Band spectral regression with both deterministic and stochastic trends is considered. It is shown that trend removal by regression in the time domain prior to band spectral regression can lead to biased and inconsistent estimates in models with frequency dependent coefficients. Both semiparametric and nonparametric regression formulations are considered, the latter including general systems of two-sided distributed lags such as those arising in lead and lag regressions. The bias problem arises through omitted variables and is avoided by careful specification of the regression equation. Trend removal in the frequency domain is shown to be a convenient option in practice. An asymptotic theory is developed and the two cases of stationary data and cointegrated nonstationary data are compared. In the latter case, a levels and differences regression formulation is shown to be useful in estimating the frequency response function at nonzero as well as zero frequencies.

In Corbae, Temzelides, and Wright (2001) (hereafter, CTW) we proposed a new version of the framework that uses bilateral matching to model the exchange process, and in particular to model the use of money as a medium of exchange. Our version does not have agents meeting exogenously and at random, but rather has agents meeting endogenously. That is, agents are matched at each date subject to a stability condition that requires, roughly, that no agents prefer to be paired with each other or to be unmatched, rather than to be paired with the partners they get along the equilibrium path. While similar in spirit to the cooperative matching concept introduced by David Gale and Lloyd Shapley (1962), we had to generalize their framework to dynamic models because we are interested in monetary economics. Here we present a version of the solution concept in CTW, specialized in some ways but also generalized to include extrinsic uncertainty (sunspots). We then discuss some applications of endogenous matching models to issues that have previously been addressed using random matching, including the existence of sunspot equilibria and the efficiency of inside versus outside money. One of our main goals is to show how endogenous matching is a useful alternative to random matching. This may be interesting to those who think that bilateral trade is a reasonable friction upon which to build a theoretical foundation for monetary economics but perhaps think that random matching is an extreme and unrealistic simplification. Another goal is to provide examples where it makes a difference for substantive results how we model the matching process, and also examples where it does not. I. Endogenous Matching