I would like to simulate data from an unconditional growth model with one fixed (level 2) covariate $x_{i}$, cross-correlation $\rho_c$ and autocorrelation $\rho_a$.

The model should have a random intercept $\beta_{0i}$ with fixed effect $\beta_{0}$ and variance $u_{0i} \sim N(0,U_{0})$ and a random slope $\beta_{1i}$ with variance $u_{1i} \sim N(0,U_{1})$. The value of the fixed effect ($\beta_{1}$) should follow from $\rho_{c}$. Hence, the model is: $y_{it} = \beta_{0} + u_{0i} + (\beta_{1i} + u_{1i}) x_{i} + e_{it}$

I tried to simulate via time-series data (following this answer), which controls the auto- and cross-correlations pretty well for the fixed effect of the correlation. However, given that I have random slopes, I have to sample such time series data with a random cross-correlation coefficient, which I call $\rho_{ci} = \rho_{c} + u_{ri}$.

Given random slope variance $U_1$ and autocorrelation $\rho_{a}$, is there a way to simulate data from a growth model with random cross-correlation $\rho_{ci}$? And how do I find the true value of fixed $\beta_{1}$? Or is there a way to transform a sample of random slopes $\beta_{1i} \sim N(\beta_{1}, U_{1})$ to a sample of $\rho_{ci}$ with the correct variance of $\rho_{c}$? Alternatively, can I sample directly from some distribution $\rho_{ci}$ with mean $\rho_{c}$ and variance $U_{c}$? If so, how do I find $U_{c}$ given $U_{1}$ and what should the distributional form be?

To add: I should be able to find the unbiased fixed and random effects via a linear mixed effects model (via lmer() in the lme4-package for example).


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