Hirschi and Gottfredson have argued that using social correlates to explain desistance is misguided, as the factors that explain crime or its absence are constant across the life course (Hirschi & Gottfredson, 1983, 1984, 1994; Gottfredson & Hirschi, 1988, 1990). They showed that a similar age-crime curve has been found in many studies in different times and places. In most of these studies, cross-sectional data have been used, which excludes the possibility of making appropriate inferences regarding changes over time.
Two demographic variables appear to be especially important for the explanation of juvenile delinquency: age and sex. According to Hirschi and Gottfredson (1983), "the age distribution of crime cannot be accounted for by any variable or combination of variables currently available to criminology" (p. 554). Shavit and Rattner (1988) have presented data that confirm Hirschi and Gottfredson's position, whereas Tittle and Ward (1993) have also provided support. With regard to the relation between sex and crime, Gottfredson and Hirschi (1990) asserted that sex differences appear to be invariant across time and space, with males committing more offenses than females. Hagan (1998) endorsed this conclusion, stating that sex is the best predictor of criminality, of all the available demographic variables. D'Unger, Land, and McCall (2002) used a nonparametric mixed Poisson model to study the differences between male and female trajectories. Their tentative conclusion is that patterns are similar for males and females, with lower overall offending levels for females. Self-report studies show that the more serious the offense, the greater the disproportionately (Adler, Mueller, & Laufer, 1998). Overbeek, Vollebergh, Meeus, Engels, and Luijpers (2001) found that a relatively stable pattern of offending is a relatively strong predictor of the rate of later offenses. VanderValk, Spruijt, de Goede, Maas, and Meeus (2005) applied the latent growth curve (LGC) model with a quadratic component and found that the development of delinquency does not differ by gender. However, Hoyt and Scherer (1998) concluded their extensive review concerning female juvenile delinquency with the statement that the research results are not conclusive. Females have a slight lead in development during adolescence, and the difference in timing of puberty (Palmert & Boepple, 2001) may have significant implications for the development of adolescent delinquent behavior. Adolescent females also tend to associate with males who are somewhat older (Harrington Cleveland, 2003). Males and females may therefore differ on such measures as peak ages of committing the offense. Even though this difference has not been reported in earlier studies (Van der Ende & Verhulst, 2005; D'Unger, Land, & McCall, 2002), we could expect that general developmental differences are also reflected in the development of delinquent behavior. An interesting question is whether the differences between males and females are constant over their respective developmental curves; that is, whether sex differences in delinquency are invariant across age. Main differences can concern the proportion of offenders, the frequency of offense, and the age at which delinquency reaches a peak level. The model presented here is based on the autoregressive differential equation, which features parameter estimates that are independent of the chosen origin and measurement interval.
Model. An autoregression model states that current delinquency is dependent on prior delinquency (Dijkum & Landsheer, 2000). For both males and females, autoregression parameters are estimated, and these parameters can be compared to describe the developmental differences. Our main question is whether different parameters are necessary to describe the development, and, if so, what parameters are needed. An autoregressive model in continuous time can be described in an elegant way by a differential equation. …