ECON 337 ECONOMETRICS II LECTURE 6 NON NORMAL DISTURBANCES JULY 1 2022 INTRODUCTION Normality is one of the most important assumptions in the CLRM The assumption here is that the residuals or the error terms must be normally distributed with a mean of zero and constant variance That is 0 2 Recall that the normal distribution is symmetric and bell shaped This is shown by the curve below It is the normality assumption that makes it possible for econometricians to make inferences about the estimated coefficients and predictions based on the sample data To see this we note that since the disturbance term is normally distributed by assumption then the DV is also normally distributed by assumption 0 1 1 Since the OLSE is linear in the DV i e linear in BLUE it follows that the OLSE is normally distributed and in which case the tests of hypotheses could be based on normal distribution provided that the standard errors SEs of the coefficients are known a situation that rarely happens However if we replace the SEs by estimated values from sample observations then the tests are based on t distribution as we saw in Econ 336 Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II The key questions then becomes i What are the consequences of non normal disturbances ii What are the formal checks of non normal disturbances CONSEQUENCES OF NON NORMAL DISTURBANCES 1 The OLSE remain unbiased In other words the unbiasedness of the OLSE is not affected by the violation of the normality assumption 2 Hypotheses tests are no longer valid in small samples although they are still valid in large samples Hence non normality affects the validity of hypotheses tests on estimated coefficients and on predictions in small sample situations that are based on normal distribution TESTING FOR NORMALITY Tests for normality in econometric models just like tests for heteroscedasticity and autocorrelation are based on the analysis of residuals A residual value is a measure of how much a regression line vertically misses a data point Regression lines are the best fit of a set of data You can think of the lines as averages a few data points will fit the line and others will miss There are 3 main tests for normality 1 Graphical method 2 Analysis of standardized residuals 3 Jarque Bera test Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II 1 Graphical Method This entails to draw either box plot or a histogram of the residuals A residual plot has the Residual Values on the vertical axis the horizontal axis displays the independent variable A residual plot is typically used to find problems with regression Some data sets are not good candidates for regression including Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II Heteroscedastic data points at widely varying distances from the line Data that is non linearly associated Data sets with outliers These problems are more easily seen with a residual plot than by looking at a plot of the original data set Ideally residual values should be equally and randomly spaced around the horizontal axis as shown above Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II Should either of them exhibit a reasonably symmetric distribution then it is assumed that the assumption of normality is valid If your plot looks like any of the following images then your data set is probably not a good fit for regression Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II Alternatively normality could be detected graphically by inspecting a normal plot of residuals Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II 2 Analysis of Standardized Residuals Another simple and commonly used procedure for checking for normality is to calculate the standardized residuals The process or Standardized residuals is exactly the same as process of standardizing any variable That is It is important to remember that the sum of all OLS residuals always equal to zero if the regression model has an intercept implying that the mean of the residuals is also equal to zero Accordingly the standardized residual is simply the ratio of the residual to the standard deviation of all the residuals That is If all the standardized residuals lie between 2 and 2 it is concluded that the normality assumption holds Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219 ECON 337 ECONOMETRICS II 3 Jarque Bera Test This is one of the most popular tests for normality It is a large sample or asymptotic test The Jarque Bera test is based on the analysis of the skewness and kurtosis of the residuals to see if they are in conformity with the normal distribution Whereas the coefficient of skewness indicates the extent to which a particular distribution is skewed the coefficient of kurtosis indicates the peakedness of the distribution The coefficient of skewness for a normal distribution is zero indicating that it is symmetric and the coefficient of kurtosis for a normal distribution is equal to 3 Accordingly the Jarque Bera test is a joint test of whether coefficient of skewness and coefficient of kurtosis of the residuals are 0 and 3 respectively The Jarque Bera test is based on the Chi square 2 distribution with 2 degrees of freedom The Jarque Bera statistic which is commonly given by many econometric software packages like STATA R Python Eviews and SPSS is computed as 2 2 6 3 2 24 Where is the skewness of the residuals is the kurtosis of the residuals 3 is commonly referred to as excess kurtosis and is the number of observations used in estimation Most econometric software packages automatically report the values of 2 statistic associated with Jarque Bera test upon running OLS regressions The Decision Rule is to reject the null hypothesis on normality if the values of the computed value of 2 is greater than 2 2 where is the level of significance and to accept the null hypothesis the values of the computed value of 2 is less than 2 2 Note that Jarque Bera test is one sided test Notes Prepared by Wanja M Douglas wmuthinji pt chuka ac ke 0711 653 219