Autocorrelation II Lecture 21 Lecture 21

Today’s plan Durbin’s h-statistic Finite Distributed Lags Koyck Transformations and Adaptive Expectations Seasonality Testing in the presence of higher order serially correlated forms. Lecture 21

Returning to the Durbin-Watson Last time we talked about how to test for autocorrelation using the Durbin-Watson test We found autocorrelation in the data in L_20.xls We used this figure: 2 4 H1 dL dU H0:  =0 Reject null Accept null 4-dU 4-dL d = 0.331 1.475 Lecture 21

Generalized least squares (3) Need an estimate of  : we can transform the variables such that: where: Known as Cochrane-Orcutt transformation. Estimating equation (3) allows us to estimate in the presence of first-order autocorrelation Lecture 21

Problems 1) 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 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: Lecture 21

Problems (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): v = square of the standard error on the coefficient (g) of the lagged endogenous variable Lecture 21

Durbin’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 the 5% level of significance L21.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 21

A note on consistency Model with lagged endogenous variable and first-order serially correlated error may be mis-specified. Yt = b0 + b1Yt-1 + ut and ut = rut-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 21

Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et Why 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 + b0Xt + b1Xt-1 +…+bkXt-k + et where k = any large number less than t-2 can eliminate coefficients b1 to bk by using a t-test number of lags included is ad-hoc Lecture 21

Problems for OLS 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 21

Why 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 21

Ad-hoc nature of lags What can we do? Two approaches Koyck transformation Adaptive expectations Different implications on the assumptions about economic processes will end up with the same estimating equation looking only at the end product, we won’t be able to tell the Koyck transformation from adaptive expectations Lecture 21

Koyck transformation Model: Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et 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: Where 0 <  < 1 j = 1,…k Lecture 21

Koyck transformation (2) So, Can use the expression for bj 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 21

Koyck transformation (3) Why is this transformation useful? Allows us to take the ad-hoc lag series and condense it into a lagged endogenous variable now we only lose one observation due to the lagged endogenous variable the  given by the estimation gives the coefficient of autocorrelation Problem: by construction, we have first-order autocorrelation use Durbin h-statistic but estimating equation might be mis-specified! Lecture 21

Adaptive expectations Another way to approach the problem of the ad-hoc nature of lags Can use the example of trying to measure the natural rate of unemployment In 1968, Friedman estimated the equation: Yt = a + bXt* + ut where Xt* = natural rate of unemployment Lecture 21

Adaptive expectations (2) Using adaptive expectations we have that Xt* - Xt-1* =  (Xt - Xt-1*) Can rewrite the equation: Xt* - (1 -  )Xt-1* =  Xt Using a lag operator where: LXt = Xt-1 L2Xt = Xt-2 where Xt* = expectation Xt = observed 0 <  < 1 Lecture 21

Adaptive expectations (3) We can then rewrite :  Xt = (1 - L )Xt* where  = (1-  ) This can be rewritten as: now we have the natural rate of unemployment in terms of the observed rate of unemployment Lecture 21

Adaptive expectations (3) Substituting into the model we get: Upon further multiplication and substitution we arrive at: this looks very similar to that for the Koyck transformation where Lecture 21

Problems with the approaches For the lagged endogenous variables in the ad-hoc lag structure, we are uncertain as to which economic model of agent behavior underlies the estimating equation We have 1st-order autocorrelation by the construction of the model use the Durbin h-statistic 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 21

Other topics Seasonality and the use of dummy variables in time series models. Trends and their use in time series models Testing and correcting in the presence of higher orders of serial correlation. Lecture 21