Recent work on econometric detection mechanisms has shown the effectiveness of recursive procedures in identifying and dating financial bubbles in real time. These procedures are useful as warning alerts in surveillance strategies conducted by central banks and fiscal regulators with real‐time data. Use of these methods over long historical periods presents a more serious econometric challenge due to the complexity of the nonlinear structure and break mechanisms that are inherent in multiple‐bubble phenomena within the same sample period. To meet this challenge, this article develops a new recursive flexible window method that is better suited for practical implementation with long historical time series. The method is a generalized version of the sup augmented Dickey–Fuller (ADF) test of Phillips et al. (“Explosive behavior in the 1990s NASDAQ: When did exuberance escalate asset values?” International Economic Review 52 (2011), 201–26; PWY) and delivers a consistent real‐time date‐stamping strategy for the origination and termination of multiple bubbles. Simulations show that the test significantly improves discriminatory power and leads to distinct power gains when multiple bubbles occur. An empirical application of the methodology is conducted on S&P 500 stock market data over a long historical period from January 1871 to December 2010. The new approach successfully identifies the well‐known historical episodes of exuberance and collapses over this period, whereas the strategy of PWY and a related cumulative sum (CUSUM) dating procedure locate far fewer episodes in the same sample range.

Causal relationships in econometrics are typically based on the concept of predictability and are established by testing Granger causality. Such relationships are susceptible to change, especially during times of financial turbulence, making the real-time detection of instability an important practical issue. This article develops a test for detecting changes in causal relationships based on a recursive evolving window, which is analogous to a procedure used in recent work on financial bubble detection. The limiting distribution of the test takes a simple form under the null hypothesis and is easy to implement in conditions of homoskedasticity and conditional heteroskedasticity of an unknown form. Bootstrap methods are used to control family-wise size in implementation. Simulation experiments compare the efficacy of the proposed test with two other commonly used tests, the forward recursive and the rolling window tests. The results indicate that the recursive evolving approach offers the best finite sample performance, followed by the rolling window algorithm. The testing strategies are illustrated in an empirical application that explores the causal relationship between the slope of the yield curve and real economic activity in the United States over the period 1980-2015. 6 Recall that the duration of the causality episode is 0.2. When the minimum window f 0 exceeds that duration, the regression contains a mix of causal and non-causal observations. 7 They are obtained by matching the SV model to real exchange rate data (Shephard, 1996).

This article provides the limit theory of real‐time dating algorithms for bubble detection that were suggested in Phillips, Wu, and Yu (PWY; International Economic Review 52 [2011], 201–26) and in a companion paper by the present authors (Phillips, Shi, and Yu, 2015; PSY; International Economic Review 56 [2015a], 1099–1134. Bubbles are modeled using mildly explosive bubble episodes that are embedded within longer periods where the data evolve as a stochastic trend, thereby capturing normal market behavior as well as exuberance and collapse. Both the PWY and PSY estimates rely on recursive right‐tailed unit root tests (each with a different recursive algorithm) that may be used in real time to locate the origination and collapse dates of bubbles. Under certain explicit conditions, the moving window detector of PSY is shown to be a consistent dating algorithm even in the presence of multiple bubbles. The other algorithms are consistent detectors for bubbles early in the sample and, under stronger conditions, for subsequent bubbles in some cases. These asymptotic results and accompanying simulations guide the practical implementation of the procedures. They indicate that the PSY moving window detector is more reliable than the PWY strategy, sequential application of the PWY procedure, and the CUSUM procedure.