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Re: st: problem with squared term
From
Michael Norman Mitchell <[email protected]>
To
[email protected]
Subject
Re: st: problem with squared term
Date
Sat, 20 Mar 2010 11:32:15 -0700
Dear Prabhat
The coefficient for kavgtemp is the linear effect of average
temperature **when all other variables are held constant at zero**. This
influences the size of the coefficient when a variable is interacted
with (multiplied by) other variables. In this case, it is when kavgtemp
is multiplied by itself, forming the squared term. So, kavgtemp reflects
the instantaneous linear slope when average temperature is equal to 0.
The because of the squared term, the linear slope will change over
the values of average temp. So, perhaps you might want to see the linear
slope when the average temp is at the mean. Using the "auto" dataset,
here is an example showing weight predicting mpg. The first example is
like yours, where the coefficient for weight changes, and the second
example uses centering around the mean.
. ***
. * Example 1
. clear
. sysuse auto
(1978 Automobile Data)
. generate wt1k = weight / 1000
. generate wt1ksq = wt1k*wt1k
.
. regress mpg wt1k
Source | SS df MS Number of obs
= 74
-------------+------------------------------ F( 1, 72) =
134.62
Model | 1591.99024 1 1591.99024 Prob > F =
0.0000
Residual | 851.469221 72 11.8259614 R-squared =
0.6515
-------------+------------------------------ Adj R-squared =
0.6467
Total | 2443.45946 73 33.4720474 Root MSE =
3.4389
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
wt1k | -6.008687 .5178782 -11.60 0.000 -7.041058
-4.976316
_cons | 39.44028 1.614003 24.44 0.000 36.22283
42.65774
------------------------------------------------------------------------------
. regress mpg wt1k wt1ksq
Source | SS df MS Number of obs
= 74
-------------+------------------------------ F( 2, 71) =
72.80
Model | 1642.522 2 821.261002 Prob > F =
0.0000
Residual | 800.937455 71 11.2808092 R-squared =
0.6722
-------------+------------------------------ Adj R-squared =
0.6630
Total | 2443.45946 73 33.4720474 Root MSE =
3.3587
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
wt1k | -14.15806 3.883535 -3.65 0.001 -21.90161
-6.414512
wt1ksq | 1.324401 .6257594 2.12 0.038 .0766722
2.57213
_cons | 51.18308 5.767884 8.87 0.000 39.68225
62.68391
------------------------------------------------------------------------------
. ***
. * Example 2, center wt1k
. summarize wt1k
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
wt1k | 74 3.019459 .7771936 1.76 4.84
. generate cwt1k = wt1k - r(mean)
. generate cwt1ksq = cwt1k*cwt1k
.
. regress mpg cwt1k
Source | SS df MS Number of obs
= 74
-------------+------------------------------ F( 1, 72) =
134.62
Model | 1591.99025 1 1591.99025 Prob > F =
0.0000
Residual | 851.469214 72 11.8259613 R-squared =
0.6515
-------------+------------------------------ Adj R-squared =
0.6467
Total | 2443.45946 73 33.4720474 Root MSE =
3.4389
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
cwt1k | -6.008687 .5178782 -11.60 0.000 -7.041058
-4.976316
_cons | 21.2973 .3997628 53.27 0.000 20.50038
22.09421
------------------------------------------------------------------------------
. regress mpg cwt1k cwt1ksq
Source | SS df MS Number of obs
= 74
-------------+------------------------------ F( 2, 71) =
72.80
Model | 1642.52201 2 821.261005 Prob > F =
0.0000
Residual | 800.93745 71 11.2808092 R-squared =
0.6722
-------------+------------------------------ Adj R-squared =
0.6630
Total | 2443.45946 73 33.4720474 Root MSE =
3.3587
------------------------------------------------------------------------------
mpg | Coef. Std. Err. t P>|t| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
cwt1k | -6.160112 .5108358 -12.06 0.000 -7.178689
-5.141534
cwt1ksq | 1.324401 .6257594 2.12 0.038 .0766721
2.57213
_cons | 20.50813 .5398843 37.99 0.000 19.43163
21.58463
------------------------------------------------------------------------------
Note, now, the coefficient for cwt1k is the linear effect of weight
on mpg when weight is at the average. If you choose a higher or lower
value for the centering (say 1sd above the mean, or 1sd below the mean),
you will get different values.
I hope this helps,
Michael N. Mitchell
See the Stata tidbit of the week at...
http://www.MichaelNormanMitchell.com
On 2010-03-20 10.42 AM, Prabhat wrote:
Dear all,
I have come across a strange problem. I am trying to estimate the
coefficients for temperature and rainfall using WLS, where my
dependent variable is rice yield.
Now, when I am including the square term for the average temperature,
I am getting a very high and impossible estimate for the temperature
variable. It should be somewhere between 100-300, but after including
square term I am getting -3500.
I have just included OLS results here.
Any comment will be appreciated.
Regards,
Prabhat Barnwal
International University of Japan
. regress kyrice kavgtemp kavgtempsq ktotppt ktotpptsq ksdtemp
Source | SS df MS Number of obs = 735
-------------+------------------------------ F( 5, 729) = 13.16
Model | 21551254.8 5 4310250.95 Prob> F = 0.0000
Residual | 238840813 729 327628.001 R-squared = 0.0828
-------------+------------------------------ Adj R-squared = 0.0765
Total | 260392067 734 354757.585 Root MSE = 572.39
------------------------------------------------------------------------------
kyrice | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
kavgtemp | -3591.648 855.2294 -4.20 0.000 -5270.655 -1912.642
kavgtempsq | 67.01803 15.34115 4.37 0.000 36.89993 97.13613
ktotppt | 1.053627 .5836936 1.81 0.071 -.0922936 2.199548
ktotpptsq | -.0005872 .0003753 -1.56 0.118 -.001324 .0001496
ksdtemp | -196.4774 56.57082 -3.47 0.001 -307.5386 -85.41624
_cons | 49759.5 11888.92 4.19 0.000 26418.9 73100.1
------------------------------------------------------------------------------
-> . regress kyrice kavgtemp ktotppt ktotpptsq ksdtemp
Source | SS df MS Number of obs = 735
-------------+------------------------------ F( 4, 730) = 11.39
Model | 15298828.3 4 3824707.07 Prob> F = 0.0000
Residual | 245093239 730 335744.163 R-squared = 0.0588
-------------+------------------------------ Adj R-squared = 0.0536
Total | 260392067 734 354757.585 Root MSE = 579.43
------------------------------------------------------------------------------
kyrice | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
kavgtemp | 143.2145 22.12018 6.47 0.000 99.78776 186.6413
ktotppt | .9560253 .5904461 1.62 0.106 -.2031498 2.1152
ktotpptsq | -.00057 .0003799 -1.50 0.134 -.0013158 .0001758
ksdtemp | -213.2879 57.13459 -3.73 0.000 -325.4556 -101.1202
_cons | -2107.959 622.1736 -3.39 0.001 -3329.422 -886.4957
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