*Bounty: 100*

Consider a simple panel data (or multilevel model) with random effects. For context, consider a wage regression, where the dependent variable $ln(y_{it})$ is the natural log of wage, where the wage is measured in £ per hour. The regression to be estimated is:

$$ln(y_{it})= X_{it}beta + zeta_{i} + eta_{t} + epsilon_{it}$$

where $zeta$ and $eta$ represent individual heterogeneity and year effects, respectively, and $epsilon_{it}$ is white noise (or idiosyncratic error).

You estimate the above model, and obtain an estimate for the random effects. I have three related questions.

**Question 1:**

Which is the dimension/units of both error components? Do they have the same units as the dependent variable? (which actually has no units, because logarithm is dimentionless). If so, is there a formal proof of this?

**Question 2:**

If the answer is yes to Q1, then, does it mean that $exp(zeta_i)$ and $exp(eta_t)$ are measured in £ per hour?

**Question 3:**

But then, how can we go back to the theory? For instance, my theory could assume that workers are paid according to their productivity. Therefore, you can somehow split the pay to wages in terms of something like

$$ y_{it} = omega_t h_{it} $$

where $h_{it}$ is productivity (output per hour) and $omega_t$ is the pay rate per product unit, i.e. £ per output, which combined give £ per hour. Thus, if one wanted to use such a wage regression to find those two elements, it seems impossible to do so, because all we are measuring is always in the same units than the left-hand side variable. We can therefore never go back to the theory.

To put it differently, say the answer to Q1 is yes (as I expect so to be). Then, let’s exponentiate the regression:

$$ y_{it} = exp(X_{it}beta) exp(zeta_i) exp(eta_i) exp(

epsilon_{it}) $$

So, $y_{it}$ is measured in £ per hour. How do we get the same units from the right-hand side? If the exponential of the two random effects (and the error term) are measured in £ per hour (Q2), then it’s up to $exp(X_{it}beta)$ to balance the units of the equation. But for this to be the case, the units of the latter would have to be $left(dfrac{hour}{£}right)^2$, which looks totally arbitrary. Furthermore, how can we ever go back to the theory and write the resulting estimates in terms of productivity and pay per unit of output? (Q3)

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