Tests of market efficiency are two tests. One, a test of efficiency, and two, a test of the model used. This is leads to the great divide in finance. There is one school, the behavioralist view, which states that a failure of a model is based on irrationality. There are mistakes caused by a deviation from rationality, behavioral biases. The other school states that markets are rational, rather it is the model that is the problem. If we have a model failure it is because we have not modeled risk properly. Any anomaly is based on our inability to correctly measure risk. Hence, there are two camps or schools of thought, the risk pricing camp and the behavioral camp.
Many of the factors that are studied in finance cane be structured around these two camps. The value factor can be thought of as price for value risk or it can be thought of as a behavioral issue. The same can be said for something like the momentum factor. The challenge for the efficient markets hypothesis is whether there is a good risk story for market anomalies or whether it is necessary to fall back on a behavioral story.
Can markets be irrational or inefficient? Yes, it is possible, but it is rarer than often thought and the first view should be that markets are rational but prices are not modeled correctly. Anomalies exist relative to our core theory and modeling. Markets are reasonably efficient, but this is not the same as perfect efficient. Market usually use all information, but the pricing of this information may not be properly models which offers opportunities for investors. Efficiency is a theory which must be tested constantly. The theory is based on the assumption that markets are competitive, and it is hard to make money in a competitive marketplace.
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