note12 - 1 Chapter 12 Serial Correlation Examples of time...

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1 Chapter 12. Serial Correlation Examples of time series models Static Phillips curve - tradeoff between inflation and unemployment rate (or GDP growth rate) Expectation augmented Phillips curve Aggregate consumption function Monetary policy reaction function Remark. Since we are considering over-time events, we use time subscript t . Let a simple time series model be specified as Suppose the Gauss-Markov assumptions are satisfied: (1) for all t and s (2) for all t and s (3) for all t s and k Then, OLS estimator of parameters are BLUE, and what we have learned before apply to this model. In many applications of time series model, assumption (3) is often violated. A shock in current period can have a spill-over effect to the next period, and is correlated with . A simple form of this correlation can be specified as where the random term is assumed to satisfy the Gauss-Markov assumption: (1) for all t and s (2) for all t and s (3) for all t s and k We say that follows the first order autoregressive process, abbreviated by AR(1). We will explain later the
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2 reason for the condition that the value of D must be between -1 and 1. It is clear that and are correlated if , violating assumption (3). This means that the OLS estimator is not the best among all linear unbiased estimators even if it is unbiased. To check whether the OLS estimator is unbiased or not, we need to check the mean of . Noting that , , and etc., repeated substitutions give Since is assumed for all s , it is easy to see that . Therefore, OLS estimator is unbiased. What about the variance of ? Is it homoskedastic? What about the correlation between and ? Between and ? To compute the variance, recall that error terms are not correlated with . Therefore, The variance of does not depend on time t and stays the same, i.e., homoskedastic. Now consider the covariance between and : where the last equality is due to the fact that is not correlated with . A similar operation leads to This indicates that the AR(1) error terms satisfy previous assumptions of zero mean and homoscedasticity, but violates the assumption of uncorrelatedness. What do we do if we know D ? Consider a regression model with AR(1) errors
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3 As a base case, suppose that the autocorrelation coefficient D is known. From , we can write The transformed regression equation thus satisfies all the assumptions necessary for the LSE of to be BLUE. This is a form of Generalized Least Squares estimator. Note that the transformed variables start from . If we wish to retain the first observation, then we take the following transformation Feasible GLS when D is unknown Since the correlation coefficient D is unknown in practice, we cannot use the GLS described above. To implement it we need an estimator of D , which will be used in the place of D . This procedure is called a feasible GLS. If error terms
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note12 - 1 Chapter 12 Serial Correlation Examples of time...

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