Welcome to The Log of Gravity page.

Reference (with open access):

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. These and other issues are also discussed in this short review. 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.

• Summary: Researchers often have to model nonnegative dependent variables (potentially with many zeros), and different approaches have been used to estimate models for this kind of data. Typical approaches include using log-linearized models and Tobit-type estimators. In the Log of Gravity, we challenged these long-established practices and proposed as an alternative the use of the Poisson pseudo maximum likelihood (PPML) estimator of Gourieroux et al. (1984). The PPML estimator is easy to implement and is extraordinarily well suited for the estimation of models for this kind of data and is especially suitable for the estimation of gravity equations..

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

• Google Scholar Classic papers in Economics:
  The Log of Gravity tops the list of Google Scholar's Classic papers in Economics.

• Incidental parameters:
  Charbonneau (2012, p. 41) claimed that "a Poisson model with two fixed effects does suffer from the incidental parameter problem", and this claim has recently been echoed by other researchers. However, Charbonneau's claim is based on an example with N = T = 2, which cannot be informative about the existence of an incidental parameter problem because this problem is asymptotic in nature. In contrast, Fernández-Val and Weidner (2014, example 2, p. 16) have proved that a Poisson model with two fixed effects does not suffer from the incidental parameter problem as long as the sizes of the two sets of fixed effects grow at the same rate and the regressors are strictly exogenous or predetermined. Therefore, under very general conditions, inference based on the estimation by ppml of gravity equations including both importer and exporter fixed effects will not be affected by an incidental parameter problem. Remarkably, Martin Weidner and Tom Zylkin have shown that PPML is also largely immune to the incidental parameter problem when the model contains three sets of fixed effects.

• 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.

• Stata commands and details on convergence issues:
  Sérgio Correia, Paulo Guimarães and Tom Zylkin have written the ppmlhdfe command for Stata that uses state-of-the-art methods to deal with the problems discussed below and allows the use of multiple sets of fixed effects. We strongly recommend that you use the ppmlhdfe command to estimate models by PPML.
  
  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, use the ppmlhdfe command or the ppml command described 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 ppmlhdfe command or the ppml command that we have written (see below); c) of course, you can just use other software. Further details on convergence issues can be found in this Stata Journal paper.
  2. We have written a ppml command for Stata that bypasses most of these problems. To install the command just type "ssc install ppml"; please read the help file carefully before using the command. Even using the ppml command you will have convergence problems if the base category of any of the dummy variables in the model is a prefect predictor of zero observations in the dependent variable. An example of this problem and a solution are described here. Note that ppml has no advantage over the more recent ppmlhdfe command.

• Testing competing models for non-negative data with many zeros:
  We have developed a simple test to choose between competing models for non-negative data with many zeros; the paper is available here. Stata code to implement the test is available, type "ssc install hpc"; the code and data used in the paper are available here.

• Instrumental Variables:
  We have written a crude Stata command 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. This estimator is now implemented in Stata; see the ivpoisson command and the add option (the default).

• 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 use the method implemented here.

• 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 PPML and panel data to estimate the trade effect of the euro; an earlier version is available here.

• Related work by other authors:
  

  1. Santos Silva and Winkelmann study the performance of PPML under misspecification and show that the estimator has some interesting properties.
  2. Breinlich, Novy, and Santos Silva consider the effects of aggregation in gravity models estimated by PPML.
  3. Mauricio "Pachá" Vargas Sepúlveda replicates the results of the Log of Gravity in R.
  4. James Anderson, Mario Larch, and Yoto Yotov have recently used PPML for a different purpose; their paper can be seen here.
  5. Thibault Fally has provided an additional motivation for using PPML; his interesting paper can be found here.
  6. This short and free book by Yoto Yotov, Roberta Piermartini, José-Antonio Monteiro, and Mario Larch discusses state-of-the-art methods to estimate gravity equations and provides practical examples.
  7. Tom Zylkin has done some excellent work on the estimation of gravity equations using PPML; some of his work is cited above, but it is worth checking his home page for updates.
  
  

• FAQs & myths about the Log of Gravity

  1 - Why can't we just use the log-linear model with robust/clustered standard errors?
    - Using robust/clustered standard errors will only affect the estimated standard errors, but will have no effect at all on the estimates of the parameters. Therefore, the log-linear model will generally be invalid with or without the robust/clustered standard errors. PPML delivers estimates of the parameters that are consistent under very general conditions; of course, robust/clustered standard errors should also be used with PPML.
  2 - Does 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.
  3 - Does 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. Notice also that under- and over-dispersion are only defined if the dependent variable is a count, which is not the case in most trade applications,
  4 - 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.
  5 - 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.
  6 - Why not use other count data models like the negative-binomial or zero inflated models?
   - As noted in 5 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!
  7 - Can Vuong's tests for non-nested hypotheses be used to choose between the PPML and the ZIP?
    - No! First of all, as noted in 6 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.
  8 - Can we use the AIC and BIC to choose between PPML and other methods or models?
    - No, the AIC and BIC are based on the likelihood function and are not valid if at least one of the models is estimated by pseudo maximum likelihood, as in the case of PPML.
  9 - Can PPML be used when there is over-dispersion in the data?
    - Yes! As noted in 3 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.