2_Probability and Statistics

Data types o cross sectional o time series o panel o

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Unformatted text preview: tabases? Data types: o Cross-sectional o Time series o Panel o Financial data (often a combination of the above) VER. 9/11/2012. © P. KOLM 8 At the Heart: Data & Models (2/3) Models – “All models are wrong, but some are useful” (Box (1976)) Model are small imitations of the real thing Role of assumptions: the most important ones are not in the equations1 Suitability, robustness and sensitivity Almost nothing is exogenous: beware of feedback effects Avoid confirmation bias2 (selectively presenting data favorable to us) How do we incorporate “soft data”? Parameterizing the models o Nonparametric methods o Semiparametric methods o Parametric methods VER. 9/11/2012. © P. KOLM 9 At the Heart: Data & Models (3/3) Learn to challenge data and models From where is the data? Is there more? What are the assumptions? Are they really valid? Why? What is the objective behind the method we use? Why? o Do we achieve the objective? Well-defined process o Do we have control of and understand the impact of every step? Why? Are we missing something – anything? If we had faster computers what would we do? Continuously re-examine and reevaluate models: Do they and will they continue to perform? Why? If you run out of questions, just continue asking “Why? Why? Why?” VER. 9/11/2012. © P. KOLM 10 Basic Probability VER. 9/11/2012. © P. KOLM 11 Random Variables X is a random variable if it represents a random draw from some population3 A discrete random variable can take on only selected values A continuous random variable can take on any value in an interval on the real line Associated with each random variable is a probability distribution VER. 9/11/2012. © P. KOLM 12 Random Variables - Examples The outcome of a P (Heads) = P (Tails)= coin toss – a discrete random variable with 1 2 The stock return in the Black-Merton-Scholes option pricing model – a continuous random variable drawn from a lognormal distribution VER. 9/11/2012. © P. KOLM 13 Expected Value of X – E(X) The expected value (or the expectation) is a probability weighted average E (X ) is the mean of the distribution of X, denoted by mX Let f (x i ) be the probability that X=xi, then n mX = E (X ) = å x i f (x i ) i =1 VER. 9/11/2012. © P. KOLM 14 Variance of X – Var(X) The variance of X is a measure of the dispersion of the distribution 2 Var(X) (also denoted by sX ), is the expected value of the squared deviations from the mean, so 2 sX º Var (X ) º E éê(X - mX )2 ùú ë û VER. 9/11/2012. © P. KOLM 15 More on Variance The square root of Var (X ) is the standard deviation of X, denoted by Std(X ) 2 or sX Var (X ) can alternatively be written in terms of a weighted sum of squared deviations n E éê(X - mX )2 ùú = å (x i - mX )2 f (x i ) ë û i =1 VER. 9/11/2012. © P. KOLM 16 Covariance – Cov(X,Y) Covariance between two random variables, X and Y, is a measure of how much they are expected to change together o If positive, then both move up or down together o If negative, then if X is high, Y is low, and vice versa Cov(X ,Y ) (also denoted by sXY ), is the expected value of the product of the mean deviations sXY º Cov(X ,Y )...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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