Lecture Ten 1 Lecture Part I Regression properties of OLS estimators assumptions of OLS pathologies of OLS diagnostics for OLS Part II Experimental Method 2 Properties of OLS Estimators Unbiased E b b Note y i a b x i e i And summing over observations i and dividing by n y a b x e so subtractin g y i y b x i x e i e Recall the estimator for the slope is n n i 1 i 1 b y i y x i x x i x 2 3 And substituting in this expression for the estimator the expression for y i y n n i 1 i 1 b b x i x e i e x i x x i x 2 n n i 1 i 1 b b e i e x i x x i x 2 E b b And taking expectations n n Note b E b b b x i x e i e x i x i 1 2 i 1 VAR b E b E b E x i x e i e x i x 2 2 n n i 1 i 1 4 So n VAR b 2 x i x 2 i 1 The dispersion in the estimate for the slope depends upon unexplained variance and inversely on the dispersion in x the estimate the unexplained mean square is used for the variance of e 5 Other Properties of Estimators Efficiency makes optimum use of the sample information to obtain estimators with minimum dispersion Consistency As the sample size increases the estimator approaches the population parameter 6 Outline Regression The Assumptions of Least Squares The Pathologies of Least Squares Diagnostics for Least Squares 7 Assumptions Expected value of the error is zero E e 0 The error is independent of the explanatory variable E e x Ex 0 The errors are independent of one another E e i e j 0 i not equal to j The variance is homoskedatic E e i 2 E e j 2 The error is normal with mean zero and 2 variance sigma squared 18 4 Error Variable Required Conditions The error is a critical part of the regression model Four requirements involving the distribution of must be satisfied The probability distribution of is normal The mean of is zero E 0 The standard deviation of is for all values of x The set of errors associated with different values of y are all independent 9 The Normality of E y x3 The standard deviation remains constant 0 1x3 E y x2 0 1x2 but the mean value changes with x 0 1x1 E y x1 From the the first first three three assumptions assumptions we we have have From x1 normally distributed distributed with with mean mean yy isis normally E y 00 11x x and and aa constant constant standard standard E y deviation deviation x2 x3 Pathologies Cross section data error variance is heteroskedatic Example could vary with firm size Consequence all the information available is not used efficiently and better estimates of the standard error of regression parameters is possible Time series data errors are serially correlated i e auto correlated Consequence inefficiency 11 Pathologies Cont Explanatory variable is not independent of the error Consequence inconsistency i e larger sample sizes do not lead to lower standard errors for the parameters and the parameter estimates slope etc are biased The error is not distributed normally Example there may be fat tails Consequence use of the normal may underestimate true 95 confidence intervals 12 Pathologies Cont Multicollinearity The independent variables may be highly correlated As a consequence they do not truly represent separate causal factors but instead a common causal factor 13 18 9 Regression Diagnostics I The three conditions required for the validity of the regression analysis are the error variable is normally distributed the error variance is constant for all values of x The errors are independent of each other How can we diagnose violations of these conditions 14 Residual Analysis Examining the residuals or standardized residuals help detect violations of the required conditions Example 18 2 continued Nonnormality Use Excel to obtain the standardized residual histogram Examine the histogram and look for a bell shaped diagram with a mean close to zero 15 Diagnostics Cont Multicollinearity may be suspected if the tstatistics for the coefficients of the explanatory variables are not significant but the coefficient of determination is high The correlation between the explanatory variable can then be calculated To see if it is high 16 Diagnostics Is the error normal Using EViews with the view menu in the regression window a histogram of the distribution of the estimated error is available along with the coefficients of skewness and kurtosis and the Jarque Bera statistic testing for normality 17 Diagnostics Cont To detect heteroskedasticity if there are sufficient observations plot the estimated errors against the fitted dependent variable 18 Heteroscedasticity When the requirement of a constant variance is violated we have a condition of heteroscedasticity Diagnose heteroscedasticity by plotting the residual against the predicted y y Residual The spread increases with y y Homoscedasticity When the requirement of a constant variance is not violated we have a condition of homoscedasticity Example 18 2 continued Residuals 1000 500 0 13500 500 14000 14500 15000 15500 16000 1000 Predicted Price 20 Diagnostics Cont Autocorrelation The Durbin Watson statistic is a scalar index of autocorrelation with values near 2 indicating no autocorrelation and values near zero indicating autocorrelation Examine the plot of the residuals in the view menu of the regression window in EViews 21 Non Independence of Error Variables A time series is constituted if data were collected over time Examining the residuals over time no pattern should be observed if the errors are independent When a pattern is detected the errors are said to be autocorrelated Autocorrelation can be detected by graphing the residuals against time 22 Non Independence of Error Variables Patterns in the appearance of the residuals over time indicates that autocorrelation exists Residual Residual 0 Time Note the runs of positive residuals replaced by runs of negative residuals 0 Time Note the oscillating behavior of the residuals around zero 23 Fix Ups Error is not distributed normally For example regression of personal income on explanatory variables Sometimes a transformation such as regressing the natural logarithm of income on the explanatory variables may make the error closer to normal 24 Fix ups Cont If the explanatory variable is not independent of the error look for a substitute that is highly correlated with the dependent variable but is independent of the error Such a variable is called an instrument 25 Data Errors May lead to outliers Typos may lead to outliers and looking for ouliers is a good way to check for serious typos 26 Outliers An outlier is an observation that is unusually small
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