christopher dougherty ec220 - introduction to econometrics (chapter 12) slideshow: eliminating ar(1)...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 12)Slideshow: eliminating AR(1) autocorrelation

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource]

© 2012 The Author

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Available in LSE Learning Resources Online: May 2012

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ELIMINATING AR(1) AUTOCORRELATION

ttt uXY 21 ttt uu 1

1

This sequence shows how AR(1) autocorrelation can be eliminated from a regression model. The AR(1) process is the equation at the top right. We will start with the simple regression model, top left.


ttt uXY 21 ttt uu 1

11211 ttt uXY

2

If the regression model is valid at time t, it is also valid at time t–1. For reasons that will become obvious in a moment, we have multiplied through the second equation by .


ttt uXY 21 ttt uu 1

11211 ttt uXY

112211 )1( tttttt uuXXYY

3

We now subtract the second equation from the first.


ttt uXY 21 ttt uu 1

11211 ttt uXY


ttttt XXYY 12211 )1(

4

The disturbance term now reduces to t, the innovation at time t in the AR(1) process. By assumption, this is independently distributed, so the problem of autocorrelation has been eliminated.


ttt uXY 21 ttt uu 1

11211 ttt uXY



5

There is one minor problem. The revised specification involves a nonlinear restriction. The coefficient of Xt–1 is minus the product of the coefficients of Xt and Yt–1.


ttt uXY 21 ttt uu 1

11211 ttt uXY



11 6.08.05.0100ˆ tttt XXYY

6

This means that we should not try to fit the equation using ordinary least squares. OLS would not take account of the restriction and so we would end up with conflicting estimates of the parameters.


ttt uXY 21 ttt uu 1

11211 ttt uXY



7

For example, we might obtain the equation shown. From it we could deduce estimates of 0.5 for and 0.8 for 2. But these numbers would be incompatible with the estimate of 0.6 for 2.

11 6.08.05.0100ˆ tttt XXYY


ttt uXY 21 ttt uu 1

11211 ttt uXY



tttt uXXY 33221 ttt uu 1

8

We therefore need to use a nonlinear estimation technique. Before doing this, we will extend the model to multiple regression with two explanatory variables.


ttt uXY 21 ttt uu 1

11211 ttt uXY



tttt uXXY 33221

113312211 tttt uXXY

ttt uu 1

9

The procedure is the same. Write the model a second time, lagged one time period, and multiply through by .


ttt uXY 21 ttt uu 1

11211 ttt uXY



tttt uXXY 33221

113312211 tttt uXXY

1133331222211 )1( tttttttt uuXXXXYY

ttt uu 1

10

Subtract the second equation from the first.


ttt uXY 21 ttt uu 1

11211 ttt uXY



11

Again, we obtain a model that is free from autocorrelation.

tttt uXXY 33221

113312211 tttt uXXY


ttttttt XXXXYY 133331222211 )1(

ttt uu 1


ttt uXY 21 ttt uu 1

11211 ttt uXY



tttt uXXY 33221

113312211 tttt uXXY



ttt uu 1

12

Now there are two restrictions. One involves the coefficients of Yt–1, X2t, and X2t–1.


ttt uXY 21 ttt uu 1

11211 ttt uXY



tttt uXXY 33221

113312211 tttt uXXY



ttt uu 1

13

The other involves the coefficients of Yt–1, X3t, and X3t–1.

============================================================Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000============================================================R-squared 0.999205 Mean dependent var 6.379059Adjusted R-squared 0.999145 S.D. dependent var 0.421861S.E. of regression 0.012333 Akaike info criter-5.866567Sum squared resid 0.006084 Schwarz criterion -5.704368Log likelihood 133.0645 Durbin-Watson stat 1.901081============================================================

14


Here is the output for a logarithmic regression of expenditure on housing services on income and price, assuming an AR(1) process, using EViews.


15


EViews allows two ways of specifying a regression equation. One is to list the variables, starting with the dependent variable, continuing with C for the intercept, and finishing with a list of the explanatory variables. This is fine for linear regressions.

ttttttt XXXXYY 133331222211 )1(============================================================Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000============================================================R-squared 0.999205 Mean dependent var 6.379059Adjusted R-squared 0.999145 S.D. dependent var 0.421861S.E. of regression 0.012333 Akaike info criter-5.866567Sum squared resid 0.006084 Schwarz criterion -5.704368Log likelihood 133.0645 Durbin-Watson stat 1.901081============================================================

16


The other method is to write the model as an equation, referring to the parameters as C(1), C(2), etc. This is what you should do when fitting a nonlinear model, such as the present one.


17


Here 1 has been denoted C(1).


18


, the coefficient of the lagged dependent variable, has been denoted C(2). It is also a component of the intercept in this model. The estimate of , 0.72, is quite high.


19


2, the coefficient of income, has been denoted C(3). The estimate is close to the OLS estimate, 1.03.


20


The coefficient of lagged income must then be specified as –C(2)*C(3).


21


3, the coefficient of price, has been denoted C(4). The estimate is the same as the OLS estimate, –0.48, at least to two decimal places. (This is a bit of a coincidence.)


22


The coefficient of lagged price must then be specified as –C(2)*C(4).


23


The only problem with this method of fitting the AR(1) model is that specifying the model in equation form is a tedious task and it is easy to make mistakes.


============================================================Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003Included observations: 44 after adjusting endpoints Convergence achieved after 21 iterations ============================================================ Variable Coefficient Std. Error t-Statistic Prob.============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000============================================================R-squared 0.999205 Mean dependent var 6.379059Adjusted R-squared 0.999145 S.D. dependent var 0.421861S.E. of regression 0.012333 Akaike info criter-5.866567Sum squared resid 0.006084 Schwarz criterion -5.704368Log likelihood 133.0645 F-statistic 16757.24Durbin-Watson stat 1.901081 Prob(F-statistic) 0.000000============================================================

24


Since the AR(1) specification is a common one, most serious regression applications provide some short-cut for specifying it easily. In the case of EViews, AR(1) estimation is invoked by adding AR(1) to the list of explanatory variables.


=============================================================Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000======================================================================================================================== Variable Coefficient Std. Error t-Statistic Prob.============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000============================================================

25


The constant is an estimate of 1.


26


The income coefficient is the estimate of the elasticity with respect to current income..

ttttttt XXXXYY 133331222211 )1(=============================================================Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000======================================================================================================================== Variable Coefficient Std. Error t-Statistic Prob.============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000============================================================

27


The price coefficient is the estimate of the elasticity with respect to current price.


28


The coefficient of AR(1) is an estimate of .


29


The coefficients of lagged income and lagged price are not reported because they are implicit in the estimates of , 2, and 3.


Copyright Christopher Dougherty 2011.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

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The content of this slideshow comes from Section 12.3 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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Individuals studying econometrics on their own and who feel that they might

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EC212 Introduction to Econometrics

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20 Elements of Econometrics

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11.07.25

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christopher dougherty ec220 - introduction to econometrics (chapter 12) slideshow: eliminating ar(1)...

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