Banka Slovenije Working Papers



Integrated Modified Least



Squares Estimation and



(Fixed-b) Inference for



Systems of Cointegrating



Multivariate Polynomial



Regressions



Authors: Sebastian Veldhuis, Martin Wagner



December 2024



Collection: Banka Slovenije Working Papers

Title: Integrated Modified Least Squares Estimation and (Fixed-b)

Inference for Systems of Cointegrating Multivariate

Polynomial Regressions

Authors: Sebastian Veldhuis, Martin Wagner

Issue: December 2024

Year: 2024

Place of publication: Ljubljana

Issued by:

Banka Slovenije

Slovenska 35,

1505 Ljubljana, Slovenija

www.bsi.si

Electronic edition:

https://www.bsi.si/en/publications/research/banka-slovenije-

working-papers

Te views expressed in this paper are solely the responsibility of

the author and do not necessarily reflect the views of Banka

Slovenije or the Eurosystem.The figures and text herein may

only be used or published if the source is cited.

© Banka Slovenije



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univerzitetni knjižnici v Ljubljani

COBISS.SI-ID 219690499

ISBN 978-961-7230-11-6 (PDF)





Integrated Modified Least Squares




Estimation and (Fixed-b) Inference for



Systems of Cointegrating Multivariate



Polynomial Regressions



Sebastian Veldhuis1 1,2,3 and Martin Wagner



1 Department of Economics, University of Klagenfurt

2 Bank of Slovenia, Ljubljana

3 Institute for Advanced Studies, Vienna



Abstract

We consider integrated modified ordinary and generalized least squares estimation

for systems of cointegrating multivariate polynomial regressions, i. e., systems of re-

gressions that include deterministic variables, integrated processes and products of

non-negative integer powers of these variables as regressors. The stationary errors

are allowed to be correlated across equations, over time and with the regressors.

The necessity to consider integrated modified generalized least squares estimation

arises in case of estimation under restrictions, which in general implies that ordinary

and generalized least squares estimators cease to be identical. We discuss in detail

hypothesis testing for the unrestricted and restricted estimators. Furthermore, we

develop asymptotically pivotal fixed-b inference, which is shown to be available only

in the case of full design for up-to-the-intercept-identical hypotheses tested in all

equations in systems with identical regressors in all equations.



JEL Classification: C12, C13, C32



Keywords: Integrated Modified Estimation, Cointegrating Multivariate Polyno-

mial Regression, Fixed-b Inference, Generalized Least Squares



1





Povzetek




Obravnavamo integrirano modificirano navadno in posploˇ seno ocenjevanje najmanjˇ sih

kvadratov za sisteme kointegrirajoˇ cih multivariatnih polinomskih regresij, tj. sisteme

regresij, ki kot regresorje vkljuˇ cujejo deterministiˇ cne spremenljivke, integrirane pro-

cese in produkte nenegativnih celoˇ stevilskih moˇ ci teh spremenljivk. Dovoljeno je, da

so stacionarne napake korelirane med enaˇ cbami, skozi ˇ cas in z regresorji. Potreba

po upoˇ stevanju integriranega modificiranega posploˇ senega ocenjevanja po metodi naj-

manjˇ sih kvadratov se pojavi v primeru ocenjevanja z omejitvami, kar na sploˇ sno pomeni,

da obiˇ cajne in posploˇ sene cenilke po metodi najmanjˇ sih kvadratov niso veˇ c identiˇ cne.

Podrobno obravnavamo testiranje hipotez za neomejene in omejene cenilke. Poleg tega

razvijamo asimptotiˇ cno pivotalno inferenco s fiksno-b, ki je na voljo le v primeru popol-

nega naˇ crta za hipoteze, identiˇ cne do prestreznega ˇ clena, testirane v vseh enaˇ cbah v

sistemih z enakimi regresorji v vseh enaˇ cbah.



2





1 Introduction




This paper considers integrated modified (IM) least squares estimation – both OLS

and GLS – for systems of cointegrating multivariate polynomial regressions (SCMPRs).

These are systems of regressions that include deterministic variables, integrated processes

and products of (non-negative) integer powers of these variables as regressors. The error

terms, assumed to be jointly stationary across equations, are allowed to be correlated

– both serially and across equations – and the stochastic regressors are allowed to be

endogenous. Thus, the paper extends the analysis of univariate cointegrating multivari-

ate polynomial regressions (CMPRs) of Vogelsang and Wagner (2024) to the systems

setting.1



Integrated modified ordinary least squares (IM-OLS) estimation, introduced for cointe-

grating linear regressions in Vogelsang and Wagner (2014), has several key advantages

compared to other modified least squares estimators used in cointegrating regression

analysis: First, estimation is tuning-parameter-free. IM-OLS requires the estimation

of a conditional long-run covariance matrix for inference only. Second, IM estimation

allows performing fixed-b inference in cointegrating regressions (see Vogelsang and Wag-

ner, 2014, Section 5). Fixed-b inference is designed to capture the impact of kernel and

bandwidth choices, required for estimating the above-mentioned conditional long-run

covariance matrix, on the sampling distributions of test statistics.2 Third, and this is a

very important conceptual advantage, IM estimation can be extended straightforwardly

to allow for the inclusion not only of powers of integrated processes as regressors (the

CPR case), but also of arbitrary non-negative integer products of integrated processes

as regressors.3



1 This paper fulfills a similar (but more encompassing) extension-to-systems-of-equations role as Wagner

(2023) fulfills for Wagner and Hong (2016). These two earlier papers – discussing fully modified least

squares estimation – only consider (systems of) cointegrating polynomial regressions (CPRs), in which

cross-products (of non-negative integer powers) of integrated processes are not included as regressors.

Note that Wagner (2023) contains a typo in the definition of ˆ + M below equation (7). The u and v

subscripts in the ˆ ∆-terms in the definition of ˆ M u need to be switched, e. g., in the i-th row ˆ ∆u ivj

needs to be replaced by ˆ ∆ v , . . . , m M j u for j = 1 . The sentence should then continue with: “and ˆ iv

defined analogously, with ˆ ∆v , . . . , n, = 1, . . . , m replaced by ˆ j u , i = 1 j ∆ i, j = 1 .” i v j v , , . . . , m i

2 Fixed-b analysis of spectral estimators has been introduced by Neave (1970). It has been developed

into an alternative framework for (robust) inference for stationary regressions in Kiefer and Vogelsang

(2005).

3 Note that this overcomes, admittedly only for the case of multivariate polynomials, the additive se-

parability between integrated regressors that is most often assumed in the nonlinear cointegration

literature. For a more encompassing discussion of this aspect see Vogelsang and Wagner (2024,

Section 1).



3





One important (economic) application of CMPRs, i. e., of regressions involving cross-


products of integrated regressors, are so-called Translog functions, see, e. g., Christensen

et al. (1971) or the example in Remark 1. A second important application is RESET-

type specification testing (originally introduced by Ramsey, 1969) of cointegrating (mul-

tivariate polynomial) regressions using the version of Thursby and Schmidt (1977), where

(non-negative integer) powers and cross-products of powers of the regressors in the orig-

inal regression are included in an augmented test regression. See the discussion in

Vogelsang and Wagner (2024, Sections 2.4 and 3) for the single-equation case.4



Considering systems of equations, useful to, e. g., analyze Translog cost or production

function systems with several outputs, adds some further aspects compared to the single-

equation setting discussed in Vogelsang and Wagner (2024). First, see also the cor-

responding discussion in Wagner (2023), systems of equations necessitate a detailed

consideration of generalized least squares estimators, in this paper integrated modified

generalized least squares (IM-GLS). This stems from the well-known fact that OLS-

and GLS-type estimators coincide in general only (for any positive definite symmetric

weighting matrix) in systems with identical regressors in all equations and without pa-

rameter constraints. Second, the scope of fixed-b inference needs to be investigated in

more detail than in the single-equation case. It turns out that – in addition to full design,

required also in Vogelsang and Wagner (2024) – fixed-b inference is only available for

up-to-the-intercept-identical hypotheses tested in all equations in systems with identical

regressors in all equations. Whilst this is, of course, restrictive it includes, e. g., fixed-b

RESET-type specification testing for systems of equations with identical regressors in

all equations under both the null specification and in the augmented test regression.5



This short paper is organized as follows: Section 2 discusses, organized in four sub-

sections, the setup and assumptions, unrestricted estimation and inference, estimation

and inference under restrictions and fixed-b inference. Section 3 briefly summarizes

and concludes. All proofs are relegated to the appendix. MATLAB code for IM-OLS/GLS

estimation and inference, including fixed-b inference – which necessitates (the generation

of) specification-dependent critical values (that additionally depend upon kernel function

and bandwidth) – is available upon request.



4 Note that, as already discussed in Wagner and Hong (2016, Section 2.3) and Vogelsang and Wagner

(2024, Section 2.4) additional integrated regressors (and their non-negative integer powers and cross-

products) can be included in the augmented test regression.

5 The limited scope of fixed-b inference implies that there is no need to consider fixed-b inference for IM-

GLS estimators, since IM-OLS and IM-GLS coincide in settings where fixed-b inference is available.



4





2 Theory




2.1 Setup and Assumptions



We start with considering unrestricted systems of cointegrating multivariate polynomial

regressions (SCMPRs) where all equations include the same set of regressors:



yt = ΘZt + ut, t = 1, . . . , T, (1)

x t = xt−1 + vt,



with ′ ′ i i1 im 0 y t := ( y 1 t , . . . , y nt ) , Z t := ( z 1 t , . . . , z |I| |I| t ) , with z it = t x · · · x for i = 1 , . . . , 1 t mt

and n×|I| i j h,i h=1,...,n,i∈I R non-negative integers for j = 0 , . . . , m , Θ = [ θ ] ∈ and x := t

( ′ x , . . . , x ). The regressors z , i = 1, . . . , |I| are ordered, e. g., by lexicographic

1t mt it

ordering of the multi-indices i := (i 0 , . . . , im) from a multi-index set I indexing all

regressors. To avoid perfect multi-collinearity by construction, we assume that no multi-

index i is contained more than once in I.



The results in this paper can be established under the same assumptions, adapted

to multivariate yt, as used in, e. g., Vogelsang and Wagner (2024, Footnote 10) and

we, therefore, abstain from positing a detailed set of assumptions. As is common in

the cointegrating regression literature, we also exclude cointegration amongst the m

integrated regressors {xt}t∈ as well as multi-cointegration in the system. Defining Z

{ ′ ′ ′ η } {} t t ∈ := ( u , v ) , a functional central limit theorem holds:

Z t t t∈Z



− B Xu(r) 1 ⌊rT ⌋ !

T /2 1/2 η ⇒ B ( r ) = = ΩW (r), (2)

t B v(r) =1 t



for 0 ≤ r ≤ 1, with W (r) denoting (n + m)-dimensional standard Brownian motion and

by assumption positive definite long-run covariance matrix:



Ω ! ∞ uu Ω

Ω = uv X ′ := E( η t −j η), (3)

Ω t vu vv j Ω

=−∞



partitioned conformably with η t. When Ωuv ̸= 0, the regressors are endogenous and

the setting also allows for relatively unrestricted forms of serial correlation of the errors

{ 1/2 1/2 ′ η t t∈ . Using, e. g., the Cholesky decomposition of Ω } = Ω Z vv vv (Ω vv ) , one can write (2)

more specifically as:



5





B ! ! ! 1 / 2 − 1 / 2 ′ u u· ( r) Ω v Ωuv (Ω vv ) W u


B := , (4) 1 / 2 ( ·v (r)

v vv v r ) 0 Ω W (r)



with Ω −1 u·v := Ω uu Ω − uv ΩΩ the (innovation) covariance matrix of B r) := B (r) vv vu u·v ( −

u

Ω −1 6 uvΩ B ( r ).

vv v



Remark 1. To exemplify the setting and notation, consider the following simple example

(with n = m = 2) of a two-output firm – with output “proper” Yt and (unwanted output)

emissions Et – using a Translog production function with the two input factors capital

K 7 t and labor L t. Including additionally (equation-specific) intercepts leads to:



ln 2 Y = θ + θ ln K + θ ln L + θ (ln K )

t 1,(0,0,0) 1,(0,1,0) t 1,(0,0,1) t 1,(0,2,0) t

+ 2 θ 1,(0,0,2) t (ln L ) + θ1,(0,1,1) t ln K ln Lt + u1t,

ln 2 E t = θ 2,(0,0,0) 2,(0,1,0) t + θ ln K + θ 2,(0,0,1) t ln L + θ 2,(0,2,0) t (ln K )

+ 2 θ 2,(0,0,2) t (ln L ) + θ2,(0,1,1) t ln K ln Lt + u2t,



with the coefficients double-indexed by first the equation index h = 1, 2 and second

the multi-index i i i 1 02 i = ( i , i , i ) ∈ I corresponding to regressor z = t (ln K ) (ln L )

0 1 2 it t t

with I = {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 2, 0), (0, 0, 2), (0, 1, 1)} and |I| = 6. In matrix

notation, the above system can be written more compactly as in (1):



6 Reduced rank of Ω u· corresponds to multi-cointegration, excluded in this paper due to the assumption v

that Ω is positive definite.

7 When considering emissions as one of the outputs, it is usual to also include energy as production factor.

We abstain from including additional production factors as well as a measure of the technology level

merely for algebraic brevity. In some applications, the inputs are disaggregated into five input factors:

capital, labor, energy, materials and services, labelled as KLEMS. For the 27 member countries of

the European Union, the EU-KLEMS project provides corresponding annual data over the period

1995–2020 for 23 industries. The short sample period makes this data set potentially unsuitable

for cointegration analysis (of the long-run behavior), but this question will be investigated in detail

elsewhere.



6





  1


 t    ln K

" # " # " #

ln   Y θ θ θ θ θ θ ln L u

t 1,(0,0,0) 1,(0,1,0) 1,(0,0,1) 1,(0,2,0) 1,(0,0,2) 1,(0,1,1)  t  1t = + ,

ln   2 E t 2,(0,0,0) 2,(0,1,0) 2,(0,0,1) 2,(0,2,0) 2,(0,0,2)  θ θ θ θ θ θ2 ,(0,1,1) t  (ln K ) u 2

t

 

| {z } | {z }  2 (ln L )  | {z } = y t =Θ  t  = u t

ln K t ln Lt

| {z }

=Zt

" # " # " #

ln Kt ln Kt−1 v1t

= + .

ln L t ln Lt−1 v2t

| {z } | {z } | {z }

=xt =xt−1 =vt



This two-output Translog example thus corresponds to a system without restrictions on

the parameter matrix, since all regressors are included in both equations.



Remark 2. The setting considered in this paper is closely related to the seemingly

unrelated cointegrating polynomial regression (SUCPR) setting considered in Wagner

et al. 8 (2020) and Knorre and Wagner (2024). A difference to those papers is the

consideration of cointegrating multivariate polynomial regressions, i. e., the inclusion of

cross-product terms (and that the two papers mentioned both consider fully modified

rather than integrated modified estimation). Given our setting, a system of seemingly

unrelated multivariate cointegrating polynomial regressions would be of the form:



y ′ = z θ + u , . . . , n, (5) , h = 1

ht ht h ht

xht = xh,t−1 + vht,



where ′ i i imh z ht := ( z h 1 t , . . . , z ) z = x · · · x for h |I with components of the form t 0 1 h | t hit h 1 t hm h t

h = 1, . . . , n and i = 1, . . . , |I | h. While in the “classical” seemingly unrelated regressions

(SUR) setting of, e. g., Zellner (1962), no explicit assumptions are posited about the

relationship between the regressors in the different equations, the seemingly unrelated

cointegrating regression literature often assumes that the regressors are disjoint between

the equations, with exceptions being Moon and Perron (2004) for the cointegrating linear

case and Knorre and Wagner (2024) for the cointegrating polynomial case. As Knorre



8 See also the discussion in Wagner (2023, Remark 2).



7





and Wagner (2024), we refer to common regressors as regressors occurring in at least


two equations.9



The equation system (5) can be rewritten in the form given in (1): Combine all distinct

elements of m x ht , h = 1 , . . . , n into a vector x t R ∈. In the case of no common integrated

regressors ′ ′ ′ P n , x t 1 := ( x , . . . , x ) , with m = m , and in the case that all integrated t nt h=1 h

regressors are common – as in the example in Remark 1 x t := x1t = · · · = xnt , with

m 10 = m 1 = · · · = m n. To define the joint regressor vector combine all distinct elements

of i i im 0 { z } · · · hit into a vector Z with elements z = t x 1 x with a corre-h =1 ,...,n,i =1 ,..., |I h | t it 1 t mt

spondingly defined multi-index set I. Clearly, elements of the vector Z t that occur as

regressors only in one or some equations, imply block-zero restrictions in rows of Θ.



Consider as a simple example a system of two countries described by Cobb-Douglas

production functions with again capital K t and labor Lt as production factors and with

a common technology level 11 A t , assumed for simplicity here as observable:



9 Note for completeness that, of course, Zellner (1962) also discusses the case where the regressors are

assumed to be identical across all equations. For this setting he shows the important and widely-

used result that OLS and GLS estimation coincide in the case that there are no restrictions on the

parameters. This result is, as is also well known, e. g., the “basis” for equation-by-equation OLS

estimation of unrestricted vector autoregressive models.

10 One key difference between Wagner et al. (2020) and Knorre and Wagner (2024) is that the former

paper excludes common regressors and the latter paper closes that gap. Both papers contain de-

tailed discussions of group-wise pooling, i. e., of groups of coefficients being identical across groups of

equations. This allows testing numerous specification-related hypotheses as well as performing cor-

respondingly restricted estimation. A second difference between the two papers is that Knorre and

Wagner (2024) provide a discussion for the general multiple integrated regressors case, whereas the

discussion in Wagner et al. (2020) considers a more stylized and, thus, algebraically more accessible

setting with only one integrated regressor and its powers per equation.

To complete the discussion, panel cointegrating (polynomial) regression settings are also closely re-

lated, see, e. g., de Jong and Wagner (2022) for pooled estimation or Wagner and Reichold (2023)

for group-mean estimation. In this case, one usually assumes that the regressors in a setting like (5)

are disjoint across equations (with some papers allowing for common regressors or factors in the

cointegrating linear case) and that the coefficients are pooled, i. e., identical across equations. In line

with the assumptions posited in classical panel analysis, the literature often assumes cross-sectional

independence and potentially cross-sectional i.i.d. behavior (see, e. g., Phillips and Moon, 1999). Fur-

thermore, it is customary, as in standard panel analysis, to include individual- and time-specific (fixed

or random) effects.

11 α β Commencing from Y = exp( θ ln A u ) K L taking logarithms leads, as is well known, to a line-

t t t t t

ar cointegrating relationship, i. e., ln Yt = θ + ln At + α ln Kt + β ln Lt + ut, if one assumes that

lnA t, ln Kt, ln Lt are integrated (but not cointegrated) and that ut is stationary.



8





 


1

  ln A

  t

" # " #   " #



ln Y 1t θ1,1 1 α1 β1 0 0  ln K  1 t u1t  =  + ,   ln Y 2 t θ 2 , 1 1 0 0 α 2 β 2  ln L 1 t  u 2 t

 

| {z } | {z }  | {z }  ln K =yt =Θ 2  t =ut 

ln L2t

| {z }

=Zt

x t = xt−1 + vt,



with ′ ′ ′ 12 x = (ln A , ln K , ln L , ln K , ln L ) , i. e., Z = (1 , x ) . In the notation of (1),

t t 1t 1t 2t 2t t t

the elements of i i 0 i i5 1 2 Z t are of the form z it = t (ln A t) (ln K 1t) · · · (ln L 2t) . Since the Cobb-

Douglas system corresponds to a system of (log-)linear cointegrating relationships, the

multi-indices i are given by (0, 0, . . . , 0) (the intercept) or i0 = 0 and exactly one ij = 1

for j = 1, . . . , 5 (the five integrated regressors), i. e., I = {(0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0),

. . . , 13 (0 , 0 , 0 , 0 , 0 , 1) } and |I| = 6.



The example illustrates several important aspects: As mentioned above, rewriting sys-

tems of seemingly unrelated cointegrating regressions into the systems format considered

in this paper leads to block-zero restrictions in rows of the coefficient matrix Θ, in this

case for the two equation-specific variables capital and labor. Common integrated regres-

sors (across all equations), in this example ln At, do not lead to row-wise zero restrictions.

Since we assume in this example that ln At enters the production function one-to-one,

the corresponding elements in Θ are restricted to be equal to one. However, one may

want to estimate the corresponding coefficients, θ 1,A and θ2,A say, to test whether the

coefficients are identical (and/or equal to one). Other hypotheses that one may want

to test in this example are constant returns to scale, i. e., α + β = 1, either in one or

both equations, or the same (in case of constant returns to scale) factor shares in both

countries, i. e., 14 α = α . Note that OLS- and GLS-type estimation do, in general,

1 2

12 The ordering of the stochastic regressors in x chosen here, with first the common regressor and then

t

the country-specific regressors ordered by equation is, of course, only one possibility. Knorre and

Wagner (2024), e. g., order common regressor(s) last.

13 Linking back to the standard notation defined above, e. g., α1 1 corresponds to θ . ,(0,0,1,0,0,0)

14 In the Translog case, testing for constant returns to scale involves several hypotheses. In the example of

Remark 1 these are (omitting the equation index): θ(0,1,0) +θ(0,0,1) = 1, θ(0,2,0) + θ(0,0,2) + θ(0,1,1) = 0

and θ(0,2,0) = θ(0,0,2).

One can also perform specification testing, using, e. g., a Translog system as a more general alterna-

tive to a Cobb-Douglas system and test the corresponding zero restrictions to the squared production

factors and the cross-product of the two production factors. In fact, one interpretation of the Translog



9





not coincide for this system, if one imposes the restrictions on Θ for estimation. This


exemplifies the need for considering both OLS- and GLS-type estimators.



Remark 3. In (1) we allow only for polynomial time trends, i. e., terms corresponding

to multi-indices of the form (i ≥ 0 , 0 , . . . , 0) for i 0 0. However, more general deterministic

components can, of course, be included, e. g., in a regression model of the form:



yt = ΘDDt + ΘZt + ut, (6)

x t = xt−1 + vt,



with Zt containing only zit with multi-indices where minj=1,...,m ij > 0. In this case, it

suffices to assume for p D t R ∈ that there exists a sequence of p × p scaling matrices AD

and a p-dimensional vector of functions D(z) such that for 0 ≤ r ≤ 1 it holds that:

Z r

lim 1/2 ′ T A D D ⌊rT ⌋ = D ( r ) with 0 < D ( z ) D ( z )dz < ∞. (7)

T →∞ 0



It is also possible to have elements of a more general Dt-vector included in the cross-

product terms, provided asymptotic multi-collinearity is excluded.



2.2 Estimation and Inference



IM-OLS estimation is in fact nothing but OLS estimation of the partial sum version of

equation (1) that is augmented by the original integrated regressors:



Sy Z u = Θ S + Γ x + S, t = 1, . . . , T, (8)

t t t t

= Φ ˜ Z u S + S,

t t



with y Pt Z u Z Z′ ′ ′ ∈ S := y , S , S defined analogously, ˜ S := ( S , x ) and Φ := (Θ , Γ) t j =1 j t t t t t

R . Stacking all observations, equation (8) can be written as: n ×(|I|+m)



S y Z u = Θ S + Γ X + S, (9)

= Φ ˜Z u S + S,



function is to consider it as a second-order Taylor approximation to an unknown more general pro-

duction function, see, e. g., Christensen et al. (1973) or Denny and Fuss (1977). This interpretation,

of course, leads to the potential consideration of higher-order Taylor approximations.



10





with y y y Z Z Z u u u S := ( S , . . . , S ), S := ( S , . . . , S ), X := ( x ), S := ( 1 1 , . . . , x S , . . . , S )


T T 1 T 1 T

and ˜Z Z ˜Z S := ( ˜ S , . . . , S ). Exactly as discussed in a closely related context in Wagner 1 T

(2023, Remark 1) and in fact known since Zellner (1962), see Footnote 9, for unre-

stricted systems of equations (that are linear in parameters) with identical regressors

in all equations, OLS estimation coincides (algebraically) with GLS estimation for any

(regular) weighting matrix. Consequently, with identical regressors in all equations and

without parameter restrictions, it suffices to consider the system version of the single

equation 15 IM-OLS estimator for CMPRs discussed in Vogelsang and Wagner (2024).

The IM-OLS estimator ˆ Φ is defined as the OLS estimator of Φ in (9), i. e.,:



Φ := ( ˆ y ˜Z′ Z ˜Z′ −1 S S )( ˜ S S ). (10)



The discussion of the asymptotic properties of the IM-OLS estimator requires the de-

finition of two quantities: First, the scaling matrix sequence AIM := diag(AIM,Θ, Im)

with i i i 0 A 1 IM · · · , Θ a diagonal matrix with the entry corresponding to regressor t x xm 1 t mt

T j=1 i − P m ( i 0 +( 2+1 2)

given by j )/ / . Second, the properly scaled partial sum process cor-

responding to the Second, the limit process corresponding to the regressors Zt, i. e.,

Z 1/2 ′ ( r ) := lim T →∞ T A ≤ Z ≤ for 0 r 1, with Z ( r ) := ( z ( r ) , . . . , z ( r )),

IM,Θ ⌊rT ⌋ 1 |I|

z i i i 0 1 · · · i ( r ) := r B v 1 ( r ) B v ≤ m ( r ) m for 0 ≤ r 1, i = 1, . . . , |I| and Bvj (r) denoting

the j-th component of B v(r).



Proposition 1. Let the data be generated by (1) with appropriate assumptions in place.

Define ∗ − 1 16 Φ := (Θ , Ω uv vv Ω ) , then as T → ∞ it holds that:

Z Z 1 1 −1

( ˆ ∗ −1 1/2 ′ ′ Φ − Φ ) A ⇒ IM u· Ω v u· W ( s ) f ( s ) ds f ( s ) f ( s )ds (11)





v


0 0

Z Z 1 − 1 1

= Ω1/2 ′ ′ u·v dW u·v ( s )[ F (1) − F ( s )] f ( s ) f ( s )ds ,

0 0



where:



" # R r r Z Z ( s ) ds

f 0 ( r ) := , F (r) := f(s)ds. (12)

Bv(r) 0



15 Later, when discussing hypothesis testing and estimation under restrictions, it is convenient to consider

vectorized version(s) of (9), either vectorized by y Z′ observation , i. e., vec( S ) = ( ˜ S ⊗ I )vec(Φ) + n

vec( u y′ Z′ ′ u′ ˜ S ) or vectorized by equation , i. e., vec( S ) = ( I ⊗ S )vec(Φ ) + vec( S).

n

16 R 1 ′ R 1 To detail notation: The ( i, j )-element of dW ( s )[ F (1) − F ( s )] is given by[F (1) − u·v j

0 0

F j(s)]dWu·v,i(s).



11





As indicated in Footnote 15, for hypothesis testing and estimation under restrictions, it


is convenient to consider the vectorized (by equation) version of the IM-OLS estimator

Φ defined in (10). Defining ˆ ′ ∗ ∗′ ϕ := vec(Φ ) and ϕ := vec(Φ), this leads to:



ϕ ˆ Z ˜ Z′ −1 Z y′ Z ˜Z′ −1 ˜ ( ˜ S S ) ( ˜ S S ) = ( I ⊗ ( ˜ S S ) )( I ⊗ S )vec(S ) (13) := vec Z y′

n n



and:



( − 1 ˆ ⊗ A ) ϕ − ϕ (14)

I ∗

n IM

Z 1 −1 ! Z 1

⇒ 1/2 ′ ′ (Ω u·v I ⊗ |I|+m )vec f ( s ) f ( s ) ds [ F (1) − F ( s )] dW u·v ( s ) .

0 0



Conditional upon Wv (r), the limiting distribution given in (14) is normal with zero mean

and (conditional) covariance matrix:



Z 1 −1

V ′ IM u := Ω ·v f ⊗ ( s ) f ( s )ds (15)

0

Z 1 ! Z 1 − 1



× ′ ′ F (1) − F ( s ) F (1) − F ( s ) ds f ( s ) f ( s )ds .

0 0



Given a consistent estimator ˆ ′ ′ ′ Ω u·v of Ω u·v , based on ˆ η := (ˆ u , v ), with ˆ u the OLS

t t t t

residuals of (1), an – up to scaling – estimator of VIM immediately follows by simply

using the sample counterparts of the expressions appearing in the limit given in (15),

i. e.,:



V ˆ ′ − ′ ′ − Z ˜ Z 1 Z ˜ Z1 IM := ˆ Ω u · v ⊗ ( ˜ S S ) CC ( ˜ S S ), (16)



with S ˜Z ˜Z ˜Z ˜Z S S − C := ( c P t ˜ Z S 1 , . . . , c T ), c t := S S for t = 1 , . . . , T , S := S and S = T t − 1 t j =1 j 0

0. By construction, when ˆ Ωu·v → Ωu·v in probability, which holds under standard

assumptions on kernels and bandwidths (see, e. g., Jansson, 2002), it follows that (In ⊗

A−1 −1 ) ˆ V ( I ⊗ A) ⇒ V .

IM IM n IM IM



The limiting distribution given in (14), in conjunction with the estimator ˆ V IM given

in (16), allows for asymptotic standard inference for testing (linear) restrictions on ϕ

under two assumptions on the restrictions matrix, R say, that are detailed (for the

single-equation case) in Vogelsang and Wagner (2024, Section 2.2): The first relates to

the fact that the parameter vector ˆ ϕ contains elements that converge at different rates,

which has some implications for hypotheses that lead to standard inference (encoded



12





in the matrix AR below). The second assumption on R, not explicitly stated in the


proposition, is that none of the hypotheses tested involves elements of Γ, which is not

estimated consistently.17



Proposition 2. Let the data be generated by (1) with appropriate assumptions in place

and assume that long-run covariance estimation is performed consistently. Consider s

linearly independent linear restrictions collected in:



H ′ : R vec (Φ) = Rϕ = r, (17)

0



with s×(|I|+m)n s R ∈ R of full row rank, r ∈ R and suppose that there exists a matrix

sequence s×s A ∈ R R such that:



lim −1 ∗ A R ( I n A ⊗ ) = R, (18)

R IM

T →∞



with ∗ s×(|I|+m)n R ∈ R of full row rank s. Then, it holds under the null hypothesis for

T → ∞ that the Wald-type statistic:



τ ˆ ˆ −1 ′ ′ ˆ

W IM s := ( R ϕ − r ) R V R ( R ϕ − r ) ⇒ O, (19)



with ˆ V IM as defined in (16) and Os denoting a chi-squared distributed random variable

with s degrees of freedom.



In the case that s = 1, it holds under the null hypothesis for T → ∞ that the t-type

statistic:



τt := ⇒ Z p, (20) ′ ˆ R R ˆ ϕ − r

VIMR



with Z denoting a univariate standard normally distributed random variable.



2.3 Estimation and Inference Under Restrictions



As discussed in Wagner (2023, Section 2.3), the cointegrating regression literature rarely

considers restricted least squares estimation, with one exception being the seemingly



17 ′ More formally, with K denoting the so-called commutation matrix, this means that for ϕ = vec(Φ) =

K ′ ′ ′ ′ ′ ′ vec(Φ) = K (vec(Θ) , vec(Γ) ) it has to hold that Rϕ = RK (vec(Θ) , vec(Γ) ) is of the form

RK = (Rvec(Θ), 0s×nm).



13





unrelated regressions (SUR) cointegration literature, see, e. g., Moon (1999), Moon and


Perron (2004), Park and Ogaki (1991) or Wagner et al. (2020). When not all equations

include the same set of regressors, OLS- and GLS-type estimation, in general, cease to be

algebraically (and asymptotically) equivalent.18 Potential choices concerning weighting

matrices in seemingly unrelated cointegrating regression systems are discussed in Park

and Ogaki (1991), see also Wagner (2023). IM-GLS estimation adds one additional

formal aspect to the discussion: The errors in the partial sum regression are integrated

and, therefore, weighting matrices cannot be directly related to covariance or long-run

covariance matrices of the error process, but rather to the first differences of the errors,

motivating the Park and Ogaki (1991) choices −1 −1 W = Ω or W = Ω also in the IM

uu u·v

setting.19 Clearly, restricted IM-OLS estimation is contained as the special case with

W ˆ 20 = W = I n .



To obtain a closed-form solution for the restricted estimator we consider, analogously to

hypothesis testing above, only affine restrictions on the parameter vector, i. e.,:



ϕ ′ = vec(Φ) = Dφ + d, (21)



with (|I|+m)n×g g D ∈ R of full column rank, φ ∈ R , arranged by equation similarly to

ϕ (|I|+m)n 21 , and d ∈ R . Given the mentioned fact that only the parameters in Θ are

estimated consistently, we only consider restrictions on Θ and do not consider restrictions

involving elements of Γ. As above, we need once again to posit an asymptotic rank

condition on the constraint matrix, i. e., we need to assume that there exists a matrix

sequence g ×g A D R ∈ such that:



lim − 1 ∗ ( I ⊗ A ) DA = D, (22)

n IM D

T →∞



with ∗ (|I|+m)n×g D ∈ R of full column rank.



18 We refer to GLS estimation for any variant of weighted least squares estimation and not – as in,

e. g., the classical Zellner (1962) setting – when weighting takes place with the inverse of the error

covariance matrix.

19 We use the notation W for weighting matrices for obvious reasons and are confident that no confusion

with Wiener processes, always written with argument, i. e., as W (r), will occur. 20 −1 −1 To be precise, the nT × nT weighting matrices considered in these cases are Ω ⊗ I , Ω ⊗ I and

uu T u·v T

In ⊗ IT , respectively. Note that all GLS results presented in this paper consider weighting matrices

of the form ˆ W ⊗ IT . From an algebraic perspective, one could, in principle, consider also “full”

W ˆ nT ×nT ∈ R weighting matrices.

21 As is well known, the explicit formulation of restrictions used in (21) is equivalent to the implicit for-

mulation Rϕ = r used in the discussion of the Wald-type test. Starting from the explicit formulation,

denote with (|I|+m)n×((|I|+m)n−g) ′ D ⊥ R ∈ a matrix of full column rank that fulfills D D = 0. Then ⊥

R ′ ′ = D , r = D d and s = (|I| + m)n − g.

⊥ ⊥



14





Proposition 3. Let the data be generated by (1) with appropriate assumptions in place


and ϕ fulfilling the (explicit) restrictions posited in (21). Furthermore, assume that there

exists a matrix sequence AD such that condition (22) holds. The restricted integrated

modified generalized least squares (IM-GLS) estimator ˆ ϕ R of ϕ with symmetric weighting

matrix sequence ˆ W is defined as:



ϕ ˆR := Dφ ˆ + d, (23)



with:



φ Z ˜Z −1 ′ ′ ˜

ˆ := (D ( ˆ W ⊗ S S )D (24)



× ′ ˜ Z y′ ˆ ˜ Z ˜ vec S S W − ( ˆ W ⊗ S S ) d .

D Z′



For ∗ ∗ ∗ ˆ φ such that ϕ = Dφ + d , it holds for T → ∞ and W → W > 0 that:



Z 1 − 1

A−1 ∗ ∗′ ′ ∗ ( ˆ φ − φ ) ⇒ D W ⊗ f ( s ) f ( s ) ds D (25)

D

0

Z 1

× ∗′ ′ D vec F (1) − F ( s ) dB u·v ( s )W .

0



The limiting distribution of φ ˆ given in (25) is – conditional upon Wv (r) – normal with

zero mean and covariance matrix:



V −1 −1 := A BA, (26)

IM ,R



with:

Z 1

A ∗′ ′ ∗ := D W ⊗ f ( s ) f ( s ) ds D, (27)

0

Z 1

B ∗′ ′ ∗ := D W Ω ⊗ u·v W F (1) − F ( s ) F (1) − F ( s ) ds D. (28)

0



An estimator of V IM,R is readily available, analogously to (16) and, therefore, asymp-

totically chi-squared or standard normal inference on φ follows, under conditions (22)

and (30), similarly to Proposition 2:22



22 Clearly, as already discussed below (21), since the elements of φ that correspond to elements of Γ are

not estimated consistently, the hypotheses are only allowed to involve entries of φ corresponding to

elements of Θ. Furthermore, note that the limiting distribution of ˆ ϕR is, by construction, singular

unless D is a regular matrix.



15





Proposition 4. Let the data be generated by (1) with appropriate assumptions in place


and assume that long-run covariance estimation is performed consistently. Let the pa-

rameter vector φ in ϕ = Dφ+d with condition (22) in place fulfill sφ linearly independent

restrictions, i. e.,:



H0 : Rφφ = rφ, (29)



with s ×g s φφ R φ R ∈ with full row rank s φ and r φ R ∈ . Furthermore, assume that there

exists a matrix sequence A ∈ s ×s φφ φ R such that:



lim −1 ∗ A R φ φ A D = R (30)

φ

T →∞



exists and has full row rank ˆ s φ and that W → W > 0 in probability. Then, it holds

under the null hypothesis for T → ∞ that the Wald-type statistic:



′ ˆ −1 − 1 ˆ ˆ

τ −1 ′ := ( R φ ˆ − r ) R A B A R (R φ ˆ − r ) ⇒ O , (31)

W ,R φ φ φ φ φ φ sφ



with Osφ denoting a chi-squared distributed random variable with sφ degrees of freedom

and:



A ˆ ′ ˜Z ˜Z′ := D ( ˆ W ⊗ S S)D, (32)

B ˆ ′ ˆ ˆ ′ := D ( ˆ W Ω ⊗ u · v W CC)D. (33)



In the case that sφ = 1, a t-type statistic that is asymptotically standard normally dis-

tributed can be defined analogously to Proposition 2.



Remark 4. To illustrate matters, let us consider the Cobb-Douglas system from Re-

mark 2 again. For this example, we have:

" #



Φ = θ1,1 1 α1 β1 0 0 γ1,1 . . . γ1,5 2×11 ∈ R, θ 2 , 1 1 0 0 α 2 β 2 γ 2 , 1 . . . γ 2 , 5



with γh,j , h = 1, 2(= n) and j = 1, . . . , 5(= m), denoting the coefficients in Γ. Since there

are 16 free parameters in Φ, using the notation as defined in (21), this implies that D ∈

R , 22 ×16 16 22 φ ∈ R and d ∈ R . More specifically, φ = (θ1,1 , α1 , β1 , γ1,1 , . . . , γ1,5 , θ2,1 , α2 , β2 ,

γ ′ ′ , . . . , γ ) and d = (0 , 1 , 0 , . . . , 0 , 1 , 0 , . . . , 0) with the two 1-entries in the second and

2,1 2,5

thirteenth coordinate. The rows of the matrix D are all of the form (i) one 1-entry, rela-

ting an element of φ, i. e., one of the unknown parameters, to an unrestricted coordinate



16





of ϕ, and all other entries equal to zero, or (ii) zeros only, corresponding to coordinates


in 23 ϕ that are fixed (either at zero or one in this example).

Next, consider restrictions on the elements of φ. The joint null hypothesis of constant

returns to scale in both countries and equal factor shares corresponds to H 0 : α1 + β1 =

1 24 , α 2 + β 2 = 1 , α 1 = α 2 :



    0 1 1 0 . . . 0 0 0 0 0 . . . 0 1

Rφ φ =  0 0 0 0 . . . 0 0 1 1 0 . . . 0  φ =  1  = rφ.    

0 1 0 0 . . . 0 0 −1 0 0 . . . 0 0



For this example, condition (22) is satisfied with −1/2 −1 A IM 5 = diag( T , T I, I5 ), AD =

I −1/2 −1 ∗ ∗ 2 diag( ⊗ T , T I 2 , I 5 ) and D = D . That D = D stems directly from the fact that

in each of the hypotheses considered only coefficients with the same convergence rate

are involved. This is a “standard testing” situation in which the rank constraint (22) is

satisfied trivially. The same observation holds true also for testing constant returns to

scale of Translog production functions (see Footnote 14).



Remark 5. Note that, exactly as discussed in Wagner (2023, Remark 1), for restrictions

of the form D = I ⊗ D n, with an asymptotic rank condition of the form (22) holding

for a full rank limiting matrix ∗ ∗ D = I n , the IM-GLS estimator coincides with the ⊗ D

IM-OLS estimator for any positive definite symmetric weighting matrix ˆ W .



2.4 Fixed-b Inference



As mentioned in the introduction, one advantage of the IM-OLS estimator introduced

for single-equation cointegrating linear regressions in Vogelsang and Wagner (2014) and

extended to the single-equation CMPR setting in Vogelsang and Wagner (2024) is that

it can be used for asymptotically pivotal fixed-b inference. In the CMPR setting, see

Vogelsang and Wagner (2024, Corollary 1 and Proposition 3), asymptotically pivotal

fixed-b inference requires full design of the regression model. Full design means that the

limit process Z(r) can be written as Z(r) = Π ZZW (r), with ΠZ a regular matrix and

Z 25 ( r ) a functional of standard Brownian motions.

W



23 This occurs in coordinates (and rows of D) two, five, six and 13 to 15 of ϕ. 24 Equivalently, one can also take β = β as third linearly independent restriction.

1 2

25 Full design of (S)CMPRs can always be achieved by adding regressors, see the (single-equation) dis-

cussion in Vogelsang and Wagner (2024).



17





The system CMPR setting considered in this paper adds another complexity to asymp-


totically pivotal fixed-b inference: The key quantity in fixed-b inference is a modified

estimator ˆ u Ω u·v,M of Ω u·v constructed from modified residuals ˆ S as defined below. In

t,M

the system case considered, this long-run covariance matrix is now, obviously, an n × n

matrix rather than, as in Vogelsang and Wagner (2014, 2024), a scalar. With respect

to VIM, this implies that (using a lower case letter for a scalar quantity) the variance

scaling factor in the test statistic is not of the form ωu·v times a matrix but, see (15),

given by the Kronecker product of Ωu·v and a matrix, M say. This implies, see the proof

of Proposition 5, that a sufficient condition for asymptotically pivotal fixed-b inference

is that the restrictions matrix s×(|I|+m)n R ∈ R – in Rϕ = r – fulfills R = I ⊗ R, with

n

R ∈ s/n×(|I|+m) s R , with s := a (positive) integer. This allows to perform fixed-b

R n

inference for testing up-to-the-intercept-identical hypotheses in all equations in systems

with identical regressors in all equations. Whilst this is clearly restrictive, it, e. g., in-

cludes RESET-type specification testing for SUCPR systems with identical regressors in

all equations in both the null specification and the augmented regression.



Proposition 5. Let the data be generated by (1) and assume that full design prevails.

Consider 26 s = s n linearly independent restrictions collected in: R



H ′ : R vec (Φ) = (I ⊗ R)ϕ = r, (34)

0 n



with s (|I|+ ) s R × mRn R ∈ R of full row rank s , r ∈ and suppose that there exists a

R R

matrix sequence A s × RsR R ∈ R such that:



lim −1 ∗ A R A R (35) = ,

T →∞ R IM



with ∗ s ×(|I|+m) R R ∈ R of full row rank sR. Then, it holds under the null hypothesis for

T → ∞ that the fixed-b Wald-type statistic:



τ ˆ ′ ˆ ′ ˆ ′ −1 −1

W ,b IM , := ( − R ϕ r ) R VM R (Rϕ − r) ⇒ Z (Q(P ) ⊗ I )Z (36) s s , R n R s R n



with ˆ ˆ ˆ ˆ V IM , M defined similarly as V IM in (16) , but with Ω u · v replaced by Ωu·v,M, defined

in (37) below, and ZsRn an sRn-dimensional standard normally distributed random vec-



26 As in Proposition 2, we assume that none of the hypotheses tested involves elements of Γ, compare

Footnote 17. This requires the last m columns of R to be zero.



18





tor independent of Q(P ). The precise form of Q(P ) depends on the specification of the


SCMPR 27 (1) , the kernel function k ( · ) and the bandwidth-to-sample-size ratio 0 < b ≤ 1 .



It is key for asymptotically pivotal fixed-b inference that Zs and Q(P ) in (36) are in-R n

dependent random variables. The necessity to achieve independence implies (for exactly

the same reason as discussed in detail in Vogelsang and Wagner, 2014, 2024) that, as

indicated above, Ω u y ˆ Z u · v cannot be estimated using the IM-OLS residuals ˆ S := S − Φ ˜ S t t t

and ˆu u ˆu u S := ( ˆ S , . . . , S ). Instead, orthogonalized modified residuals, ˆ S , have to 1 T t, M

be used to annihilate (nuisance-parameter-dependent) correlation. These are given

by ˆu ⊥′ ⊥ ⊥′ −1 ⊥ ⊥ ˜Z′ Z ˜Z′ −1 ˜Z S := ˆ S I − − M u T M ( M M ) M , with M := M ( I T S ( ˜ S S ) S),

M P P P T − ˜ Z t 1j ˜Z := ( M 1 T t j , . . . , M ) and M − := t S S for t = 1, . . . , T . The =1 j j=1 s=1 s

required modified estimator of Ωu·v is now defined as:



ˆ − X X i T T | − | 1 j u

Ω u′ u ·v,M := T k ∆ ˆ S , ∆ ˆ S (37)

B i,M j,M

i=2 j=2

⇒ 1/2 1/2′ Ω u·v Q ( P )Ω u·v ,



with kernel function k(·) and bandwidth B = bT for some 0 < b ≤ 1.



Remark 6. Let us illustrate the scope of fixed-b inference with the Translog example

discussed in Remark 1 and the Cobb-Douglas example in Remarks 2 and 4. The two-

outputs Translog example is an unrestricted system, i. e., a system with identical regres-

sors with unrestricted coefficients in both equations. It is therefore possible to perform

fixed-b inference for testing whether the system is in fact Cobb-Douglas, corresponding

to the coefficients to the squared production factors and the cross product being zero

in both equations, or whether both outputs are produced with constant returns to scale

(with the corresponding hypotheses given in Footnote 14). For the Cobb-Douglas exam-

ple fixed-b inference is not available since the starting point is, due to country-specific

production factors, a restricted system of equations (with block-zero restrictions in the

parameter matrix Θ.



27 Given the comparably limited scope for fixed-b inference in the SCMPR setting, we abstain from

explicitly stating and defining all necessary quantities. The stochastic process P (r) is the multivariate

analogue of P (r) as defined in Vogelsang and Wagner (2024, Proposition 3). The key difference is

that Wu·v(r) is now an n-dimensional rather than a scalar process. The other elements constituting

P(r) – g(r), G(r), h(r) and H (r) – are exactly as in Vogelsang and Wagner (2024, Corollary 1 and

Proposition 3). Furthermore, the form of the functional(s) Q(P ) is exactly as given above Vogelsang

and Wagner (2024, Proposition 3), conveniently defined there already for the multivariate case.



19





Remark 7. Note that if the restrictions considered in this subsection are not rejected,


the discussion in Footnote 21 clarifies that the corresponding restricted estimation prob-

lem is, unsurprisingly, subject to the type of restrictions discussed in Remark 5. This

is a situation in which IM-GLS coincides with IM-OLS, or, in other words, the fixed-b

discussion in this paper is (algebraically) confined to IM-OLS.



3 Summary and Conclusions



This paper has extended integrated modified least squares estimation theory from the

single equation cointegrating multivariate polynomial regression setting considered in Vo-

gelsang and Wagner (2024) to the systems case. It is important to note that the CMPR

and SCMPR setting allow to overcome the ubiquitous additive separability assumption

between integrated regressors in the nonlinear cointegrating regression literature, ad-

mittedly only for polynomial functions. Whilst this is, of course, a very special class

of functions it (i) includes the relevant case of Translog-type (systems of) equations,

(ii) allows for RESET-type specification testing and (iii) is an important starting point

for multivariate Sieve-type (or Taylor) approximation of more general (differentiable)

functions.



Considering systems of equations adds two aspects compared to single equation analysis.

First, it necessitates the detailed consideration of not only ordinary least squares but also

generalized least squares estimators. This stems from the well-known fact that OLS and

GLS estimators only necessarily coincide in equation systems with identical regressors

across all equations and without parameter constraints (as shown in Zellner, 1962).

Whilst such an unrestricted system might be a valid starting point for empirical analysis,

the result of, e. g., specification testing or the imposition of restrictions stemming from

economic theory will typically be restricted systems. Consequently, we discuss in detail

inference for both unrestricted and restricted IM estimators. It furthermore turns out

that the scope of fixed-b inference is relatively limited. In addition to full design of the

regression system – a necessary condition also in the single-equation setting in Vogelsang

and Wagner (2024) – it turns out that fixed-b inference is only available for up-to-the-

intercept-identical hypotheses tested in all equations in systems with identical regressors

in all equations. Despite this being very restrictive, it allows, e. g., for RESET-type

specification testing for SUCPR systems with identical regressors in all equations in

both the null specification and the augmented regression.



20





The methods developed in this paper are going to be used in future work to estimate


Translog-type production function systems, both at the aggregate (multi-)national level

as well as at the sectoral level.28 In addition, it may also be interesting to extend the

SUR-type theory of Knorre and Wagner (2024) from the SCPR to the SCMPR case

considered in this paper.



Acknowledgements



The authors gratefully acknowledge financial support from the Jubil¨ aumsfonds of the

Oesterreichische Nationalbank (Grant no. 18692). This paper was written whilst the

second author gratefully enjoyed the hospitality of the Robert Schuman Centre for Ad-

vanced Studies of the European University Institute during a Simone Veil Fellowship in

the first half of 2024. The views expressed in this paper are solely ours and not neces-

sarily those of the Bank of Slovenia or the European System of Central Banks. On top

of this the usual disclaimer applies.



References



Christensen, L.R., D.W. Jorgenson and L.J. Lau (1971). Conjugate Duality and the

Transcendental Logarithmic Production Function. Econometrica 39, 255–256.



Christensen, L.R., D.W. Jorgenson and L.J. Lau (1973). Transcendental Logarithmic

Production Frontiers. Review of Economics and Statistics 55, 28–45.



de Jong, R.M. and M. Wagner (2022). Panel Cointegrating Polynomial Regression Anal-

ysis and an Illustration with the Environmental Kuznets Curve. Econometrics and

Statistics, in press: https://doi.org/10.1016/j.ecosta.2022.03.005.



Denny, M. and M. Fuss (1977). The Use of Approximation Analysis to Test for Separa-

bility and the Existence of Consistent Aggregates. American Economic Review 67,

404–418.



28 In the context of such an empirical analysis we will also delve into the finite sample performance

of the developed methods more systematically. Cursory simulation evidence suggests that in terms

of performance effectively the “usual kind of results” can be expected (compare, e. g., Wagner and

Hong, 2016; Wagner et al., 2020; Knorre and Wagner, 2024; Vogelsang and Wagner, 2024), also with

respect to the impact of the number of equations in relation to the sample size.



21

Jansson, M. (2002). Consistent Covariance Matrix Estimation for Linear Processes.

Econometric Theory 18, 1449–1459.



Kiefer, N.M. and T.J. Vogelsang (2005). A New Asymptotic Theory for

Heteroskedasticity-Autocorrelation Robust Tests. Econometric Theory 21, 1130–

1164.



Knorre, F. and M. Wagner (2024). Fully Modified OLS Estimation and Inference for

Seemingly Unrelated Cointegrating Polynomial Regressions with Common Inte-

grated Regressors. Mimeo.



Moon, H.R. (1999). A Note on Fully-Modified Estimation of Seemingly Unrelated Re-

gression Models with Integrated Regressors. Economics Letters 65, 25–31.



Moon, H.R. and B. Perron (2004). Efficient Estimation of the SUR Cointegration Re-

gression Model and Testing for Purchasing Power Parity. Econometric Reviews 23,

293–323.



Neave, H.R. (1970). An Improved Formula for the Asymptotic Variance of Spectrum

Estimates. Annals of Mathematical Statistics 41, 70–77.



Park, J.Y. and M. Ogaki (1991). Seemingly Unrelated Canonical Cointegrating Regres-

sions. Mimeo.



Phillips, P.C.B. and H.R. Moon (1999). Linear Regression Limit Theory for Nonstation-

ary Panels. Econometrica 67, 1057–1111.



Ramsey, J.B. (1969). Tests for Specification Errors in Classical Linear Least-Squares

Regression Analysis. Journal of the Royal Statistical Society B 31, 350–371.



Thursby, J.G. and P. Schmidt (1977). Some Properties of Tests for Specification Error

in a Linear Regression Model. Journal of the American Statistical Association 72,

635–641.



Vogelsang, T.J. and M. Wagner (2014). Integrated Modified OLS Estimation and Fixed-b

Inference for Cointegrated Regressions. Journal of Econometrics 178, 741–760.



Vogelsang, T.J. and M. Wagner (2024). Integrated Modified OLS Estimation and Fixed-

b Inference for Cointegrating Multivariate Polynomial Regressions. IHS Working

Paper Series 53.



22

Wagner, M. (2023). Fully Modified Least Squares Estimation and Inference for Systems

of Cointegrating Polynomial Regressions. Economics Letters 228, 111186.



Wagner, M. and S.H. Hong (2016). Cointegrating Polynomial Regressions: Fully Modi-

fied OLS Estimation and Inference. Econometric Theory 32, 1289–1315.



Wagner, M. and K. Reichold (2023). Panel Cointegrating Polynomial Regressions:

Group-Mean Fully Modified OLS Estimation and Inference. Econometric Reviews

42, 358–392.



Wagner, M., P. Grabarczyk and S.H. Hong (2020). Fully Modified OLS Estimation and

Inference for Seemingly Unrelated Cointegrating Polynomial Regressions and the

Environmental Kuznets Curve for Carbon Dioxide Emissions. Journal of Econo-

metrics 214, 216–255.



Zellner, A. (1962). An Efficient Method of Estimating Seemingly Unrelated Regressions

and Tests for Aggregation Bias. Journal of the American Statistical Association 57,

348–368.



Appendix: Proofs



Proof of Proposition 1. The result presents the system version of the IM-OLS estimator

and its asymptotic properties derived for the single-equation CMPR setting with n = 1 in

Vogelsang and Wagner (2024, Proposition 1) and follows upon combining the individual

equation results. ■



Proof of Proposition 2. Under the null hypothesis and condition (18) on AR, it holds

that:



− 1 ˆ − ⊗ ⊗ − ⇒ Y

A −1 −1 ∗ ∗ ( R ϕ r ) = A R ( I n A IM n ) ( I A )( ˆ ϕ ϕ ) R ,

R R IM



with ∗ Y denoting the random variable (limiting distribution) given in (14). RY is

under the null hypothesis – conditional upon Wv (r) – normally distributed with zero

mean and covariance matrix ∗ ∗′ R V R. Under condition (18), it furthermore holds that

IM

A−1 ˆ ′ −1′ ∗ ∗′ R V IM R A ⇒ R V R. Combining the two results now immediately leads to the R R IM

asymptotic chi-squared distribution for τW as defined in (19) by noting that conditional



23





convergence to a chi-squared distribution that is (by definition) independent of Wv(r)


amounts to unconditional convergence. ■



Proof of Proposition 3. Centering of the IM-OLS estimator, compare Proposition 1,

takes place around Φ∗. Therefore, considering:



′ ˜ Z ′ ′ ′ ˜ Z ∗′ ′ ∗ ˜ Z ˜ ˆ Z′ ˜ Z ˜ ⊗ ⊗

D Z ∗ vec S S Φ W = D ( ˆ W S S ) ϕ = D ( ˆ W S S )(Dφ + d),



implies:



φ Z ˆ ∗ ′ ˜Z ˜Z ′ − 1 ′ ˜Z y ∗ ˜ ′

ˆ − φ = D ( ˆ W ⊗ S S )D (D vec S (S − Φ S ) W (38)



= ′ ˜Z ˜Z′ ′ ˜Z u −1 ′ ˆ − 1

D ( ˆ W ⊗ S S )D D vec S (S − ΩuvΩ X) W . vv



With condition (22) and ˆ W → W > 0 in probability in place, it follows from straight-

forward calculations that:

Z 1 −1

A−1 ∗ ∗′ ′ ∗ ( ˆ φ − φ ) ⇒ D W ⊗ f ( s ) f ( s ) ds D (39)

D

0

Z 1

× ∗′ ′ D vec f ( s ) B u ·v ( s )dsW ,

0



with the result as given in the main text in (25) following by partial integration. ■



Proof of Proposition 4. The result follows analogously to the result for the Wald-type

statistic for linear hypotheses on ϕ derived in Proposition 2. An additional complication

is that two asymptotic full rank conditions, one related to the matrix D relating φ and

ϕ, given in (22), and one related to the restrictions matrix Rφ, given in (30), have to be

fulfilled. Also, of course, ˆ A and ˆ B need to be properly scaled to converge. ■



Proof of Proposition 5. As in the proof of Vogelsang and Wagner (2014, Lemma 2), it is

easiest to establish the asymptotic behavior of the modified residuals ˆu S by noting

⌊rT ⌋,M

that they are equivalently given as the OLS residuals of the regression of y Z S on ˜ S and

t t

M 1 ⌋ / 2 P ⌊ rT 1/2 u ⇒ t . Based on this observation, it can be shown that T ∆ ˆ S Ω ( ), with t t, M u · v P r =2

P (r) defined similarly to (28) in Vogelsang and Wagner (2024), with the only difference

being that 29 W u·v ( r ) is now an n -dimensional process rather than a scalar process. The

−1

29 R r R 1 ′ R 1 ′ To be precise, P ( r ) := dW u·v ( s ) − dW u ·v ( s )[ H (1) − H ( s )] h ( s ) h ( s )ds h(r). Note that

0 0 0

H(r) is – which requires full design – a functional of standard Brownian motions.



24





second important ingredient for asymptotically pivotal fixed-b inference is independence


of ∗ ∗′ −1/2 ∗ P ( r ) – as input in Q ( P ) – and Z = ( R V R ) ( RY). This can be shown

sRn IM

analogously to the n = 1 case in the proof of Vogelsang and Wagner (2024, Proposition 3),

in particular (57)–(59).30 Write V IM IM u· as defined in (15) for brevity as V = Ω ⊗ M and v

consider – to conclude the proof – the asymptotic behavior of the modified covariance

estimator which is the central term in the fixed-b Wald-type statistic τW,b defined in (36):



A− 1 ˆ ′ −1′ ∗ 1/2 1/2′ V I ⊗ R R IM,M R A ⇒ ( R n ) Ω u ·v Q ( P )Ω u ·v ( ⊗ MI ⊗ R ) (40) R ∗′

n

= Ω1/2 1/2′ ∗ ∗′ u·v Q ( P )Ω u·v ⊗ ( R MR)

= (Ω ∗ ∗′ 1/2 ∗ ∗′ 1/2′ u ·v ⊗ R MR ) ( Q ( P ) ⊗ I sR ) (Ω u ·v ⊗ R MR )

= ( ∗ ∗′ 1/2 ∗ ∗′ 1/2′ R V IM s R ) ( Q ( P ) ⊗ I ) ( R V R ).

R IM



Combining the parts defining τW,b establishes the result. ■



30 Since Y, as given in (14), can – in the case of full design – be written as

1 / 2 − 1 ′ R 1 ′ − 1 R 1 ′ 1/2 Ωu ·v u· Π ⊗ ( g(s)g(s) ds) vec [G(1) − G(s)]dW v (s) , with Π := diag(Π Z , Ωvv 0 ), the rel- 0

evant component for showing independence is vec [G(1) − R 1 ′

0 G(s)]dWu·v(s) .



25