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  3. Multiple Regression Analysis The general linear regression with  k  explana-tory variables is just an extension of the sim-ple regression as follows(1)  y i  =  β 0  + β 1 x i 1  + ··· + β k x ik  + u i . Because(2)  ∂y i ∂x ij =  β  j  j  = 1 ,...,k , coefficient  β  j  indicates the marginaleffect of variable  x  j , and indicates the amount y  is expected to change as  x  j  changes byone unit and other variables are kept con-stant (ceteris paribus).The multiple regression opens up several ad-ditional options to enrich analysis and makemodeling more realistic compared to the sim-ple regression. 1  Example 3.1: Consider the hourly wage example. En-hance the model as(3) log( w ) =  β 0  + β 1 x 1  + β 2 x 2  + β 3 x 3 , where  w  = average hourly earnings,  x 1  = years of ed-ucation (educ),  x 2  = years of labor market experience(exper), and  x 3  = years with the current employer(tenure). Dependent Variable: LOG(WAGE)Method: Least SquaresDate: 08/21/12 Time: 09:16Sample: 1 526Included observations: 526VariableCoefficientStd. Errort-StatisticProb. C0.2843600.1041902.7292300.0066EDUC0.0920290.00733012.555250.0000EXPER0.0041210.0017232.3914370.0171TENURE0.0220670.0030947.1330700.0000R-squared0.316013 Mean dependent var1.623268 Adjusted R-squared0.312082 S.D. dependent var0.531538S.E. of regression0.440862 Akaike info criterion1.207406Sum squared resid101.4556 Schwarz criterion1.239842Log likelihood-313.5478 Hannan-Quinn criter.1.220106F-statistic80.39092 Durbin-Watson stat1.768805Prob(F-statistic)0.000000 For example the coefficient 0.092 means that, hold-ing exper and tenure fixed, another year of educationis predicted to increase wage by approximately 9.2%.Staying another year at the same firm (educ fixed, ∆ exper= ∆ tenure=1) is expected to result in a salaryincrease by approximately 0 . 4%+2 . 2% = 2 . 6%. 2  Example 3.2: Consider the consumption function C   =  f  ( Y  ), where  Y   is income. Suppose the assump-tion is that as incomes grow the marginal propensityto consume decreases.In simple regression we could try to fit a level-logmodel or log-log model.One possibility also could be β 1  =  β 1 l  + β 1 q Y  , where according to our hypothesis  β 1 q  <  0. Thus theconsumption function becomes C   =  β 0  +( β 1 l  + β 1 q Y  ) Y   + u =  β 0  + β 1 l Y   + β 1 q Y   2 + u This is a multiple regression model with  x 1  =  Y   and x 2  =  Y   2 .This simple example demonstrates that we can mean-ingfully enrich simple regression analysis (even thoughwe have essentially only two variables,  C   and  Y  ) andat the same time get a meaningful interpretation tothe above polynomial model.The response of   C   to a one unit change in  Y   is now ∂ C ∂ Y   =  β 1 l  +2 β 1 q Y  . 3  EstimationIn order to estimate the model we replace theclassical assumption 3 as3. None of the independent variables is con-stant, and no observation vector of any in-dependent variable can be written as a lin-ear combination of the observation vectorsof any other independent variables.The estimation method again is the OLS,which produces estimates ˆ β 0 ,  ˆ β 1 ,...,  ˆ β k  by min-imizing(4) n  i =1 ( y i − β 0 − β 1 x i 1 −···− β k x ik ) 2 with respect to the parameters.Again the first order solution is to set the( k  +1) partial derivatives equal to zero.The solution is straightforward although theexplicit form of the estimators become com-plicated. 4
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