Econ 140 Autocorrelation Lecture 19 Lecture 19 1 Today s plan Econ 140 Durbin s h statistic Autoregressive Distributed Lag model Finite Lags Koyck Transformation Testing in the presence of higher order serially correlated forms Seasonality Note should also look at Chapter 13 of the book Stock and Watson as well as indicated parts of Chapter 12 from the reading list Lecture 19 2 Returning to the Durbin WatsonEcon 140 Last time we talked about how to test for autocorrelation using the Durbin Watson test We found autocorrelation in the model in L 18 xls Yt a bXt et DW test gave figure of 0 331 DL critical value 1 475 Reject H0 0 Indication of positive first order autocorrelation Note no lagged regressors in the model If we reject null need an estimate for for generalized least squares estimation Lecture 19 3 Generalized least squares Econ 140 Need an estimate of we can transform the variables such that where Yt a bX t et 3 Yt Yt Yt 1 Known as Cochrane Orcutt transformation Notice that describes the relationship between neighboring errors in the model Estimating equation 3 allows us to estimate in the presence of first order autocorrelation Lecture 19 4 Problems Econ 140 1 The model presented by Yt a bX t et may still have some autocorrelation the D W test doesn t tell us anything about this we have to retest the model 2 We may lose information when we lag our variables to get around this information loss we can use the Prais Winsten formula to transform the model Y1 1 2Y1 X 1 1 2 X 1 Lecture 19 5 Problems 2 Econ 140 3 We might want to include a lagged endogenous variable in the model Yt a bX 1 gYt 1 et including the lagged endogenous variable Yt 1 biases the Durbin Watson test towards 2 this means it s biased towards the null of no autocorrelation in this instance we ll use Durbin s h statistic 1970 n h 1 nv Lecture 19 v square of the standard error on the coefficient g of the lagged endogenous variable 6 Durbin s h statistic Econ 140 Durbin s h statistic is normally distributed and is approximated by the z statistic standard normal null hypothesis H0 0 the null can be rejected at say the 5 level of significance L19 xls has example Problems with the h statistic the product nv must be less than one where n of observations if nv 1 the h statistic is undefined Lecture 19 7 A note on consistency Econ 140 Model with lagged endogenous variable and first order serially correlated error may be mis specified Yt b0 b1Yt 1 ut and ut ut 1 et If so presence of first order serial correlation may induce omitted variable bias Need to include additional lagged endogenous variable term Yt a0 a1Yt 1 a2Yt 2 et Lecture 19 8 Why lags Econ 140 This mainly relates to macroeconomic models economic events such as consumer expenditure production or investment for instance consumer expenditure this year may be related to consumer expenditure last year In a general distributed lag model Yt a 1Yt 1 2Yt p b0Xt b1Xt 1 bkXt q et where p q lag length note problems for degrees of freedom can eliminate coefficients by using a t test or joint test using F Lecture 19 9 Why lags 2 Econ 140 Number of lags included is ad hoc Test on Causality does the X cause Y by using the Granger causality test F test on b1 to bq equaling zero Known as an ADL p q autoregressive distributed lag model of order p on dependent q on independent Lags lead to severe problems for ordinary least squares loss of information degrees of freedom independent variables X are highly correlated multicollinearity problem Lecture 19 10 Why lags are useful Econ 140 Psychological reasons behavior is habit forming so things like labor market behavior and patterns of money holding can be captured using lags Technological reasons a firm s production pattern Institutional unions Multipliers short run and long run multipliers how to read finite distributed lags in a model Lecture 19 11 Ad hoc nature of lags Econ 140 What can we do Two approaches Transform the model e g Koyck Use of information criterion Both approaches have costs and benefits Lecture 19 12 Koyck transformation Econ 140 Model Yt a b0Xt b1Xt 1 bkXt k et Note no lagged variables on the dependent variable The Koyck transformation suggests that the further back in time we go the less important is that factor for instance information from 10 years ago vs information from last year The transformation suggests b j b0 j Lecture 19 Where 0 1 j 1 k 13 Koyck transformation 2 b1 b0 So Econ 140 and b2 b0 2 Can use the expression for b j to rewrite the model Yt a b0 Xt Xt 1 2Xt 2 kXt k et 4 this imposes the assumption that earlier information is relatively less important Lagging the equation and multiplying it by we get Yt 1 a b0 Xt 1 2Xt 2 kXt k et 1 5 Subtracting 5 from 4 we get Yt a 1 b0Xt Yt 1 vt where vt et et 1 Lecture 19 14 Koyck transformation 3 Econ 140 Why is this transformation useful Allows us to take the ad hoc lag series on independent variable and condense it into a lagged endogenous variable now we only lose one observation due to the lagged endogenous variable the given by transform provides estimate of Problem by construction we have first order autocorrelation use Durbin h statistic but estimating equation might be mis specified Lecture 19 15 Information Criterion Econ 140 Determining the order of autoregression inclusion of lagged values of Y or the lag length for the variables in the model the order p and q for the ADL Same formula for both known as Bayes or Schwartz Information Criterion BIC p ln SSR p T p 1 lnT T As lags on dependent variable increase up to pmax SSR sum of squared residuals decreases The term starting p 1 increases Trade off one against the other need BIC at a minimum Same principle for q lags on independent variable Lecture 19 16 Problems with the approaches Econ 140 How do we know model of economic behavior represented by Koyck actually occurs Estimating form of model can have other interpretations e g adaptive expectations Koyck gives 1st order autocorrelation by the construction of the model use the Durbin h statistic If autocorrelation detected transform model and estimate by GLS Yt 1 and et 1 ut 1 are sure to be correlated E X e 0 this leads to biased estimates we ll deal with this using instrumental variables and simultaneous equations Lecture 19 17 Problems with the approaches Econ 140 ADL model gives no idea of how many lags should be included despite the BIC What do we do if the number of observations does not allow optimal lag lengths to be included Problem
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