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or if the variables are in logarithms (where the difference of a logged variable is approximately equal to its proportional growth) z z ( Z ) - ( Z ) g t t t t-1 Z - = - 1 ln ln Let us now apply these ideas. Suppose we have the following econometric model: ln ln ln ln ln t 1 2 t 3 t-1 4 t- 5 t-1 t Y = + X + X + X + Y + u α α α α α 1 In long run static equilibrium, all terms involving differences will be zero. Furthermore, we set the disturbance term in equilibrium to its expected value. So we have 0 = + 0 + 0 + X + Y or Y = - - X 1 2 3 4 5 1 5 4 5 α α α α α α α α α ln ln ln ln as the static long run equilibrium solution. In practice, we shall replace the parameter values in this expression by their estimated values. (H): SEQUENTIAL TESTING AND THE SIGNIFICANCE LEVEL OF HYPOTHESIS TESTS Notes still to be written on this. (I): THE ROLE OF MISSPECIFICATION TESTING IN THE GENERAL-TO-SPECIFIC SEARCH PROCEDURE We have argued that our objective is to simplify an original 'general' or unrestricted model into a parsimonious and interpretable 'specific' model. Such a final model should be an adequate approximation to the true, but unknown, data generating process (DGP). However, a precondition for placing any confidence in the regression results is that the estimated model should exhibit no evidence of statistical misspecification . Why is this so? In order for us to be confident that the estimator does have good properties and that inference is valid, the validity of a set of assumptions we make in carrying out estimation needs to be checked. This is one interpretation of what is being done in misspecification testing. It is to this topic that we turn next. So in conclusion, we state that any final model should be such that it: (i) satisfies all criteria listed in section 1 (ii) is 'parsimonious' (iii) is parameterised in a form that has an intuitive economic interpretation. CONCLUSION We have argued that, when considering relationships between variables, one can distinguish between a long-run or equilibrium relationship between the variables the short term dynamics which govern adjustment of variables when out of equilibrium Usually, a researcher will wish to know (or rather to estimate) parameters of both the long-run equilibrium and the short term adjustment processes. However, in the next set of notes, we consider the case where the information sought is only about the parameters of an underlying long run relationship. REQUIRED READING: Thomas, Ch 11: Estimating dynamic models Thomas, Ch 12: Choosing the appropriate model Thomas, Ch 13 (section 13.3 and 13.4 only) 7
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