Efficient Estimation by Fully Modified GLS with an Application to the Environmental Kuznets Curve

Hanno Reuvers; Yicong Lin

Efficient Estimation by Fully Modified GLS with an Application to the Environmental Kuznets Curve

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read FM-GLS makes nonlinear cointegration inference chi-square.

desk verdict A credible and useful FM-GLS for cointegrating polynomial regressions; the asymptotic core holds up, but the empirical application and under-covering intervals keep it from being a clean yes. read the letter →

arxiv 1908.02552 v2 pith:L3KZQEIE submitted 2019-08-07 econ.EM stat.ME

Yicong Lin , Hanno Reuvers This is my paper · ORCID

classification econ.EMstat.ME MSC 62M1062P2062F12

keywords CointegratingPolynomialRegressionSeeminglyUnrelatedFullyModifiedEstimationGeneralizedLeastSquaresCholeskyBlockDecompositionBandedInverseAutocovarianceMatrixCointegrationTestingEnvironmentalKuznetsCurve

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a feasible fully modified generalized least squares (FM-GLS) estimator for seemingly unrelated cointegrating polynomial regressions, where cointegrating relations contain deterministic trends, stochastic trends, and integer powers of integrated regressors. The paper's central claim is that, after correcting second-order biases, the normalized FM-GLS estimator converges to a zero-mean Gaussian mixture distribution, so Wald tests are asymptotically chi-square under the null. This matters because existing fully modified estimators in this setting require leads-and-lags augmentation or kernel-based long-run covariance estimation, and they tend to understate parameter uncertainty in applications. The paper also constructs multivariate KPSS-type cointegration tests based on filtered residuals. In the Environmental Kuznets Curve application, FM-GLS supports the inverted-U hypothesis but produces wider confidence intervals than FM-SUR or FM-SOLS, with closer-to-nominal coverage in a calibrated simulation.

What carries the argument

The central object is the Modified Cholesky Block Decomposition (MCBD) of the inverse autocovariance matrix, $\Sigma_u^{-1}=M_u'S_u^{-1}M_u$, built from linear minimum MSE predictors of the innovations; banding the sub-diagonal blocks yields the Banded Inverse Autocovariance Matrix (BIAM), which is estimated directly from first-stage OLS residuals rather than by inverting a large covariance matrix. The MCBD lets the GLS estimator be read as weighted least squares applied to filtered data, and the same estimated VAR coefficients feed the long-run covariance estimators and the second-order bias correction that removes the endogeneity and serial-correlation terms from the limiting distribution. Rate conditions on the banding parameter, inherited from fitting VARs of increasing order, make the estimation error asymptotically negligible.

What would settle it

Run the paper's Setting B at the high-persistence, high-endogeneity corner (eigenvalues in [0.8,0.95], theta=0.5) with T=100 and record the Monte Carlo distribution of the residual-distance measure and of the normalized FM-GLS estimator; if the distance is not bounded or the empirical distribution fails to converge to the claimed zero-mean Gaussian mixture as T grows, Assumption 2 or the limit in Theorem 3(c) is false.

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Extended reading notes

Core claim

The paper's central discovery is a feasible estimator whose normalized error converges to a zero-mean Gaussian mixture. In notation, under the paper's assumptions, $G_T^{-1}(\hat\beta_{\text{FGLS}}^+ - \beta)$ converges weakly to $\left(\int_0^1 J(r)\Omega_{uu}^{-1}J(r)'dr\right)^{-1} \int_0^1 J(r)\Omega_{uu}^{-1}dB_{u.v}(r)$, a distribution that is normal with zero mean conditional on the driving Brownian motion. Because the conditional variance is consistently estimable, the Wald statistic in Theorem 4 is asymptotically $\chi^2_k$. This is the first fully modified estimator for the SUCPR framework that obtains the mixed normal limit without leads-and-lags augmentation, using instead direct estimation of the inverse autocovariance matrix and an explicit second-order bias correction. The paper further claims the same machinery delivers multivariate KPSS-type tests whose null limit is $\int_0^1 \|W(r)\|^2 dr$, free of nuisance parameters.

Load-bearing premise

The load-bearing premise is that first-stage least-squares residuals remain within a bounded distance of the true innovations in overall size; the paper asserts this is mild and satisfied by least squares, but it is assumed rather than proven from primitive conditions, and the consistency of the banded inverse covariance estimate together with the FM-GLS normal-mixture limit both depend on it.

Editorial extensions

If this is right

Applied researchers can test linear hypotheses on individual coefficients of a seemingly unrelated cointegrating polynomial regression with conventional chi-square critical values, without leads-and-lags augmentation or simulation-based critical values.
FM-GLS should dominate the existing FM-SOLS and FM-SUR estimators in mean squared error, with the gains increasing in serial correlation and endogeneity; the simulations show the gains persist even for infeasible versions, pointing to the GLS weighting as the source.
The three multivariate KPSS-type tests give a nuisance-parameter-free null distribution, so the cointegration specification of a system can be checked without tabulating new critical values.
In the Environmental Kuznets Curve data, all three estimators support an inverted-U relation for all six countries, but confidence intervals differ sharply in width; the FM-GLS intervals are wider and their empirical coverage is closest to the nominal 95% level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension the paper leaves implicit is to build the BIAM from a first-stage residual that is itself bias-corrected rather than OLS; the high-persistence, high-endogeneity corner of the simulations is where such a variant would show whether Assumption 2 or the second-order correction is the limiting factor.
Because the BIAM controls parameter proliferation by banding, the construction may extend to panels with cross-section dimension growing with T, but the fixed-n rate conditions in Assumptions 3 and 5 would need to be reworked for that regime.
The reported power loss of the prefiltered KPSS test suggests a size-power tradeoff that could be explored by choosing the filter order or mixing filtered and unfiltered residuals; this is not pursued in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The bottom line: this is a solid, incremental paper. The FM-GLS estimator is a genuine addition to the seemingly unrelated cointegrating polynomial regression toolkit, and the asymptotic machinery checks out. The infeasible GLS limit coincides with MSUR, as the authors note, so the real novelty is the feasible estimator built on banded inverse autocovariance estimation plus the bias corrections. That part is well executed and the mixed normal limit (3.13) is derived from primitive OLS rates, not assumed. I do not see a load-bearing flaw in the central theory; Assumption 2 is actually proved in the proof of Theorem 3(c), so that concern disappears on close reading.

What they do well: the BIAM construction is computationally clean and avoids leads-and-lags; the Wald test in Theorem 4 is standard and works; the Monte Carlo is honest and relatively comprehensive. It includes the known failure case (high persistence, high endogeneity, T=100) and reports it openly. That kind of transparency counts.

The soft spots are in the empirical application and, to a lesser extent, the manuscript's completeness. All three cointegration tests reject the maintained quadratic EKC specification at 5%, and the authors proceed anyway. They justify this by comparability with Wagner et al. (2020), but it does mean the reported EKC coefficients come from a model the paper itself rejects. The confidence intervals also undercover significantly—89% for FM-GLS at nominal 95%, worse for FM-SOLS and FM-SUR—and the authors are upfront about it, even calibrating the scaling factors. Still, the applied reader should treat the EKC numbers as illustrative rather than decisive. The proofs depend on supplementary lemmas S1-S4 that are not fully in the preprint, and no code or replication data are shipped. These are addressable requirements for a journal version, not deep flaws.

The citation pattern is fine. The self-citation to Beutner et al. (2019) is for the covariance parametrization tool, not to inflate novelty.

Bottom line: the paper deserves a serious referee. The theory is a modest but sound step forward, the simulations support the claims, and the empirical caveats are at least acknowledged. I would send it to a good econometrics journal and ask the authors to make the supplement available and to strengthen the empirical discussion, possibly by presenting the specification tests as evidence against the quadratic model rather than continuing as if they had passed.

Referee Report

0 major / 6 minor

Summary. The paper proposes a Fully Modified GLS estimator for seemingly unrelated cointegrating polynomial regressions with deterministic trends and integer powers of I(1) regressors. The inverse autocovariance matrix of the errors is estimated directly through a modified Cholesky block decomposition based on first-stage OLS residuals, and the estimator applies a second-order bias correction so that the scaled estimation error converges to a zero-mean mixed Gaussian distribution. The paper also develops multivariate KPSS-type cointegration tests, studies their size and power by simulation, and applies the methods to a six-country Environmental Kuznets Curve model. The central theoretical result is Theorem 3(c), equation (3.13), giving the limiting distribution of FM-GLS, and Theorem 4, which derives a chi-square Wald statistic.

Significance. If the asymptotic results are correct, the paper makes a useful contribution by providing a feasible fully modified estimator for multivariate cointegrating polynomial regressions that avoids leads-and-lags augmentation and supports standard chi-square inference. The supplement contains the auxiliary lemmas underlying the consistency of the banded inverse autocovariance estimator, and the proof of Theorem 3(c) derives the previously flagged Assumption 2 from standard OLS rates rather than simply assuming it. The simulations compare feasible and infeasible estimators, and the paper is unusually candid about finite-sample failures, including the large MSE for high persistence at T=100 and the below-nominal coverage of empirical confidence intervals. The theoretical core is internally consistent: the bias terms cancel by construction in the definition of the FM-GLS correction, and the claimed mixed normal limit follows from standard cointegrating polynomial regression asymptotics.

minor comments (6)

[Section 3.2, Assumption 2] Assumption 2 is stated as a mild condition satisfied by OLS residuals, but the verification is deferred to the proof of Theorem 3(c). Since Theorem 2 already relies on Assumption 2, the verification should be moved to Section 3.2 or explicitly referenced there so that the feasibility of the GLS estimator does not depend on a later proof.
[Proof of Theorem 3(c)] The displayed bound for the OLS residuals reads ||hat u - u||_2 <= ||G_T^{-1}(hat beta_OLS - beta)|| ||G_T Z'Z G_T||, but the correct intermediate factor is ||Z G_T|| = ||G_T Z'Z G_T||^{1/2}, not the full norm ||G_T Z'Z G_T||. The conclusion O_p(1) is unaffected because ||G_T Z'Z G_T|| = O_p(1), but the inequality as written should be corrected.
[Assumption 3 and Theorem 5(b)] Assumption 3 is printed as "1/q_T + q_T^3/T T -> 0"; the intended condition is 1/q_T + q_T^3/T -> 0, and Theorem 5(b) similarly appears to omit a division sign in the requirement q_T/b_T + b_T/T -> 0. These typos should be fixed.
[Section 5, Table 5] All three cointegration tests reject the null of cointegration at the 5% level, yet the paper proceeds to estimate and interpret the EKC model. The authors disclose this, but the framing should make explicit that the empirical application is illustrative and cannot serve as evidence in favor of the quadratic EKC specification, given the formal rejection.
[References] The reference list contains two entries, Ing, Chiou, and Guo (2016a) and (2016b), with the same title and journal information. The duplication should be reconciled or the two distinct contributions should be clearly distinguished.
[Title and Section 1] The title uses "Efficient Estimation," but the paper does not establish a formal efficiency lower bound or compare the limit distribution to a semiparametric efficiency bound. Consider qualifying the claim, for example by referring to efficiency in the GLS sense, to avoid overstating the theoretical result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FM-GLS limit is derived from convergence arguments, and the sole self-citation is non-load-bearing.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 1 derives the infeasible GLS limit from the Beveridge-Nelson decomposition and standard weak convergence for cointegrating polynomial regressions; the statement that this limit coincides with Wagner et al. (2020)'s MSUR is a cross-check, not an input. Theorem 2 proves consistency of the BIAM estimator from Lemma S3 and truncation bounds, and the proof of Theorem 3(c) derives Assumption 2 from first-stage OLS rates rather than assuming it: 'Since the residuals {ˆut} are obtained by first stage OLS, we get ||ˆu−u||2 ≤ ||G−1T (ˆβOLS −β)||2 ||GT Z′ZGT || = Op(1).' The bias correction in (3.12) is constructed so that hat(B)+ estimates B+ = B_{epsilon-eta} - B_{vu}, but the mixed normal limit (3.13) follows from the convergence of each stochastic integral and the consistency of hat(B)+, not from the definition alone; this is the standard fully modified construction, not a reduction of the target to its input. The only self-citation, Beutner et al. (2019), is used as a pointer for applying the modified Cholesky block decomposition to time series; the decomposition itself is attributed to Pourahmadi (1999), Kim and Zimmerman (2012), and Kohli et al. (2016), and no theorem in the paper relies on Beutner et al. for its proof. The empirical EKC section explicitly acknowledges that the country sample was selected in Wagner et al. (2020) because it displays EKC behaviour, and the reported under-coverage of confidence intervals is a stated limitation rather than a fitted claim. No equation reduces by construction to a fitted target, and no load-bearing result is imported from a self-citation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard time series assumptions plus two tuning parameters and the unproved Assumption 2. No new physical or economic entities are introduced. The most consequential choices are the banding parameter and block size, both selected by data-driven rules rather than fixed constants.

free parameters (3)

banding parameter q_T = data-driven, e.g., H=[2 T^{1/4}], l0=[T/5] in Supplement S4
Controls how many VAR lags are used in the BIAM estimator. Any sequence satisfying q^3/T -> 0 is allowed asymptotically, but finite-sample values are chosen by subsampling and risk minimization rather than derived from the model.
block size b_T for KPSS tests = data-driven by minimum volatility rule of Romano and Wolf
Used in the subsampling cointegration tests. Asymptotic theory requires b_T/T -> 0, but the finite-sample choice affects size and power and is selected by a data-dependent rule.
autocovariance truncation r_T for long-run covariance estimator = Assumption 5: r_T = O(q_T), r_T q_T^3/T -> 0
Tuning parameter for the BIAM-based one-sided long-run covariance estimator in (3.7). No automatic selector is specified in the main text.

assumptions (6)

domain assumption Assumption 1: i.i.d. innovations ζ_t with positive definite covariance, finite 2r moments, invertible VAR(∞) errors with sum j||A_j||_F < infinity.
Standard linear process assumptions that justify the FCLT, Beveridge-Nelson decomposition, and VAR approximation used throughout the proofs.
domain assumption Assumption 1(c): det(D(1)) != 0, so there is no cointegration among the I(1) regressors, and no cross-sectional cointegration is allowed.
Rules out unit roots in the regressor innovation process and cointegration across equations, which are needed for the stated convergence rates and mixed normal limits.
ad hoc to paper Assumption 2: first-stage OLS residuals satisfy ||\hat u - u||_2 = O_p(1).
The paper states this is mild and satisfied by least squares residuals but does not prove it from primitive conditions. It is load-bearing for Lemma S3 and Theorem 2.
ad hoc to paper Assumptions 3 and 5: banding parameter q_T satisfies q_T^3/T -> 0 and r_T satisfies r_T q_T^3/T -> 0 with r_T = O(q_T).
Rate conditions on tuning parameters that ensure the BIAM covariance estimator and long-run covariance estimators are consistent.
domain assumption Assumption 4: kernel estimators of long-run covariance matrices are consistent.
Used for FM-SOLS and FM-SUR comparisons; standard conditions are cited from Andrews (1991) and Newey-West (1994).
standard math Baxter's inequality, the First Moment Bound Theorem of Findley and Wei (1993), and de Jong (2002) for nonlinear transformations of integrated processes.
External mathematical tools used in the supplementary proofs to control VAR approximation errors and partial sum moments.

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Cite this review

Pith. "Pith review of Efficient Estimation by Fully Modified GLS with an Application to the Environmental Kuznets Curve." pith.science (2026). https://pith.science/paper/L3KZQEIE

@misc{pith2026190802552,
  author       = {Pith},
  title        = {Pith review of: Efficient Estimation by Fully Modified GLS with an Application to the Environmental Kuznets Curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3KZQEIE}},
  note         = {Machine review of arXiv:1908.02552}
}

read the original abstract

This paper develops the asymptotic theory of a Fully Modified Generalized Least Squares estimator for multivariate cointegrating polynomial regressions. Such regressions allow for deterministic trends, stochastic trends and integer powers of stochastic trends to enter the cointegrating relations. Our fully modified estimator incorporates: (1) the direct estimation of the inverse autocovariance matrix of the multidimensional errors, and (2) second order bias corrections. The resulting estimator has the intuitive interpretation of applying a weighted least squares objective function to filtered data series. Moreover, the required second order bias corrections are convenient byproducts of our approach and lead to standard asymptotic inference. We also study several multivariate KPSS-type of tests for the null of cointegration. A comprehensive simulation study shows good performance of the FM-GLS estimator and the related tests. As a practical illustration, we reinvestigate the Environmental Kuznets Curve (EKC) hypothesis for six early industrialized countries as in Wagner et al. (2020).

Figures

Figures reproduced from arXiv: 1908.02552 by the authors.

**Figure 1.** Empirical size of the joint Wald tests H0 : β1,4 = β2,4 = · · · = βn,4 = −0.3, where βi,4 denote the coefficients in front of x 2 it. The Wald-SOLS test (green) and Wald-SUR test (blue) are based on Proposition 2 by Wagner et al. (2020). The Wald-FGLS test (red) is found in Theorem 4. We vary the serial correlation parameter ρ1 from 0 to 0.9 while keeping ρ2 = ρ3 = ρ4 = 0.8 fixed. The cross-sectional dimension is n … view at source ↗

**Figure 2.** Empirical size of the joint Wald tests H0 : β1,4 = β2,4 = · · · = βn,4 = −0.3, where βi,4 denote the coefficients in front of x 2 it. The Wald-SOLS test (green) and Wald-SUR test (blue) are based on Proposition 2 by Wagner et al. (2020). The Wald-FGLS test (red) is found in Theorem 4. We vary the endogeneity parameter ρ2 from 0 to 0.9 while keeping ρ1 = ρ3 = ρ4 = 0.8 fixed. The cross-sectional dimension is n = 3. 33… view at source ↗

**Figure 3.** Empirical size-corrected power of the single-equation Wald tests H0 : β1,4 = −0.3 where β1,4 is the coefficient in front of x 2 1t . We consider n = 3, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}. The WaldSOLS test (green) and Wald-SUR test (blue) are based on Proposition 2 by Wagner et al. (2020). The Wald-FGLS test (red) is found in Theorem 4. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Empirical size-corrected power of the single-equation Wald tests H0 : β1,4 = −0.3 where β1,4 is the coefficient in front of x 2 1t . We consider n = 5, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}. The WaldSOLS test (green) and Wald-SUR test (blue) ar…

**Figure 5.** Figure 5: Empirical size-corrected power of the joint Wald tests H0 : β1,4 = β2,4 = · · · = βn,4 = −0.3 where βi,4 are the coefficients in front of x 2 it. We consider n = 3, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}. The Wald-SOLS test (green) and Wald-SUR t…

**Figure 6.** Figure 6: Empirical size-corrected power of the joint Wald tests H0 : β1,4 = β2,4 = · · · = βn,4 = −0.3 where βi,4 are the coefficients in front of x 2 it. We consider n = 5, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}. The Wald-SOLS test (green) and Wald-SUR t…

**Figure 7.** Figure 7: The plots of the residuals ˆut,S OLS = yt − Ztβb+ S OLS (top), ˆut,S UR = yt − Ztβb+ S UR (middle), and ˆut,FGLS = yt − Ztβb+ FGLS (bottom) for the empirical study. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_7.png]

**Figure 8.** Figure 8: The fit of the FM-SOLS, FM-SUR, and FM-GLS estimates. S3.2 Simulation DGP The following procedure was used to obtain a simulation DGP that closely mimics the data characteristics. (a) Fit VAR(p) models (1 ≤ p ≤ 8) to the series{uˆt,FGLS } and {∆xt} individually. The B…

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

goodness of ﬁt

Abadir, K. M. and J. R. Magnus (2005). Matrix Algebra. Cambridge University Press. Anderson, T. W. and D. A. Darling (1952). Asymptotic theory o f certain “goodness of ﬁt” criteria based on stochastic processes. The Annals of Mathematical Statistics 23 , 193–212. Andrews, D. W. K. (1991). Heteroskedasticity and autocorre lation consistent covariance matri...

work page 2005
[2]

follows from Lemma 1(a) and∑∞ j=q+1 j1/2 ‖ ‖ ‖A j ‖ ‖ ‖F = o(q−1/2). Using (A.2) and ‖ ‖ ‖S(q)−Σ ηη ‖ ‖ ‖→ 0 (see (S1.17) in the supplementary material), we obtain T∑ t=1 ( Aq(L)Z ′ t GT )′ S−1(q) ( Aq(L)Z ′ t GT ) = T∑ t=1 GT ZtΩ −1 uu Z ′ t GT + Op (q−1/2) ⇒ ∫ 1 0 J (r)Ω −1 uu J (r)′dr. (A.4) We rewrite the second part in (A.1) as T∑ t=1 ( Aq(L)Z ′ t GT...

work page 2016
[3]

■ Proof of Theorem 1 The premultiplication by Mu(q) applies a linear ﬁlter whereas S−1 u (q) im- plies weighting

for the ﬁnal inequality, we derive ∑q j=1 ‖ ‖ ‖~A j(q) ‖ ‖ ‖F ≤ ∑q j=1 j ‖ ‖ ‖A j(q) ‖ ‖ ‖F ≤ ∑q j=1 j ‖ ‖ ‖A j(q) − A j ‖ ‖ ‖F +∑q j=1 j ‖ ‖ ‖A j ‖ ‖ ‖F ≤ C ∑q j=1 j ‖ ‖ ‖A j ‖ ‖ ‖F ≤ C. ■ Proof of Theorem 1 The premultiplication by Mu(q) applies a linear ﬁlter whereas S−1 u (q) im- plies weighting. Since the behaviour of the ﬁrst q ≪ T elements does not...

work page 1988
[4]

The Wald-SOLS test (green) and Wald-SUR test (blue) are bas ed on Proposition 2 by Wagner et al

n = 3 n = 5 ρ Wald-SOLS Wald-SUR Wald-FGLS Wald-SOLS Wald-SUR Wald-FGL S Panel A: Single-equation test T = 100 0 11.75 13.44 9.89 14.30 17.84 13.44 0.3 13.25 15.09 9.92 15.32 19.59 13.55 0.6 16.13 19.03 7.54 18.80 27.65 11.86 0.8 20.56 26.60 4.70 26.72 43.11 10.13 T = 200 0 9.02 10.00 7.48 9.90 12.00 8.47 0.3 9.96 10.93 7.24 11.32 13.62 8.81 0.6 12.68 14....

work page 2020
[7]

(2) By Assumption 1, for any 1 ≤ k, i ≤ n, vkt and ηit are Near Epoch 24 Dependent in L4-norm on {[η ′ t , ε′ t]′} t∈Z of size −1 and arbitrary size, respectively

j] ηit ⏐ ⏐ ⏐≤ CT −3/2 ∑T t=1 |ηit| = op(1). (2) By Assumption 1, for any 1 ≤ k, i ≤ n, vkt and ηit are Near Epoch 24 Dependent in L4-norm on {[η ′ t , ε′ t]′} t∈Z of size −1 and arbitrary size, respectively. A small variation on Theorem 17.9 from Davidson (1994) shows that {vitηit} are L2-NED of size −1. The i.i.d. as- sumption on {[η ′ t , ε′ t]′}allows ...

work page 1994
[8]

The speciﬁc reason is as follows. By the binomial expansio n (below (A.3)), we have T −( j+1)/2 T∑ t=1 ∆x j ktηit = T −( j+1)/2 j−1∑ m=0 ( j m ) T∑ t=1 xm kt−1v j−m kt ηit = jT −( j+1)/2 T∑ t=1 x j−1 kt−1vktηit + op(1) = jΣ ǫkηi 1 T T −1∑ t=1 ( xkt √ T )j−1 + j√ T 1√ T T −1∑ t=1 ( xkt √ T )j−1 ( vktηit − Σ ǫkηi ) + op(1) ⇒ jΣ ǫkηi ∫ 1 0 B j−1 vk (r)dr, wh...

work page 2002
[9]

13 We conclude that ‖ ‖ ‖ˆMu(q)−Mu(q) ‖ ‖ ‖= Op (√ q3/T )

max 1≤ℓ≤q ‖ ‖ ‖ˆA(ℓ) − A(ℓ) ‖ ‖ ‖ 2 = Op (q3 T ) , where the ﬁnal step follows from Lemma S3. 13 We conclude that ‖ ‖ ‖ˆMu(q)−Mu(q) ‖ ‖ ‖= Op (√ q3/T ) . The di ﬀerence ˆS −1 u (q) − S−1 u (q) forms a symmetric and block diagonal matrix, hence ‖ ‖ ‖ˆSu(q) − Su(q) ‖ ‖ ‖= max {‖ ‖ ‖ˆS(0) − S(0) ‖ ‖ ‖, max1≤ℓ≤q ‖ ‖ ‖ˆS(ℓ) − S(ℓ) ‖ ‖ ‖ } . By Assumption 2, ‖ ...

work page 2015
[10]

Given Theorem 2, we have GT Z ′ ˆΣ −1u (q)ZG T = GT Z ′Σ −1 u (q)ZG T + op(1) and it converges weakly to the expression in (A.4)

From (3.12), the deﬁnition of the FM-GLS estimator, we have G−1 T (ˆβ + FGLS − β ) = ( GT Z ′ ˆΣ −1u (q)ZG T )−1 [ GT Z ′ ˆΣ −1u (q)u − GT Z ′( IT ⊗ ˆΩ −1 uu ˆΩ uv ˆΩ −1 vv ) v − GTˆB +] . Given Theorem 2, we have GT Z ′ ˆΣ −1u (q)ZG T = GT Z ′Σ −1 u (q)ZG T + op(1) and it converges weakly to the expression in (A.4). To continue, we deﬁne ˆAq(L) = In −∑q ...

work page 2016

Show all 24 references

[11]

The column labeled FGLS contains the numerical value of the MSE of feasible FM-G LS

under error Setting B. The column labeled FGLS contains the numerical value of the MSE of feasible FM-G LS. Other MSEs are expressed relative to this benchmark. V alues above 1 indicate a better performance of feasible FM-GLS. n = 3 n = 5 (λ, ¯λ) T SOLS SUR FGLS infSOLS infSUR...

2020
[13]

The Wald-SOLS test (green) and Wald-SUR test (blue) are bas ed on Proposition 2 by Wagner et al

32 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 Figure 2: Empirical size of the joint Wald tests H0 : β1,4 = β2,4 = · · · = βn,4 = −0.3, where βi,4 ...

2020
[14]

We consider n = 3, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}

33 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0...

2020
[15]

We consider n = 5, T ∈ {100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ {0.6, 0.8}

34 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0...

2020
[16]

We consider n = 3, T ∈ { 100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ { 0.6, 0.8}

35 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0...

2020
[17]

We consider n = 5, T ∈ { 100, 200}, and ρ1 = ρ2 = ρ3 = ρ4 = ρ with ρ ∈ { 0.6, 0.8}

36 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0.286 -0.284 -0.282 -0.28 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.3 -0.298 -0.296 -0.294 -0.292 -0.29 -0.288 -0...

2020
[18]

At a signiﬁcance level of 5%, the null hypothesis of cointegration is rejected when ‘’rej ection rule’ is less than 5%

38 Table 5: Outcomes for the joint tests of cointegration in the empiric al study. At a signiﬁcance level of 5%, the null hypothesis of cointegration is rejected when ‘’rej ection rule’ is less than 5%. KS OLS KS UR KBIAM Statistic 16.54 12.66 8.19 Rejection Rule (in %) 0.00 0...

2012
[19]

We observe that ui,t = ∑∞ j=0 rowi (C j )η t− j

Recall C(L) = [A(L)]−1 = ∑∞ j=0 C jL j with C0 = In, and ∑∞ j=0 j ‖C j‖F < ∞. We observe that ui,t = ∑∞ j=0 rowi (C j )η t− j. By Proposition 10.2(b) of Hamilton (1994), abso- lute summability of the coe ﬃcient matrices {C j}∞ j=0 implies ∑∞ s=0 |γu,k(s)| < ∞ where γu,k(s) = 4...

1994
[20]

Overall, this gives ‖ ‖ ‖ ‖ ‖ ‖ ‖ 1 T − q T −1∑ t=q ˆut(q)ˆut(q)′ − E(ut(q)ut(q)′) ‖ ‖ ‖ ‖ ‖ ‖ ‖= Op ( q √ T ) + Op ( q√ T ) = op(1)

Because 1 T −q ∑T −1 t=q ‖ut(q)‖= Op(q) by Markov’s inequality, we conclude Ib = Op ( q/ √ T ) . Overall, this gives ‖ ‖ ‖ ‖ ‖ ‖ ‖ 1 T − q T −1∑ t=q ˆut(q)ˆut(q)′ − E(ut(q)ut(q)′) ‖ ‖ ‖ ‖ ‖ ‖ ‖= Op ( q √ T ) + Op ( q√ T ) = op(1). (S1.11) Now observe that E(ut(q)ut(q)′)is a le...

1987
[21]

Collecting all the results, we have Ia ≤ C √ Kq, Ib ≤ C √ Kq, and thus ‖ ‖ ‖Fu(q) − Fu ‖ ‖ ‖1 ≤ C √ Kq

It is likewise straightforward to derive ∑T −1− j i=1 ‖Ai(i + j) − Ai(q)‖2 F ≤ CKq. Collecting all the results, we have Ia ≤ C √ Kq, Ib ≤ C √ Kq, and thus ‖ ‖ ‖Fu(q) − Fu ‖ ‖ ‖1 ≤ C √ Kq. For ‖Mu(q) − Mu‖∞, we are bounding the maximum absolute row sums. For an arbitr ary (nT ×...

1952
[22]

It suﬃces to prove ‖ ‖ ‖ˆΣ ξ(qT ) ‖ ‖ ‖= Op(1)

Finally, showing ‖ ‖ ‖ˆΣ ξ(qT )Q1 ‖ ‖ ‖= Op(1) will complete the proof after a straightforward comparison of the established stochastic orders. It suﬃces to prove ‖ ‖ ‖ˆΣ ξ(qT ) ‖ ‖ ‖= Op(1). Weyl’s inequality (e.g. pages 40 and 46 in Tao (2012)) an d Theorem 2 imply ⏐ ⏐ ⏐ ⏐λm...

2012
[23]

The BIC criterion select the V AR(1) speciﬁcation for both series (Table 7)

to the series { ˆut,FGLS } and {∆xt} individually. The BIC criterion select the V AR(1) speciﬁcation for both series (Table 7). St ore the coeﬃcient matrices ˆAu and ˆAv as well as the residual series { ˆη t} and { ˆεt}, respectively. (b) Stack ˆζt = [ ˆη ′ t , ˆε′ t]′ and com...

2008
[24]

does not lead to qualitatively di ﬀerent results. When we implement the minimum volatility rule as mentioned i n Section 3 to select b, the values of tuning parameters are adopted from Wagner and Hong (2016), see their online supple- mentary material. 50

2016
[323]

Newey, W. K. and K. D. West (1994). Automatic lag selection in covariance matrix estimation. The Review of Economic Studies 61 , 631–653. Nordhaus, W. D. (2013). The Climate Casino: Risk, Uncertainty, and Economics for a W arming World. Y ale University Press. 20 Nyblom, J. an...

1994
[383]

Romano, J. P . and M. Wolf (2001). Subsampling intervals in au toregressive models with linear time trend. Econometrica 69, 1283–1314. Saikkonen, P . (1992). Estimation and testing of cointegrat ed systems by an autoregressive approx- imation. Econometric Theory 8 , 1–27. Shin...

2001
[502]

Lin, and S

Beutner, E., Y . Lin, and S. Smeekes (2019). GLS estimation an d conﬁdence sets for the date of a single break in models with trends. Working Paper. Bickel, P . J. and E. Levina (2008). Regularized estimation of large covariance matrices. The Annals of Statistics 36 , 199–227....

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

goodness of ﬁt

Abadir, K. M. and J. R. Magnus (2005). Matrix Algebra. Cambridge University Press. Anderson, T. W. and D. A. Darling (1952). Asymptotic theory o f certain “goodness of ﬁt” criteria based on stochastic processes. The Annals of Mathematical Statistics 23 , 193–212. Andrews, D. W. K. (1991). Heteroskedasticity and autocorre lation consistent covariance matri...

work page 2005

[2] [2]

follows from Lemma 1(a) and∑∞ j=q+1 j1/2 ‖ ‖ ‖A j ‖ ‖ ‖F = o(q−1/2). Using (A.2) and ‖ ‖ ‖S(q)−Σ ηη ‖ ‖ ‖→ 0 (see (S1.17) in the supplementary material), we obtain T∑ t=1 ( Aq(L)Z ′ t GT )′ S−1(q) ( Aq(L)Z ′ t GT ) = T∑ t=1 GT ZtΩ −1 uu Z ′ t GT + Op (q−1/2) ⇒ ∫ 1 0 J (r)Ω −1 uu J (r)′dr. (A.4) We rewrite the second part in (A.1) as T∑ t=1 ( Aq(L)Z ′ t GT...

work page 2016

[3] [3]

■ Proof of Theorem 1 The premultiplication by Mu(q) applies a linear ﬁlter whereas S−1 u (q) im- plies weighting

for the ﬁnal inequality, we derive ∑q j=1 ‖ ‖ ‖~A j(q) ‖ ‖ ‖F ≤ ∑q j=1 j ‖ ‖ ‖A j(q) ‖ ‖ ‖F ≤ ∑q j=1 j ‖ ‖ ‖A j(q) − A j ‖ ‖ ‖F +∑q j=1 j ‖ ‖ ‖A j ‖ ‖ ‖F ≤ C ∑q j=1 j ‖ ‖ ‖A j ‖ ‖ ‖F ≤ C. ■ Proof of Theorem 1 The premultiplication by Mu(q) applies a linear ﬁlter whereas S−1 u (q) im- plies weighting. Since the behaviour of the ﬁrst q ≪ T elements does not...

work page 1988

[4] [4]

The Wald-SOLS test (green) and Wald-SUR test (blue) are bas ed on Proposition 2 by Wagner et al

n = 3 n = 5 ρ Wald-SOLS Wald-SUR Wald-FGLS Wald-SOLS Wald-SUR Wald-FGL S Panel A: Single-equation test T = 100 0 11.75 13.44 9.89 14.30 17.84 13.44 0.3 13.25 15.09 9.92 15.32 19.59 13.55 0.6 16.13 19.03 7.54 18.80 27.65 11.86 0.8 20.56 26.60 4.70 26.72 43.11 10.13 T = 200 0 9.02 10.00 7.48 9.90 12.00 8.47 0.3 9.96 10.93 7.24 11.32 13.62 8.81 0.6 12.68 14....

work page 2020

[5] [7]

(2) By Assumption 1, for any 1 ≤ k, i ≤ n, vkt and ηit are Near Epoch 24 Dependent in L4-norm on {[η ′ t , ε′ t]′} t∈Z of size −1 and arbitrary size, respectively

j] ηit ⏐ ⏐ ⏐≤ CT −3/2 ∑T t=1 |ηit| = op(1). (2) By Assumption 1, for any 1 ≤ k, i ≤ n, vkt and ηit are Near Epoch 24 Dependent in L4-norm on {[η ′ t , ε′ t]′} t∈Z of size −1 and arbitrary size, respectively. A small variation on Theorem 17.9 from Davidson (1994) shows that {vitηit} are L2-NED of size −1. The i.i.d. as- sumption on {[η ′ t , ε′ t]′}allows ...

work page 1994

[6] [8]

The speciﬁc reason is as follows. By the binomial expansio n (below (A.3)), we have T −( j+1)/2 T∑ t=1 ∆x j ktηit = T −( j+1)/2 j−1∑ m=0 ( j m ) T∑ t=1 xm kt−1v j−m kt ηit = jT −( j+1)/2 T∑ t=1 x j−1 kt−1vktηit + op(1) = jΣ ǫkηi 1 T T −1∑ t=1 ( xkt √ T )j−1 + j√ T 1√ T T −1∑ t=1 ( xkt √ T )j−1 ( vktηit − Σ ǫkηi ) + op(1) ⇒ jΣ ǫkηi ∫ 1 0 B j−1 vk (r)dr, wh...

work page 2002

[7] [9]

13 We conclude that ‖ ‖ ‖ˆMu(q)−Mu(q) ‖ ‖ ‖= Op (√ q3/T )

max 1≤ℓ≤q ‖ ‖ ‖ˆA(ℓ) − A(ℓ) ‖ ‖ ‖ 2 = Op (q3 T ) , where the ﬁnal step follows from Lemma S3. 13 We conclude that ‖ ‖ ‖ˆMu(q)−Mu(q) ‖ ‖ ‖= Op (√ q3/T ) . The di ﬀerence ˆS −1 u (q) − S−1 u (q) forms a symmetric and block diagonal matrix, hence ‖ ‖ ‖ˆSu(q) − Su(q) ‖ ‖ ‖= max {‖ ‖ ‖ˆS(0) − S(0) ‖ ‖ ‖, max1≤ℓ≤q ‖ ‖ ‖ˆS(ℓ) − S(ℓ) ‖ ‖ ‖ } . By Assumption 2, ‖ ...

work page 2015

[8] [10]

Given Theorem 2, we have GT Z ′ ˆΣ −1u (q)ZG T = GT Z ′Σ −1 u (q)ZG T + op(1) and it converges weakly to the expression in (A.4)

From (3.12), the deﬁnition of the FM-GLS estimator, we have G−1 T (ˆβ + FGLS − β ) = ( GT Z ′ ˆΣ −1u (q)ZG T )−1 [ GT Z ′ ˆΣ −1u (q)u − GT Z ′( IT ⊗ ˆΩ −1 uu ˆΩ uv ˆΩ −1 vv ) v − GTˆB +] . Given Theorem 2, we have GT Z ′ ˆΣ −1u (q)ZG T = GT Z ′Σ −1 u (q)ZG T + op(1) and it converges weakly to the expression in (A.4). To continue, we deﬁne ˆAq(L) = In −∑q ...

work page 2016