Since oil prices are typically governed by nonlinear and chaotic behavior, it's become rather difficult to capture the dominant properties of their fluctuations. In recent years, unprecedented interest emerged on the decomposition methods in order to capture drifts or spikes relatively to this data. Together, our understanding of the nature of crude oil price shocks and their effects on the stock market returns has evolved noticeably. We accommodate these findings to investigate two issues that have been at the center of recent debates on the effect of crude oil shocks on the stock market returns of five developed countries (USA, UK, Japan, Germany and Canada). First, we analyze whether shocks and or volatility emanating from two major crude oil markets are transmitted to the equity markets. We do this by applying, the Haar A Trous Wavelet decomposition to monthly real crude oil series in a first step, and the trivariate BEKK Markov Switching GARCH model to analyze the effect of the smooth part on the degree of the stock market instability in a second step. The motivation behind the use of the former method is that noises and erratic behavior often appeared at the edge of the signal, can affect the quality of the shock and thus increase erroneous results of the shock transmission to the stock market. The proposed model is able to circumvent the path dependency problem that can influence the prediction's robustness and can provide useful information for investors and government agencies that have largely based their views on the notion that crude oil markets affect negatively stock market returns. Second, under the hypothesis of common increased volatility, we investigate whether these states happen around the identified international crises. Indeed, the results show that the A Haar Trous Wavelet decomposition method appears to be an important step toward improving accuracy of the smooth signal in detecting key real crude oil volatility features. Additionally, apart from UK and Japanese cases, the responses of the stock market to an oil shock depend on the geographic area for the main source of supply whether from the North Sea or from the North America (as we take two oil benchmarks WTI and Brent respectively). (C) 2011 Elsevier Ltd. All rights reserved.