Psychological measures frequently show trait-like properties, and the ontological status of stable psychological traits has been discussed for decades. We argue that these properties can emerge from causal dynamics of time-varying processes, which are omitted from the analysis model, potentially leading to the estimation of traits that are, at least in part, illusory. Theories positing the importance of a large set of dynamic psychological causes across development are consistent with the existence of illusory traits. We show via simulation that even a linear system with many processes can generate a covariance matrix with trait-like properties. We then attempt to examine how illusory traits affect our conclusions drawn from a common statistical model which assumes stable traits to analyze longitudinal panel data --- a random-intercept cross-lagged panel model (RI-CLPM). We find that the RI-CLPM sometimes falsely detects the existence of traits in the presence of omitted processes, even when the data generating model does not include any traits. However, in this scenario, the RI-CLPM estimates less causally biased autoregressive and cross-lagged effects than an analysis model which does not assume traits (i.e., the cross-lagged panel model). Results indicate that the detection of trait variance should not be inferred as strong evidence for the existence of time-invariant trait causes. On the other hand, even when traits are illusory, statistical models assuming stable traits may sometimes be useful for causal inference.