EC 761 Fall 1998
Available from the class homepage:
http://fmwww.bc.edu/ec-c/f98/ec761.f98.html
This handout reproduces the Stata documentation for the 'colldiag' command and
then illustrates, with a constructed regressor, how increasing correlation
among regressors increases the maximal condition index and generates
diagnostics for its identification.
-------------------------------------------------------------------------------
help for colldiag (STB-25: sg32.1)
-------------------------------------------------------------------------------
Collinearity Diagnostics
------------------------
colldiag [nocons]
Description
-----------
colldiag calculates and displays the matrix of variance decomposition
proportions for the independent variables in a linear regression model. This
command must follow a call to fit.
Remarks
-------
In the case of orthogonal predictors, the variance decomposition proportion
matrix would be an identity matrix. One should examine the dependencies of
the variances on the principal components, by focusing on the decomposition
of the variables associated with high condition numbers. The condition number
is a measure of the dependence of the independent variables. Typical values
used are (n* = ) 10, 15, or even 30. As, you look at the row associated with
the high condition numbers, you should note the variance decomposition propor-
tions that are higher than some threshold value (like p*=.50).
You should note the following:
1) The independent variable will have a degraded coefficient because
of a near dependency if it is one of two or more variates with
variance-decomposition proportions in excess of some threshold value
p*, such as .50. The number of near dependencies is the number of
condition numbers greater than the threshold value n*.
2) Those variates whose aggregate variance-decomposition proportion
exceed the threshold value p* are involved in at least one of
the dependencies. The aggregate is formed over the competing condition
numbers (condition numbers of the same order of magnitude that exceed
the threshold value n*).
3) A dominating dependency occurs when the condition number is an
order of magnitude larger than the other condition numbers. This can
obscure information about the variate's simultaneous involvement in a
weaker dependency. In this case, additional analysis is warranted to
investigate the relationships of all potentially involved variates.
Example
-------
We have data on men involved in a physical fitness course. The purpose of
the study is to model the oxygen uptake rate by the age, weight, time
to run one and a half miles, the heart rate while resting, heart rate while
running, and the maximum heart rate while running.
de
Contains data from fitness.dta
Obs: 31 (max= 50172) Fitness data
Vars: 7 (max= 99) 16 Nov 1994 15:47
Width: 28 (max= 200)
1. age float %9.0g
2. weight float %9.0g
3. oxy float %9.0g
4. runtime float %9.0g
5. rstpulse float %9.0g
6. runpulse float %9.0g
7. maxpulse float %9.0g
Sorted by:
fit oxy age weight runtime rstpulse runpulse maxpulse
Source | SS df MS Number of obs = 31
---------+------------------------------ F( 6, 24) = 22.43
Model | 722.543528 6 120.423921 Prob > F = 0.0000
Residual | 128.837947 24 5.3682478 R-squared = 0.8487
---------+------------------------------ Adj R-squared = 0.8108
Total | 851.381475 30 28.3793825 Root MSE = 2.3169
------------------------------------------------------------------------------
oxy | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
age | -.2269738 .0998375 -2.273 0.032 -.4330282 -.0209194
weight | -.0741774 .0545932 -1.359 0.187 -.1868521 .0384974
runtime | -2.628653 .3845622 -6.835 0.000 -3.42235 -1.834955
rstpulse | -.0215336 .0660543 -0.326 0.747 -.1578629 .1147957
runpulse | -.3696278 .1198529 -3.084 0.005 -.6169921 -.1222634
maxpulse | .3032171 .1364952 2.221 0.036 .0215049 .5849294
_cons | 102.9345 12.40326 8.299 0.000 77.33541 128.5335
------------------------------------------------------------------------------
We are somewhat concerned that there may be a dependency among the pulse
variables and investigate this with the new diagnostic tool.
colldiag
Proportion of variance associated with the decomposition
Cond |
Number | age weight runtime rstpulse runpulse maxpulse _cons
---------+--------------------------------------------------------------------
1 | 0.0002 0.0002 0.0002 0.0003 0.0000 0.0000 0.0000
19.2909 | 0.1463 0.0104 0.0252 0.3906 0.0000 0.0000 0.0022
21.5007 | 0.1501 0.2357 0.1286 0.0281 0.0012 0.0012 0.0006
27.6212 | 0.0319 0.1831 0.6090 0.1903 0.0015 0.0012 0.0064
33.8292 | 0.1128 0.4444 0.1250 0.3648 0.0151 0.0083 0.0013
82.6376 | 0.4966 0.1033 0.0975 0.0203 0.0695 0.0056 0.7997
196.786 | 0.0621 0.0228 0.0146 0.0057 0.9128 0.9836 0.1898
If we use 30 as our value for n* and .50 as our threshold for p*, then we see
that points 2 and 3 from above are exhibited in our output. The competing
dependency is for the condition numbers 33.8292 and 82.6376 which are of the
same order of magnitude and both exceed our threshold value of 30. Aggregating
the variance-decomposition proportions, we note that age (.1128+.4966=.6014),
weight (.4444+.1033=.5477), and the constant (.0013+.7997=.8010) are involved
in a competing dependency. We also note that we have a dominating dependency
with a condition number greater than 196 and involving the runpulse and
maxpulse variables.
Since we have 3 near dependencies (3 condition numbers greater than n*=30), we
should be able to express 3 of our independent variables in terms of the
remaining 4. How do we choose the variates for which to solve? Beginning with
the largest condition number, we see that we should choose either runpulse or
maxpulse. Since maxpulse has the remainder of it variance determined in a more
removed dependency, we can choose it as our first dependent variable in the
auxiliary regression. Now, since we are not as interested in the constant
term, we may choose the weight and age as our remaining pivots.
Author
------
James W. Hardin, Stata Corporation
stata@stata.com
See Also
--------
STB: STB-25 sg32.1, STB-24 sg32
Manual: [5s] fit
On-line: fit, vif if installed
EXAMPLE OF COLLINEARITY INCREASING WITH CORRELATION OF REGRESSORS
do ":Keewaydin:Desktop Folder:colldiag.do"
. * colldiag EC761 cfb 8902
. set matsize 800
. use ":Keewaydin:Stata:auto.dta"
(1978 Automobile Data)
. gen pr=price/1000
. gen e = 100*invnorm(uniform())
. gen mpga=mpg+e
. corr mpg mpga
(obs=74)
| mpg mpga
--------+------------------
mpg| 1.0000
mpga| 0.1570 1.0000
. fit pr displ mpg
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 2, 71) = 13.35
Model | 173.587101 2 86.7935503 Prob > F = 0.0000
Residual | 461.478281 71 6.4996941 R-squared = 0.2733
---------+------------------------------ Adj R-squared = 0.2529
Total | 635.065382 73 8.69952578 Root MSE = 2.5494
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0105088 .0045855 2.292 0.025 .0013657 .019652
mpg | -.1211832 .0727884 -1.665 0.100 -.2663193 .0239528
_cons | 6.672765 2.29972 2.902 0.005 2.087254 11.25828
------------------------------------------------------------------------------
. fit pr displ mpg mpga
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 3, 70) = 8.96
Model | 176.159644 3 58.7198815 Prob > F = 0.0000
Residual | 458.905737 70 6.55579625 R-squared = 0.2774
---------+------------------------------ Adj R-squared = 0.2464
Total | 635.065382 73 8.69952578 Root MSE = 2.5604
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0103511 .0046121 2.244 0.028 .0011526 .0195497
mpg | -.1177836 .0733031 -1.607 0.113 -.2639819 .0284148
mpga | -.001717 .0027409 -0.626 0.533 -.0071835 .0037496
_cons | 6.648921 2.309938 2.878 0.005 2.041896 11.25595
------------------------------------------------------------------------------
. lincom mpg+mpga
( 1) mpg + mpga = 0.0
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | -.1195005 .0731512 -1.634 0.107 -.2653961 .026395
------------------------------------------------------------------------------
. colldiag
Proportion of variance associated with the decomposition
Cond |
Number | displ mpg mpga _cons
---------+--------------------------------------
1 | 0.0098 0.0040 0.0021 0.0021
1.67485 | 0.0012 0.0000 0.9532 0.0000
3.74772 | 0.2611 0.0653 0.0439 0.0015
16.3562 | 0.7278 0.9307 0.0007 0.9964
. gen mpgb=mpg+e/10
. corr mpg mpgb
(obs=74)
| mpg mpgb
--------+------------------
mpg| 1.0000
mpgb| 0.5358 1.0000
. fit pr displ mpg mpgb
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 3, 70) = 8.96
Model | 176.159645 3 58.7198816 Prob > F = 0.0000
Residual | 458.905737 70 6.55579624 R-squared = 0.2774
---------+------------------------------ Adj R-squared = 0.2464
Total | 635.065382 73 8.69952578 Root MSE = 2.5604
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0103511 .0046121 2.244 0.028 .0011526 .0195497
mpg | -.1023308 .0790545 -1.294 0.200 -.26 .0553384
mpgb | -.0171697 .0274091 -0.626 0.533 -.0718354 .037496
_cons | 6.648921 2.309938 2.878 0.005 2.041896 11.25595
------------------------------------------------------------------------------
. lincom mpg+mpgb
( 1) mpg + mpgb = 0.0
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | -.1195005 .0731512 -1.634 0.107 -.2653961 .026395
------------------------------------------------------------------------------
. colldiag
Proportion of variance associated with the decomposition
Cond |
Number | displ mpg mpgb _cons
---------+--------------------------------------
1 | 0.0056 0.0022 0.0134 0.0013
3.19432 | 0.1292 0.0033 0.2548 0.0010
6.02763 | 0.1645 0.1147 0.7129 0.0163
18.5765 | 0.7006 0.8799 0.0189 0.9814
. gen mpgc=mpg+e/100
. corr mpg mpgc
(obs=74)
| mpg mpgc
--------+------------------
mpg| 1.0000
mpgc| 0.9832 1.0000
. fit pr displ mpg mpgc
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 3, 70) = 8.96
Model | 176.159648 3 58.7198828 Prob > F = 0.0000
Residual | 458.905734 70 6.55579619 R-squared = 0.2774
---------+------------------------------ Adj R-squared = 0.2464
Total | 635.065382 73 8.69952578 Root MSE = 2.5604
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0103511 .0046121 2.244 0.028 .0011526 .0195497
mpg | .0521968 .2862681 0.182 0.856 -.5187469 .6231405
mpgc | -.1716973 .2740909 -0.626 0.533 -.7183543 .3749596
_cons | 6.648921 2.309938 2.878 0.005 2.041896 11.25595
------------------------------------------------------------------------------
. lincom mpg+mpgc
( 1) mpg + mpgc = 0.0
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | -.1195006 .0731512 -1.634 0.107 -.2653961 .026395
------------------------------------------------------------------------------
. colldiag
Proportion of variance associated with the decomposition
Cond |
Number | displ mpg mpgc _cons
---------+--------------------------------------
1 | 0.0052 0.0002 0.0002 0.0012
3.72551 | 0.2208 0.0013 0.0015 0.0001
17.2937 | 0.7714 0.0110 0.0195 0.9743
56.1544 | 0.0026 0.9875 0.9787 0.0244
. gen mpgd=mpg+e/1000
. corr mpg mpgd
(obs=74)
| mpg mpgd
--------+------------------
mpg| 1.0000
mpgd| 0.9998 1.0000
. fit pr displ mpg mpgd
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 3, 70) = 8.96
Model | 176.159594 3 58.7198647 Prob > F = 0.0000
Residual | 458.905788 70 6.55579697 R-squared = 0.2774
---------+------------------------------ Adj R-squared = 0.2464
Total | 635.065382 73 8.69952578 Root MSE = 2.5604
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0103511 .0046121 2.244 0.028 .0011526 .0195497
mpg | 1.597456 2.744571 0.582 0.562 -3.876417 7.071329
mpgd | -1.716956 2.740911 -0.626 0.533 -7.18353 3.749617
_cons | 6.64892 2.309938 2.878 0.005 2.041895 11.25595
------------------------------------------------------------------------------
. lincom mpg+mpgd
( 1) mpg + mpgd = 0.0
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | -.1195005 .0731512 -1.634 0.107 -.2653961 .026395
------------------------------------------------------------------------------
. colldiag
Proportion of variance associated with the decomposition
Cond |
Number | displ mpg mpgd _cons
---------+--------------------------------------
1 | 0.0052 0.0000 0.0000 0.0012
3.74763 | 0.2240 0.0000 0.0000 0.0001
17.4837 | 0.7688 0.0002 0.0002 0.9978
554.506 | 0.0020 0.9998 0.9998 0.0009
. gen mpge=mpg+e/10000
. corr mpg mpge
(obs=74)
| mpg mpge
--------+------------------
mpg| 1.0000
mpge| 1.0000 1.0000
. fit pr displ mpg mpge
Source | SS df MS Number of obs = 74
---------+------------------------------ F( 3, 70) = 8.96
Model | 176.159898 3 58.719966 Prob > F = 0.0000
Residual | 458.905484 70 6.55579262 R-squared = 0.2774
---------+------------------------------ Adj R-squared = 0.2464
Total | 635.065382 73 8.69952578 Root MSE = 2.5604
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
displ | .0103512 .0046121 2.244 0.028 .0011526 .0195497
mpg | 17.05096 27.4117 0.622 0.536 -37.61994 71.72186
mpge | -17.17046 27.40891 -0.626 0.533 -71.83581 37.49488
_cons | 6.648902 2.309937 2.878 0.005 2.041877 11.25593
------------------------------------------------------------------------------
. lincom mpg+mpge
( 1) mpg + mpge = 0.0
------------------------------------------------------------------------------
pr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | -.1194998 .0731513 -1.634 0.107 -.2653954 .0263957
------------------------------------------------------------------------------
. colldiag
Proportion of variance associated with the decomposition
Cond |
Number | displ mpg mpge _cons
---------+--------------------------------------
1 | 0.0052 0.0000 0.0000 0.0012
3.74956 | 0.2243 0.0000 0.0000 0.0001
17.4885 | 0.7677 0.0000 0.0000 0.9984
5543.13 | 0.0029 1.0000 1.0000 0.0003
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end of do-file
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