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Econometrics-I-2 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 2. Regression and Projection
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Statistical Relationship Objective :  Characterize the stochastic  relationship between a variable and a set of  'related' variables  Context:   An inverse demand equation,  P =   α   +   β Q  +   γ Y, Y = income.  Q and P are two  obviously related random variables.  We are  interested in studying the relationship between P  and Q. By ‘relationship’ we mean (usually) covariation.   (Cause and effect is problematic.)
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Bivariate Distribution - Model for a Relationship Between Two Variables We might posit a bivariate distribution for Q and P,  f(Q,P)  How does variation in P arise?  With variation in Q, and  Random variation in its distribution.  There exists a conditional distribution f(P|Q) and a  conditional mean function, E[P|Q].  Variation in  P  arises  because of  Variation in the mean,  Variation around the mean,  (possibly) variation in a covariate, Y.
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Implications Regression  is the conditional mean There is always a conditional mean It may not equal the structure implied by a theory What is the implication for least squares estimation? LS always estimates regressions LS does not necessarily estimate structures Structures may not be estimable – they may not be  identified .
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Conditional Moments The conditional mean function is the   regression function . P  =  E[P|Q]  +  (P - E[P|Q])  =   E [P|Q] +  ε E[ ε |Q] = 0 = E[ ε ].  Proof:  (The Law of iterated expectations) Variance of the conditional random variable = conditional variance,  or the   scedastic function . A “trivial relationship” may be written as P = h(Q) +  ε , where the  random variable  ε =P-h(Q) has zero mean by construction.  Looks  like a regression “model” of sorts.  An extension:  Can we carry  Y  as a parameter in the bivariate  distribution?  Examine  E [P|Q,Y]
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Sample Data (Experiment)
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50 Observations on P and Q Showing Variation of P Around E[P]
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Variation Around E[P|Q] (Conditioning Reduces Variation)
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Means of P for Given Group Means of Q
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Another Conditioning Variable E[P|Q,Y=1] E[P|Q,Y=2]
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Conditional Mean Functions No requirement that they be "linear" (we will  discuss what we mean bylinear) No restrictions on conditional variances
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Projections and Regressions We explore the difference between the linear projection  and the conditional mean function y  =   α   +   β x  +   ε   where   ε    x,  E( ε |x)  =  0              Cov(x,y)  =  Cov(x, α )  +   β Cov(x,x)  +  Cov(x, ε )   
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