Extracting marginal effects

[Jack Jameson]

Hi all,

There may be a straightforward answer for this question, but I can't find it anywhere. I am specifying a simple OLS regression of the following form:

Y = alpha + Beta₁ * A + Beta₂ * A²+ Beta₃ * (A - B)²

where A and B are continous variables, and "alpha" and "beta" are model parameters.

In the past I have been creating a new variable z = (A - B)² in order to run this using the

reg

command. However, I want to plot the marginal effect of A on Y using

marginsplot

, which shouldn't ignore the fact that z is a function of A. In addition, I cannot reorder the terms because I am specifically interested in analysing the Beta₃ coefficient.

Any idea of how to go out this problem?

Thanks,
Jack

[Andrew Musau]

ssc install f_able, replace
help f_able

[John Mullahy]

f_able, suggested in #2 by Andrew Musau, is a nice program. If for some reason that doesn't suffice you could try either nl or gmm, e.g.

cap preserve
cap drop _all

set obs 100
set seed 2345

gen A=runiform()
gen B=runiform()
gen y=1 + A + A^2 + (A-B)^2 + rnormal(0,1)

nl (y = {alpha} + {beta1}*A + {beta2}*A^2 + {beta3}*(A-B)^2), variables(A B)
margins, dydx(*)

cap restore

Results:

. nl (y = {alpha} + {beta1}*A + {beta2}*A^2 + {beta3}*(A-B)^2), variables(A B)

Iteration 0:  residual SS =  82.20916
Iteration 1:  residual SS =  82.20916


      Source |      SS            df       MS
-------------+----------------------------------    Number of obs =        100
       Model |  11.789461          3  3.92982025    R-squared     =     0.1254
    Residual |  82.209157         96  .856345381    Adj R-squared =     0.0981
-------------+----------------------------------    Root MSE      =   .9253893
       Total |  93.998617         99  .949480983    Res. dev.     =   264.1974

------------------------------------------------------------------------------
           y | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      /alpha |   1.202186   .3333567     3.61   0.000     .5404785    1.863894
      /beta1 |   1.518958   1.422263     1.07   0.288    -1.304211    4.342127
      /beta2 |  -.2374228    1.39287    -0.17   0.865    -3.002248    2.527403
      /beta3 |   .6301616   .5753816     1.10   0.276    -.5119618    1.772285
------------------------------------------------------------------------------
Note: Parameter alpha is used as a constant term during estimation.

. margins, dydx(*)

Average marginal effects                                   Number of obs = 100
Model VCE: GNR

Expression: Fitted values, predict()
dy/dx wrt:  A B

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
           A |   1.213612   .3485474     3.48   0.000     .5304712    1.896752
           B |    .085941   .0784701     1.10   0.273    -.0678576    .2397396
------------------------------------------------------------------------------

[Andrew Musau]

Taking John's example, I have argued elsewhere that in cases where variables are not explicitly present in the estimated model, one can also manually compute the derivative and use the -expression()- option of margins. Given:

$$y = \beta_0 + \beta_1 A + \beta_2 A^2 + \beta_3 (A-B)^2$$

The partial derivative is:

$$\frac{\partial y}{\partial A} = \beta_1 + \left(2\beta_2\times A\right)+ \left(2\beta_{3}\times(A-B)\right).$$

cap preserve
cap drop _all

set obs 100
set seed 2345

gen A=runiform()
gen B=runiform()
gen y=1 + A + A^2 + (A-B)^2 + rnormal(0,1)
gen Z= A-B
gen Z2= (A-B)^2

regress y c.A##c.A Z2
margins, expression(_b[A]+ (2*_b[c.A#c.A]*A) + (2*_b[Z2]*Z))

Res.:

 . regress y c.A##c.A Z2

      Source |       SS           df       MS      Number of obs   =       100
-------------+----------------------------------   F(3, 96)        =      4.59
       Model |  11.7894607         3  3.92982022   Prob > F        =    0.0048
    Residual |  82.2091566        96  .856345382   R-squared       =    0.1254
-------------+----------------------------------   Adj R-squared   =    0.0981
       Total |  93.9986173        99  .949480983   Root MSE        =    .92539

------------------------------------------------------------------------------
           y | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
           A |   1.518958   1.422263     1.07   0.288    -1.304211    4.342127
             |
     c.A#c.A |  -.2374228    1.39287    -0.17   0.865    -3.002248    2.527403
             |
          Z2 |   .6301615   .5753816     1.10   0.276    -.5119619    1.772285
       _cons |   1.202186   .3333567     3.61   0.000     .5404785    1.863894
------------------------------------------------------------------------------

.
. margins, expression(_b[A]+ (2*_b[c.A#c.A]*A) + (2*_b[Z2]*Z))
warning: option expression() does not contain option predict() or xb().

Predictive margins                                         Number of obs = 100
Model VCE: OLS

Expression: _b[A]+ (2*_b[c.A#c.A]*A) + (2*_b[Z2]*Z)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _cons |   1.213612   .3485474     3.48   0.000     .5304712    1.896752
------------------------------------------------------------------------------

.

[Jack Jameson]

Dear Andrew Musau and John Mullahy ,

Thank you very much for your detailed and helpful answers. I ended up running with Andrew Musau 's last option.

Just a quick follow up: any idea how to overlay marginsplots for different expressions? Eg:

- expression 1: _b[A]+ (2*_b[c.A#c.A]*A) + (2*_b[Z2]*Z)
- expression 2: _b[A]+ (2*_b[c.A#c.A]*A)

Thanks,
Jack

[Jack Jameson]

Edit: the -combomarginsplot- command worked fine