# Fixed Effects Variation

Suppose I wish to run the following gravity regression: $$Y_{ijt}=\boldsymbol{X_{ijt}'\beta+}ij+i_{t}+j_{t}+\epsilon_{ijt}$$ where the LHS is bilateral trade between $i$ and $j$ in year $t,$ the first part of the RHS denotes control variables, specific to $i$ and $j$. The term $ij$ is the pair fixed effects term, which is a fixed effect for a particular term. The third term is the importer-time fixed effect, and the fourth term is the exporter fixed effect. Importantly, within each year, I have bilatereal information between various possible combinations of importers and exporters. My questions are as follows: 1) Are the coefficients on $ij$, $i_{t}$ and $j_{t}$ identified? My first inkling is yes. That is because, assuming a balanced panel, the total number of observations are: $$T\times I\times J$$ whereas the number of coefficients to be identified as: $$IJ+IT+JT$$ which is less than $TIJ$ if: $$IJ\left(1-\frac{1}{T}\right)>I+J$$ which should generally be the case.

2) On what basis is $\beta$ identified? What variation is being used? The $ij$ pair fixed effects would imply this is being identified from within country pair- over time variation in $\boldsymbol{X.}$ What role does $i_{t}$ and $j_{t}$ play?

Thanks!

It isn't enough to have a large number of observations. You need multiple observations of each $i$ and $j$ for multiple time periods and doing trade with multiple countries. If, for example, if countries $k$ and $l$ only trade with each other, it wouldn't be possible to distinguish between $k$, $l$, and $lk$. In additions, you need multiple observations because you want to identify a time subscript on $i$ and $j$. If you have many countries, many periods, and many trade relationships, and the data are not sparse then this may not matter much.

You may find this paper of use: Gravity or dummies? the limits of identification in gravity estimations (Hornok 2011)

This paper argues that identification of trade policy effects with a gravity equation that includes country-time dummies to control for the theoretical Multilateral Trade Resistances (MTR) is severely limited. In most cases heterogeneous policy effects, i.e. more than one policy dummies, cannot be identified separately, because the policy dummies and the country-time dummies are perfectly collinear. Although a single policy dummy can b e identified, the estimate may not b e meaningful, because country-time dummies absorb to o much of the useful variation of the data. Standard estimation techniques often do not reveal these problems. The pap er demonstrates these arguments by taking four typical research questions on the effect of a trade policy, checking the identifiability of the corresponding policy effect and deriving the estimates. Empirical exercise on estimating the trade effects of EU enlargement complements the analytical findings.

And Estimating the gravity model without gravity using panel data (Westerlund and Wilhelmsson 2009)

"...$\alpha_{ij}$ which is unidentified in the fixed effects formulation of the model. In order to identify $\alpha_{ij}$, a random effects assumption is needed. But such assumptions are generally not satisfied in practice..."

@ChinG, here is the basic gravity idea: Newton's law of universal gravitation tells us that the force of gravity is $$F = \frac{G M_1 M_2}{R^{2}_{1,2}}$$ which implies $$f = g + m_1 + m_2 - 2 \cdot r_{1,2}$$ (where lowercase letters are the logs of upper case ones). In the trade equation, the force of gravity is replaced with the volume of trade. Usually, $M_1$ represents the economic size of economy one (say measured by $GDP_1$ and in your regression analogous to $I_t$), $M_2$ represents the economic size of economy two ($GDP_2$ and analogous to $J_t$), and $r_{1,2}$ is the geographic or effective economic distance between the two countries ($\alpha_{i,j}$). GDP isn't exactly the right measure because countries differ in fraction of the economy devoted to trad-able output. Think of it as analogous to the importance of that country in global trade (in an absolute and not relative sense).

• Hi @Bkay, thanks a lot for your response. In your explanation, is it possible to indicate what the coefficients on j,i and ij are intuitively measuring? – ChinG Jul 10 '18 at 14:35
• Let me know if this update is what you needed. – BKay Jul 10 '18 at 17:10
• Sorry @BKay, I think I was not clear in my question. My question is generalizable to any question with multiple (twoway or threeway fixed effects). My question more specifically is: I am searching for the intuition behind what the fixed effects terms are measuring, not in a gravity setting but more generally- and moreover, how they are identified in the present context. So for instance, j is perhaps identified by calculating the average trade flows where j is an exporter. I'm not able to properly put my finger on the precise source of variation. How is i and j separately identified from ij? – ChinG Jul 10 '18 at 17:31