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Unformatted text preview: PAM 3100 Multiple Regression Analysis Multiple Regression Analysis: Estimation Fall 2010 Michael Lovenheim mfl55@cornell.edu In the two variable model, we often run into the problem that the assumption of E[U  X]=0 is unreasonable. The problem is there are variables in U that are correlated with X: Cigarette tax elasticities and antismoking sentiment. Isolating the effect of education spending on student outcomes? Is time to degree influenced by collegiate resources? In all of these examples, the two variables model cannot identify the policy effects of interest. What we want is to control for these other factors to isolate the causal effect of each policy. Independent Variable (i) (ii) (iii)0.3560.3610.177 (0.015) (0.015) (0.022) 0.0630.154 (0.059) (0.070)0.692 (0.054) 5.824 5.183 6.749 (0.058) (0.608) (0.694) Log Cigarette Taxes Log Per Capita Income State AntiSmoking Sentiment Constant Dependent Variable: Log Cigarette Sales per Capita Preferred Model: u Smoking Anti Capita Income Tax Capita Sales + + + + = ) ( ) / ln( ) ln( ) / ln( 3 2 1 Regression of ln(tax) on State AntiSmoking Sentiment gives a coefficient of 1.317. Independent Variable (i) (ii) (iii) 0.020 0.018 0.014 (0.001) (0.001) (0.001)0.0510.035 (0.008) (0.008)0.0700.066 (0.007) (0.007)0.0360.003 (0.006) (0.006)0.009 (0.001) 4.366 5.013 5.520 (0.015) (0.035) (0.041) Math Test Percentile Constant Dependent Variable: Time To Degree (in Years) StudentFaculty Ratio Mother's Education (Years) Father's Education (Years) Parental Income ($) u Mathscore Income Fathed Mothed SF TTD + + + + + + = 5 4 3 2 1 Independent Variable (i) (ii) (iii)0.4320.4420.057 (0.200) (0.205) (0.053) 0.6390.699 (0.705) (0.293) 0.0660.097 (0.114) (0.031)0.691 (0.273) 0.088 (0.275) 1.041 (0.027) Constant 12.093 5.161 4.434 (0.214) (7.410) (2.414) Log K12 Expenditures per Student Dependent Variable: Log State College Enrollment Log Average InState Tuition Log State Income per Capita State Unemployment Log K12 StudentTeacher Ratio Log 18 Year Old Population The population model with k dependent variables is written Where Y is the dependent variable, each X is an independent variable, and U is the error term. This population model implies the following conditional expectation function: We write this, to reduce notation, as: U X X X Y k k + + + + + = 2 2 1 1 k k k X X X X X X Y E + + + + = 2 2 1 1 2 1 ] ,..., ,  [ k k X X X X Y E + + + + = 2 2 1 1 ]  [ The multiple variable model has assumptions similar to the 2 variable model: 1) E[ U ] = 0 2) E[ U  X 1 ,,X k ] = 0 The second assumption means that the error term is uncorrelated with each of the independent variables. But recall, now U does not include the independent variables....
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