Testing Panel Data Regression Models with Spatial Error Correlation

3.2 Marginal and Conditional Tests for λ = 0

Figure 1 plots the frequency of rejections in 2000 replications for testing λ = 0, i.e., zero
spatial error correlation. Figure 1 reports these frequencies for various values of N = 25; 49
and T = 3; 7, for both Rook and Queen weight matrices.  Marginal tests for H0: λ = 0

(assuming ¾^ = 0) as well as conditional tests for Hd: λ = 0 (assuming ¾_t > 0) are plotted
for various values of λ. As clear from the graphs, marginal tests can have misleading size
when ½ is large (0:5 or 0:8). Marginal tests also have lower power than conditional tests for
½ > 0:2 and 0:2 ∙ λ ∙ 0:8. This is true whether we use LM or LR type tests. This difference
in power is quite substantial for example when ½ = 0:8 and λ = 0:6. This phenomena persists
even when we increase N or T. However, it is important to note that marginal tests still
detect that something is wrong when ½ is large.

3.3  Marginal and Conditional Tests for σ²_μ = 0

Table 2 gives the frequency of rejections in 2000 replications for the marginal LR and LM
tests for H0 : σ²_μ = 0 (assuming λ = 0). The results are reported only when σ²_μ = 0 for
N = 25; 49 and T = 3; 7 for both the Queen and Rook weight matrices. Table 2 shows
that at the 5% level, the size of the two-sided LM test (LMG) for H₀^b (compared to its one
sided counterpart LM₁) could be missleading, especially when A is large. For example, for
the Queen weight matrix when N = 49; T = 7 and A = 0:9, the frequency of rejection for
LMG is 50:4% whereas the corresponding one-sided LM (LM1) has a size of 7:6%: The two-
sided likelihood ratio (LRG) test for H₀^b performs better than its two-sided LM counterpart
(LMG). However, in most experiments, LRG underestimates its size and is outperformed by
its one-sided LR alternative (LR1).

Table 2 also gives the frequency of rejections in 2000 replications for the conditional LR
and LM tests (LR* and LM* ) for Hθ : σ²_μ = 0 (assuming A = 0). These were derived in
Section 2.2. The results are reported only when σ²_μ = 0 for N = 25; 49 and T = 3; 7 for
both the Queen and Rook weight matrices. For most experiments, the conditional LM and
LR tests have size not significantly different from 5%. For cases where A is large, conditional
tests have better size than marginal tests. For example, when the weight matrix is Queen,
N = 49; T = 3 and A = 0:9, the frequency of rejections at the 5% significance level, when the
null is true, is 11:4% and 12% for LMi and LRi compared to 4:9% and 4:3% for LM'* and
LR^* .

M

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