D [∫W W]^{-1} ∫W dV,
mixed-normal at the super-consistent rate.

Theorem 2 gives self-normalised central-limit theorems that the implementation
exploits for inference:

(Σ z_t z_t)^{1/2} (α̂ - α) →_D N(0, σ²I);

(Σ x_t x_t)^{1/2} (β̂ - β) →_D N(0, σ²I).

This means individual z-tests, joint Wald tests and confidence intervals
based on the standard sandwich V = σ̂² (XX)^{-1} (with
σ̂² = (1/n)∑e_t²) are asymptotically valid for both blocks, although
the β-block intervals shrink at the super-consistent n-rate.

mixi01_acl complements mixi01_fmols (Phillips-1995 FM-OLS):
both target the same data-generating process, but FM-OLS applies an explicit
kernel correction whereas ACL exploits joint convergence to argue that OLS
already suffices for valid inference. Compare the two on your data and
choose the framework that better fits your maintained assumptions.

Options

Integration classification

i1(varlist) declares which regressors are I(1) (nonstationary).
Aliased to i1vars().

i0(varlist) declares which regressors are I(0) (stationary).
Aliased to i0vars().

auto runs an Augmented Dickey-Fuller (ADF) test on every regressor and
classifies a series as I(0) if the test rejects at 5% level, otherwise I(1).
Cannot be combined with explicit i1() or i0() lists.

If neither classification list nor auto is supplied, all regressors are
treated as I(1).

Deterministic terms

noconstant omits the intercept.

trend(#) adds a polynomial deterministic trend:
trend(1) = linear, trend(2) = quadratic; default is trend(0).

Reporting

level(#) sets the confidence level for displayed intervals; default 95.

Remarks

Inference. Standard errors reported by mixi01_acl are
sqrt(diag(σ̂² (XX)^{-1})). For a Wald test on the I(1) block
(β), Peng-Dong Theorem 2 implies

(β̂ - β_0) (Σ x_t x_t) (β̂ - β_0) / σ̂²
→_D χ²(d_1).

The matrix Σ x_t x_t is stored as e(Sxx) and Σ z_t
z_t as e(Szz), so users wanting the exact self-normalised
test can construct it from e() directly.

Strong vs weak correlation. Peng-Dong allow z_t = ∑φ_j ε_ + η_t
with ε the same shocks driving the I(1) innovations. Standard FM-OLS
(Phillips 1995) is also valid in this case but requires kernel/bandwidth
choices. ACL is parameter-free.

Trend. If trend(1) or trend(2) is supplied, the trend terms
are appended after the I(0) block and treated as additional regressors in
(XX). Inference on trend coefficients uses the same V matrix.

Examples

Mixed I(1)/I(0) ACL regression

. mixi01_acl y x1 x2 z1, i1(x1 x2) i0(z1)

Auto-classification by ADF

. mixi01_acl y x1 x2 x3 x4, auto

With linear trend

. mixi01_acl y x1 x2 z1, i1(x1 x2) i0(z1) trend(1)

Specification check via residual ADF (Peng-Dong Sec. 5)

. mixi01_acl y x1 x2 z1, i1(x1 x2) i0(z1) cointtest adflags(4)

Compare with FM-OLS

. mixi01_acl y x1 x2 z1, i1(x1 x2) i0(z1)

. mixi01_fmols y x1 x2 z1, i1(x1 x2) i0(z1) kernel(bartlett)

Stored results

mixi01_acl stores the following in e():

Scalars

e(N) number of observations

e(r2) R-squared

e(r2_a) adjusted R-squared

e(rmse) root MSE = sqrt(σ̂²)

e(sigma2) σ̂² = (1/n)∑ê_t² (Peng-Dong Corollary 1)

e(n_i1) number of I(1) regressors

e(n_i0) number of I(0) regressors

e(trend) trend degree

e(level) confidence level

e(adf_resid_t) residual ADF statistic (only if cointtest)

e(adf_resid_p) p-value of residual ADF (only if cointtest)

e(adf_resid_lags) lag length used in residual ADF (only if cointtest)

Macros

e(cmd) mixi01_acl

e(depvar) dependent variable name

e(i1vars) I(1) regressor list

e(i0vars) I(0) regressor list

e(properties) b V

Matrices

e(b) coefficient vector (β, α, deterministics)

e(V) variance-covariance = σ̂²(XX)^{-1}

e(XX) XX

e(Sxx) Σ x_t x_t (self-normalisation for β)

e(Szz) Σ z_t z_t (self-normalisation for α)

e(iorder) row vector of integration orders (1 = I(1), 0 = I(0)/det)

Functions

e(sample) marks the estimation sample

References

Peng, Z. and C. Dong (2021). Augmented cointegrating linear models with
possibly strongly correlated stationary and nonstationary regressors.
SSRN Working Paper No. 3943779.

Dong, C. and O. Linton (2018). Additive nonparametric models with time
variable and both stationary and nonstationary regressors.
Journal of Econometrics, 207(1), 212–236.

Park, J. Y. and P. C. B. Phillips (2001). Nonlinear regressions with
integrated time series. Econometrica, 69(1), 117–161.

Phillips, P. C. B. (1995). Fully modified least squares and vector
autoregression. Econometrica, 63(5), 1023–1078.

Wang, Q. (2015). Limit Theorems for Nonlinear Cointegrating Regression.
World Scientific.

Author

Dr Merwan Roudane

Department of Economics (Independent Researcher)

merwanroudane920@gmail.com

Also see

Master help — mixi01.

Sibling commands — mixi01_fmols, mixi01_fmvar,
mixi01_fmiv, mixi01_svar, mixi01_vecm,
mixi01_irf, mixi01_test.