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Statistics review for finance students

Jamal Munshi, Sonoma State Univesity, 1992
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LOGICAL POSITIVISM
Reality is determinate and exists independent of us or our experience and this reality is logical and rational and is therefore subject to logical (Aristotle) and mathematical (Descartes) analyysis.

EMPIRICISM
Follows from logical positivism. If reality exists independent of us and is logical, then we may subject theories to logical analysis and use our observations to test the testable implications of theory (Ho: the theory is correct) and continue to accept the theory on the basis of its logical correctness until we find evidence that is inconsistent with the theory (reject Ho).

THE SCIENTIFIC METHOD
A widely accepted method of applying empiricism. It consists of the following components.

  • Observations (data) = instance of sensation and realization.
  • Law = logical generalization of patterns of observations which may be used to predict the outcome of unobserved events.
  • Theory = Logically ordered sets of laws that postulate causality. Leads to understanding.
  • Knowledge = An internally logically consistent set of theories that explain the universe (reality). It is the realization of truth. A logical apparatus of knowing and predicting "consistent" with the existing base of knowledge.

Theories are statements of truth based on logical and mathematical analysis. Whether they really are true may be tested using empirical methods. First a "testable implication" of the theory is mathematically derived. If this is not possible then the theory is not testable. For a theory to be empirically testable there must exist some observable 'sensations' that are inconsistent with the theory. Once such a testable implication is identified then we set up our Ho and Ha as: Ho: the theory is correct Vs. Ha: it is not. Then we take data (i.e. observe or experience sensations). The data must be independent of the theory for the p-value computations to make sense. We use the data to test the hypothesss at some alpha level. If we reject Ho then we have found evidence that the theory is false.

EXAMPLE
Early work by Batchelior (Paris) in the 19th century and again by Osborne (Missouri) in the 1950s' led to a Law that movement of stock prices is random.

The theory implies that any information that is publicly available may not be used to make abnormal profits in the stock market. In 1968 the Laws of stock price movements were formalized into a logical set of arguments called the Theory of Efficient Markets by Eugene Fama (Chicago). The theory has withstood many tests and after decades of testing and failure to reject, it has gained widespread acceptance. Its principal progenitors were recently awarded the Nobel Prize in Economics.

INFERENTIAL STATISTICS
Classical (or Neyman-Pearson) inferential statistical techniques are the tools the scientist uses to generalize observations into truth only after a Theory has been constructed and logically analyzed.

REALISM
Realists claim that reject Ho does not imply causality They point out that what is implied to be the cause in the theory MIGHT be the cause but nothing in the empiricists methods is able to determine that it IS the cause. Without knowing causality there can be no knowledge. Realism rejects the notion that all of reality is determinate and subject to human logic and observation. It accepts the empiricists Laws as generalizations of patterns of sensation but does not accept prima facia the Theories about what is really going on out there. Realists also reject the notion that reality is independent of us. In controlled experiments designed to establish causality, the realists argue, the experimenter's meddling changes reality.