1. WAR, POWER-LAWS & MLE Miles D. Townes The George Washington University mdtownes@gwu.edu Draft, links, and STATA do-files available at http://home.gwu.edu/~mdtownes/MLE Abstract: Richardson (1948) discovered a power-law relationship between the number of deaths in a “fatal quarrel” and the relative frequency of such quarrels. Of late this study has attracted renewed interest from IR scholars, but often the power-law relationship is estimated by OLS regression. In this study, I demonstrate why OLS is inappropriate, using MLE and simulation techniques to arrive at correct estimates of the power-law parameters. I also extend these techniques to interpretation of the power-law result, towards a better understanding of what Richardson's Law means for International Relations and international conflict.

2. Power Laws This can also be relationship can also be described as log p(x) = c – α log x So why not use OLS regression? Because it introduces bias: “First, the errors are hard to estimate because they are not well-described by the usual regression formulas, which are based on assumptions that do not apply in this case. Second, a fit to a power-law distribution can account for a large fraction of the variance even when the fitted data do not follow a power law... And third, the fits extracted by regression methods do not satisfy basic requirements on probability distributions, such as normalization, and hence cannot be correct.” (Clauset et al, 2007; 22) Power-laws are described by the distribution:

3. Adapting techniques suggested by Clauset et al (2007), I generated a simulated dataset of 1000 observations with a know α = 1.5. The differences are two-fold: first, OLS in fact estimates not α, but slope of the distribution function, -(α – 1). Even correcting for the unit difference, the OLS estimate is still biased. Though the difference appears small, nonetheless the 95% confidence interval for OLS (-1.4967, -1.4943) excludes the true value of α = 1.5, per the simulated data. This is a critical problem for correct interpretation of the estimation results. Simulation MLE and OLS estimates for simulated power-law data with α = 1.5

4. Richardson's Law

5. Correlates of war These graphs plot the year and magnitude of observed conflicts in three COW datasets for war, which I also combine into a single dataset: All Wars.

6. MLE results Comparison of ML estimations for three COW datasets plus combined dataset These results suggest that the datasets reflect three distinct processes, but there is nonetheless good reason to think that these differences reflect coding artifacts from COW and not systematic differences among the types of conflict.

7. Next Steps -Further verification of STATA syntax. -Calculation of p-value for each COW dataset. -Using power-law data to debunk “long cycles” of war. -Using α to test: Are there differences by continent/region in α ? Are there differences in α by time period? Are great-powers more violent than lesser powers? Are nuclear-power interactions less violent than those among non-nuclear powers? Do international organizations exerting a pacifying effect?

8. Testing the Power-Law Clauset et al suggest five steps to test whether a distribution fits a power law: 1. Determine the best fit of the power-law to the data, including both alpha and xmin 2. Calculate the KS statistic for goodness-of-fit for step 1 3. Simulate steps 1 and 2 for a large number of synthetic datasets with alpha and xmin the same as step 1. 4. Calculate the p-value as the fraction of KS statistics for the synthetic data whose value exceeds the KS statistic for the real data. 5. If the p-value is sufficiently small, the power-law distribution can be rejected. Using this test for the All Wars dataset, p =.83 I cannot reject the power-law distribution – in fact, it looks like a solid fit.