No abstract

SUMMARYIn the last two decades, fiscal sustainability has been tested through the use of non-stationary time series analysis. Two different approximations can be found in the literature: first, a univariate approach that has focused on the stochastic properties of the stock of debt and, second, a multivariate one that has focused on the long-run properties of the flows of expenditures and revenues, i.e., in the stochastic properties of the deficit. In this paper we unify these approaches considering the stock-flow system that fiscal variables configure. Our approach involves working in an I(2) stochastic processes framework. Given the possibility of the existence of regime shifts in the sustainability of US deficit that the literature has pointed out, we develop a new statistic that can be applied to test several types of I(2) cointegration and multicointegration relationships allowing for regime shifts. To test for these kinds of changing long-run relationships we propose the use of a residual-based Dickey-Fuller class of statistic that accounts for one structural break. We show that consistent estimates of the break fraction can be obtained through the minimization of the sum of squared residuals when there is I(2) cointegration. The finite sample performance of the proposed statistic is investigated by Monte Carlo simulations. The econometric methodology is applied to assess whether the US fiscal deficit and debt are sustainable.

An extended and improved theory is presented for marked and weighted empirical processes of residuals of time series regressions. The theory is motivated by 1step Huber-skip estimators, where a set of good observations are selected using an initial estimator and an updated estimator is found by applying least squares to the selected observations. In this case, the weights and marks represent powers of the regressors and the regression errors, respectively. The inclusion of marks is a non-trivial extention to previous theory and requires refined martingale arguments.

A uniform weak consistency theory is presented for the marked and weighted empirical distribution function of residuals. New and weaker su¢ cient conditions for uniform consistency are derived. The theory allows for a wide variety of regressors and error distributions. We apply the theory to 1-step Huber-skip estimators. These estimators describe the widespread practice of removing outlying observations from an intial estimation of the model of interest and updating the estimation in a second step by applying least squares to the selected observations. Two results are presented. First, we give new and weaker conditions for consistency of the estimators. Second, we analyze the gauge, which is the rate of false detection of outliers, and which can be used to decide the cut-o¤ in the rule for selecting outliers.

We show that the cumulated sum of squares statistic has a standard Brownian bridge–type asymptotic distribution in nonlinear regression models with (possibly) nonstationary regressors. This contrasts with cumulated sum statistics which have been previously studied and whose asymptotic distribution has been shown to depend on the functional form and the stochastic properties, such as persistence and stationarity, of the regressors. A recursive version of the test is also considered. A local power analysis is provided, and through simulations, we show that the test has good size and power properties across a variety of situations.