QE Regression

QE Regression

1. The CEF

   • Of Yi given Xi, is the expectation of Yi with Xi fixed
   • 'How the mean of Yi changes as you change Xi
   • Combinations of PDFs for different values of Xi
   • Yi = CEF + residual
   • PROOFS
     • Residual MEAN INDEPENDENT, and ORTHOGONAL to Xi
       • Bit 'explained by' Xi, bit that isn't
     • Decomposition of variance...
     • That the CEF is the BEST PREDICTOR ot Yi 'in the least squares sense
   • 1. Yi=CEF+ei 2.varYi=varCEF+varei 3,CEF LSBP of Yi
   • CEF residuals are necessarily MI and ORTH, but LRF ones are't!

2. Linear regression and CEF

   • We don't know b0 or b1
   • Values which min exp loss
   • PROOFS
     • Why B0, B1 min
     • B1 = Cov(Yi,Xi)/Var(Xi) B0 = E[Yi}-B1E[Xi}
     • 1, LRF BL PREDICTOR of Yi 2. LRF BL ARROXIMATION
   • If CEF is normal, LRF recovers it EXACTLY
   • 1. LRM BLP, like CEF best unrestricted predictor 2.Equal to CEF if CEF linear 3. Best linear approximation to CEF even if CEF non-linear

3. FIT

   • Derive SSR, TSS, ESS

4. Multiple regression

   • FWL theorem gets us from single to multiple
   • MRM: 1. linear model of/approx to CEF, 2. Tells you how E[Yi] varies with (X1i, X2i...)
   • THE SLOPE COEFFCIEINTS TELL YOU HOW THE CE OF YI VARIES WITH XI HOLDING OTHER VARIABLES CONSTANT

5. ELASTICITY = dy/dx(y/x)

6. Using LOGS?

   • JI: Logs an APPROXIMATION
   • MARGINAL effects/elasticities

7. WITH A SAMPLE