lecture9 - LECTURE 9 Introductory Econometrics...

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LECTURE 9 Introductory Econometrics Autocorrelation November 26, 2013 1 / 30
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O N PREVIOUS LECTURES I We discussed the specification of a regression equation I Specification consists of choosing: 1. correct independent variables 2. correct functional form 3. correct form of the stochastic error term I We talked about the choice of independent variables and their functional form I We started to talk about the form of the error term - we discussed heteroskedasticity 2 / 30
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O N TODAY S LECTURE I We will finish the discussion of the form of the error term by talking about autocorrelation (or serial correlation ) I We will learn I what is the nature of the problem I what are its consequences I how it is diagnosed I what are the remedies available 3 / 30
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N ATURE OF AUTOCORRELATION I Observations of the error term are correlated with each other Cov ( ε i , ε j ) 6 = 0 , i 6 = j I Violation of one of the classical assumptions I Can exist in any research study in which the order of the observations has some meaning - most frequently in time-series data I Particular form of autocorrelation - AR ( p ) process: ε t = ρ 1 ε t - 1 + ρ 2 ε t - 2 + . . . + ρ p ε t - p + u t I u t is a classical (not autocorrelated) error term I ρ k are autocorrelation coefficients (between -1 and 1) 4 / 30
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E XAMPLES OF PURE AUTOCORRELATION I Distribution of the error term has a autocorrelation nature I First order autocorrelation ε t = ρ 1 ε t - 1 + u t I positive serial correlation: ρ 1 is positive I negative serial correlation: ρ 1 is negative I no serial correlation: ρ 1 is zero I positive autocorrelation very common in time series data I e.g.: a shock to GDP persists for more than one period I Seasonal autocorrelation ε t = ρ 4 ε t - 4 + u t 5 / 30
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E XAMPLES OF IMPURE AUTOCORRELATION I Autocorrelation caused by specification error in the equation: I omitted variable I incorrect functional form I How can misspecification cause autocorrelation in the error term? I Recall that the error term includes the omitted variables, nonlinearities, measurement error, and the classical error term. I If we omit a serially correlated variable, it is included in the error term, causing the autocorrelation problem. I Impure autocorrelation can be corrected by better choice of specification (as opposed to pure autocorrelation). 6 / 30
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A UTOCORRELATION X Y 7 / 30
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C ONSEQUENCES OF AUTOCORRELATION I Estimated coefficients ( b β ) remain unbiased and consistent I Variance of b β increases I serially correlated error term causes the dependent variable to fluctuate in a way that the OLS estimation procedure attributes to the independent variable I OLS tend to underestimate the variance of b β I autocorrelation increases the variances of the estimates in a way that is masked by OLS estimates The same consequences as for the heteroskedasticity 8 / 30
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V ARIANCE OF OLS UNDER AUTOCORRELATION I Consider the model y = X β + ε I Suppose that ε t = ρε t - 1 + u t I OLS estimate is b β = ( X 0 X ) - 1 X 0 y I Variance of the estimate is Var ± b β ² = ( X 0 X ) - 1 X 0 Ω X ( X 0 X ) - 1 , where Ω = ( ε ) =
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