Identification and Testing Structural Changes at Unknown Locations with Wavelets

我的研究计划的主要目标是设计强大的统计测试，可用于识别小样本线性回归模型参数中的结构断裂（变化）。结构性断裂是指基础统计过程的平均值、持续性和/或方差的意外变化。这种断裂的存在导致了预测错误和经济和金融活动的错误模型。结构断裂可能影响模型的部分或全部参数。例如，有些可能会导致某一日期后序列相关性的改变，有些可能会改变趋势的倾向，有些可能会导致潜在波动性的变化。制度变化（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断裂的原因。从原始数据中可以很容易地识别出一些中断，但由于摩擦的性质和经济的波动性，有些中断可能是潜在的。在回归框架中，断点解释了两个或多个变量之间关系本质的变化。小波变换是设计强大的结构变化测试的理想工具集，因为它在局部时间窗口中工作，并且具有强大的时频局部化特征。我的原始小波框架的机制可以用以下方式描述：小波分解产生与数据点数量相等的局部加权差（小波系数）。将每个小波系数平方得到其大小，并计算平方系数的样本平均值。我期望这个样本平均值等于1 / 2的方差加上1 / 2的协方差在零下，当没有结构变化时。然而，当存在一个或多个结构断裂时，小波系数平方的样本平均值应该明显偏离零平均值。对小波系数平方的样本平均值进行中心化和标准化，得到无结构变化的零分布。结构性断裂可以是永久性的，也可以是暂时性的。如果结构性断裂是永久性的，那么在过程的平均值、持久性或方差中就会有一个无限期持续的永久性变化。在暂时中断中，平均值、持久性或方差从其空值移动，但最终恢复到空值。无论这种中断是暂时的还是永久性的，它们可能突然发生，也可能逐渐发生。为了捕捉这种可能性，我将考虑突然和逐渐平滑的结构断裂。我关注光滑的多重结构断裂有两个原因。首先，大多数经济和金融数据在一个时间窗口内表现出渐进的结构性变化，而最突然的结构性变化是例外，而不是规律。其次，我的框架包罗万象，允许突然的变化。我的种子蒙特卡罗模拟表明，相对于它们的标称尺寸，存在最小的经验尺寸扭曲，与现有的结构断裂测试相比，存在显著的功率改进。识别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前兆。对这种变化的诊断需要具有接近其名义对应和实质性力量的经验规模的检验，否则我们可能无法正确和及时地确定基本关系。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