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Autocorrelation: Remedies. Aims and Learning Objectives. By the end of this session students should be able to: Use the generalised least squares procedure to deal with autocorrelation Understand the various ways  can be estimated Describe other ways of dealing with the

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Presentation Transcript
slide2

Aims and Learning Objectives

  • By the end of this session students should be able to:
  • Use the generalised least squares procedure to deal
  • with autocorrelation
  • Understand the various ways  can be estimated
  • Describe other ways of dealing with the
  • autocorrelation problem
slide3

Introduction

We said in Lecture 13 that when the errors are

correlated the OLS estimators are inefficient

(they are LUE rather than BLUE)

In order to remedy the situation we need to know

something about the nature of the interdependence

between the disturbance terms

slide4

Regression Model

Yt = 1 + 2X2t + 3X3t + Ut

Cov (Ut, Ut-s)

or E(Ut, Ut-s) = 0

No autocorrelation:

Cov (Ut, Ut-s)  0

or E(Ut, Ut-s)  0

Autocorrelation:

slide5

Remedies

  • Respecification: Include lagged variables and
  • dummies (particularly if working with seasonal
  • data)
  • Generalised Least Squares
  • Newey-West Robust Standard Errors
slide6

Generalised Least Squares

AR(1) :

Ut = Ut1 + t

substitute

in for Ut

Yt = 1 + 2X2t + 3X3t + Ut

Yt = 1 + 2X2t +3X3t +Ut1 + t

Now we need to “get rid” of Ut1

(continued)

slide7

Yt = 1 + 2X2t +3X3t +Ut1 + t

Yt = 1 + 2X2t + 3X3t + Ut

lag the

errors

once

Ut = Yt1 - 2X2t3X3t

Ut1 = Yt1 1 - 2X2t-1- 3X3t-1

Yt = 1 + 2X2t +3X3t +

Yt11 - 2X2t-13X3t-1+ t

(continued)

slide8

Yt = 1 + 2X2t +3X3t +

Yt11 - 2X2t-13X3t-1+ t

slide9

Problems estimating this model:

1. One observation is used up in creating the transformed (lagged) variables leaving only (n1) observations for estimating the model.

2. The value of  is not known. We must find some way to estimate it.

slide10

Estimating Unknown  Value

  • Estimated  from OLS residuals
  • Estimated  from Durbin-Watson d Statistic
  • Cochrane-Orcutt Method for estimating 
slide11

et = et1 + t

et = Yt - 1 - 2X2t - 3X3t

Estimating  from OLS residuals

First, use least squares to estimate the model:

^

^

^

Yt = 1 + 2X2t + 3X3t + et

The residuals from this estimation are:

^

^

^

Next, estimate the following by least squares:

Use this  to run (estimated) GLS by substituting it into

slide12

^

d  2(1)

Estimating  from Durbin-Watson d Statistic

Recall from lecture 13:

Therefore

Use this  to run (estimated) GLS by substituting it into

slide13

et = et1 + t

Estimating  Using the Cochrane-Orcutt Procedure

More accurate method for obtaining 

Step 1: estimate the regression and obtain the

residuals

Step 2: estimate

Step 3: Use this  to run (estimated) GLS by

substituting it into

^

slide14

Estimating  Using the Cochrane-Orcutt Procedure

Step 4: The previous steps are repeated (iterated)

until further iteration results in little change

in  (we say it has converged)

^

This involves substituting the values of obtained

in step 3 into the original regression (estimated in

step 1)

slide15

Other Remedies

  • Re-specification:
  • Including other variables and their lags may remove
  • autocorrelation arising from misspecification
  • Newey-West Robust Standard Errors:
  • Focuses on adjusting the standard errors
  • of the estimates
  • (These procedures can be implemented in Microfit)
slide16

Summary

In this lecture we have:

1. Discussed how the GLS procedure can be used

to remove problems associated with

autocorrelated disturbances

2. Discussed practical ways of estimating 

3. Outlined alternative methods for dealing with

autocorrelation