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Unformatted text preview: CHAPTER 14 AUTOCORRELATION: WHAT HAPPENS IF ERROR TERMS ARE CORRELATED? QUESTIONS 14.1. ( a ) The correlation between the current value of the error with its own past value(s). ( b ) The correlation between the current value of the error with its immediate past value. ( c ) The correlation between observations over space rather than over time. Note : Some authors use the term serial correlation for correlation observed in time series data [i.e., in the sense defined in ( a )] and autocorrelation for correlation observed in cross-section data [in the sense defined in ( c )]. 14.2. Although in general an AR(m) scheme can be used, the AR(1) scheme has been found to be quite useful in many time series analysis. With the AR(1) scheme, many properties of the OLS estimators can be easily established. 14.3. The consequences are: (1) The OLS estimators are unbiased, but are not efficient. (2) The conventionally estimated standard errors of OLS estimators are biased. (3) As a result, the conventionally computed t and F tests are unreliable, the conventional estimator of 2 σ is biased, and the conventionally computed 2 R may not represent the true 2 R . 14.4. The method of generalized difference equation will produce BLUE estimators, provided the first-order autocorrelation parameter, ρ , is known or can be estimated. Also, remember to transform the first observation on the dependent and explanatory variables a la Prais-Winsten if the sample size is small. 14.5. These methods are: (1) The first difference method, where it is assumed that ρ = 1 (2) ρ estimated from the Durbin-Watson d as: 2 1 ρ / d- ≈ 121 (3) ρ estimated from the regression t t t v e ˆ e + =- 1 ρ (4) The Cochrane-Orcutt iterative procedure (5) The Cochrane-Orcutt two-step method (6) Durbin's two-step method (7) Hildreth-Lu search procedure (8) Maximum Likelihood method. 14.6. (1) The graphical method : There are no particular assumptions made. We simply plot the residuals from an OLS regression chronologically or plot the current residuals on the residuals in the previous time period, if the AR(1) scheme is assumed. (2) The Durbin-Watson test : This test is based on several assumptions, such as ( i ) an intercept term is included in the model; ( ii ) X variables are non- stochastic (fixed in repeated sampling); ( iii ) AR(1) autoregressive scheme; ( iv ) no lagged values of the dependent variable are included as explanatory variables. (3) The runs test : This is a non-parametric test. 14.7. On the Durbin-Watson d test’s assumptions, see part (2) of Question 14.6....
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- Regression Analysis, Autocorrelation