There is a well-known equation in international economics, called the gravity equation. This equation expresses imports from a country as a function of importer size, exporter size, and distance. In particular, $$ X_{ij}=\frac{Y_{i}Y_{j}}{\tau_{ij}} $$ Here, the LHS measures the value of imports by $i$ from~$,$$\tau_{ij}$ represents distance between $i$ and $j,$$Y_{i}$ represents $i's$ size and~$Y_{j}$ represents $j's$ size., Now, in a regression framework, one estimates this equation in additive form: $$ lnX_{ij}=lnY_{i}+lnY_{j}-ln\tau_{ij}+\epsilon_{ij} $$ where $\epsilon_{ij}$ is a random measurement regression error term. The OLS predicted values one obtains are defined as $\widehat{lnX_{ij}}$. As always, the hats denote the orthogonal projection of $X_{ij}$ onto the space spanned by $Y_{i},Y_{j}$ and $\tau_{ij}.$Without getting into the details of the procedure, one wishes to add over all the trading partners for a country $i,$ after exponentiating this expression. In other words, total\emph{ predicted} imports by $i$ are $$ \sum_{j\neq i}exp\left(\hat{\beta}_{1}lnY_{i}+\hat{\beta}_{2}lnY_{j}+ln\tau_{ij}\right) $$ The expression above is quite complicated. If there were no exponential, this would be easily simplified. Can this be approximated by a linear function, or any function that allows me to say collect terms that don't involve $j$ , and just make this expression more tractable? I feel that this question is not apt for the economics forum, as the question is not about the economics.
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For large $N$, $\sum_{j=1}^Nu_j\approx\int_0^Nu_jdj$ can be a good approximation, but whether the integral's more tractable than the sum when $u_j=\exp(a_j+b_j+c_j)$ depends on $a_j+b_j+c_j$. – J.G. Mar 11 '20 at 20:30
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Perhaps a taylor series approximation could do the trick? Actually maybe not, because the derivative of the exponential is also the exponential.. – ChinG Mar 11 '20 at 20:33
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Maybe we can help you more if you show how this problem came up. – J.G. Mar 11 '20 at 20:34
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Right, will try to modify it accordingly. – ChinG Mar 11 '20 at 20:35
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Are you trying to estimate the $\beta_k$ ? But why are they not just $1$, according to the initial equation ? If true, you can resort to the Levenberg-Marquardt algorithm. – Mar 11 '20 at 20:58
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Why does $\ln \tau_{ij}$ has a plus sign ? – Mar 11 '20 at 20:59
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Should be minus, thanks. – ChinG Mar 11 '20 at 21:01
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@YvesDaoust, they should be in theory, but don't always hold in the data. – ChinG Mar 11 '20 at 21:01
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Could you answer my first question ? – Mar 11 '20 at 21:02
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@YvesDaoust- Yes, the \beta are not 1 in the data. The estimation itself is not the problem, it is just the simplification of the final expression I am after. – ChinG Mar 11 '20 at 21:03
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I think I can go with a Maclaurin series expansion – ChinG Mar 11 '20 at 21:14
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Isn't it more traditional to apply least-squares to the exponents used here, i.e. to minimize the product of the exponentials instead of their sum? – J.G. Mar 11 '20 at 22:14