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This study investigated the efficiency of the apartment market in the Housing Speculative Zones by testing whether the market follows stochastic processes like white noise, martingale, or random walk. Market efficiency is commonly tested with the martingale process, but this study employed more test methods to obtain robust results. For the study data, the KB monthly apartment price indices from April 2013 to July 2019 and the Korea Appraisal Board transaction-based quarterly apartment price indices from the first quarter of 2006 to the third quarter of 2019 were used. The Ljung-Box (LB) test, generalized spectral test, and variance ratio test were adopted for analyzing the white noise, the martingale differential sequence, and the random walk process, respectively. The results show that the apartment market is generally inefficient except for the four districts in Seoul (Gangnam, Seocho, Songpa, and Gangdong), where most people are very attentive to the movements of the market. This study is significant in that it revealed that some submarkets in the overall inefficient housing market can become efficient.