Volume 12, Issue 1

We provide methods for the empirical analysis of a class of two-player repeated games with i.i.d. shocks, allowing for non-Markovian strategies. The number of possible equilibria in these games is large and, usually, theory is silent about which equilibrium will be chosen in practice. Thus, our method remains agnostic about selection among these multiple equilibria, which leads to partial identification of the parameters of the game. We propose a profiled likelihood criterion for building confidence sets for the structural parameters of the game and derive an easily computable upper bound on the critical value. We demonstrate good finite-sample performance of our procedure using a simulation study. We illustrate the usefulness of our method by studying the effect of repealing the Wright Amendment on entry and exit into Dallas airline markets and find that the static game approach overestimates the negative effect of the law on entry into these markets.

We examine the conditional logit estimator for binary panel data models with unobserved heterogeneity. A key assumption used to derive the conditional logit estimator is conditional serial independence (CI), which is problematic when the underlying innovations are serially correlated. A Monte Carlo experiment suggests that the conditional logit estimator is not robust to violation of the CI assumption. We find that higher persistence and smaller time dimension both increase the magnitude of the bias in slope parameter estimates. We also compare conditional logit to unconditional logit, bias corrected unconditional logit, and pooled correlated random effects logit.

We consider a multiplicative decomposition of the financial returns to improve the density forecasts of financial returns. The multiplicative decomposition is based on the identity that financial return is the product of its absolute value and its sign. Advantages of modeling the two components are discussed. To reduce the effect of the estimation error due to the multiplicative decomposition in estimation of the density forecast model, we impose a moment constraint that the conditional mean forecast is set to match with the sample mean. Imposing such a moment constraint operates a shrinkage and tilts the density forecast of the decomposition model to produce the improved maximum entropy density forecast. An empirical application to forecasting density of the daily stock returns demonstrates the benefits of using the decomposition and imposing the moment constraint to obtain the improved density forecast. We evaluate the density forecast by comparing the logarithmic score (LS), the quantile score (QS), and the continuous ranked probability score (CRPS). We contribute to the literature on the density forecast and the decomposition models by showing that the density forecast of the decomposition model can be improved by imposing a sensible constraint in the maximum entropy framework.

This paper provides a user’s guide to the general theory of approximate randomization tests developed in Canay, Romano, and Shaikh (2017a. “Randomization Tests under an Approximate Symmetry Assumption.” Econometrica 85 (3): 1013–30) when specialized to linear regressions with clustered data. An important feature of the methodology is that it applies to settings in which the number of clusters is small – even as small as five. We provide a step-by-step algorithmic description of how to implement the test and construct confidence intervals for the parameter of interest. In doing so, we additionally present three novel results concerning the methodology: we show that the method admits an equivalent implementation based on weighted scores; we show the test and confidence intervals are invariant to whether the test statistic is studentized or not; and we prove convexity of the confidence intervals for scalar parameters. We also articulate the main requirements underlying the test, emphasizing in particular common pitfalls that researchers may encounter. Finally, we illustrate the use of the methodology with two applications that further illuminate these points: one to a linear regression with clustered data based on Meng, Qian, and Yared (2015. “The Institutional Causes of china’s Great Famine, 1959–1961.” The Review of Economic Studies 82 (4): 1568–611) and a second to a linear regression with temporally dependent data based on Munyo and Rossi (2015. “First-day Criminal Recidivism.” Journal of Public Economics 124: 81–90). The companion R and Stata packages facilitate the implementation of the methodology and the replication of the empirical exercises.

This paper investigates the identification and estimation of the quantile treatment effect in a difference in differences (DID) setting when treatment is provided only to qualified individuals at a certain point in time and the qualification is time-varying. The time-varying qualification may affect an outcome variable even when the actual effect of treatment is zero. We show how to account for this “movers effect” bias and propose the quantile treatment effect on “in-stayers” that are qualified both before and after the treatment. The estimate is identified under three main assumptions: (i) panel data availability, (ii) a distributional common trend assumption conditional on covariates, and (iii) a copula stability assumption. We then apply our method to estimate the effects of an increase in the benefits of the Supplemental Nutrition Assistance Program (SNAP) on recipients’ food expenditure shares. The results show significant heterogeneity and highlight the importance of accounting for time-varying qualification.

Virtually every seasonal adjustment software includes an ensemble of tests for assessing whether a given time series is in fact seasonal and hence a candidate for seasonal adjustment. However, such tests are certain to produce either agreeing or conflicting results, raising the questions how to identify the most accurate tests and how to aggregate the results in the latter case. We suggest a novel random forest-based approach to answer these questions. We simulate seasonal and non-seasonal ARIMA processes that are representative of the macroeconomic time series analysed regularly by the Bundesbank. Treating the time series’ seasonal status as a classification problem, we use the p -values of the seasonality tests implemented in the seasonal adjustment software JDemetra+ as predictors to train conditional random forests on the simulated data. We show that this aggregation approach avoids the size distortions of the JDemetra+ tests without sacrificing too much power compared to the most powerful test. We also find that the modified QS and Friedman tests are the most accurate ones in the considered ensemble.

We revisit the Helmert transformation, and provide a useful and simple derivation of the joint distribution of the sample mean and the sample variance in samples from independently and identically distributed normal random variables. Our derivation is distinguished by concreteness, very little abstractness, and should be appealing to beginning students of statistics, and to both beginning and advanced students of econometrics. We also highlight one fruitful application of the Helmert transformation in panel data econometrics. The Helmert transformation can be used to eliminate the fixed effects in the estimation of fixed effects models, and we briefly review this application of the transformation in the panel data context.

The standard approximation of a natural logarithm in statistical analysis interprets a linear change of p in ln( X ) as a (1 + p ) proportional change in X , which is only accurate for small values of p . I suggest base-(1 + p ) logarithms, where p is chosen ahead of time. A one-unit change in log 1 + p ( X ) is exactly equivalent to a (1 + p ) proportional change in X . This avoids an approximation applied too broadly, makes exact interpretation easier and less error-prone, improves approximation quality when approximations are used, makes the change of interest a one-log-unit change like other regression variables, and reduces error from the use of log(1 + X ).