2. THE MODEL

Consider the following panel data regression model:

where y_ti is the observation on the ith region for the tth time period, X_ti denotes the k × 1 vector of observations on the nonstochastic regressors and u_ti is the regression disturbance. Here α is a scalar and β is a k × 1 vector of slope parameters. In vector form, the disturbance vector of (1) is assumed to have random region effects and SAR remainder disturbances:

with

where u_t′ = (u_t1,…,u_tN) and ε_t and ν_t are similarly defined. Here μ′ = (μ₁,μ₂,…, μ_N) denote the vector of random region effects, which are assumed to be iid(0,σ_μ²), λ is the scalar SAR coefficient with |λ| < 1, whereas ν_ti ∼ iid(0,σ_ν²). We assume that μ and ν are independent. The term W_N is an N × N weighting matrix with zero elements across the diagonal and equal elements (1/(N − 1)) off the diagonal. In other words, the disturbance for each unit is related to an average of the (N − 1) disturbances of the remaining units. Such a weighting matrix was recently considered by Kelejian and Prucha (2002), Lee (2002), and Kelejian et al. (2005) and would naturally arise if all units are neighbors to each other and there is no other reasonable or observable measure of distance between them. Kelejian and Prucha (2002) consider the case of a spatially lagged dependent variable model with a row normalized weighting matrix with equal elements. They show that OLS and two-stage least squares (2SLS) are inconsistent unless panel data are available. Kelejian et al. (2005) give exact small-sample results that corroborate the earlier asymptotic findings. In addition, they demonstrate that for the spatial panel data case with fixed effects across time, OLS and 2SLS are both inconsistent. Lee (2002), on the other hand, shows that OLS can be consistent in an economic spatial environment where each unit can be influenced aggregately by a significant portion of units in the population.

One can rewrite (3) as

where B_N = I_N − λW_N. The model (1) can be rewritten in matrix notation as

where y is of dimension NT × 1, ι_NT is a vector of ones of dimension NT, X is NT × k, u is NT × 1, and Z = (ι_NT,X). Here X is assumed to be of full column rank, and its elements are assumed to be bounded in absolute value. The disturbance term can be written in vector form as

where ν′ = (ν₁′,ν₂′,…, ν_T′) and u is similarly defined, ι_T is a vector of ones of dimension T and I_T is an identity matrix of dimension T, and [otimes ] denotes the Kronecker product. Under these assumptions, the variance-covariance matrix of u can be written as

Kelejian and Prucha (2002) showed that W_N can be written as

where J_N is a matrix of ones and I_N is an identity matrix, both of dimension N. Defining J_N = J_N /N and E_N = I_N − J_N, and replacing J_N and I_N by their equivalent terms NJ_N and (E_N + J_N) and collecting like terms (see Wansbeek and Kapteyn, 1982), one gets

with

Therefore

Note that W_N is symmetric and

where c₁ = (N − 1)²/(N − 1 + λ)² and c₂ = 1/(1 − λ)².

Replacing J_T and I_T by their equivalent terms TJ_T and (E_T + J_T) in (7), one gets

Collecting like terms, we obtain the spectral decomposition of Ω,

Hence

and

Note that if λ = 0, so that there is no spatial autocorrelation, then c₁ = c₂ = 1 and Ω in (13) reduces to the familiar variance-covariance matrix of the random effects panel data model given by Ω = (Tσ_μ² + σ_ν²)(J_T [otimes ] I_N) + σ_ν²(E_T [otimes ] I_N) (see Baltagi, 2005). Premultiplying the regression model in (5) by

from (15), one gets a similar transformation to the one suggested by Fuller and Battese (1973) for the random effects panel data model. In fact

will have typical elements y_ti* = (y_ti − θ₁ y_t. − θ₂ y_.i + θ₃ y_..) where y_t. denotes the sample average over regions, y_.i denotes the sample average over time, and y_.. denotes the average over the entire sample. The θ's can be easily obtained from the corresponding (c₁,c₂,σ_μ²,σ_ν²) parameters. For example, θ₁ is the coefficient corresponding to (I_T [otimes ] J_N), which can be verified to be

, θ₂ is the coefficient corresponding to (J_T [otimes ] I_N), which can be verified to be

, and θ₃ is the coefficient corresponding to (J_T [otimes ] J_N), which can be verified to be

. If λ = 0, then

, which reduces y_ti* to (y_ti − θ₂ y_.i). This is the familiar Fuller and Battese (1973) random effects transformation that allows us to obtain GLS as weighted least squares.

If σ_μ² = 0, i.e., the case of no random effects, the variance-covariance matrix in (7) reduces to

which from (11) reduces to

Hence

and

so that the typical element of

is y_ti* = (y_ti − θ₁ y_t.) where

. Note that as

.

A special case of this model is the cross-section spatial regression model with (T = 1) and equal weight matrix given by (8). In this case, one can show that OLS is equivalent to GLS as long as there is a constant in the regression. To prove this, note that the model in (5) becomes

where Z_N = (ι_N, X_N) and γ = (α,β)′. Here y_N is a vector of observations on the dependent variable and ι_N is a vector of ones, both of dimension N, and X_N is an N × K matrix of observations on the K explanatory variables. The disturbance vector is assumed to follow a spatial autoregressive process

where W_N is an N × N equal weighting matrix given in (8). In fact, one can prove that the Zyskind (1967) necessary and sufficient condition for OLS to be equivalent to GLS on (20) is satisfied. This calls for P_ZΣ = ΣP_Z, where Z = Z_N = (ι_N, X_N) is the matrix of regressors in (20) and Σ = E(u_Nu_N′) = σ_ν²Ω is the variance-covariance matrix of the disturbances given in (21). It is straightforward to show that Ω = c₁ E_N + c₂ J_N and

because P_Zι_N = ι_N, P_Z J_N = J_N, and P_Z E_N = P_Z − J_N. Similarly,

Hence, P_ZΩ = ΩP_Z.

In fact, another necessary and sufficient condition for OLS to be equivalent to GLS, which relies on Ω⁻¹, is given by Milliken and Albohali (1984), and this condition calls for Z′Ω⁻¹(I_N − P_Z) = 0. Here

and using the fact that J_N(I_N − P_Z) = 0, we get

Hence

because Z′P_Z = Z′. Note that

and this, in general, is not equal to the usual formula for

computed by regression packages, i.e., σ_ν²(Z′Z)⁻¹, unless λ = 0, which is the case of no spatial correlation.

For the spatial panel regression with equal weights, this result is not necessarily true as long as σ_μ² > 0. Two special cases where OLS on (5) is the same as GLS, i.e.,

, are the following: (i) the trivial case where σ_μ² = 0 and λ = 0 (in fact, when λ = 0, c₁ = c₂ = 1, θ₁ = 0, and y_ti* = y_ti and GLS reduces to OLS); also, (ii) when the matrix of regressors X is invariant across time, i.e., when X = ι_T [otimes ] X_N, with X_N being the N × k matrix of exogenous regressors that is invariant across time. To show this, we prove that the Zyskind (1967) necessary and sufficient condition for OLS to be equivalent to GLS on (5) is satisfied. This calls for P_ZΩ = ΩP_Z, where Z = ι_T [otimes ] Z_N = ι_T [otimes ] (ι_N, X_N) is the matrix of regressors in (5) and Ω is the variance-covariance matrix of the disturbances given in (17). It is straightforward to show that P_Z = J_T [otimes ] P_ZN and that P_ZNι_N = ι_N, P_ZN J_N = J_N, and P_ZN E_N = P_ZN − J_N. Hence

Similarly,

Hence, P_ZΩ = ΩP_Z.

In fact, another necessary and sufficient condition for OLS to be equivalent to GLS, which relies on Ω⁻¹, is given by Milliken and Albohali (1984), and this condition calls for Z′Ω⁻¹(I_TN − P_Z) = 0. Using (18), and the fact that J_N(I_N − P_ZN) = 0, we get

Hence

because ι_T′J_T = ι_T′ and Z_N′P_ZN = Z_N′.

An anonymous referee pointed out that an alternative derivation of these results can be obtained using Lemma 2.1 of Magnus (1982). This lemma was derived in the context of a multivariate error component model, but when applied to the model considered in this paper, one common theme is the following: if W_N = α₁ A_N + α₂(I_N − A_N) with α₁ ≠ α₂, and A_N is a symmetric idempotent matrix, then all powers of W_N (also negative powers), its eigenvalues, and determinant can all be easily calculated. Here, we have I_N − λW_N = β₁ A_N + β₂(I_N − A_N), with β₁ = 1 − λα₁ and β₂ = 1 − λα₂. Assuming that λα₁ ≠ 1 and λα₂ ≠ 1, we get (I_N − λW_N)⁻¹ = (1/β₁)A_N + (1/β₂)(I_N − A_N). Hence Ω = (I_N − λW_N)⁻² = (1/β₁²)A_N + (1/β₂²)(I_N − A_N). Applying the Milliken and Albohali (1984) necessary and sufficient condition that calls for Z′Ω⁻¹(I_N − P_Z) = 0, we get Z′Ω⁻¹(I_N − P_Z) = (β₁² − β₂²)Z′A_N(I_N − P_Z) = 0, which is [hArr ] Z′A_N(I_N − P_Z) = 0. In the special case where A_N = J_N, we see that OLS is equivalent to GLS if and only if Z′ι_N = 0 or (I_N − P_Z)ι_N = 0, i.e., if and only if the regressors are measured in deviations from the mean or the regression contains a constant.

3. EXTENSIONS

So far, we have considered the spatial autoregressive (SAR) specification for the disturbances. An alternative specification is the spatial moving average (SMA) specification. Anselin (2003) classifies the spatial covariance structure induced by SAR as global and that by SMA as local. In this case, the model given by (1) and (2) is the same, but the disturbances in (3) become

In vector form, the panel disturbances in (6) become

with variance covariance matrix

because W_N given by (9) is symmetric. In fact,

where d₁ = (N − 1 − λ)²/(N − 1)² and d₂ = (1 + λ)².

Replacing J_T and I_T by their equivalent terms TJ_T and (E_T + J_T) in (34), and collecting like terms, we obtain

Hence, we get the same results for the SMA specification as for the SAR specification when the weight matrix is equal. The only difference between (36) and (13) is d₁ and d₂ rather than c₁ and c₂. Similarly, Ω⁻¹ and Ω^−1/2 are the same as (14) and (15) with d₁ and d₂ replacing c₁ and c₂. The same holds true for the Fuller and Battese type transformation described following (15). Note that if λ = 0, so that there is no spatial autocorrelation, then d₁ = d₂ = 1 and Ω in (36) reduces to the familiar variance-covariance matrix of the random effects panel data model. If σ_μ² = 0, i.e., the case of no random effects, the variance-covariance matrix in (34) reduces to

Hence, Ω⁻¹ and Ω^−1/2 are the same as in (18) and (19) with d₁ and d₂ replacing c₁ and c₂. The Fuller and Battese transformation given following (19) is now

with typical element y_ti* = (y_ti − θ₁ y_t.) where

. Note that as

.

Kelejian and Robinson (1995) considered an alternative spatial error components (SEC) specification that differs from the SAR and SMA specifications. In this case, the model given by (1) and (2) is the same, but the disturbances in (3) become

where ψ_t is an (N × 1) vector of spillover error components. The two component vectors ψ_t and ν_t are assumed to consist of independent and identically distributed (i.i.d.) terms with respective variances σ_ψ² and σ_ν² and are uncorrelated. In vector form, the panel disturbances in (6) become

with variance-covariance matrix

because W_N given by (9) is symmetric. In fact,

where b₁ = (σ_ν² + (λ²σ_ψ²/(N − 1)²)) and b₂ = (σ_ν² + λ²σ_ψ²). The rest of the derivations are the same as before with b₁ and b₂ replacing c₁ and c₂. Note that as

.

For the cross-section (T = 1) spatial regression model with SMA or SEC disturbances and an equal weight matrix, one can easily show that OLS is equivalent to GLS as long as there is a constant in the model; the proof is left to the reader. For the spatial panel regression with equal weights, this result is not necessarily true as long as σ_μ² > 0. Two cases where OLS is the same as GLS are once again (i) the trivial case where σ_μ² = 0 and λ = 0 or (ii) when the matrix of regressors X is invariant across time. The proofs for the SMA and SEC cases are the same as that for the SAR case and are left for the reader. The key for these results is the fact that (11) for SAR, (35) for SMA, and (41) for SEC are all linear combinations of E_N and J_N. These matrices are idempotent, are orthogonal to each other, and sum to I_N. Alternatively, one can apply Lemma 2.1 of Magnus (1982) to obtain the same results.

Next, we consider an alternative panel data model with spatially correlated error components suggested by Kapoor et al. (2005). The regression model is the same as (1), but the SEC structure given by (2) and (3) becomes

with

where μ and ν are the same as before, i.e., iid(0,σ_μ²) and iid(0,σ_ν²) independent of each other and among themselves. Performing the spatial Cochrane–Orcutt type transformation on (5) one gets

with

and y*(λ), X*(λ), and Z*(λ) defined similarly. The variance-covariance matrix Ω_ε = E(εε′) = (Tσ_μ² + σ_ν²)(J_T [otimes ] I_N) + σ_ν²(E_T [otimes ] I_N) is the usual panel data error components random effects variance-covariance matrix. GLS on (44) can be obtained as OLS on the Fuller–Battese transformed equation (44), i.e., after premultiplying it by σ_νΩ_ε^−1/2 = (E_T [otimes ] I_N) + (σ_ν /σ₁)(J_T [otimes ] I_N), where

. In this case,

For the equal weight matrix given in (8), (I_N − λW_N) is given following (9) and can be rewritten as (I_N − λW_N) = a₁ E_N + a₂ J_N, where a₁ = (N − 1 + λ)/(N − 1) and a₂ = (1 − λ). Substituting this expression in (46) yields

Once again, we see that for the Kapoor et al. (2005) specification, with an equal weight matrix, GLS on (44) can be written as a weighted combination similar to that following (15), i.e., y_ti** = a₁(y_ti − θ₁ y_t. − θ₂ y_.i + θ₃ y_..) where y_t. denotes the sample average over regions, y_.i denotes the sample average over time, and y_.. denotes the average over the entire sample. The θ's can be easily obtained from the corresponding (a₁,a₂,σ_μ,σ_ν) parameters. For example, θ₁ is the coefficient corresponding to (I_T [otimes ] J_N), which can be verified to be (1 − (a₂ /a₁)), θ₂ is the coefficient corresponding to (J_T [otimes ] I_N), which can be verified to be (1 − (σ_ν /σ₁)), and θ₃ is the coefficient corresponding to (J_T [otimes ] J_N), which can be verified to be (1 − (a₂ /a₁) − (σ_ν /σ₁) + (a₂σ_ν /a₁σ₁)). If λ = 0, a₁ = a₂ = 1, then θ₁ = θ₃ = 0 and θ₂ = (1 − (σ_ν /σ₁)), which reduces y_ti** to (y_ti − θ₂ y_.i). This is the familiar Fuller and Battese (1973) random effects transformation.

If σ_μ² = 0, i.e., the case of no random effects, then σ_ν = σ₁, θ₂ = θ₃ = 0, and θ₁ = (1 − (a₂ /a₁)), which reduces y_ti** to a₁(y_ti − θ₁ y_t.). In fact, y**(λ) from (47) reduces to

Note that a₁θ₁ = (a₁ − a₂). Therefore, y_ti** is a weighted combination of y_ti and y_t.. For this model, OLS is the same as GLS for the trivial case where σ_μ² = 0 and λ = 0. The more interesting case where OLS on (44) is equivalent to GLS is the case where the matrix of regressors X is invariant across time. The proof is along the same lines as before and is left for the reader. In fact, one can easily show that when X = ι_T [otimes ] X_N, with X_N being the N × k matrix of exogenous regressors that is invariant across time, both the Zyskind (1967) and the Milliken and Albohali (1984) necessary and sufficient conditions are satisfied for this model.