[Gretl-users] Re: resampling with or without replacement

Sure without replacement is not for bootstrap purposes. It was necessary for me when I tried to write estimation procedures for generalized dynamic factor models: Forni, M., M. Hallin, M. Lippi and P. Zaffaroni (2015) 'Dynamic factor models with infinite-dimensional factor spaces: One-sided representations', Journal of Econometrics 185(2): 359-371. Forni, M., M. Hallin, M. Lippi and P. Zaffaroni (2017) 'Dynamic factor models with infinite-dimensional factor space: Asymptotic analysis', Journal of Econometrics 199(1): 74-92. For a number of quantities (parameters, impulse response function and other), their estimators (in the way of estimating the common components and factors) strongly depend on the ordering of the cross-section. We are not interested in such matrices per se but only insofar as they enter the impulse-response functions and their estimators. The authors argue that although the population impulse-response functions are permutation-equivariant, their estimators are not. Simulations by Forni et al. (2017) provide convincing evidence that, selecting a small number of permutations at random and averaging the corresponding estimators of the impulse-response functions leads to rapidly stabilizing results and a substantial reduction of the expected Mean Square Estimation Error (MSE). More simply, say I have 100 series, in any particular uninformative order (no identification of any kind). I estimate some quantities, then I change (shuffle) the variables and I re-estimate the quantities,...do it 100 times and average the estimates to get the necessary results. Yiannis