Economics 105: Statistics

Economics 105: Statistics GH 20 due Thur Modeling exercise … any questions Brief discussion of how I grade … I curve ! But not until the end of semester

The Multiple Regression Model Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi) Multiple Regression Model with k Independent Variables: Population slopes Random Error Y-intercept • Endogenous explanatory variables

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?

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)

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

Measures of Variation • Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error Sum of Squares where: = Average value of the dependent variable Yi = Observed values of the dependent variable i = Predicted value of Y for the given Xi value

Measures of Variation (continued) Y Yi   Y SSE= (Yi-Yi)2 _ SST=(Yi-Y)2  _ Y  _ SSR = (Yi -Y)2 _ Y Y X Xi

Goodness of Fit • The coefficient of determination is the portion of the total variation in Y that is explained by variation in X • Also called r-squared and denoted r2 (or R2)

Examples of Approximate R2 Values Y R2= 1 Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X X R2= 1 Y X R2= 1

Examples of Approximate R2Values Y 0 < R2< 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X X Y X

Examples of Approximate R2Values R2= 0 Y No linear relationship between X and Y: The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) X R2= 0

Standard Error of the Estimate • The variation of observations around the sample regression line is estimated by • an unbiased estimator of stddev of error term • where SSE = error sum of squares • n = sample size • K = number of slope beta parameters • Also called “standard error of the model,” or “root mean squared error” (RMSE). Book calls SYX.

Comparing Standard Errors • Seis a measure of the variation of observed Y values around the regression line Y Y X X • The magnitude of Seshould always be judged relative to the variation of the Y values in the sample data (measured by SY, the sample standard deviation of the actual Y values) • Closer to 0, than to sY , the better the fit

The coefficient of determination, , in simple regressions of the form, is equal to the square of the correlation coefficient. • Provides a link between correlation and regression • “Multiple R” is • In multiple regression context, • It is another, less commonly used, measure of strength of the relationship between the dependent var and the independent (explanatory) vars. “Multiple R”

Goodness of Fit • One should not place too much importance on obtaining a high R2 • If all else is equal, a model with a higher R2 explains a higher fraction of the variance • The model has more explanatory power • Dependent var must be the same to compare • However, R2 can be influenced by factors such as the nature of the data • Cross-sectional data on individual people: .1 to .2 • Cross-sectional data on firms, counties, cities, countries, states: .4 to .6 • Time-series data: > .80

Economics 105: Statistics