This is a fundamental question I have, and so far I have not found a convincing answer.

My theory is this: $$ w_{ijt} = \theta_{jt} a_{ij} $$ where $w_{ijt}$ is wage of worker $i$ in occupation $j$ in period $t$, $\theta_{jt}$ is the "efficiency wage" paid to every worker in occupation $j$, in period $t$, and $a_{ij}$ is the ability of worker $i$ in occupation $j$. (Since workers are observationally identified by $i$ and $t$, we could define $w_{ijt}=w_{it}$)

Now, we observe wages but not ability nor "efficiency wage". So, to estimate these two components, I use a two-way error component model (taking logs):

$$ \log(w_{it}) = \log(\theta_{jt}) + \log(a_{ij}) + e_{it} $$ Notice that this decomposition is possible because each component varies across different dimensions.

Now, from my estimations I get $\hat{\theta_{jt}}$ and $\hat{a_{ij}}$.

The question: How can I know that the units of measurement of my estimated components are the same than those of my theory?

Does the answer change if I extend my estimation with:

$$ \log(w_{it}) = X\beta + \log(\theta_{jt}) + \log(a_{ij}) + e_{it} $$ where $X$ are other determinant of wages?

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    $\begingroup$ It's really not clear to me what you're doing in your complex example (you seem to be calling it regression in your title, but it isn't). Can you give a small numerical example, and if possible a link of the kind of model you're talking about? $\endgroup$ – Glen_b Aug 20 '16 at 10:14
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    $\begingroup$ two mistakes at least, first, you forgot to include the error term in your first example. The second, your complex example is not regression, it is matrix decomposition. If I understand your question well, you should forget about regression and stick to decomposition. Lets start, for any matrix A there is at least one decomposition called SVD. It is just re arranging the space and then not make any change in units. But there is no guarantee that it provided the same unit as aX in your example. $\endgroup$ – TPArrow Aug 20 '16 at 10:41
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    $\begingroup$ @luchonacho Can you write the model you're talking about in your panel model and indicate what exactly is observed? In the panel data models I've seen both y and X are observed, but your post says X is not observed. If you're doing matrix decomposition as TPArrow suggests could you please be explicit about that. If you're not doing it, can you also be clear that you aren't? $\endgroup$ – Glen_b Aug 20 '16 at 10:51
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    $\begingroup$ @luchonacho I advise you to stick to one of the questions and make is rich with providing examples or make it clearer. $\endgroup$ – TPArrow Aug 20 '16 at 11:03
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    $\begingroup$ You seem to be posting essentially the same question in two different places. Please see the help center which says "Please note, however, that cross-posting is not encouraged on SE sites. Choose one best location to post your question. Later, if it proves better suited on another site, it can be migrated." $\endgroup$ – Glen_b Aug 20 '16 at 11:21

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