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Re: st: problem with squared term


From   Michael Norman Mitchell <Michael.Norman.Mitchell@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: problem with squared term
Date   Sat, 20 Mar 2010 12:06:01 -0700

Dear Prabhat

  These are excellent followup questions...

1. Why am I not getting this problem with ktotppt and ktotpptsq (here
totppt is total rainfall)?

  I think there are two reasons... 1) that the ktotpptsq effect is much smaller (and is not significant), so it means that there is very little curvature, and/or 2) possibly because the *zero* value for ktotppt is not as far from the mean as it was for temperature.

2. The coefficient on cw1ksq is significant and positive. So, if I get
similar result for my variable i.e. kavgtemp, shold I say that it is
"inverted U" kind of relation (and not linear) ? I mean it is straight
forward but still would like to confirm if there is some trick.

UCLA ATS has put together a great page on exactly this question. You can see it at

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/curves.htm

  I hope this helps.

Best regards,

Michael N. Mitchell
See the Stata tidbit of the week at...
http://www.MichaelNormanMitchell.com

On 2010-03-20 11.50 AM, Prabhat wrote:
Thank you so much Michae!
I think your analysis captures it very well.

However, I have two new queries-
1. Why am I not getting this problem with ktotppt and ktotpptsq (here
totppt is total rainfall)?
2. The coefficient on cw1ksq is significant and positive. So, if I get
similar result for my variable i.e. kavgtemp, shold I say that it is
"inverted U" kind of relation (and not linear) ? I mean it is straight
forward but still would like to confirm if there is some trick.

Thanks again!

Regards,
Prabhat


On Sun, Mar 21, 2010 at 3:32 AM, Michael Norman Mitchell
<Michael.Norman.Mitchell@gmail.com>  wrote:
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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