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Exploring Models in Economics: Understanding Key Statistical Concepts

This project focuses on the application of statistical modeling in economics, particularly in examining the influence of various factors on outcomes. Topics include the impact of SAT scores on grades, the effectiveness of police on crime rates, the relationship between education and wages, and the effect of school start times on academic achievement. We will explore issues of endogenous explanatory variables and biases like omitted variable bias and measurement errors in regression analysis. Students will learn to interpret Ordinary Least Squares (OLS) estimators and how to construct sample regression equations.

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Exploring Models in Economics: Understanding Key Statistical Concepts

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  1. Economics 105: Statistics Any questions? GH #19 due Friday. Introduce Modeling Exercise group project

  2. Modeling Exercise examples • What is the effect of your roommate’s SAT scores on your grades? The effect of studying? • Do police reduce crime? • Does more education increase wages? • What is the effect of school start time on academic achievement? • Does movie violence increase violent crime?

  3. Endogenous Explanatory Variable • Causes of endogenous explanatory variables include … • Wrong functional form • Omitted variable bias … occurs if both the • Omitted variable theoretically determines Y • Omitted variable is correlated with an included X • Errors-in-variables (aka, measurement error) • Sample selection bias • Simultaneity bias (Y also determines X)

  4. Stochastic Linear Models Assumptions (1) Simple regression vs. Multiple regression Linear function, plus error Variation in Y is caused by , the error (as well as X) (2) Sources of error Idiosyncratic, “white noise” Measurement error on Y Omitted relevant explanatory variables … why?

  5. Y E[Y] = 0+ 1X X Stochastic Linear Models Assumptions (3) Homoskedasticity (4) No autocorrelation (5) Errors and the explanatory variable are uncorrelated (6) Errors are normally distributed

  6. Stochastic Linear Models Assumptions so far imply Need to estimate population intercept & slope Take a sample of data & obtain the sample regression line

  7. Sample Regression Equation(Prediction Line) The sample regression line equation provides an estimate of the population regression line Estimated (or predicted) Y value for observation i Estimate of the regression intercept Estimate of the regression slope Value of X for observation i Other notation: The individual random error terms ei have a mean of zero

  8. Sample Regression Equation chosen in sample not chosen in sample Y estimated error for X3 (residual) Observed Value of Y for X3 e3 ε3 Predicted Value of Y for X3 X X3

  9. Sample Regression Equation Y Residual, ei, is the prediction error Positive errors Negative errors X

  10. Derivation of OLS Estimators Select to minimize SSE Set first partial derivatives = 0 Results are

  11. OLS Estimators

  12. OLS Example: Scatterplot

  13. OLS calculations “by hand” File is in P:\Economics\Eco 105 (Statistics)\lec_simple reg.xls

  14. Sample regression line

  15. OLS Residuals (Excel output)

  16. The effect of X on Y (from regressing “Y on X”) Interpretation of OLS Parameters • For one-unit change in X, the average value of Y changes by 1 units • intercept

  17. Unbiased estimator • Efficiency of an estimator • Intuition for when var is smaller • We won’t know , so we’ll need to estimate it Properties of OLS Estimator • Gauss-Markov Theorem • Under assumptions (1) - (5) [don’t need normality of errors], is B.L.U.E. of

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