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  2. The Simple Regression Model2.1 DefinitionTwo (observable) variables ” y ” and ” x ”. y  =  β  0  + β  1 x + u. (1)Equation (1) defines the  simple regressionmodel  .Terminology: y x Dependent variable Independent variableExplained variable Explanatory variableResponse variable Control variablePredicted variable PredictorRegressand Regressor 1  Error term  u i  is a combination of a numberof effects, like:1. Omitted variables: Accounts the effectsof variables omitted from the model2. Nonlinearities: Captures the effects of nonlinearities between  y  and  x . Thus, if thetrue model is  y i  =  β  0  + β  1 x i  + γx 2 i  + v i , andwe assume that it is  y i  =  β  0  + β  1 x + u i , thenthe effect of   x 2 i  is absorbed to  u i . In fact, u i  =  γx 2 i  + v i .3. Measurement errors: Errors in measuring y  and  x  are absorbed in  u i .4. Unpredictable effects:  u i  includes also in-herently unpredictable random effects. 2  2.2 Estimation of the model, OLSGiven observations ( x i ,y i ),  i  = 1 ,...,n , weestimate the population parameters  β 0  and β 1  of (1) making the followingAssumptions (classical assumptions):1.  y  =  β 0  + β 1 x + u  in the population.2.  { ( x i ,y i ) :  i  = 1 ,...,n }  is a random sampleof the model above, implying uncorrelatedresiduals:  C ov( u i ,u  j ) = 0 for all  i   =  j .3.  { x i , ,i  = 1 ,...,n }  are not all identical,implying   ni =1 ( x i  −  ¯ x ) 2 >  0.4.  E [ u | x ] = 0 for all  x  (zero average error),implying  E [ u ] = 0 and  C ov( u,x ) = 0 .5.  V ar[ u | x ] =  σ 2 for all  x , implying V ar[ u ] =  σ 2 (homoscedasticity).Here  | x  means “conditional on  x ”, that is,we restrict our sample space to this partic-ular value of   x . The practical implication incalculations and derivations is that we cantreat  x  as nonrandon, for the price that ourresult will hold for that particular value of   x only. Otherwise the calucation rules for con-ditional expectations and variances are iden-tical to their unconditional counterparts. 3  The goal in the estimation is to find valuesfor  β  0  and  β  1  that the error terms is as smallas possible (in suitable sense).Under the classical assumptions above, theOrdinary Least Squares (OLS) that minimizesthe residual sum of squares of the error terms u i  =  y i  −  β  0  −  β  1 x i  produces optimal estimatesfor the parameters (the optimality criteria arediscussed later).Denote the sum of squares as f  ( β  0 ,β  1 ) = n  i =1 ( y i  −  β  0  −  β  1 x i ) 2 . (2)The first order conditions (foc) for the mini-mum are found by setting the partial deriva-tives equal to zero. Denote by ˆ β  0  and ˆ β  1  thevalues satisfying the foc. 4
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