# ecmc3

Categories
Published

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Description
econometrc
Transcript
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

Jul 23, 2017

#### Quiz Safety Activity Mobility

Jul 23, 2017
Search
Similar documents
Tags