Intro%20to%20Multiple%20Regression%20MD%20verstion

Intro%20to%20Multiple%20Regression%20MD%20verstion -...

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1 Applied Business Tools ECO 6416 Introduction to Multiple Regression Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4 Measures of fi 2 4. Measures of fit 5. Sampling distribution of the OLS estimator 6. Hypothesis testing in multiple regression 7. Dummy variables Introduction to Multiple Regression ± Last time we saw how failing to control for important explanatory variables can lead us to draw evidently incorrect conclusions from a regression model. – Using time series data, simple (bivariate) regression Using time series data, simple (bivariate) regression seemed to imply that Mexican imports “cause” US manufacturing wages to fall manufacturing wages to fall. – A multiple regression indicated that the apparent A multiple regression indicated that the apparent negative association between US wages and Mexican imports was more likely the result of declining unionization over the sample period. ± Illustrates “omitted variable bias” ± Regression of wages on imports biased by omission of unionization rate Omitted Variable Bias ± Two requirements for omitted variable bias: – 1. The omitted independent variable is correlated with 1. The omitted independent variable is correlated with the included independent variable(s). ± Declining unionization correlated with rising imports. – 2. The omitted independent variable is a determinant of 2. The omitted independent variable is a determinant of the dependent variable. ± Unionization affects manufacturing wages. ± When these two conditions hold, the omitted variable biases the coefficients estimated in a regression of the dependent variable on included independent variable(s). – That is, the sample regression results no longer match That is, the sample regression results no longer match the population regression on average across all possible samples. Review: Population Bivariate Linear Regression Model Y i = β 0 + 1 X i + u i , i = 1,…, n X is the independent variable or regressor Y is the dependent variable 5 0 = intercept 1 = slope u i = the regression error The regression error consists of omitted factors, or possibly measurement error in the measurement of Y . In general, these omitted factors are other factors that influence Y , other than the variable X Example ± Unit of observation = elementary school districts in California ± Y = Standardized test score, average of reading & math in 5 th grade ( TESTSCR ) ± X = Student = Student-teacher ratio, average in teacher ratio, average in district ( STR ) ± n = 420
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2 Observations Observations on on Y and X ; the population regression line; and the regression error (the “error term”): 7 OLS regression: STATA output regress testscr str, robust Regression with robust standard errors Number of obs = 420 F( 1, 418) = 19.26 Prob > F = 0.0000 R-squared = 0.0512 Root MSE = 18.581 ------------------------------------------------------------------------- 8 | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------+---------------------------------------------------------------- str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671 _cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057 ------------------------------------------------------------------------- n TestScore = 698.9 – 2.28 × STR OLS Review Questions 1.
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