Empirical macroeconomists who analyze typical time series, such as GDP, interest rates, stock returns have to worry about the structural change. Possibilities of structural change over time and the properties of structural change parameters are the focus of my dissertation. It includes the choice of break point estimators between using level shift model and trend shift model, break tests robust to I(0)/I(1) errors, and the estimation of break numbers.I. Break Point Estimates for a Shift in Trend: Levels versus First DifferencesIn the first chapter I analyze the estimation of an unknown break point in a univariate trend shift model under I(1) errors by minimizing the sum of squared residuals. Two break point estimators are considered, one is from the original trend shift model and the other is from its first difference, a mean shift model with I(0) errors. Simulations show a discrepancy between existing asymptotic theories and finite sample distributions of the break point estimators. To achieve a closer approximation, I derive an asymptotic theory for the break point estimators assuming the break magnitude is within a T^{-1/2} neighborhood of zero.The break point estimator from the trend break model converges to its true value at rate T^{1/2} under I(1) errors, while the break point estimator of the first difference model converges at rate T under I(0) errors. Given this fact, many researchers would think they should use the estimator that converges to the faster rate. However I show that when the break magnitude is small relative to the noise magnitude, the break point estimator from the trend shift model may have thinner tails and concentrates more on the true break point than that from the first difference transformation. That indicates a preference of the break point estimator from the level model.II. Fixed-b Analysis of LM Type Tests for a Shift in MeanWe analyze lagrange multiplier (LM) tests for a shift in mean of a univariate time series at an unknown date. We consider a class of LM statistics based on nonparametric kernel estimates of the long run variance and we develop a fixed-b asymptotic theory for the statistics. We provide results for the case of I(0) and I(1) errors and use the fixed-b theory to explain finite sample null rejection probabilities and finite sample power of the LM tests.We show that the choice of bandwidth has a large impact on the size and power of the tests. In particular we find that larger bandwidths lead to non-monotonic power whereas smaller bandwidths give tests with monotonic power. The fixed-b theory suggests that, for a given statistic, kernel and significance level, there exists a ``robust" and width such that the fixed-b asymptotic critical value is the same for both I(0) and I(1) errors. In the case of the supremum statistic, the robust bandwidth LM test has good power that is monotonic whereas the power of the mean statistic is non-monotonic.III. Consistency of Break Point Estimator under Misspecification of Break NumberIn this chapter, I discuss the inconsistency of sequential trend break point estimators in the presence of underspecification of the number of breaks. The analysis of models with level shifts has been documented by researchers under comprehensive settings such as allowing a time trend in the model. Despite the consistency of break point estimators of level shifts, there are few papers on the consistency of trend shift point estimators under misspecification. My simulation study and asymptotic analysis show that the trend break point estimators do not converge to the true breaks points under most conditions when the number of estimated breaks is smaller than the true number of breaks. This inconsistency leads to a potential power loss for testing for multiple trend breaks. Taking first difference is proposed to deal with this problem under certain circumstances.
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