Lecture 21 1

Econ 140Econ 140

Autocorrelation IILecture 21

Lecture 21 2

Econ 140Econ 140Today’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 3

Econ 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 data in L_20.xls• We used this figure:

2 40

H1

dL dU

H0: =0Reject null Accept null

4-dU 4-dL

Reject nullH1

d = 0.3311.475

Lecture 21 4

Econ 140Econ 140Generalized 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

(3) * ***ttt ebXaY

1*

ttt YYY

Lecture 21 5

Econ 140Econ 140Problems

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:

* ***ttt ebXaY

12*

1

12*

1

1

1

XX

YY

Lecture 21 6

Econ 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):

ttt egYbXaY 11

nvnh

1

v = square of the standard error on the coefficient (g) of the lagged endogenous variable

Lecture 21 7

Econ 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 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 8

Econ 140Econ 140A 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 = 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 21 9

Econ 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 + 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 10

Econ 140Econ 140Problems 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 11

Econ 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 21 12

Econ 140Econ 140Ad-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 13

Econ 140Econ 140Koyck 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:j

j bb 0 Where 0 < < 1j = 1,…k

Lecture 21 14

Econ 140Econ 140Koyck 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 getYt = a(1- ) + b0Xt + Yt-1 + vt where vt = et - et-1

and 20201 bbbb

Lecture 21 15

Econ 140Econ 140Koyck 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 16

Econ 140Econ 140Adaptive 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 17

Econ 140Econ 140Adaptive expectations (2)

• Using adaptive expectations we have thatXt* - 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

Xt* = expectationXt = observed0 < < 1

where

Lecture 21 18

Econ 140Econ 140Adaptive 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

tt XL

X

1

*

Lecture 21 19

Econ 140Econ 140Adaptive 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

ttt uXL

baY

1

tttt vYXbaY 11

where 11 ttt uuv

Lecture 21 20

Econ 140Econ 140Problems 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 21

Econ 140Econ 140Other 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.