Welcome to The Log of Gravity page.

Reference: Santos Silva, J.M.C. and Tenreyro, Silvana (2006), The Log of Gravity, The Review of Economics and Statistics, 88(4), pp. 641-658.

An early version of the paper can be found at CEP/LSE (and an even earlier version at Boston Fed).

In this page you can find the data set used in the paper, codes to extend some of the results in the paper, and other useful information on the implementation of the PPML estimator.

If you cannot find here the answer to your question about The Log of Gravity, please do not hesitate to contact the authors; we will be only too pleased to help.

• The dataset (in xls and dta formats) is available here.

• Convergence problems:

  1. We are aware of two situations where you may find that (at first) the PPML estimator has trouble to converge. Fortunately, there are simple solutions for both cases:
     i) You may have problems using PPML when the dependent variable has many zeros and the model includes many dummies. A typical example of this is when panel data is used to estimate gravity equations including time-varying importer and exporter fixed effects. A simple explanation for why this may lead to convergence problems can be found here and this Economics Letters paper provides the more technical details. To by-pass the problem, see the ppml command below.
     ii)  Stata's poisson command is not very good at dealing with numerical problems and the algorithm may not converge either if the dependent variable has very large values or if the regressors are highly collinearity or have different scales. In this case there are several possible solutions: a) you may be able to solve the problem just by re-scaling or re-centring the problematic variables; b) rather than using the poisson command you can use the ppml command that we have written (see below); c) of course, you can just use other software; one of us is a great fan of TSP.
  2. Further details on convergence issues can be found in this Stata Journal paper (an earlier version is available here).
  3. We have written a ppml command for Stata that bypasses most of these problems. To install the command just type "findit ppml" (or "net sj 11-2") in Stata and follow the links; please read the help file carefully before using the command. Note: ppml does not have an option to include country-pair fixed effects; if you really want to include them you may want to consider Timothy Simcoe's xtpqml Stata command.

• Instrumental Variables:
  We have written a crude Stata command (recently updated) to estimate the IV version of PPML. This estimator was originally described in Windmeijer, F. and Santos Silva, J.M.C. (1997), "Estimation of Count Data Models with Endogenous regressors: An Application to Demand for Health Care," Journal of Applied Econometrics, 12(3), pp. 281-294. Here is an example of how to use the command.

• RESET test:
  Here is a sample of the code to perform the test.

• R-squared:
  If you want to compute the R-squared for a model estimated by PPML, you can used the method implemented here.

• PPML performance with many zeros:
  Simulation evidence on the excellent performance of the PPML estimator when the data has many zeros can be found in this Economics Letters paper.

• Warning:
  Be advised that there are several papers purporting to introduce estimators that improve on the PPML. While it is of course possible to find estimators that outperform PPML in very specific conditions, and under reasonably strong distributional assumptions, to our knowledge, all the proposed alternatives to PPML are either simply invalid or valid only under implausibly strong distributional assumptions. Therefore we stand by the claim that PPML has all the characteristics needed to be the workhorse for the estimation of constant-elasticity models such as the gravity equation. If you believe to have evidence that another estimator generally outperforms PPML in this context please do let us know; we would be delighted to acknowledge that.

• We have written a short reply to "The log of gravity revisited".

• If you want to compute 'undertrading' and 'overtrading' after fixed-effects regressions with panel data, you need to obtain a set of residuals with zero mean. Here is how to do it.

• In this Annual Review of Economics paper we use the PPML estimator to estimate the trade effect of the euro; an earlier version is available here.

• Thibault Fally has recently provided an additional motivation for using PPML; his interesting paper can be seen here.

• Seven FAQ's & myths about the Log of Gravity

  1 - Does the PPML perform poorly when the data has a large proportion of zeros?
    - No, this is an unfortunate myth! In this paper, we have provided ample evidence that the estimator works very well even when the proportion of zeros is very large.
  2 - Does the PPML assume equi-dispersion?
    - Not at all! This is another unfortunate myth. PPML is optimal when the conditional variance is proportional (not necessarily equal) to the conditional mean. This allows both for under- and over-dispersion. Even if the conditional variance is not proportional to the conditional mean, the PPML will still be consistent.
  3 - How can a continuous variable like exports have a Poisson distribution?
    - It can not! However, PPML does not require the data to follow a Poisson distribution (that is why it is a pseudo-maximum likelihood estimator and not a maximum likelihood estimator). In fact, all that is needed for the PPML estimator to be consistent is that the conditional mean of the variate of interest is correctly specified.
  4 - Why use a count data estimator when the dependent variable is continuous?
    - The estimator proposed in the Log of Gravity is simply a weighted non-linear least squares estimator. It turns out that the with the proposed weights, the first-order conditions for this estimator are identical to those of the Poisson pseudo-maximum likelihood regression. Therefore, the fact that we recommend the used of a count data estimator for the gravity equation is just a fortunate coincidence that allows the use of a well-known regression method which is widely available is econometric and statistics software.
  5 - Why not use other count data models like the negative-binomial or zero inflated models?
   - As noted in 4 above, the fact that we recommended that gravity equations should be estimated with a count data model is essentially a coincidence and there is no guarantee that other count data models would be adequate to estimate gravity equations. For example, both the negative-binomial and the zero-inflated regression models have the important drawback of not being invariant to the scale of the dependent variable. That is, measuring trade in dollars or in thousands of dollars will lead to different estimates of the elasticities of interest!
  6 - Can Vuong's tests for non-nested hypotheses be used to choose between the PPML and the ZIP estimators?
    - No! First of all, as noted in 5 above, it makes little sense to estimate a gravity equation using a zero-inflated model. In any case, Vuong's test is appropriate to compare models estimated by maximum likelihood, and it cannot be used when at least one of the competing models is estimated by other method. Therefore, the test cannot be used when one of the models is estimated by pseudo-maximum likelihood, as in the case of the PPML.
  7 - Can PPML be used when there is over-dispersion in the data?
    - Yes! As noted in 2 above, PPML is consistent, and can even be optimal, when there is under- or over-dispersion. Actually, it does not even make much sense to test for over-dispersion when estimating by PPML because the estimator makes no assumption about it.

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Last updated on 28 November 2012