# Mixed Effects model with quadratic term in fixed, but not in random part

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random intercept and slope).

$$y_{ij} = \beta_1 + \beta_2 x_{ij} + \beta_3 x_{ij}^2 + \zeta_{1j} + \zeta_{2j}x_{ij} + \epsilon_{ij}$$

## Question

Is the model mis-specified? Is it the case that the quadratic term has to be included also in the random part?

## References

Looking a bit around, looks like there are mixed views: