Omitted Variable Bias 12 14 13 10 22 PM Occurs when 2 conditions are true 1 Omitted variable is correlated with the included regressor and 2 The omitted variable is a determinant of the dependent variable OVB formula the Pxu Ou Ox is the bias in B1 that persists even in large samples 1 OVB is a problem whether sample size is large or small o B hat is not a consistent estimator of B1 when there is OVB 2 Size of bias depends on correlation between the regressor and the error term o Pxu o Larger the absolute value of it the larger the bias 3 Directon of the bias in B hat depends on whether X and u are positively or negatively correlated o negative correlation positive bias and vis versa control variables 1 or more independent variables in the multiple regression model Coefficient interpretation in this model is different than when X1 was only regressor B1 is the effect on Y of a unit change in X1 holding X2 constant or controlling for X2 Also called partial effect Intercept B0 determines how far up the Y axis the population regression line starts Homoscedastic Variance of the conditional distribution of u is constant Otherwise it is heteroskedastic Idea behind Ordinary Least Squares Estimators of the coefficients B0 B1 Bk that minimize the sum of the squared mistakes in OLS regression line Straight line constructed using the OLS estimators Result would be the predicted value OLD residual predicted value For the ith observation is the difference between Yi and its OLS Two advantages to multiple regression 1 Quantitative estimate of the effect of a unit decrease 2 Readily extends to more than 2 regressors so the multiple regression can be used to control for measurable factors other than just the percentage of English learners B1 and B2 have joint normal distribution 3 commonly used summary statistics in multiple regression 1 Standard error of the regression 2 Regression R2 3 Adjusted R2 or R2 with bar over it all 3 of these tell how well the OLS estimate of the multiple regression line describes or fits the data SER line Estimates the standard deviation of the error term ui Is measure of spread of the distribution of Y around the regression here SSR is the sum of squared residuals for SER in chapter 4 single regression we divided by n 2 to adjust for downward bias 2 was for the intercept and regression line n k 1 adjusts for the downward bias introduced by estimating k 1 coefficients k slope coefficients plus the intercept small n negligible degrees of freedom adjustment Fraction of the sample variance of Yi explained by or predicted by R2 is also 1 minus the fraction of the variance of Yi not explained by R2 the regressors the regression R2 ESS TSS 1 SSR TSS R2 increases whenever a regressor is added UNLESS the estimated explained sum coefficient on the added regressor is exactly In practice it is extremely unusual for an estimated coefficient to be exactly 0 o Thus in general the SSR will decrease when a new regressor is added o THIS MEANS THAT R2 GENERALLY INCREASES and never decreases when a new regressor is added Adjusted R2 Simple increase in R2 by adding a variable seemingly inflates the estimate of how well the regression fits the data Can correct for this is by reducing R2 by some factor Adjusted R2 is a modified version of the R2 that does not necessarily increase when a new regressor is added o second portion is ratio of the sample variance of the OLS residuals three things to know about adjusted R2 o 1 n 1 n k 1 is always greater than 1 so adjusted R2 is always less than R2 o 2 Adding a regressor has 2 opposite effects on adjusted R2 SSR falls increasing the adjusted Other hand the factor n 1 n k 1 increases Therefore whether the adjusted increases or decreases depends on which of these 2 effects is stronger o 3 Adjusted R2 can be negative this happens when the regressors taken together reduce the sum of squared residuals by such a small amount that this reduction fails to offset the factor n 1 n k 1 4 assumptions extended to allow for multiple regressors Assumption 1 Expected value has mean of 0 Assumption 2 are iid Assumption 3 Large outliers are unlikely Nonzero finite 4th moments Finite kurtosis Assumption 4 no perfect multicollinearity Impossible to calculate OLS estimator if perfect Is perfect if one of the regressors is a perfect linear function of the other regressors Produces division by 0 in the OLS formulas When perfect multicollinearity occurs it often reflects a logical mistake in choosing the regressors or some previously unrecognized feature of the data set Solution modify the regressors to eliminate the problem Under the least squares assumptions the OLS estimators are unbiased and consistent estimators in the linear multiple regression model Large samples it will be normal distribution for the joint distribution dummy variable trap when multiple binary dummy variables are used s regressors will lead to perfect multicollinearity otherwise avoid this by exclusiding some of the binary variables from the multiple regression so only G 1 of the G binary variables are included as regressors o this basically lets the intercept stay CAN include all G binary regressors if the intercept is omitted o More conventional to leave the constant term imperfect multicollinearity two or more of the regressors are highly correlated o there is a linear function of the regressors that is highly correlated with another regressor unlike perfect multico imperfect does NO pose any problems for the theory of the OLS estimators sampling uncertainty must be quantified Hypothesis Tests and CI in Multiple Regression12 14 13 10 22 PM F statistic Standard Errors for the OLS Estimators Square root of variance of the sample average was the SE of predicted B1 Key idea is same for SE the large sample normality of the estimators and the ability to estimate consistently the SD of their sampling distribution are the same whether one has 1 2 or 12 regressors o Actual formula is stated as a matrix Hypothesis Tests for a Single Coefficient reminder standard errors are below the coefficients in the parentheses these test rely on the large sample normal approximation to the distribution of the OLS estimator only guaranteed to work in large samples 7 2 Test of Joint Hypotheses joint hypothesis A hypothesis that imposes 2 or more restrictions on the regression coefficients One at a time method rejects the null when it is true and gives too many too many chances If any ONE of the equalities under the