Added variable non-significant but model fit better? I ran two nested models: Model 1 without an interaction term, and model 2 with the interaction term. The interaction term turned out to be non-significant in model 2, but the model fit index (wald value) of model 2 is significantly better than model 1.
Any idea how or why a non-significant added interaction term can contribute to a better model fit? Any thoughts are appreciated!
Background info：I use STATA to run dynamic panel modeling (xtbond2). I compared the wald values of model 1 and model 2 based on the chi-square table and found the improvement of the value is larger than the threshold value when df=1, so this improvement is statistically significant.
 A: Almost every statistical software gives p-values in a regression model summary that are based on Wald testing. When you compared model 2 and model 1 using a chi-squared table, you compared them using a likelihood ratio test, which gives a different answer, particularly when the sample size is small.
I do not know STATA, but here is a simulation in R. Notice that the sample size of $25$ gives different p-values, but the sample size of $250,000$ gives nearly identical p-values. Indeed, asymptotics are at play.
library(lmtest)
set.seed(2021)
N <- 25
x1 <- runif(N)
x2 <- runif(N)
y <- x1 + x2 + 0*x1*x2+ rnorm(N, 0.2)
L_full <- lm(y ~ x1 + x2 + x1*x2)
L_reduced <- lm(y ~ x1 + x2)
summary(L_full)
lmtest::waldtest(L_full, L_reduced) # Wald
lmtest::lrtest(L_full, L_reduced) # Likelihood ratio
N <- 250000
x1 <- runif(N)
x2 <- runif(N)
y <- x1 + x2 + 0*x1*x2+ rnorm(N, 0.2)
L_full <- lm(y ~ x1 + x2 + x1*x2)
L_reduced <- lm(y ~ x1 + x2)
summary(L_full)
lmtest::waldtest(L_full, L_reduced) # Wald
lmtest::lrtest(L_full, L_reduced) # Likelihood ratio

In your case, according to Wald, the interaction is insignificant, and model 2 has no better fit; according to the likelihood ratio, the interaction is significant, and model 2 has a better fit. There is no contradiction.
