ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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: 𝜀~𝑁(0,𝜎2) 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). 𝑦𝑖=𝛽0+ 𝛽1𝑥1+ 𝜀𝑖 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).ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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 Hypotheses tests are no longer valid in small samples although they are still valid in large samples.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 testECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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:ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 - 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.ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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.ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 Alternatively, normality could be detected graphically by inspecting a normal plot of residuals.ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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, 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑖𝑧𝑒𝑑 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙=𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙− 𝑚𝑒𝑎𝑛𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠 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, 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑖𝑧𝑒𝑑 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙=𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠 If all the standardized residuals lie between -2 and 2, it is concluded that the normality assumption holds.ECON 337-ECONOMETRICS II Notes Prepared by Wanja M. Douglas [email protected] 0711 653 219 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 ( 𝜒2) distribution with 2 degrees of freedom. The Jarque-Bera statistic, which is commonly given by many econometric software packages (like