Towards Optimal Moment Estimation in Streaming and Distributed Models

One of the oldest problems in the data stream model is to approximate the p-th moment ||X||_p^p = sum_{i=1}^n X_i^p of an underlying non-negative vector X in R^n, which is presented as a sequence of poly(n) updates to its coordinates. Of particular interest is when p in (0,2]. Although a tight space bound of Theta(epsilon^-2 log n) bits is known for this problem when both positive and negative updates are allowed, surprisingly there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is O(epsilon^-2 log n) bits, while the lower bound is only Omega(epsilon^-2 + log n) bits. Recently, an upper bound of O~(epsilon^-2 + log n) bits was obtained under the assumption that the updates arrive in a random order. We show that for p in (0, 1], the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of O~(epsilon^-2 + log n) bits for estimating |X |_p^p. Our techniques also give new upper bounds for estimating the empirical entropy in a stream. On the other hand, we show that for p in (1,2], in the natural coordinator and blackboard distributed communication topologies, there is an O~(epsilon^-2) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies G, obtaining an O~(epsilon^2 log d) max-communication upper bound, where d is the diameter of G. Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an Omega(epsilon^-2 log n) bit lower bound for p in (1,2] for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter

International Business Cycles with Mutliple Input Investment Technologies

Backus, Kehoe, and Kydland (International Real Business Cycles, JPE, 100 (4), 1992) documented several discrepancies between the observed post-war business cycles of developed countries and the predictions of a two-country, complete-market model. The main discrepancy dubbed as the quantity anomaly, that cross-country consumption correlations are higher than that of output in the model as opposed to the data, has remained a central puzzle in international economics. The main thesis of this paper is that when the standard two-country model with traded and non-traded goods and complete ¯nancial markets, as in Stockman and Tesar (Tastes and Technology in a Two Country Model of the Business Cycles: Explaining International Comovements, 85 (1), AER, 1995) is extended to include capital goods sectors that utilize both traded and non-traded goods as intermediates, and when the non-traded aggregate is reclassi¯ed to include distribution and transportation services, the model produces the correct ordering of the cross-country correlations of consumption and output.International business cycles; Quantity anomaly; Distribution costs; Cross-country correlations.

Distribution Costs and International Business Cycles

Backus, Kehoe, and Kydland (International Real Business Cycles, JPE, 100(4),1992) documented several discrepancies between the observed post-war business cycles of developed countries and the predictions of a two-country, complete-market model. The main discrepancy termed as the “quantity anomaly†that cross-country consumption correlations are higher than that of output in the model as opposed to data, has remained a central puzzle in international economics. In order to resolve this puzzle mainly two strategies: restrictions on asset trade, and introducing non-traded goods in the model, have been employed by researchers. While these extensions have been successful in closing the gap to some extent, the ordering of correlations has stayed unchanged: consumption correlations still exceed that of output. This paper attempts to resolve the quantity puzzle by introducing non-traded distribution costs in the retailing of traded goods. In a standard two-good model traded output and traded consumption, by definition, are identical goods. With distribution costs, traded output and consumption are two distinct entities as each unit of final traded consumption good incorporates a unit of traded good and a fixed amount of non-traded goods. Thus, effectively, the model with distribution costs can be viewed as a model without distribution costs but with a modified utility function that has a substantially stronger complementarity between traded and non-traded goods. In a simple two-good extension of the Backus, Kehoe, and Kydland model, it is shown that the cross-country consumption and output correlations are 0.55 and 0.30, respectively, whereas with distribution costs consumption correlation reduces to 0.09, output correlation to 0.23. Incorporating distribution costs, in addition, improves the model’s performance in matching the volatility of real exchange rates and the correlation of net exports with output. These improvements are achieved without sacrificing the model's performance in any other dimension.open economy business cycles; quantity puzzle; distribution costs