2.1. The Model

The model we consider is the same as that in Phillips and Moon (1999). For completeness, we briefly describe the data generating process. The panel data model is based on the vector integrated process

with common initialization Z_i,0 = 0 for all i. The zero initialization is maintained for simplicity. All the results in the paper hold if we assume

We partitioned the m-vectors Z_i,t and U_i,t into m_y and m_x components (m = m_x + m_y) as Z_i,t′ = (Y_i,t′,X_i,t′) and U_i,t′ = (U_yi,t′,U_xi,t′). The error term U_i,t is assumed to be generated by the random coefficient linear process

where (i) {C_i,t} is a double sequence of m × m random matrices across i and t; (ii) the m-vectors V_i,t are independent and identically distributed (i.i.d.) across i and t with EV_i,t = 0, EV_i,tV_i,t′ = I_m, and EV_a,i,t⁴ = v⁴ for all i and t, where V_a,i,t is the ath element of V_i,t. (iii) C_i,s and V_j,t are independent for all i,j,s,t.

Let C_a,i,s be the ath element of vec(C_i,s) and σ_kas = EC_a,i,s^k. We make two further assumptions on the random coefficients.

Assumption 1 (Random coefficient condition). C_i,s is i.i.d. across i for all s.

Assumption 2 (Summability condition).

.

Assumptions 1 and 2 are the same as Assumptions 1(i) and 2(ii) of Phillips and Moon (1999). Note that their Assumptions 1(ii) and 2(i) are both implied by their Assumption 2(ii), so there is no need to state their Assumptions 1(ii) and 2(i) here. Assumption 1 and the assumption that V_i,t is i.i.d. imply cross sectional independence, an assumption that may be restrictive for some economic applications. However, because of the lack of natural ordering, there is no completely satisfactory and general way of modeling cross-sectional dependence, although some important progress has been made (see Conley, 1999; Phillips and Sul, 2003; Andrews, 2003). In this paper, we follow the large panel data literature and maintain the assumption of cross-sectional independence.

Let

. Under Assumptions 1 and 2, we can prove the following lemma, which ensures the integrability of the terms that appear frequently in our development.

LEMMA 1. Let Assumptions 1 and 2 hold; then

Under Assumptions 1 and 2, the processes U_i,t admit the following Beveridge-Nelson decomposition almost surely:

Using this decomposition and following Phillips and Solo (1992), we can prove that

where W_i(r) is a standard Brownian motion with

signifies the weak convergence conditional on

, the sigma field generated by the sequence {C_i,t}_t=0^∞.

To give a rigorous definition of the preceding conditional weak convergence, we expand the probability space in such a way that the partial sum process

can be represented almost surely and up to a negligible error in terms of a Brownian motion W_i(r) that is defined on the same probability space. Such an expansion can be justified using the Hungarian construction (e.g., Shorack and Weller, 1986). We will proceed as if the probability space has been expanded in the rest of the paper. Let

; then a formal definition of the conditional weak convergence in (2.5) is that

for all continuous and bounded functionals on D[0,1].

2.2. Definition and Estimation of Long-Run Average Relationship

Let Ω_i be the LRV matrix of Z_i,t conditional on

. It is well known that Ω_i is proportional to the conditional spectral density matrix f_UiUi(λ) of U_i,t evaluated at the origin, i.e., Ω_i = 2πf_UiUi(0). Partitioning Ω_i conformably, we have

By Lemma 1(c), Ω_i is integrable and

which is called the long-run average variance matrix of Z_i,t. Following a classical regression approach, we can analogously define a long-run regression coefficient between Y and X by β = Ω_yxΩ_xx⁻¹. For more discussion on this analogy, see Phillips and Moon (2000).

To construct an estimate of β, we first estimate Ω_i as follows:

where U_i,t = Z_i,t − Z_i,t−1, K(·,·) is a kernel function. When K(x,y) depends only on x − y, i.e., K(x,y) is translation invariant, we write K(x,y) = k(x − y). In this case,

reduces to

From the preceding formulation, it is clear that

is the usual kernel LRV estimator using the full bandwidth. It should be noted that translation invariant kernels are commonly used in the estimation of the LRV matrix. We consider the kernels other than the translation invariant ones to include some existing estimators of the long-run average relationship as special cases. This will be made clear in Section 2.3.

Based on the previous estimate, we can estimate Ω by

The long-run average relationship parameter β can then be estimated by

which is called the LRV-based estimator.

Note that the LRV-based estimator

depends on the observations Z_i,t only through their first-order difference. Therefore, when the model contains individual effects such that

where Z_i,0⁰ = 0 and U_i,t follows the linear process defined in (2.3), the LRV-based estimator

can be computed exactly the same as before. In other words, the LRV-based estimator is robust to the presence of the individual effects.

2.3. Relationship between New and Existing Estimators

Phillips and Moon (1999) show that both PLS and FE estimators are consistent and asymptotically normal. In this section, we examine the relationships between the LRV-based estimator and the PLS and FE estimators.

The PLS estimator is

Some simple algebraic manipulations show that

where

Hence, the PLS estimator is a special case of the LRV-based estimator. Note that the kernel for the PLS estimator depends on T. If we replace K_PLS,T(s,t) by K_PLS(s,t) = 1 − (s ∨ t), then we get an asymptotically equivalent estimator

. In view of (2.9), we see that

is an LRV-based estimator with kernel K(s,t) = 1 − (s ∨ t).

We now consider the FE estimator, namely,

where

. Again, some algebraic manipulations yield

where

The kernel function K_FE,T(s,t) depends on T. As before, we can replace K_FE,T(s,t) by K_FE(s,t) = min(s,t) − st to obtain an estimator

that is asymptotically equivalent to

. The resulting estimator

is an LRV-based estimator with kernel K(s,t) = min(s,t) − st.

In summary, the existing estimators or their asymptotically equivalent forms are special cases of the LRV-based estimator. The underlying LRV estimators use kernels that are not translation invariant. This sharply contrasts with the usual LRV estimators where translation invariant kernels are commonly used.

4.1. Sequential Limit Theory and Joint Limit Theory

The following theorem establishes the consistency of

as either (T,n → ∞)_seq or (T,n → ∞).

THEOREM 6. Let Assumptions 1–4 hold; then

as either (T,n → ∞)_seq or (T,n → ∞).

Remark.

is consistent even though

is inconsistent. This is not surprising as

equals μΩ_i plus a noise term. Although the noise in the time series estimation is strong, we can weaken the strong effect of noise by averaging across independent individuals. This is reflected in Theorem 6(a) and (b), which show that

are respective consistent estimates of Ω_xx and Ω_yx up to a multiplicative scalar.

Now we proceed to investigate the asymptotic distribution of

. We consider the sequential asymptotics first and then extend the result to the joint asymptotics. To get a definite joint limit, we need to control the relative rate of expansion of the two indexes. Write

. Theorem 6 describes the asymptotic behavior of

under the sequential and joint limits. Under Assumption 4, Ω_xx has full rank, which implies that

converge to μ⁻¹Ω_xx⁻¹. Therefore, it suffices to consider the limiting distribution of

.

Under the sequential limit, we first let T → ∞ for fixed n. The intermediate limit is

where

C_yi(1) is the m_y × m matrix consisting of the first m_y rows of C_i(1), and C_xi(1) is the m_x × m matrix consisting of the last m_x rows of C_i(1). In view of Lemma 5, the mean of the summand is

and the covariance matrix Θ is E vec(Q_i)vec(Q_i)′. An explicit expression for Θ is established in the following lemma.

LEMMA 7. Let Assumptions 1–4 hold. Then Θ is equal to

where K_{my mx} is the m_y m_x × m_y m_x commutation matrix.

The sequence of random matrices C_yi(1)Ξ_i C_xi′(1) − βC_xi(1)Ξ_i C_xi′(1) is i.i.d. (0,Θ) across i. From the multivariate Linderberg–Levy theorem, we then get, as n → ∞,

Combining (4.4) with the limit lim

, we establish the sequential limit in the following theorem.

THEOREM 8. Let Assumptions 1–4 hold. Then, as (T,n → ∞)_seq,

where Θ_LRV is

We now show that the limiting distribution continues to hold in the joint asymptotics as (T,n → ∞). Write

as

where

and

Because of Lemma 3, the term b_nT vanishes under the sequential limit. However, under the joint limit, we need to exercise some control over the relative expansion rate of (T,n) so that b_nT vanishes as (T,n → ∞). When this occurs, the term

will deliver the asymptotic distribution as (T,n → ∞).

Using Lemma 3, we have

because the O(·) terms in the summand are independent across i. Therefore, to eliminate the asymptotic bias, we need to assume the two indexes pass to infinity in such a way that

. Under this condition, we can prove the following theorem, which provides the asymptotic distribution under the joint limit.

THEOREM 9. Let Assumptions 1–4 hold. Then, as (T,n → ∞) such that

,

Remarks.

(i) For the PLS estimator, K(r,s) = 1 − (r ∨ s). Therefore,

. Hence, the PLS estimator satisfies, under both the sequential and joint limits,

(ii) For the FE estimator, K(s,t) = min(s,t) − st. In this case, it is easy to see that

. So

. Hence

has the limiting distribution given in (4.12) and (4.13) but with

replaced by

. Once again, the asymptotic result is consistent with Phillips and Moon (1999).

(iii) The efficiency of

depends only on μ⁻²δ². The smaller μ⁻²δ² is, the more efficient the estimator is. This is because the sum of the last two terms in (4.6) is

which is positive semidefinite. Therefore,

is more efficient than

. But

is less efficient than

. In Section 4.2, we consider a class of new kernels that have smaller κ values.

If we assume that C_i,t are the same across individuals, then Ω_i = Ω and β_i = β for some β and all i. In this case, Ω_yxi − βΩ_xxi = 0. As a consequence, Θ_LRV reduces to

and we obtain the following corollary.

COROLLARY 10. Let Assumptions 1–4 hold. If C_i,t =_a.s C_t where C_t is an m × m nonrandom matrix for all t, then, as (T,n → ∞)_seq, or as (T,n → ∞) with

,

Remarks.

(i) The corollary generalizes a result of Kao (1999). He considers the homogeneous spurious regression and shows that under the sequential limit, the FE estimator satisfies (4.14) with

.

(ii) Note that the matrix Ω_xx⁻¹ [otimes ] (Ω_yy − Ω_yxΩ_xx⁻¹Ω_xy) is positive semidefinite. Therefore, the efficiency of

depends only on μ⁻²δ² regardless of whether C_i,t is heterogeneous or not.

4.2. LRV-Based Estimator with Exponentiated Kernels

In this section, we exponentiate some commonly used kernels and investigate the asymptotic properties of the LRV-based estimators that these exponentiated kernels deliver.

We first consider the sharp kernels defined by k(x) = k_Bart^ρ(x), where k_Bart(·) is the Bartlett kernel and

. These kernels, as so defined, exhibit a sharp peak at the origin. Sharp kernels are positive semidefinite, as they are equal to the products of the positive semidefinite kernels. To see this, we may use equation (A.11) in the Appendix and represent the Bartlett kernel by

Then

So, for any function g(x) ∈ L²[0,1], we have

which implies that k_Bart²(r − s) is indeed positive semidefinite. Iterating the previous procedure leads to the positive semidefiniteness of k_Bart^ρ(r − s) for any

.

For sharp kernels, the Parzen characteristic exponent is q* = 1 and k₁ = ρ. The value of κ is κ = 1/(ρ + 1). Therefore, κ is a decreasing function of the exponent ρ. In principle, we can choose ρ to make κ as small as possible. However, the finite-sample performance can be hurt when ρ is too large for a moderate time series dimension. This is because the bias of

increases as ρ increases, as shown by Lemma 3. In fact, when

, the asymptotic distribution of

under the joint limit is

where b = −2πα(ρ + 1)(Ω_xx⁻¹ [otimes ] I_my)vec(Ef_UyiUxi⁽¹⁾ − βEf_UxiUxi⁽¹⁾). Therefore, the squared asymptotic bias b′b is increasing in ρ while the asymptotic variance is decreasing in ρ. This observation implies that there exists an optimal ρ that minimizes the mean squared errors. The optimal ρ depends on the ratio α and the average spectral density of U_i. We can estimate the optimal ρ along the lines of Andrews (1991), but we do not pursue this analysis in the present paper.

Next, we consider the steep kernels defined by k(x) = (k_PR(x))^ρ where k_PR(x) is the Parzen kernel. These kernels decay to zero as x approaches one. The speed of decay depends on ρ. The larger ρ is, the faster the decay and the steeper the kernel. Steep kernels are positive semidefinite because the Parzen kernel is positive semidefinite. The difference between the sharp kernels and the steep kernels is that the former are not differentiable at the origin whereas the latter are. For steep kernels, the Parzen characteristic exponent is q* = 2 and k₂ = 6ρ. The value of κ can be calculated using numerical integration. They are given in Table 1 for ρ = 1,…,6. Obviously, κ decreases as ρ increases. This is expected because (k_PR(x))^ρ₁ ≤ (k_PR(x))^ρ₂ if ρ₁ ≥ ρ₂. Therefore, the steep kernel can deliver an LRV-based estimator

that is more efficient than

, as long as the exponent is greater than 1 (see Table 1).

The values of κ for some kernels

When the steep kernel is employed, the dominating bias of

is independent of the exponent. If (n,T → ∞) such that

, then the asymptotic distribution of

is

where b = −2πα(Ω_xx⁻¹ [otimes ] I_my)vec(Ef_UyiUxi⁽¹⁾ − βEf_UxiUxi⁽¹⁾). This limiting distribution seems to imply that we can choose ρ to make κ as small as possible without inflating the asymptotic bias. This is true in large samples. But in finite samples, a large κ may lead to a poor performance. The reason is that the second dominating bias term in

is T⁻²2πk₂ Ef_UiUi⁽²⁾, which depends on k₂. As a consequence, the asymptotic bias of

under the joint limit is

The O(·) term vanishes when (n,T → ∞) such that

. But in finite samples, the O(·) term may have an adverse effect on the performance of

. Nevertheless, the effect is expected to be small, especially when T is large.

Finally, we may take powers of the kernels in

and obtain more efficient estimates. Although Assumption 3 does not cover exponentiated kernels of this sort. Theorems 8 and 9 go through without modification.

Table 1 summarizes the values of κ for different exponentiated kernels. The table clearly shows that for a given “mother” kernel, the value of κ decreases as the exponent increases. Recall that the smaller κ is, the more efficient the LRV-based estimator is. We can thus conclude that a larger exponent (ρ) gives rise to a more efficient estimator.