what does Wald test of spatial terms mean in spregress

[Zhongying Gan]

Hello,

I'm using Stata 16, and before posting this question I have read "STATA SPATIAL AUTOREGRESSIVE MODELS REFERENCE MANUAL RELEASE 16" thoroughly but couldn't find answers to this question. I also tried to Google the answer but couldn't find any.

Every time I use spregress, at the end of the results there will be a line saying "Wald test of spatial terms: ****"

The following are two examples from the manual:

My question is:
1. What is the null hypothesis of the Wald test of spatial terms
2. When do I reject the null hypothesis
3. What is the number in the parentheses after chi2? In the first screenshot, it's chi2(1). In the second screenshot, it's chi2(2)
4. In the second screenshot, the regression already takes into account the possible spatial correlation of error terms by adding errorlag(W), why does the estimation still perform Wald test of spatial terms?
5. Where may I find Stata documentations on the Wald test of spatial terms

I would really appreciate it if you could share with me your insights. Thank you.

[Jose Cardoso]

I have the same questions. There is nothing in the Stata files about this test.

[Maxence Morlet]

I don't know much about spatial regression, but I can help with the Wald test (see Hayashi, 2000):

1. H0 of the Wald test is whatever you want it to be, but it has to be a linear combination; Wald test W = (Rb - r)(RAvar(b)R' )^-1(Rb - r)

where b is your estimate of parameter Beta, R is the linear combination of b estimates, r is what these are equal to under the null, Avar is asymptotic variance. Under the null, this statistic is asymptotically chi-squared distribtued.

2. You reject the null at the alpha% level when the Wald stat exceeds the (1-alpha) quantile of the chi squared distribution (you can look it up in a table, or compute the associated p-value).

3. That number is the only parameter of the chi2 distribution; the number of restrictions you've imposed under the null.