Identification and Testing Structural Changes at Unknown Locations with Wavelets

用小波识别和测试未知位置的结构变化我的研究计划的主要目标是设计强大的统计检验，用来在小样本中识别线性回归模型参数的结构突变(变化)。结构性突变是指基本统计过程的平均值、持久性和/或方差发生意外变化。这种突变的存在会导致预测错误和错误指定经济和金融活动的模型。结构中断可能会影响模型的部分或所有参数。例如，一些可能会导致某一日期之后序列相关性的改变，一些趋势倾向的变化，以及一些可能导致潜在波动性的变化。体制变革(改革)、政治动荡、新经济的出现、经济和金融灾难可能是经济和金融时间序列中出现这种结构性突变的原因。一些突变可能很容易从原始数据中识别出来，但由于摩擦的性质和经济的波动性，一些突变可能是潜在的。在回归框架中，突变说明了两个或更多变量之间关系性质的变化。小波变换是设计功能强大的结构变化测试的理想工具集，因为它在本地时间窗口运行，并且具有强大的时间/频率局部化特性。我最初的小波框架的机制可以用以下方式来描述：小波分解产生局部加权差(小波系数)，其数量等于数据点的数量。我对每个小波系数进行平方，得到其大小，并计算平方系数的样本平均值。当没有结构变化时，我期望这个样本平均值等于零值下过程的方差的一半加上协方差的一半。然而，当存在一个或多个结构突变时，平方小波系数的样本平均应该显著偏离零平均。我对平方小波系数的样本平均值进行居中和标准化处理，从而在不改变结构的情况下得到零分布。结构性中断可以是永久性的，也可以是暂时性的。如果结构性突变是永久性的，那么过程的平均值、持续性或方差就会有无限期的永久性变化。在临时中断中，均值、持久性或方差从其空值转移，但最终恢复为其空值。无论这种中断是暂时的还是永久性的，它们都可能突然发生，也可能逐渐发生。为了抓住这种可能性，我将考虑到突然和渐进的平稳结构性中断。我关注平稳的多重结构性突破有两个原因。首先，大多数经济和金融数据在一个时间窗口内呈现出渐进的结构性变化，而最突然的结构性变化是例外，而不是规则。其次，我的框架包罗万象，并允许突然的变化。我的种子蒙特卡罗模拟表明，相对于其名义尺寸，存在最小的经验尺寸扭曲，并且与现有的结构断裂测试相比，功率显著提高。确定经济活动中的结构性变化是避免错误政策和设计适当补救措施的先兆。对这种变化的诊断需要检验的经验规模接近其名义规模和实质力量，否则我们可能无法正确和及时地识别潜在关系。我的研究计划的目标是通过提供强大的测试来克服这些限制。

Identification and Testing Structural Changes at Unknown Locations with Wavelets The primary goal of my research program is to design powerful statistical tests that can be used to identify structural breaks (changes) in the parameters of a linear regression model in small samples. A structural break is an unexpected shift in the mean, persistence and/or in the variance of the underlying statistical process. The presence of such breaks leads to prediction errors and mis-specified models of economic and financial activity. A structural break may affect some or all parameters of a model. Some, for instance, may lead into an altered serial correlation after a certain date, some change in the propensity of a trend and some may lead into a change in the underlying volatility. Institutional changes (reforms), political turmoil, emergence of new economies, economic and financial disasters may account for the presence of such structural breaks in the economic and financial time series. Some breaks may be readily identifiable from raw data, but some may be latent because of the nature of the frictions and volatile nature of the economy. In a regression framework, the breaks account for the change in the nature of the relationship between two or more variables altogether. The wavelet transformation is an ideal toolset for the design of powerful structural change tests because it operates in local time windows and has powerful time/frequency localization features. The mechanism of my original wavelet framework can be described in the following way: The wavelet decomposition yields localized weighted differences (wavelet coefficients) that are equal in number to the number of data points. I square each wavelet coefficient to obtain its magnitude and calculate the sample average of the squared coefficients. I expect this sample average to be equal to one-half of the variance plus one-half of the covariance of the process under the null when there is no structural change. However, the sample average of the squared wavelet coefficients should be significantly deviate from the null average when there are one or more structural breaks. I center and standardize the sample average of the squared wavelet coefficients to obtain the null distribution with no structural change. Structural breaks can be permanent or temporary in nature. If a structural break is permanent, then there is a permanent change of indefinite duration in the mean, persistence or variance of the process. In a temporary break, the mean, persistence or the variance shifts from its null value but eventually reverts to its null value. Whether such breaks are temporary or permanent in nature, they may occur abruptly or gradually. To capture such possibilities, I will allow for abrupt as well as gradual smooth structural breaks. I focus on smooth multiple structural breaks for two reasons. First, most economic and financial data exhibit gradual structural changes in a time window, and the most abrupt structural changes are exceptions rather than the rule. Second, my framework is all-encompassing and allows for abrupt changes. My seed Monte Carlo simulations indicate the presence of minimal empirical size distortions relative to their nominal sizes and significant power improvements compared to the existing structural break tests. Identification of structural changes in an economic activity is a pre-cursor to avoid erroneous policies and to design proper remedies. The diagnosis of such changes requires tests with empirical size close to its nominal counterpart and substantive power, otherwise we may not be able to identify the underlying relationship correctly and in a timely manner. The goal of my research program is to overcome such limitations by providing powerful tests with m