Transcript No Slide Title

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]
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ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
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.
1
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
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 .
2
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
We now subtract the second equation from the first.
3
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
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.
4
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
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.
5
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yˆt  100  0.5Yt 1  0.8 X t  0.6 X t 1
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.
6
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yˆt  100  0.5Yt 1  0.8 X t  0.6 X t 1
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.
7
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
We therefore need to use a nonlinear estimation technique. Before doing this, we will
extend the model to multiple regression with two explanatory variables.
8
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
Yt 1   1    2 X 2 t 1   3 X 3 t 1  ut 1
The procedure is the same. Write the model a second time, lagged one time period, and
multiply through by .
9
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
Yt 1   1    2 X 2 t 1   3 X 3 t 1  ut 1
Yt  Yt 1   1 (1   )   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1  ut  ut 1
Subtract the second equation from the first.
10
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
Yt 1   1    2 X 2 t 1   3 X 3 t 1  ut 1
Yt  Yt 1   1 (1   )   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
Again, we obtain a model that is free from autocorrelation.
11
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
Yt 1   1    2 X 2 t 1   3 X 3 t 1  ut 1
Yt  Yt 1   1 (1   )   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
Now there are two restrictions. One involves the coefficients of Yt–1, X2t, and X2t–1.
12
ELIMINATING AR(1) AUTOCORRELATION
Yt   1   2 X t  ut
ut  ut 1   t
Yt 1   1    2 X t 1  ut 1
Yt  Yt 1   1 (1   )   2 X t   2 X t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X t   2 X t 1   t
Yt   1   2 X 2 t   3 X 3 t  ut
ut  ut 1   t
Yt 1   1    2 X 2 t 1   3 X 3 t 1  ut 1
Yt  Yt 1   1 (1   )   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1  ut  ut 1
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
The other involves the coefficients of Yt–1, X3t, and X3t–1.
13
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
Here is the output for a logarithmic regression of expenditure on housing services on
income and price, assuming an AR(1) process, using EViews.
14
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
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.
15
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
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.
16
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
Here 1 has been denoted C(1).
17
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
, 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.
18
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
2, the coefficient of income, has been denoted C(3). The estimate is close to the OLS
estimate, 1.03.
19
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
The coefficient of lagged income must then be specified as –C(2)*C(3).
20
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
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.)
21
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
The coefficient of lagged price must then be specified as –C(2)*C(4).
22
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
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
============================================================
R-squared
0.999205
Mean dependent var 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
Durbin-Watson stat 1.901081
============================================================
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.
23
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample(adjusted): 1960 2003
Included 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 6.379059
Adjusted R-squared
0.999145
S.D. dependent var 0.421861
S.E. of regression
0.012333
Akaike info criter-5.866567
Sum squared resid
0.006084
Schwarz criterion -5.704368
Log likelihood
133.0645
F-statistic
16757.24
Durbin-Watson stat
1.901081
Prob(F-statistic) 0.000000
============================================================
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.
24
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
=============================================================
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
============================================================
The constant is an estimate of 1.
25
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
=============================================================
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
============================================================
The income coefficient is the estimate of the elasticity with respect to current income..
26
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
=============================================================
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
============================================================
The price coefficient is the estimate of the elasticity with respect to current price.
27
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
=============================================================
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
============================================================
The coefficient of AR(1) is an estimate of .
28
ELIMINATING AR(1) AUTOCORRELATION
Yt   1 (1   )  Yt 1   2 X 2 t   2 X 2 t 1   3 X 3 t   3 X 3 t 1   t
=============================================================
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
============================================================
The coefficients of lagged income and lagged price are not reported because they are
implicit in the estimates of , 2, and 3.
29
Copyright Christopher Dougherty 2011.
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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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11.07.25