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Unformatted text preview: Introduction to Econometrics 91 Technical Notes Introduction to Econometrics Lecture 9: Diagnostic Tests  Nonlinearity and Heteroscedasticity Introduction to Diagnostic Testing Estimating a regression equation uses N pieces of information (the observations) to compute K regression coefficients. The rest of the information generates the OLS residuals e i : although there are N residuals they contain only N  K distinct ‘pieces’ of information (since we know that Σ e i = 0, and that for each regressor { X 1 .. X K } Σ i X K i e i = 0). If a regression equation has been well specified these residuals will be (approximately) random, and we will not be able to distinguish any systematic pattern to them. If such a pattern exists (for example, if the residuals corresponding to a particular group of observations are all positive) we should exploit the information implied by the pattern to improve the regression equation. In a simple regression many such patterns are easily detected using a scatter plot of Y against X, but for multiple regression scatter diagrams are much less helpful. Diagnostic statistics are statistics which have been designed to detect certain common patterns in the residuals (just as medical diagnostic tests such as blood samples are designed to detect common diseases). Eviews and other modern computer packages will compute a number of important diagnostic statistics automatically, and other tests can be performed by running extra regressions. The aim of diagnostic testing is to detect misspecification in a regression equation, and if possible to suggest its cause. Often this can be interpreted as the exclusion of a relevant explanatory variable, so many diagnostic tests have been called variable addition tests: the test is designed to show if an omitted variable is significant. Sometimes this omitted variable has an obvious economic interpretation, but for some diagnostic tests the omitted variable concerned is designed purely to detect a particular sort of misspecification and has no economic meaning. Omitting a relevant explanatory variable almost always (with economic data) leads to biased estimates of the regression coefficients, so the ‘warning signals’ provided by diagnostic tests should not be ignored. There are two important general warnings about diagnostic testing • it is hard to apply diagnostic tests to a poor equation (ie an equation with a large residual variance) because for such equations the power of any hypothesis test is low. Hence the fact that a model passes all the tests is not necessarily a good sign. • taking a patient’s temperature allows a doctor to find out whether the patient has a wide range of illnesses, but does not provide information about the specific disease. Similarly, some diagnostic tests can detect several types of misspecification, but cannot establish the precise nature of the problem. In technical language, the test has power against a wide range of alternatives. An example of this point is technical language, the test has power against a wide range of alternatives....
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 Spring '10
 Cowell
 Economics, Econometrics, Regression Analysis, log likelihood, technical notes

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