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I want to estimate the following equation using a panel data set with countries $i$ and years $t$, preferably in R:

$$ y_{it} = \beta_1 \cdot x_{1,it} + \beta_2 \cdot x_{2,it} + \epsilon_{it} $$

However, I cannot observe $y_{it}$ directly. Instead, I can only observe $\tilde y_{it} = s_i \cdot y_{it}$, where $s_i$ is a share of $y_{it}$ that varies across countries, but not over time. $s_i$ is not observable. If it were observable, I could probably just substitute and estimate

$$ \tilde y_{it} = \beta_1 \cdot x_{1,it} \cdot s_i + \beta_2 \cdot x_{2,it} \cdot s_i + \epsilon_{it} $$

directly.

Is there a way to estimate the above relationship given my data constraints? It somehow feels like a problem that can be addressed using country dummies (i.e. a fixed-effects model) or a random-effects model, but I cannot seem to figure it out.

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  • $\begingroup$ Unfortunately, I have negative independent variables, but I might be willing to drop one (the only) negative variable if this turns out to be the only way to estimate the model. How would I do this exactly, though? If I take the log of the right-hand side, I end up with $log(... + ...)$, don't I? That wouldn't be a linear term anymore. (EDIT: This comment is the answer to another comment that was deleted in the meantime: "If all your variables are positive, you can take logs and estimate a fixed-effects model.") $\endgroup$
    – severin
    Commented Feb 23, 2015 at 12:46
  • $\begingroup$ If the first model is true, then multiplying both sides by $s_i$ we get that either $\tilde{y}_{it}=\beta_1\tilde{x}_{1,it}+...$ or $\tilde{y}_{it}=\beta_{i1}x_{1,it}+...$, where $\beta_{i1}=s_i\beta_1$. Since $s_i$ is not observable, it is only possible to estimate the second model. But this model has different coefficients for different $i$, so it is not a panel data model. If you have sufficient data you can simply estimated individual regression for each $i$. Adding individual country dummies and their interactions would result in the same model. $\endgroup$
    – mpiktas
    Commented Feb 23, 2015 at 12:58
  • $\begingroup$ @mpiktas: Thanks! Estimating individual regressions would be an option. I just wonder whether it isn't possible to somehow use the information that all $\beta$s contain the same $s_i$, to either estimate or at least pull out the $s_i$ somhow. $\endgroup$
    – severin
    Commented Feb 23, 2015 at 13:05
  • $\begingroup$ $s_i$ comes into model non-linearly, via multiplication, while the usual panel data techniques are designed to deal with linear effects. $\endgroup$
    – mpiktas
    Commented Feb 23, 2015 at 13:12

1 Answer 1

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Since you have enough of the data to estimate individual regressions, you can at least get a sense of $s_i$ in the following way.

  1. Estimate individual regressions, get the estimates $\beta_{i1},\beta_{i2}$
  2. Choose base country $k$
  3. Calculate the ratios: $\beta_{k1}/\beta_{i2}$, $\beta_{k2}/\beta_{i2}$, for all $i$.
  4. If the original hypothesis holds, these ratios should be equal to $s_k/s_i$.
  5. If the equality is feasible, multiply all countries $i$ by estimated ratios $s_k/s_i$. You get the equation:

$$y_{it}s_k = \beta_1 x_{1,it}s_k + \beta_2 x_{2,it}s_k + \varepsilon_{it}s_k$$

All of the variables are observed, so you can estimate this using panel data.

  1. Given $\beta_1$ and $\beta_2$ you can calculate $s_i$ as $\beta_{i1}/\beta_{1}$ and $\beta_{i2}/\beta_2$.

This is not a strict statistical procedure, but you at least will get a feeling whether your original hypothesis holds.

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