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Econometrics-I-2

# 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
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.
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,  but h(Q) is only E[P|Q] for one specific function. An extension:  Can we carry  Y  as a parameter in the  bivariate distribution?  Examine  E [P|Q,Y]
Sample Data (Experiment)

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50 Observations on P and Q Showing Variation of P Around E[P]
Variation Around E[P|Q] (Conditioning Reduces Variation)

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Means of P for Given Group Means of Q
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
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,

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

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