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# Re: st: problem with squared term

 From Michael Norman Mitchell To statalist@hsphsun2.harvard.edu 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
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
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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