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I like to estimate a mixed model with two Random Effects, that are independent of each other and among themselves.

I use Panel data with a nested structure (counties $j$ nested within regions $i$). The error component should decompose in the following way:

$$E_{ijt} = \alpha_i + \mu_{ij} + e_{ijt}$$

$\alpha_i$ is an unobservable region specific time-invariant effect (normal distributed)

$\mu_{ij}$ is a nested effect of county $j$ within region $i$ (normal distributed)

$e_{ijt}$ is a remainder disturbance (normal distributed)

I read somewhere that "lmer" is capable to fit such specifications with idenpendent Random Effects.

lmer(y~.... + (1|region) + (1|county)

Is the lmer-model adequate for my task?

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Yes, lme4 should be fine in here. The formula you are asking is:

y ~ 1 + (1|region) + (1|region:county)

because in model you described country is nested in region.

For an extended description and examples see here, here or here.

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