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I'm going to test for the effect of the change in market efficiency on the stock market portfolio, and, I want to know what are the main measures known in the academic literature in order to compare them and to choose the "best" one for a given market.

Till now, I found the following measures to test for market efficiency in the broad market-specific case:

  • Variance Ratio (Lo & MacKinlay, 1988)
  • Approximate Entropy (Pincus, 1991)
  • The Hurst exponent (Peters, 1994)
  • Mkt Delay (Pagano & Schwartz, 2002)

For the firm-specific case, I found that the event-study procedure is the most common to test the market efficiency.

Could you suggest other measures or other procedures to test for market efficiency in the stock market, in addition to the ones I cited above, or, alternatively, suggest which is the best one by providing a reference?

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There are different methodologies to detect a change in the market efficiency, both in the market and firm-specific cases.

In the FIRM-SPECIFIC case, the most common procedure is the event study methodology; you can find how to construct an event-study case explained in Kothari & Warner (2006), who collected all the event study methodology implemented till 2006 in the academic literature; below, you can find the paper reference and relative link:

Kothari, S. P., and Jerold B. Warner. "The econometrics of event studies." > Available at SSRN 608601 (2004).

In the BROAD MARKET case, there exist different measures/proxy for the market efficiency exploited in the academic literature to test and measure the effects of the change in market efficiency on the markets.

Below, you can find a list of the proxies developed in the academic literature to test/measure the level of market efficiency and the relative references in chronological order.

As regards the review about market efficiency test and measuring:

Bollerslev, Tim, and Robert J. Hodrick. Financial market efficiency tests. No. w4108. National bureau of economic research, 1992.

Lim, Kian‐Ping, and Robert Brooks. "The evolution of stock market efficiency over time: a survey of the empirical literature." Journal of Economic Surveys 25.1 (2011): 69-108.

As regards the market efficiency test:

  • ARMA() Model test

Amihud, Yakov, and Haim Mendelson. "Trading mechanisms and stock returns: An empirical investigation." The Journal of Finance 42.3 (1987): 533-553.

  • Variance Ratio Test/MEC()

Lo, Andrew W., and A. Craig MacKinlay. "Stock market prices do not follow > random walks: Evidence from a simple specification test." Review of financial > studies 1.1 (1988): 41-66.

  • Hurst Exponent Over-Time

Peters, Edgar E. Fractal market analysis: applying chaos theory to investment and economics. Vol. 24. John Wiley & Sons, 1994. Cajueiro, Daniel O., and Benjamin M. Tabak. "The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient." Physica A: Statistical Mechanics and its Applications 336.3 (2004): 521-537.

As regards the market efficiency measures:

  • Approximate Entropy

Pincus, Steven M. "Approximate entropy as a measure of system complexity." Proceedings of the National Academy of Sciences 88.6 (1991): 2297-2301.

Look at here to know more about the statistical properties of the approximate entropy measure.

  • Efficiency Index

L. Kristoufek and M. Vosvrda. Measuring capital market efficiency: Global and local correlations structure. Physica A, 392:184–193, 2013.

  • Market Model $R^2$

Bramante, Riccardo, Diego Zappa, and Giovanni Petrella. "On the interpretation and estimation of the market model R-square." Electronic Journal of Applied Statistical Analysis 6.1 (2013): 57-66.

THIS ANSWER WILL BE UPDATED; ANY HINT, ADVISES OR IDEA WILL BE APPRECIATED

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