AutocorrelationToday’s planReturning to the Durbin-WatsonGeneralized least squaresProblemsProblems (2)Durbin’s h-statisticA note on consistencyWhy lags?Why lags? (2)Why lags are usefulAd-hoc nature of lagsKoyck transformationKoyck transformation (2)Koyck transformation (3)Information CriterionProblems with the approachesProblems with the approachesOther topicsLecture 19 1Econ 140Econ 140AutocorrelationLecture 19Lecture 19 2Econ 140Econ 140Today’s plan•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 3Econ 140Econ 140Returning to the Durbin-Watson•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 4Econ 140Econ 140Generalized least squares•Need an estimate of : we can transform the variables such that:where: •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(3) ****tttebXaY 1*tttYYYLecture 19 5Econ 140Econ 140Problems1) The model presented by may still have some autocorrelation–the D-W test doesn’t tell us anything about this–we have to retest the model2) 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: ****tttebXaY 12*112*111XXYYLecture 19 6Econ 140Econ 140Problems (2)3) We might want to include a lagged endogenous variable in the model–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):tttegYbXaY 11nvnh1ˆv = square of the standard error on the coefficient (g) of the lagged endogenous variableLecture 19 7Econ 140Econ 140Durbin’s h-statistic•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 undefinedLecture 19 8Econ 140Econ 140A note on consistency•Model with lagged endogenous variable and first-order serially correlated error may be mis-specified.Yt = b0 + b1Yt-1 + utand 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 + etLecture 19 9Econ 140Econ 140Why lags?•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 10Econ 140Econ 140Why lags? (2)•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 [multi-collinearity problem]Lecture 19 11Econ 140Econ 140Why lags are useful•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 12Econ 140Econ 140Ad-hoc nature of lags •What can we do?•Two approaches–Transform the model (e.g. Koyck)–Use of information criterion•Both approaches have costs and benefitsLecture 19 13Econ 140Econ 140Koyck transformation•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:jjbb0Where 0 < < 1j = 1,…kLecture 19 14Econ 140Econ 140Koyck transformation (2)•So,•Can use the expression for bj to rewrite the modelYt = 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 getYt = a(1- ) + b0Xt + Yt-1 + vt where vt = et - et-1 and 20201bbbb Lecture 19 15Econ 140Econ 140Koyck transformation (3)•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 16Econ 140Econ 140Information Criterion•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
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