On the Estimation and Inference of a Cointegrated Regression in Panel Data

Chihwa Kao and Min-Hsien Chiang^{∗} Syracuse University

First Draft: May 1, 1995 First Revised Draft: January 1997 This Draft: February 27, 1997

Abstract

In this paper, we study the asymptotic distributions for least-squares (OLS), fully modified (FM), and dynamic OLS (DOLS) estimators in cointegrated regression models in panel data. We show that the OLS, FM, and DOLS estimators are all asymptotically normally distributed. However, the asymptotic distribution of the OLS estimator is shown to have a non-zero mean. Monte Carlo results examine the sampling behavior of the proposed estimators and show that

(1) the OLS estimator has a non-negligible bias in finite samples, (2) the FM estimator does not improve over the OLS estimator in general, and (3) the DOLS out-performs both the OLS and FM estimators.

Key Words: Panel Data, OLS Estimator; FM Estimator, DOLS Estimator, Heterogeneous Panels.

1 Introduction

Evaluating the statistical properties of data along the time dimension has proven to be very different from analysis of the cross-section dimension. As economists have gained access to better data with more observations across time, understanding these properties has grown increasingly important. An area of particular concern in time series econometrics has been the use of non-stationary data. With the desire to study the behavior of a cross-sectional data over time and the increasing use of panel data, one new research area is examining the properties of non-stationary time-series data in panel form. It is an intriguing question to ask: how exactly does this hybrid style of data combine the statistical elements of traditional cross-sectional analysis and time-series analysis? In particular, what is the correct way to analyze non-stationarity, the spurious regression problem, and cointegration in panel data?

∗We are thankful to three anonymous referees and Peter Phillips for pointing out several technical errors in an earlier version and providing comments that led to improvements of this paper. We also thank Peter Pedroni for his helpful comments and Bangtian Chen for his research assistance on an earlier draft of this paper. An electronic version of the paper in postscript format can be retrieved from http://web.syr.edu/~cdkao. Address correspondence to: Chihwa Kao, Center for Policy Research, 426 Eggers, Syracuse University, Syracuse, NY 13244; e-mail: cdkao@maxwell.syr.edu.

Given the immense interest in testing for unit roots and cointegration in time-series data, not much attention has been paid to testing the unit roots in panel data. The only theoretical studies as far as we know in this area are Breitung and Meyer (1994), Quah (1994), Levin and Lin (1993), Im, Pesaran, and Shin (1995), and Maddala and Wu (1996). Breitung and Meyer (1994) derived the asymptotic normality of the Dickey-Fuller test statistic for panel data with a large cross-section dimension and a small time-series dimension. Quah (1994) studied a unit root test for panel data that have simultaneously extensive cross-section and time-series variation. He showed that the asymptotic distribution for the proposed test is a mixture of the standard normal and Dickey-Fuller-Phillips asymptotics. Levin and Lin (1993) derived the asymptotic distributions for unit roots on panel data and showed that the power of these tests increases dramatically as the cross-section dimension increases. Im et al. (1995) critiqued the Levin and Lin panel unit root statistics and proposed alternatives. Maddala and Wu (1996) provided a comparison of the tests of Im et al. (1995) and Levin and Lin (1993). They suggested a new test based on the Fisher test. However, to this date, little is known about cointegration tests and estimation with regression models in panel data. Exceptions are Kao (1996), McCoskey and Kao (1996), and Pedroni (1995, 1996). In the first half of Kao (1996), he studied a spurious regression in panel data. Asymptotic properties of the least-squares (OLS) estimator and other conventional statistics were examined. Kao (1996) showed that the OLS estimator is consistent for its true value, but the t-statistic diverges so that inferences about the regression coefficient, β, are wrong with probability that goes to one as N → ∞ and T → ∞. Furthermore, Kao (1996) examined the Dickey-Fuller (DF) and the augmented Dickey-Fuller (ADF) tests to test the null hypothesis of no cointegration in panel data. McCoskey and Kao (1996) proposed further tests for the null hypothesis of cointegration in panel data. Pedroni (1995) derived asymptotic distributions for residual based tests of cointegration for both homogenous and heterogenous panels. Pedroni (1996) proposed a fully modified estimator for heterogenous panels. On the other hand, Park and Ogaki (1991) derived asymptotic distributions for cointegration coefficient estimators and related t-statistics for panel data using CCR transformations. Although they used an SUR approach rather than N dimension asymptotics, many of the issues they dealt with are similar.

In this paper, we study the limiting distributions for the ordinary least squares (OLS), fully modified (FM), and Dynamic OLS (DOLS) estimators in panel cointegrated regression models.

Section 2 introduces the model and assumptions. Section 3 develops the asymptotic theory for OLS, FM and DOLS estimators. Section 4 gives the limiting distributions of FM and DOLS estimators for heterogeneous panels. Section 5 develops the limiting distributions of the Wald statistics. Section 6 presents some Monte Carlo results to evaluate the finite sample properties of the OLS, FM, and dynamic OLS estimators. Section 7 summarizes the findings. All proofs are in the Appendix.

A word on notation. We write the integral ∫_{0}^{1} W (s)ds as ∫ W when there is no ambiguity over limits. We define Ω^{1/2} to be any matrix such that Ω = (Ω^{1/2}) (Ω^{1/2})′ . We use ∥A∥ to denote {tr (A′A)}^{1/2} , |A| to denote the determinant of A, →^{d} to denote convergence in distribution, →^{p} to denote convergence in probability, [x] to denote the largest integer ≤ x, I(0) and I(1) to signify a time series that is integrated of order zero and one, respectively, and BM (Ω) to denote Brownian motion with covariance matrix Ω.

2 The Model and Assumptions

Consider the following fixed effect panel regression:

y_{it} = α_{i} + xit^{′}β + u_{it}, i = 1, ..., N, t = 1, ...T,  (1)

where {y_{it}} are 1 × 1, β is a k × 1 vector of the slope parameters, {α_{i}} are the intercepts, and {u_{it}} are the stationary disturbance terms. We assume that {x_{it}} are k × 1 integrated processes of order one for all i, where

x_{it} = x_{it−1} + ε_{it}.

Under these specifications, (1) describes a system of cointegrated regressions, i.e., y_{it} is cointegrated with x_{it}. The initialization of this system is y_{i0} = x_{i0} = 0 for all i.

Assumption 1 {y_{it,}x_{it}} are independent across i.

Assumption 2 The cross-section dimension is a monotonic function of the time-series dimension, i.e., N = N(T), so that the law of large numbers (Theorem 6.2, Billingsley, 1986, p. 81) and the central limit theorem (Theorem 27.2, Billingsley, 1986, p. 369) for triangular arrays can be applied.

Next, we characterize the innovation vector w_{it} = (u_{it}, εit^{′})′. We assume that w_{it} is a linear process that satisfies the following assumption.

Assumption 3 (e.g., Phillips, 1995)

(a) w_{it} = Π(L)ϵ_{it} = ∑_{j=0}^{∞} Π_{j}ϵ_{it−j}, ∑_{j=0}^{∞} j^{a} ∥Π_{j}∥ < ∞, |Π(1)| ≠ 0 for some a > 1.

(b) ϵ_{it} is i.i.d. with zero mean, variance matrix Σ_{ϵ}, and finite fourth order cumulants. Assumption 3 implies that (e.g., Phillips and Solo, 1992) the partial sum process \frac{1}{\sqrt{T}} ∑_{t=1}^{[T r]} w_{it} satisfies

the following multivariate invariance principle:

\frac{1}{\sqrt{T}} ∑_{t=1}^{[T r]} w_{it} →^{d} B_{i}(r) ≡ BM_{i} (Ω) as T → ∞,  (2)

where

B_{i} = [ \begin{matrix}B_{ui} \\ B_{εi} \end{matrix} ] .

The long-run covariance matrix of {w_{it}} is given by

Ω = ∑_{j=−∞}^{∞} E (w_{ij}wi0^{′})
= Π(1)Σ_{ϵ}Π(1)′
= Σ + Γ + Γ′
≡ [ \begin{matrix}Ω_{u} & Ω_{uε} \\ Ω_{εu} & Ω_{ε} \end{matrix} ] ,

where

Γ = ∑_{j=1}^{∞} E (w_{ij}wi0^{′}) ≡ [ \begin{matrix}Γ_{u} & Γ_{uε} \\ Γ_{εu} & Γ_{ε} \end{matrix} ]  (3)

and

Σ = E (w_{i0}wi0^{′}) ≡ [ \begin{matrix}Σ_{u} & Σ_{uε} \\ Σ_{εu} & Σ_{ε} \end{matrix} ]  (4)

are partitioned conformably with w_{it}. Assumption 4 Ω_{ε} is non-singular, i.e., {x_{it}} are not cointegrated.

Define

Ω_{u.ε} = Ω_{u} − Ω_{uε}Ω_{ε}^{−1}Ω_{εu}.  (5)

Then, B_{i} can be rewritten as

B_{i} = [ \begin{matrix}B_{ui} \\ B_{εi} \end{matrix} ] = [ \begin{matrix}Ω_{u.ε}^{1/2} & Ω_{uε}Ω_{ε}^{−1/2} \\ 0 & Ω_{ε}^{1/2} \end{matrix} ] [ \begin{matrix}V_{i} \\ W_{i} \end{matrix} ] ,

where [ \begin{matrix}V_{i} \\ W_{i} \end{matrix} ] = BM (I) is a standardized Brownian motion. Define the one-sided long-run covariance

Δ = Σ + Γ
= ∑_{j=0}^{∞} E (w_{ij}wi0^{′})

with

Δ = [ \begin{matrix}Δ_{u} & Δ_{uε} \\ Δ_{εu} & Δ_{ε} \end{matrix} ] .

Remark 1 Here we assume that panels are homogeneous, i.e., the variances are constant across the crosssection units. We will relax this assumption in Section 4 to allow for different variances for different i.

3 OLS, Fully Modified, and Dynamic OLS Estimators

Let us first study the limiting distribution of the OLS estimator for equation (1). The OLS estimator of β is

β̂_{OLS} = [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (y_{it} − \overline{y}_{i})] .  (6)

It follows that \sqrt{N}T (β̂_{OLS} − β)

= [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} (\frac{1}{T} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it})]
= [\frac{1}{N} ∑_{i=1}^{N} ζ_{2iT} ]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} ζ_{1iT} ]
= [ξ_{2NT}]^{−1} \sqrt{N}ξ_{1NT} ,

where \overline{x}_{i} = \frac{1}{T} ∑_{t=1}^{T} x_{it}, \overline{y}_{i} = \frac{1}{T} ∑_{t=1}^{T} y_{it}, ζ_{1iT} = \frac{1}{T} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it}, ζ_{2iT} = \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′, ξ_{1NT} = \frac{1}{N} ∑_{i=1}^{N} ζ_{1iT}, and ξ_{2NT} = \frac{1}{N} ∑_{i=1}^{N} ζ_{2iT} . Before going into the next theorem, we need to consider some preliminary results. All limits in (a) − (d) in Lemma 1 are taken as T → ∞. Also, all limits in (e) − (f) in Lemma 1 and Theorems 1 − 4 are taken as N → ∞ and T → ∞.

Lemma 1 If Assumptions 1 − 4 hold, then

(a) ζ_{1iT} →^{d} Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2} + Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu} ≡ ζ_{1i},

(b) ζ_{2iT} →^{d} Ω_{ε}^{1/2} (∫ W̃_{i}W̃_{i}′) Ω_{ε}^{1/2} ≡ ζ_{2i},

(c) E [ζ_{1i}] = −\frac{1}{2}Ω_{εu} + Δ_{εu}, (d) E [ζ_{2i}] = \frac{1}{6}Ω_{ε},

(e) ξ_{1NT} →^{p} −\frac{1}{2}Ω_{εu} + Δ_{εu},

(f) ξ_{2NT} →^{p} \frac{1}{6}Ω_{ε},

where W̃_{i} = W_{i} − ∫ W_{i}. Remark 2 −\frac{1}{2}Ω_{εu} is due to the endogeneity of the regressor x_{it}, and Δ_{εu} is due to the serial correlation.

Thus, we have established the following theorem: Theorem 1 If Assumptions 1 − 4 hold, then

(a) T (β̂_{OLS} − β) →^{p} −3Ω_{ε}^{−1}Ω_{εu} + 6Ω_{ε}^{−1}Δ_{εu},

(b) \sqrt{N}T (β̂_{OLS} − β) − \sqrt{N}δ_{NT} →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε}) ,

where

δ_{NT} = [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − x_{it}) (x_{it} − \overline{x}_{i})′]^{−1} [\frac{1}{N} ∑_{i=1}^{N} Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}] .

Remark 3 We notice that δ_{NT} →^{p} −3Ω_{ε}^{−1}Ω_{εu} + 6Ω_{ε}^{−1}Δ_{εu}. Remark 4 The normality of the OLS estimator comes naturally. When summing across i, the nonstandard asymptotic distribution due to unit root in the time dimension is smoothed out. However, it is important to note that the OLS estimator is asymptotically biased. The asymptotic bias is

β̂_{OLS} − β ≅ \frac{δ_{NT}}{T} ≅ \frac{−3Ω_{ε}^{−1}Ω_{εu} + 6Ω_{ε}^{−1}Δ_{εu}}{T}

which decreases as T increases. Remark 5 Ω_{ε}^{−1}Ω_{u.ε} can be seen as the long-run signal-to-noise ratio. Remark 6 If w_{it} = (u_{it}, εit^{′})′ are i.i.d., then

δ_{NT} →^{p} 3Σ_{ε}^{−1}Σ_{εu}

which was examined by Kao and Chen (1995).

Chen, McCoskey, and Kao (1996) investigated the finite sample proprieties of the OLS estimator in (6), the t-statistic, the bias-corrected OLS estimator, and the bias-corrected t-statistic. They found that the bias-corrected OLS estimator does not improve over the OLS estimator in general. The results of Chen, McCoskey, and Kao (1996) suggest that alternatives, such as the FM estimator or DOLS estimator (e.g., Saikkonen, 1991; Stock and Watson, 1993) may be more promising in cointegrated panel regressions. Thus, we begin our study by examining the limiting distribution of FM estimator, β̂_{F M}. Following Pedroni (1996), we begin our study by examining the limiting distribution of the FM estimator, β̂_{F M}. In contrast to Pedroni (1996), we initially consider the case where Ω is common across members of the panel in order to focus on the role that the signal to noise ratio, Ω_{ε}^{−1}Ω_{u.ε}, can play in the asymptotic distribution of an FM estimator.

The FM estimator is constructed by making corrections for endogeneity and serial correlation to the OLS estimator β̂_{OLS} in (6). Let Ω̂_{εu} and Ω̂_{ε} are consistent estimates of Ω_{uε} and Ω_{ε}. Define

u_{it}^{+} = u_{it} − Ω_{uε}Ω_{ε}^{−1}ε_{it},
u_{it}^{+} = u_{it} − Ω̂_{uε}Ω̂_{ε}^{−1}ε_{it},
y_{it}^{+} = y_{it} − Ω_{uε}Ω_{ε}^{−1}ε_{it},

and

y_{it}^{+} = y_{it} − Ω̂_{uε}Ω̂_{ε}^{−1}ε_{it}.

Note that

[ \begin{matrix}u_{it}^{+} \\ ε & it \end{matrix} ] = [ \begin{matrix}1 & −Ω_{uε}Ω_{ε}^{−1} \\ 0 & I_{k} \end{matrix} ] [ \begin{matrix}u_{it} \\ ε & it \end{matrix} ] ,

which has the long-run covariance matrix

[ \begin{matrix}Ω_{u.ε} & 0 \\ 0 & Ω_{ε} \end{matrix} ] ,

where I_{k} is a k × k identity matrix. The endogeneity correction is achieved by modifying the variable y_{it} in (1) with the transformation

\begin{aligned}y_{it}^{+} & = & y_{it} − Ω̂_{uε}Ω̂_{ε}^{−1}ε_{it} \\ & = & α_{i} + xit^{′}β + u_{it} − Ω̂_{uε}Ω̂_{ε}^{−1}ε_{it}. \end{aligned}

The serial correlation correction term has the form

Δ̂_{εu}^{+} = ( Δ̂_{εu} Δ̂_{ε} ) ( \begin{matrix}1 \\ −Ω̂_{ε}^{−1}Ω̂_{εu} \end{matrix} )
= Δ̂_{εu} − Δ̂_{ε}Ω̂_{ε}^{−1}Ω̂_{εu},

where Δ̂_{εu} and Δ̂_{ε} are kernel estimates of Δ_{εu} and Δ_{ε}. Therefore, the FM estimator is

β̂_{F M} = [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} [∑_{i=1}^{N} (∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) y_{it}^{+} − T Δ̂_{εu}^{+})] .  (7)

Now, we state the limiting distribution of β̂_{F M}. Theorem 2 If Assumptions 1 − 4 hold, then \sqrt{N}T (β̂_{F M} − β) →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε}) . Remark 7 Note that Pedroni (1996) allowed the drifts for the integrated regressors in his cointegrated system. This paper only considers the regression in which integrated regressors do not have drifts. Also we propose the FM estimators for multiple regression.

Next, we propose a DOLS estimator, β̂_{D}, which uses the past and future values of △x_{it} as additional regressors. We then show that the limiting distribution of β̂_{D} is the same as the FM estimator, β̂_{F M}. But first, we need the following additional assumption: Assumption 5 The process {u_{it}} can be projected on to {ε_{it}} to get

u_{it} = ∑_{j=−∞}^{∞} c_{ij}ε_{it+j} + v_{it},  (8)

where

∑_{j=−∞}^{∞} ∥c_{ij}∥ < ∞,

{v_{it}} is stationary with zero mean, and {v_{it}} and {ε_{it}} are uncorrelated not only contemporaneously but also in all lags and leads. Remark 8 Assumption 5 can be guaranteed by following the conditions in Saikkonen (1991, p. 11). Remark 9 In practice, the leads and lags may be truncated while retaining Assumption 5 approximately, so that

u_{it} = ∑_{j=−q_{1}}^{q_{2}} c_{ij}ε_{it+j} + v_{it}.

This is because {c_{ij}} are assumed to be absolutely summable, i.e., ∑_{j=−∞}^{∞} ∥c_{ij}∥ < ∞.

We also need to require that q_{1} and q_{2} tend to infinity with T at a suitable rate, i.e., Assumption 6 \frac{q_{1}^{3}}{T} → 0, \frac{q_{2}^{3}}{T} → 0, and

T^{1/2} ∑_{|j|>q_{1} or q_{2}} ∥c_{ij}∥ → 0.  (9)

We then substitute (8) into (1) to have

y_{it} = α_{i} + xit^{′}β + ∑_{j=−q_{1}}^{q_{2}} c_{ij}ε_{it+j} + v_{it}.

Therefore, we obtain the DOLS of β, β̂_{D}, by running the following regression:

y_{it} = α_{i} + xit^{′}β + ∑_{j=−q_{1}}^{q_{2}} c_{ij}△x_{it+j} + v_{it}.  (10)

Next, we show that β̂_{D} has the same limiting distribution β̂_{F M} as in Theorem 2.

Theorem 3 If Assumptions 1 − 6 hold, then \sqrt{N}T (β̂_{D} − β) →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε}) .

4 Heterogeneous Panels

The paper so far assumes that the panel data are homogeneous. The substantial heterogeneity exhibited by actual data in the cross-sectional dimension severely restricts the practical applicability of such estimators. Also, the estimators in Sections 2 and 3 are not easily extended to cases of broader cross-sectional heterogeneity since the variances and biases are specified in terms of the asymptotic covariance parameters that are assumed to be shared cross sectionally. Recently, Pedroni (1996) proposed an FM estimator for heterogeneous panels. Pedroni (1996) proposed the following panel FM estimator (using his notations):

β̂_{NT}^{∗} − β = (∑_{i=1}^{N} L̂_{22i}^{−2} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i})^{2})^{−1} ∑_{i=1}^{N} L̂_{11i}^{−1}L̂_{22i}^{−1} (∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it}^{∗} − T γ̂_{i}) ,  (11)

where L̂_{i} is the lower triangular decomposition of a consistent estimator of the asymptotic covariance matrix

Ω_{i} = Ω_{i}^{0} + Γ_{i} + Γi^{′},

and where u_{it}^{∗} is given by

u_{it}^{∗} = u_{it} − \frac{L̂_{21i}}{L̂_{22i}} △ x_{it}

and the serial correlation adjustment parameter γ̂_{i} is given by

γ̂_{i} = Γ̂_{21i} + Ω̂_{21i}^{0} − \frac{L̂_{21i}}{L̂_{22i}} (Γ̂_{21i} + Ω_{i}^{0}) .

Pedroni (1996) then derived the following result (his Proposition 1.2):

T \sqrt{N} (β̂_{NT}^{∗} − β) → N (0, v) ,

where

v = \{ \begin{matrix}2 & iff \overline{x}_{i} = \overline{y}_{i} = 0 \\ 6 & else \end{matrix} .

In this section, we propose an alternative representation of the panel FM estimator for heterogeneous panels. Again, in contract to Pedroni (1996), this section only considers the regression that integrated regressors do not have drifts. Also we propose an FM estimator for multiple regression. Before we discuss the FM estimator we need the following assumptions:

Assumption 7 We assume the panels are heterogeneous, i.e., Ω_{i}, Γ_{i} and Σ_{i} are varied for different i. We also assume the invariance principle in (2), (8) in Assumption 5, and (9) in Assumption 6 still hold.

Let

x_{it}^{∗} = Ω̂_{iε}^{−1/2}x_{it},  (12)
u_{it}^{∗} = Ω̂_{iu.ε}^{−1/2}u_{it}^{+},  (13)

and

y_{it}^{∗} = Ω̂_{iu.ε}^{−1/2}y_{it}^{+},  (14)

where Ω̂_{iε} and Ω̂_{iu.ε} are consistent estimators of Ω_{iε} and Ω_{iu.ε}, respectively.

Assumption 8 Ω̂_{iε} is not singular for all i.

Then, we define the FM estimator for heterogeneous panels as

β̂_{F M}^{∗} = [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) (x_{it}^{∗} − \overline{x}_{i}^{∗})′]^{−1} [∑_{i=1}^{N} (∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) y_{it}^{∗} − T Δ̂_{iεu}^{+})] ,  (15)

where

Δ̂_{iεu}^{+} = ( Δ̂_{iεu} Δ̂_{iε} ) ( \begin{matrix}1 \\ −Ω̂_{iε}^{−1}Ω̂_{iεu} \end{matrix} )
= Δ̂_{iεu} − Δ̂_{iε}Ω̂_{iε}^{−1}Ω̂_{iεu}.

β̂_{F M}^{∗} can be written as

\sqrt{N}T (β̂_{F M}^{∗} − β)
= [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) (x_{it}^{∗} − \overline{x}_{i}^{∗})′]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} (\frac{1}{T} ∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) u_{it}^{∗} − T Δ̂_{iεu}^{+})]
= [\frac{1}{N} ∑_{i=1}^{N} ζ_{4iT} ]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} ζ_{3iT} ]
= [ξ_{4NT}]^{−1} \sqrt{N}ξ_{3NT} ,

where \overline{x}_{i}^{∗} = \frac{1}{T} ∑_{t=1}^{T} x_{it}^{∗}, ζ_{3iT} = \frac{1}{T} ∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) u_{it}^{∗} −T Δ̂_{iεu}^{+}, ζ_{4iT} = \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it}^{∗} − \overline{x}_{i}^{∗}) (x_{it}^{∗} − \overline{x}_{i}^{∗})′, ξ_{3NT} = \frac{1}{N} ∑_{i=1}^{N} ζ_{3iT}, and ξ_{4NT} = \frac{1}{N} ∑_{i=1}^{N} ζ_{4iT} . It is clear that from Lemma 1 that

ζ_{3iT} →^{d} ∫ W̃_{i}dV_{i},

ζ_{4iT} →^{d} ∫ W̃_{i}W̃_{i}′,

ξ_{4NT} →^{p} \frac{1}{6}I_{k},

and

ξ_{3NT} →^{p} \frac{1}{6}I_{k}.

It follows that

\sqrt{N}T (β̂_{F M}^{∗} − β) →^{d} N (0, 6I_{k}) .

Hence, we have established the following theorem: Theorem 4 If Assumptions 7 − 8 hold, then \sqrt{N}T (β̂_{F M}^{∗} − β) →^{d} N (0, 6I_{k}) .

The DOLS estimator for heterogeneous panels, β̂_{D}^{∗}, can be obtained by running the following regression:

y_{it}^{∗} = α_{i} + xit^{∗′} β + ∑_{j=−q_{1}}^{q_{2}} c_{ij}△x_{it+j}^{∗} + v_{it}^{∗}.

It is straightforward to show that β̂_{D}^{∗} also has the same limiting distribution as β̂_{F M}^{∗}. Theorem 5 If Assumptions 7 − 8 hold, then \sqrt{N}T (β̂_{D}^{∗} − β) →^{d} N (0, 6I_{k}) . Remark 10 Theorems 4 and 5 show that the limiting distributions of β̂_{F M}^{∗} and β̂_{D}^{∗} are free of nuisance parameters.

5 Hypothesis Testing

We now consider a linear hypothesis that involves the elements of the coefficient vector β. We show that hypothesis tests constructed using the FM and DOLS estimators have asymptotic chi-squared distributions. The null hypothesis has the form:

H_{0} : Rβ = r,  (16)

where r is a m × 1 known vector and R is a known m × k matrix describing the restrictions. A natural test statistic of the Wald test using β̂_{F M} or β̂_{D} for homogeneous panels is

W = \frac{1}{6}NT^{2} (Rβ̂_{F M} − r)′ [RΩ̂_{ε}^{−1}Ω̂_{u.ε}R′]^{−1} (Rβ̂_{F M} − r) .  (17)

For the heterogeneous panels, a natural statistic using β̂_{F M}^{∗} or β̂_{D}^{∗} to test the null hypothesis is

W^{∗} = \frac{1}{6}NT^{2} (Rβ̂_{F M}^{∗} − r)′ [RR′]^{−1} (Rβ̂_{F M}^{∗} − r) .  (18)

It is clear that W and W^{∗} converge in distribution to a chi-squared random variable with k degrees of freedom, χ_{k}^{2}, as N → ∞ and T → ∞ under the null hypothesis. Hence, we establish the following theorem:

Theorem 6 If Assumptions 1 − 8 hold, then under the null hypothesis (16),

(a) W →^{d} χ_{k}^{2},

(b) W^{∗} →^{d} χ_{k}^{2}.

Remark 11 Because the FM and the DOLS estimators have the same asymptotic distribution, it is easy to verify that the Wald statistics based on the FM estimator share the same limiting distributions with those based on the DOLS estimator.

6 Monte Carlo Simulations

To compare the performance of OLS, FM, and DOLS estimators, we conducted Monte Carlo experiments based on the design similar to Phillips and Hansen (1990) and Phillips and Loretan (1991). The data generating process (DGP) was

y_{it} = α_{i} + βx_{it} + u_{it}

and

x_{it} = x_{it−1} + ε_{it}

for i = 1, ..., N, t = 1, ...T, where

( \begin{matrix}u_{it} \\ ε & it \end{matrix} ) = ( \begin{matrix}u_{it}^{∗} \\ ε & {∗ \atop it} \end{matrix} ) + ( \begin{matrix}0.3 & −0.4 \\ θ_{21} & 0.6 \end{matrix} ) ( \begin{matrix}u_{it−1}^{∗} \\ ε & {∗ \atop it−1} \end{matrix} )

with

( \begin{matrix}u_{it}^{∗} \\ ε & {∗ \atop it} \end{matrix} ) ∼^{iid} N ([ \begin{matrix}0 \\ 0 \end{matrix} ] , [ \begin{matrix}1 & σ_{21} \\ σ_{21} & 1 \end{matrix} ]) .

We generated α_{i} from a uniform distribution, U[0, 10], and set β = 2. We allowed θ_{21} and σ_{21} to vary and considered values of {0.8, 0.4, 0.0, −0.8} for θ_{21} and {−0.8, −0.4, 0.4} for σ_{21}. Random numbers for (u_{it}^{∗}, ε_{it}^{∗}) were generated by the GAUSS procedure RNDNS. At each replication, we generated N(T + 1000) length of random numbers and then split it into N series so that each series had the same mean and variance. The first 1, 000 observations were discarded for each series. {u_{it}} and {ε_{it}} were constructed with u_{i0} = 0 and ε_{i0} = 0.

Once the estimates of w_{it}, ŵ_{it} were estimated, we used

Σ̂ = \frac{1}{NT} ∑_{i=1}^{N} ∑_{t=1}^{T} ŵ_{it}wit^{′}  (19)

to estimate Σ. Ω was estimated by

Ω̂ = \frac{1}{N} ∑_{i=1}^{N} \{\frac{1}{T} ∑_{t=1}^{T} ŵ_{it}wit^{′} + \frac{1}{T} ∑_{τ=1}^{l} ϖ_{τl} ∑_{t=τ+1}^{T} (ŵ_{it}wit−τ^{′} + ŵ_{it−τ} wit^{′})\} ,  (20)

where ϖ_{τl} is a weight function or a kernel. Using Phillips and Durlauf (1986) and the law of large numbers for triangular arrays, Σ̂ and Ω̂ can be shown to be consistent for Σ and Ω. The estimate of the long-run covariance matrix in (20) was obtained by using the procedure KERNEL in COINT 2.0 with a Bartlett window of lag length five. Results with other kernels, such as Parzen and QS kernels, are not reported, because no essential differences were found for most cases.

Next, we recorded the results from our Monte Carlo experiments that examined the finite-sample properties of the OLS estimator, β̂_{OLS}; the FM estimator, β̂_{F M}; and the DOLS estimator, β̂_{D}. The simulations were performed by a Sun SparcServer 1, 000. GAUSS 3.2.31 and COINT 2.0 were used to perform the simulations. The results we report are based on 10, 000 replications and are summarized in Tables 1 and 2. The FM estimator was obtained by using a Bartlett window of lag length five as in (20). Four lags and two leads were used for the DOLS estimator.

Table 1 reports the Monte Carlo means and standard deviations (in parentheses) of (β̂_{OLS} − β) , (β̂_{F M} − β) , and (β̂_{D} − β) for sample sizes T (N) = (20, 40, 60) . The biases of the OLS estimator, β̂_{OLS}, decrease at a rate of T. For example, with σ_{21} = −0.8 and θ_{21} = 0.8, the bias at T = 20 is − 0.201 and at T = 40 is −0.104. Also, the biases increase in θ_{21} (if θ_{21} > 0) and decrease in σ_{21}.

In general, the FM estimator, β̂_{F M}, presents the same degree of difficulty with bias as does the OLS estimator, β̂_{OLS}. For example, while the FM estimator, β̂_{F M}, reduces the bias substantially and outperforms β̂_{OLS} when θ_{21} > 0 and σ_{21} < 0, the opposite is true when θ_{21} > 0 and σ_{21} > 0, the opposite is true. Likewise, when θ_{21} = −0.8, β̂_{F M} is less biased than β̂_{OLS} for values of σ_{21} = −0.8. Yet, for values of σ_{21} = −0.4, the bias in β̂_{OLS} is less than the bias in β̂_{F M}. There seems to be little to choose between β̂_{OLS} and β̂_{F M} when θ_{21} < 0. This is probably due to the failure of the semiparametric correction procedure in the presence of a negative serial correlation of the errors, i.e., a negative MA value, θ_{21} < 0. Finally, for the cases where θ_{21} = 0.0, β̂_{F M} outperforms β̂_{OLS} when σ_{21} < 0. On the other hand, β̂_{F M} is more biased than β̂_{OLS} when σ_{21} > 0.

In contrast, the results in Table 1 show that the DOLS, β̂_{D}, is distinctly superior to the OLS and FM estimators for all cases in terms of the mean biases. Clearly, the DOLS out-performed both the OLS and FM estimators.

While the limiting theory depends on the assumption that the cross-section and time-series dimensions are comparable in magnitude, the actual panel data have a wide variety of cross-section and time-series dimensions. It is important to know the effects of the variations in panel dimensions. Table 2 considers 20 different settings for N and T, each ranging from 20 to 120. First, we notice that the cross-section dimension has significant effect on the biases of β̂_{OLS}, β̂_{F M}, and β̂_{D} when N is increased from 1 to 20. However, when N is increased from 20 to 40 and on, there is little effect on the biases of β̂_{OLS}, β̂_{F M}, and β̂_{D}. The results in Table 2 also confirm the superiority of the DOLS.

7 Conclusion

This paper derives limiting distributions for the OLS, FM, and DOLS estimators in a cointegrated regression and shows they are asymptotically normal.

We also investigated the finite sample proprieties of the OLS, FM, and DOLS estimators. Our findings are summarized as follows:

1. The OLS estimator has a non-negligible bias in finite samples.

2. The FM estimator does not improve over the OLS estimator in general.

3. The DOLS estimator may be more promising than OLS or FM estimators in estimating the cointegrated panel regressions.

Appendix

A Proof of Lemma 1

Proof. (a) and (b) are taken from the literature, i.e.,

\begin{aligned}ζ_{2iT} & = & \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′ \\ & →^{d} & Ω_{ε}^{1/2} (∫ W̃_{i}W̃_{i}′) Ω_{ε}^{1/2} = ζ_{2i}, \end{aligned}  (21)

where

W̃_{i} = W_{i} − ∫ W_{i},

and ζ_{1iT} = \frac{1}{T} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it}

→^{d} Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2} + Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}

= ζ_{1i},

where △_{εu} = Σ_{εu} + Γ_{εu}.

The row i, column i diagonal element of ∫ W̃_{i}W̃_{i}′ is ∫ W_{i}^{2} −[∫ W_{i}]^{2}, and the row i, column j off-diagonal element is

∫ W_{i}W_{j} − [∫ W_{i}] [∫ W_{j}] .

Using

E [∫ W_{i}^{2} − [∫ W_{i}]^{2}] = \frac{1}{6}

and

V ar [∫ W_{i}^{2} − [∫ W_{i}]^{2}] = \frac{1}{45},

the corresponding generalization is

E [∫ W̃_{i}W̃_{i}′] = \frac{1}{6}I_{k},

where I_{K} is a k × k identity matrix. It then follows that

E [ζ_{2i}] = \frac{1}{6}Ω_{ε}^{1/2}I_{k}Ω_{ε}^{1/2}  (22)
= \frac{1}{6}Ω_{ε}  (23)

establishing (d). To prove (c), we use the fact that

\begin{matrix}E [(∫ W̃_{i}dV_{i})] \\ = 0 \end{matrix}

and

\begin{matrix}E [∫ W̃_{i}dWi^{′}] \\ = −\frac{1}{2}I_{k} \end{matrix}

resulting in

E [ζ_{1i}] = −\frac{1}{2}Ω_{εu} + Δ_{εu}.

Finally,

ξ_{1NT} ≡ \frac{1}{N} ∑_{i=1}^{N} ζ_{1iT} →^{p} −\frac{1}{2}Ω_{εu} + Δ_{εu}

and

ξ_{2NT} ≡ \frac{1}{N} ∑_{i=1}^{N} ζ_{2iT} →^{p} \frac{1}{6}Ω_{ε}  (24)

as required for (e) and (f) by the law of large numbers for triangular arrays.

B Proof of V ar [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}] = \frac{1}{6}Ω_{u.ε}Ω_{ε}

Proof. Note that

V ar [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}] = E [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}] [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}]′
= Ω_{u.ε}Ω_{ε}^{1/2}E [(∫ W̃_{i}dV_{i}) (∫ W̃_{i}dV_{i})′] Ω_{ε}^{1/2}
= Ω_{u.ε}Ω_{ε}^{1/2} \frac{1}{6}I_{k}Ω_{ε}^{1/2}
= \frac{1}{6}Ω_{u.ε}Ω_{ε}

using E [(∫ W̃_{i}dV_{i}) (∫ W̃_{i}dV_{i})′] = \frac{1}{6}I_{k}.

C Proof of Theorem 1

Proof. (a) is immediately evident from Lemma 1.

Recall that

\sqrt{N}T (β̂_{OLS} − β) − [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} \sqrt{N} \frac{1}{N} [∑_{i=1}^{N} Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}]

= [\frac{1}{N} ∑_{i=1}^{N} ζ_{2iT} ]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} \{ζ_{1iT} − Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}\}]

= [ξ_{2NT}]^{−1} \sqrt{N}ξ_{1NT}^{∗} .

Note that the sequence {ζ_{1iT} − Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}} in

\sqrt{N} (ξ_{1NT}^{∗}) = \frac{1}{\sqrt{N}} ∑_{i=1}^{N} \{ζ_{1iT} − Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}\}

is a triangular array sequence; thus, a central limit theorem of a triangular array is needed. Recall that the variances are assumed to be the same across i,which implies that the Lindeberg condition will hold. We can readily see that the \sqrt{N}ξ_{1NT}^{∗} will converge to a normal variable with an appropriate normalization and that ξ_{2NT} will converge to \frac{1}{6}Ω_{ε} in probability by a law of large number for a triangular array. First, we note from Lemma 1 that

ζ_{1iT} − Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu} →^{d} Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}.

It follows that

\frac{1}{N} ∑_{i=1}^{N} \{ζ_{1iT} − Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}\} →^{p} E [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}] = 0

and

V ar [Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2}] = \frac{1}{6}Ω_{u.ε}Ω_{ε}

from Appendix B. Using the Slutsky theorem, we obtain

[ξ_{2NT}]^{−1} \sqrt{N}ξ_{1NT}^{∗} →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε}) .

.

Hence,

\begin{matrix}\sqrt{N}T (β̂_{OLS} − β) − \sqrt{N}δ_{NT} \\ →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε}) , \end{matrix}  (25)

proving (b), where

δ_{NT} = [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} \frac{1}{N} [∑_{i=1}^{N} Ω_{ε}^{1/2} (∫ W̃_{i}dWi^{′}) Ω_{ε}^{−1/2}Ω_{εu} + Δ_{εu}] .  (26)

Therefore, we establish Theorem 1.

D Proof of Theorem 2

The FM estimator of β can be rewritten as follows

\begin{aligned}β̂_{F M} & = & [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} [∑_{i=1}^{N} (∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) y_{it}^{+} − T Δ̂_{εu}^{+})] \\ & = & β + [∑_{i=1}^{N} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′]^{−1} [∑_{i=1}^{N} (∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it}^{+} − T Δ̂_{εu}^{+})] . \end{aligned}  (27)

First, we note that

\sqrt{N}T (β̂_{F M} − β) = [\frac{1}{N} ∑_{i=1}^{N} \frac{1}{T^{2}} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′] \sqrt{N} \frac{1}{N} ∑_{i=1}^{N} \frac{1}{T} ∑_{t=1}^{T} [(x_{it} − \overline{x}_{i}) u_{it}^{+} − Δ̂_{εu}^{+}]
= [\frac{1}{N} ∑_{i=1}^{N} ζ_{2iT} ]^{−1} [\sqrt{N} \frac{1}{N} ∑_{i=1}^{N} ζ_{1iT}^{∗∗} ]
= [ξ_{2NT}]^{−1} \sqrt{N}ξ_{1NT}^{∗∗} ,

(28)

where ζ_{1iT}^{∗∗} = \frac{1}{T} ∑_{t=1}^{T} [(x_{it} − \overline{x}_{i}) u_{it}^{+} − Δ̂_{εu}^{+}] , and ξ_{1NT}^{∗∗} = \frac{1}{N} ∑_{i=1}^{N} ζ_{1iT}^{∗∗}.

Let w_{it}^{+} = ( u_{it}^{+} εit^{′} )′ and we have

\frac{1}{\sqrt{T}} ∑_{t=1}^{[T r]} w_{it}^{+} →^{d} [ \begin{matrix}B_{ui}^{+} \\ B_{εi} \end{matrix} ] ≡ BM (Ω^{+}) as T → ∞,  (29)

where

Ω^{+} = [ \begin{matrix}Ω_{u.ε} & 0 \\ 0 & Ω_{ε} \end{matrix} ] = Σ^{+} + Γ^{+} + Γ^{+′}

and

[ \begin{matrix}B_{ui}^{+} \\ B_{εi} \end{matrix} ] = [ \begin{matrix}I & −Ω_{uε}Ω_{ε}^{−1} \\ 0 & I \end{matrix} ] [ \begin{matrix}B_{ui} \\ B_{εi} \end{matrix} ] .

Let

Δ^{+} = Σ^{+} + Γ^{+}

and

ζ_{1iT}^{+} = \frac{1}{T} ∑_{t=1}^{T} (x_{it} − \overline{x}_{i}) u_{it}^{+}.

From Lemma 1 and the consistency of Ω̂_{uε} and Ω̂_{ε}^{−1} we note that

ζ_{1iT}^{+} →^{d} ∫ B̃_{εi}dB_{ui}^{+} + Δ_{εu}^{+},

where

Δ_{εu}^{+} = ( Δ_{εu} Δ_{ε} ) ( \begin{matrix}1 \\ −Ω_{ε}^{−1}Ω_{εu} \end{matrix} )
= Δ_{εu} − Δ_{ε}Ω_{ε}^{−1}Ω_{εu}.

It follows in terms of standard Wiener processes that

ζ_{1iT}^{+} →^{d} Ω_{ε}^{1/2} (∫ W̃_{i}dV_{i}) Ω_{u.ε}^{1/2} + Δ_{εu}^{+}.

Now let

ζ_{1iT}^{∗∗} = ζ_{1iT}^{+} − Δ̂_{εu}^{+}.

Clearly, from Lemma 1 we know that

E [ζ_{1iT}^{∗∗}] = 0

and

V ar [ζ_{1iT}^{∗∗}] = \frac{1}{6}Ω_{u.ε}Ω_{ε}.

A similar argument to the proof of Theorem 1 yields

\sqrt{N}T (β̂_{F M} − β) →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε})

as required.

E Proof of Theorem 3

First we write (10) in vector from:

y_{i} = eα_{i} + x_{i}β + Z_{iq}C + v_{i}
= x_{i}β + Z_{i}D + v_{i} (say),

where y_{i} is a T × 1 vector of y_{it}; e a is T × 1 unit vector, Z_{iq} is the T × (q_{1} + q_{2}) matrix of observations on the q_{1} + q_{2} regressors △x_{it−q_{1}}, ⋯, △x_{it+q_{2}}; x_{i} is a vector of T × k of x_{it}; C is a (q_{1} + q_{2}) × 1 vector of c_{ij}; v_{i} is a T × 1 vector of v_{it}; Z_{i} is a T × (q_{1} + q_{2} + 1) matrix, Z_{i} = (e, Z_{iq}) ; and D is a (q_{1} + q_{2} + 1) × 1 vector of parameters. Let Q_{i} = I − Z_{i}(Z′_{i}Z_{i})^{−1}Zi^{′}. It follows that

(β̂_{D} − β) = [∑^{N} ; i=1 (xi^{′}Q_{i}x_{i})]^{−1} [∑^{N} ; i=1 (x′_{i}Q_{i}v_{i})] .

We then write \sqrt{N}T (β̂_{D} − β)

= [\frac{1}{N} ∑^{N} ; i=1 \frac{1}{T^{2}} (x′_{i}Q_{i}x_{i})]^{−1} [\sqrt{N} \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T} (x′_{i}Q_{i}v_{i})]

= [\frac{1}{N} ∑^{N} ; i=1 ζ_{6iT} ]^{−1} [\sqrt{N} \frac{1}{N} ∑^{N} ; i=1 ζ_{5iT} ]

= [ξ_{6NT}]^{−1} [\sqrt{N}ξ_{5NT} ] ,

where ξ_{5NT} = \frac{1}{N} ∑^{N} ; i=1 ζ_{5iT} , ζ_{5iT} = \frac{1}{T} (x′_{i}Q_{i}v_{i}) , ξ_{6NT} = \frac{1}{N} ∑^{N} ; i=1 ζ_{6iT} , and ζ_{6iT} = \frac{1}{T^{2}} (x′_{i}Q_{i}x_{i}) .

Observe that from Saikkonen (1991)

ξ_{6NT} = \frac{1}{N} ∑^{N} ; i=1 ζ_{6iT}
= \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T^{2}} (x′_{i}Q_{i}x_{i})
= \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T^{2}} (xi^{′}W_{T} x_{i}) + o_{p}(1)
= \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T^{2}} ∑_{t=q_{1}+1}^{T −q_{2}} (x_{it} − \overline{x}_{i}) (x_{it} − \overline{x}_{i})′ + o_{p}(1),

\sqrt{N}ξ_{5NT} = \sqrt{N} \frac{1}{N} ∑^{N} ; i=1 ζ_{5iT}
= \sqrt{N} \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T} (x′_{i}Q_{i}v_{i})
= \sqrt{N} \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T} (xi^{′}W_{T} v_{i}) + o_{p}(1)
= \sqrt{N} \frac{1}{N} ∑^{N} ; i=1 \frac{1}{T} ∑^{T −q_{2}} ; t=q_{1}+1 (x_{it} − \overline{x}_{i}) v_{it} + o_{p}(1),

and

\frac{1}{T} ∑^{T −q_{2}} ; t=q_{1}+1 (x_{it} − \overline{x}_{i}) v_{it} →^{d} ∫ B̃_{εi}dB_{ui}^{+},

where W_{T} = I_{T} − \frac{1}{T} ee′. Using arguments similar to those in Theorem 2, we establish the following:

\sqrt{N}T (β̂_{D} − β) →^{d} N (0, 6Ω_{ε}^{−1}Ω_{u.ε})

as required.

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Table 1: Means Biases and Standard Deviations of OLS, FM, and DOLS Estimators

\overline{σ_{21} = −0.8  σ_{21} = −0.4  σ_{21} = 0.8}

β̂_{OLS}−β β̂_{F M}−β β̂_{D}−β  β̂_{OLS}−β β̂_{F M}−β β̂_{D}−β  β̂_{OLS}−β β̂_{F M}−β β̂_{D}−β
θ_{21}= 0.8
T = 20  -.201  -.189  -.001  -.097  -.117  -.002  -.022  -.069  -.009
(.049)  (.047)  (.040)  (.032)  (.035)  (.033)  (.011)  (.016)  (.009)
T = 40  -.104  -.099  -.000  -.049  -.060  -.001  -.011  -.034  -.004
(.019)  (.017)  (.013)  (.012)  (.013)  (.011)  (.004)  (.006)  (.003)
T = 60  -.070  -.067  -.000  -.033  -.041  -.000  -.007  -.023  -.003
(.010)  (.009)  (.007)  (.007)  (.007)  (.006)  (.002)  (.003)  (.002)
θ_{21}= 0.4
T = 20  -.132  -.079  -.001  -.082  -.075  -.002  -.014  -.068  -.003
(.038)  (.026)  (.027)  (.030)  (.029)  (.031)  (.013)  (.018)  (.013)
T = 40  -.066  -.040  -.001  -.041  -.038  -.001  -.007  -.034  -.001
(.014)  (.009)  (.027)  (.011)  (.011)  (.009)  (.005)  (.006)  (.004)
T = 60  -.044  -.027  .-.000  -.027  -.025  -.001  -.005  -.023  -.001
(.007)  (.005)  (.005)  (.006)  (.006)  (.005)  (.002)  (.003)  (.002)
θ_{21}= 0.0
T = 20  -.079  -.016  .001  -.059  -.027  .002  .005  -.062  .006
(.027)  (.015)  (.017)  (.026)  (.022)  (.026)  (.016)  (.019)  (.017)
T = 40  -.039  -.008  .001  -.059  -.027  .002  .001  -.031  .003
(.009)  (.005)  (.005)  (.026)  (.022)  (.026)  (.003)  (.007)  (.005)
T = 60  -.026  -.005  .000  -.019  -.014  -.001  .001  -.021  .002
(.005)  (.003)  (.003)  (.005)  (.008)  (.008)  (.003)  (.004)  (.003)
θ_{21}= −0.8 .000
T = 20  -.029  .029  .007  -.019  .029  .007  .114  .029  .000
(.016)  (.012)  (.008)  (.017)  (.015)  (.014)  (.034)  (.027)  (.031)
T = 40  -.017  .015  .003  -.009  .015  .003  .057  .016  -.000
(.006)  (.003)  (.002)  (.006)  (.005)  (.004)  (.012)  (.009)  (.009)
T = 60  -.009  .010  .002  -.007  .010  .002  .038  .011  .000
(.003)  (.002)  (.001)  (.003)  (.002)  (.002)  (.007)  (.005)  (.005)

Note:

(a) N =T.

(b) The lag length 5 of the Bartlett windows is used for the FM estimator.

(c) The 4 lags and 2 leads are used for the DOLS estimator.

Table 2: Means Biases and Standard Deviations of OLS, FM, and DOLS Estimators for Different N and T

(N,T)  β̂_{OLS}−β  β̂_{F M}−β  β̂_{D}−β
(1,20)  -.135  -.122  -.007
(.184)  (.189)  (.297)
(1,40)  -.070  -.065  -.001
(.093)  (.092)  (.106)
(1,60)  -.047  -.043  -.001
(.063)  (.061)  (.064)
(1,120)  -.024  -.022  -.001
(.032)  (.031)  (.029)
(20,20)  -.082  -.075  -.002
(.030)  (.029)  (.031)
(20,40)  -.042  -.039  .-.001
(.016)  (.015)  (.014)
(20,60)  -.028  -.026  - .000
(.010)  (.009)  (.009)
(20,120)  -.014  -.013  - .000
(.005)  (.005)  (.005)
(40,20)  -.081  -.073  -.001
(.022)  (.021)  (.022)
(40,40)  -.041  -.038  -.001
(.011)  (.011)  (.009)
(40,60)  -.028  -.025  -.001
(.007)  (.007)  (.007)
(40,120)  -.014  -.013  -.000
(.004)  (.003)  (.003)
(60,20)  -.080  -.073  -.002
(.017)  (.017)  (.018)
(60,40)  -.041  -.038  -.001
(.009)  (.009)  (.008)
(60,60)  -.027  -.025  -.001
(.006)  (.006)  (.005)
(60,120)  -.014  -.012  -.000
(.003)  (.003)  (.003)
(120,20)  -.079  -.072  -.002
(.012)  (.012)  (.012)
(120,40)  -.041  -.037  -.001
(.006)  (.006)  (.006)
(120,60)  -.027  -.025  -.001
(.004)  (.004)  (.004)
(120,120)  -.014  -.013  -.000
(.002)  (.002)  (.002)

Note:

(a) The lag length 5 of the Bartlett windows is used for the FM estimator.

(b) The 4 lags and 2 leads are used for the DOLS estimator.