4Multiple Regression Analysis - Estimation

4Multiple Regression Analysis - Estimation - PAM 3100...

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Unformatted text preview: PAM 3100 Multiple Regression Analysis Multiple Regression Analysis: Estimation Fall 2010 Michael Lovenheim [email protected] 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 anti-smoking 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)-0.356-0.361-0.177 (0.015) (0.015) (0.022) 0.063-0.154 (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 Anti-Smoking 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 β β β β Regression of ln(tax) on State Anti-Smoking Sentiment gives a coefficient of 1.317. Independent Variable (i) (ii) (iii) 0.020 0.018 0.014 (0.001) (0.001) (0.001)-0.051-0.035 (0.008) (0.008)-0.070-0.066 (0.007) (0.007)-0.036-0.003 (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) Student-Faculty Ratio Mother's Education (Years) Father's Education (Years) Parental Income ($) u Mathscore Income Fathed Mothed SF TTD + + + + + + = 5 4 3 2 1 β β β β β β Independent Variable (i) (ii) (iii)-0.432-0.442-0.057 (0.200) (0.205) (0.053) 0.639-0.699 (0.705) (0.293) 0.066-0.097 (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 K-12 Expenditures per Student Dependent Variable: Log State College Enrollment Log Average In-State Tuition Log State Income per Capita State Unemployment Log K-12 Student-Teacher 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 + + + + + = β β β β 2 2 1 1 k k k X X X X X X Y E β β β β + + + + = 2 2 1 1 2 1 ] ,..., , | [ k k X X X X Y E β β β β + + + + = 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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4Multiple Regression Analysis - Estimation - PAM 3100...

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