I have noticed more than once that, using the same alpha (e.g., .05, two-tailed test), a parameter estimate is not significant (e.g., two-tailed p-value = .054) but its standardized estimate is (e.g., .046), whether you look at STDYX, STDY, or STD results. What causes this inconsistency and how should I handle this situation? In other words, should I report the coefficient as significant, not significant, or "it depends"? Thanks.
The sampling distributions for the two parameters differ. This is the reason the significance is not the same for both parameters. You need to decide which parameter to report and report the significance for that parameter.