Problem calculating elasticities using margins after fixed-effects regression.

[Luis Espinoza]

GOAL: To plot a semi-elasticity at different levels of another variable based on a regression with multiple fixed effects.
PROBLEM: margins does not calculate the semi-elasticities.

I. Preliminaries
I am estimating the following extended gravity equation for one year:
\[ \log{\left(\text{Trade}_{ijp} \right)} = \exp{\Bigg\{ \delta_{ik} + \delta_{jk} + \delta_{ij} + \beta_D \Big[ \log{ \left( \text{Dist}_{ij} \right) } \times \text{RS}_p \Big] + \beta_C \Big[ \mu_i \times \text{RS}_p \Big] + \beta_{CD} \Big[ \mu_i \times \log{ \left( \text{Dist}_{ij} \right) } \times \text{RS}_p \Big] \Bigg\}} \eta_{ij} \]
where:

• i and j index the exporter and importer, respectively;
• p and k index products and industries, respectively (industries may produce many products, but each product belongs to only one industry);
• The lowercase deltas represent fixed effects (exporter-industry, importer-industry and country-pair);
• RS is a binary variable indicating if the good is "specific"; and
• The lower case mu represents a proxy for the "quality" of contract enforcement institutions, normalized to take values from 0 (worse) to 1 (best).
• The error term (lower case eta) is assumed independent from the regressors with conditional expected value 1.

I ran the regression using Poisson Pseudo Maximum Likelihood (PPML), where I clustered errors by country-pair:

ppmlhdfe value 1.RS##c.mu##c.ldist, a(ctry_x#sector ctry_m#sector ctry_x#ctry_m) vce(cluster ctry_x#ctry_m) d

II. Problem with margins
I want to calculate the semi-elasticity of trade with respect to a marginal increase in the proxy for institutional quality (lower case mu) for different values of log distance. First, I saved the main percentiles of log distance in memory:

summarize ldist, detail
local mean = `r(mean)'

In my first attempt, I wrote:

margins, eydx(mu) at(ldist== (`r(p1)' `r(p5)' `r(p10)' `r(p25)' `r(mean)' `r(p50)' `r(p75)' `r(p90)' `r(p95)' `r(p99)') RS == 1) continuous

Comments:
- Since I am interested in the semi-elasticity, I use eydx.
- Since the variable mu is continuous, I add the option continuous

Although I did not get an error message, most estimates are "not estimable":

------------------------------------------------------------------------------
                |               Delta-method
                |    ey/dx   std. err.   z   P>|z|   [95% conf. interval]
-------------+----------------------------------------------------------------
mu           |
          _at |
             1 |          . (not estimable)
             2 |          . (not estimable)
             3 |          . (not estimable)
             4 |          . (not estimable)
             5 |          . (not estimable)
             6 |          . (not estimable)
             7 |          . (not estimable)
             8 |          . (not estimable)
             9 |          . (not estimable)
           10 |          . (not estimable)
------------------------------------------------------------------------------

Based on the discussion in this post, I tried to fix this by adding noestimcheck as an option:

margins, eydx(mu) at(ldist== (`r(p1)' `r(p5)' `r(p10)' `r(p25)' `r(mean)' `r(p50)' `r(p75)' `r(p90)' `r(p95)' `r(p99)') RS == 1) continuous noestimcheck

Unfortunately, this made things worse and now I get the following error message:

could not calculate numerical derivatives -- discontinuous region with missing values encountered

Could anybody help me understand what am I missing? I am using Stata 18 (SE) on Windows 11.

Thanks!

Luis

[Clyde Schechter]

Not having any understanding of trade dynamics, I'm just going to make a guess here based on general principles of regression. If this variable mu is a time-invariant attribute of ctry_m, or of ctry_x, or of the pair ctry_m#ctry_x, then it is, by linear algebra, impossible to estimate mu's marginal effect (nor any elasticity or semielasticity) in a model with those absorbed effects.

[Luis Espinoza]

EDIT TO PREVIOUS POST: The dependent variable in the first equation should have been in levels. That is, it should have been:
\[ \text{Trade}_{ijp} = \cdots \]

---

Hi Clyde,

Thank you for your response, and apologies for not being more clear in my initial post. I can think of two ways of reading your message, so let me address each one. Obviously, if your message meant something else, please help me understand it.

I. I cannot identify mu's coefficient in the baseline regression itself, let alone its calculate marginal effects
If I was working with aggregate bilateral trade data, the variation of mu (a time invariant attribute of ctry_x) would indeed be absorbed by the ctry_x fixed effect (FE) and, thus, I wouldn't be able to get an estimate of mu's coefficient.
Clarification:

1. I have bilateral trade flows by product. Thus, even when controlling for exporter-industry, importer-industry and exporter-importer fixed effects, I still can exploit the variation across products within the same country-pair and industry.
2. I am not interested in estimating the effect of mu itself, but its additional effect on a particular type of goods ("specific" goods, which in my data are those observations with RS = 1). While I cannot estimate the coefficient of mu alone (which is always absorbed by the FEs), I can estimate the three coefficients of interest because all are associated to regressors that vary by product type.

II. I can get estimates for the coefficients, but marginal effects cannot be calculated anyway
Even if I can get estimates for the coefficients (which I do), I cannot calculate the marginal effect of mu because part of it is "hidden" in the FEs:
\[ \frac{\partial E\Big[ \text{Trade}_{ijp} | \text{RS}_p = 1 \Big] }{\partial \mu_i} = \frac{\partial \delta_{ik} }{\partial \mu_i} + \frac{\partial \delta_{ij} }{\partial \mu_i} + \beta_C + \beta_{CD} \log{\text{Dist}_{ij}} \]
Clarification: I do not want to calculate the above effect, but the following discrete difference, evaluated for different values of log Dist:
\[ \frac{\partial E\Big[ \text{Trade}_{ijp} | \text{RS}_p = 1 \Big] }{\partial \mu_i} - \frac{\partial E\Big[ \text{Trade}_{ijp} | \text{RS}_p = 0 \Big] }{\partial \mu_i} = \beta_C + \beta_{CD} \log{\text{Dist}_{ij}} \]

I hope this explanation makes my goal more clear. Is it possible to accomplish my goal using margins?

[Luis Espinoza]

EDIT TO PREVIOUS POST: I am interested in estimating the second-order derivative:
\[ \frac{\partial^2 \log{E\Big[ \text{Trade}_{ijp}\Big]} }{\partial \text{RS}_p \partial \mu_i} = \frac{\partial \log{E\Big[ \text{Trade}_{ijp} | \text{RS}_p = 1 \Big]} }{\partial \mu_i} - \frac{\partial \log{E\Big[ \text{Trade}_{ijp} | \text{RS}_p = 0 \Big]} }{\partial \mu_i} = \beta_C + \beta_{CD} \log{\text{Dist}_{ij}} \]

[Andrew Musau]

ppmlhdfe is from SSC (FAQ Advice #12). If you have derived the expression correctly, then yes, you can use either margins with the -expression()- option or nlcom to calculate the derivative. Although you provide no reproducible example, here is an illustration:

Consider the model:

\[
\mathbb{E}[y \mid x_1, x_2] = \mu(x_1, x_2) = \exp\left( \beta_0 + \beta_1 x_1 + \beta_2 x_1 x_2 \right)
\]

The partial derivative of \(\mathbb{E}[y]\) with respect to \(x_1\) is

\[
\frac{\partial \mu}{\partial x_1} = \frac{d}{dx_1} \left[ \exp\left( \beta_0 + \beta_1 x_1 + \beta_2 x_1 x_2 \right) \right] = \exp\left( \beta_0 + \beta_1 x_1 + \beta_2 x_1 x_2 \right) \cdot \left( \beta_1 + \beta_2 x_2 \right)
\]

Evaluating at Sample Means:

Let \( \bar{x}_1 \) and \( \bar{x}_2 \) denote the sample means of \( x_1 \) and \( x_2 \), respectively. Then:

\[
\left. \frac{\partial \mu}{\partial x_1} \right|_{x_1 = \bar{x}_1, \, x_2 = \bar{x}_2}
= \exp\left( \beta_0 + \beta_1 \bar{x}_1 + \beta_2 \bar{x}_1 \bar{x}_2 \right) \cdot \left( \beta_1 + \beta_2 \bar{x}_2 \right)
\]

clear
input float(id y x1 x2)
  1 2      .2488 -1.1430041
  2 1  1.3352333  1.1311661
  3 1  -.4587567 -1.4932548
  4 1   1.319296   7.923935
  5 2 -2.0332499   4.529856
  6 2   -.646085 -4.5292335
  7 1   5.043857   6.302709
  8 2   5.941895   6.645915
  9 0   3.718591  -.2793825
 10 3   5.071329   2.413525
 11 1   2.602065   7.383534
 12 4   .7288715   2.838717
 13 0    2.93099  -2.412951
 14 2   .4572504  -.7622573
 15 3 -.24726455  -5.066185
 16 2 -2.0624917   7.994617
 17 1  .04221758   3.689781
 18 3  2.0986538   3.779051
 19 3  .14305335 -1.4006286
 20 0  -.8681847   2.469832
 21 4  4.1309566   .6822363
 22 2  1.9485594  1.0957936
 23 1   7.379262   8.829028
 24 2     .07606  2.9235585
 25 5   4.741378  10.108137
 26 3    .304807   3.482954
 27 1  .11917886   4.861798
 28 3  2.0989184  1.8407828
 29 2 -1.9350915   4.587946
 30 3  2.0964124   7.992496
 31 1 -1.6835184   -.412823
 32 1 -.04724126   8.475094
 33 2   5.098127  -3.600388
 34 0  1.6679398   .8006483
 35 3 -.22119495 .025044275
 36 1  1.4548622   .4852332
 37 1  4.3716683   .8371546
 38 3   .6414328   4.800642
 39 5  -5.698874  2.7759914
 40 3   -.961277   4.419597
end

ppmlhdfe y c.x1 c.x1#c.x2, cl(id) noabsorb d(ppml1)
foreach var in x1 x2{
    qui sum `var'
    local m`var'=r(mean) 
}

*USING MARGINS
margins, dydx(x1) atmeans

*USING EXPRESSION OPTION OF MARGINS
margins, expression(exp(_b[_cons]+ (_b[x1]*`mx1') +(_b[c.x1#c.x2]*`mx1' *`mx2'))*(_b[x1]+(_b[c.x1#c.x2]*`mx2'))) 

*USING NLCOM 
nlcom exp(_b[_cons]+ (_b[x1]*`mx1') +(_b[c.x1#c.x2]*`mx1' *`mx2'))*(_b[x1]+(_b[c.x1#c.x2]*`mx2'))

Res.:

. *USING MARGINS
. margins, dydx(x1) atmeans

Conditional marginal effects                                Number of obs = 40
Model VCE: Robust

Expression: Predicted mean of y, predict()
dy/dx wrt:  x1
At: x1 = 1.273711 (mean)
    x2 = 2.625667 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
          x1 |  -.1057673   .0964378    -1.10   0.273    -.2947819    .0832473
------------------------------------------------------------------------------

. 
. *USING EXPRESSION OPTION OF MARGINS
. margins, expression(exp(_b[_cons]+ (_b[x1]*`mx1') +(_b[c.x1#c.x2]*`mx1' *`mx2'))*(_b[x1]+(_b[c.x1#c.x2]*`mx2'))) 
warning: option expression() does not contain option predict() or xb().
warning: prediction constant over observations.

Predictive margins                                          Number of obs = 40
Model VCE: Robust

Expression: exp(_b[_cons]+ (_b[x1]*1.273710907436907) +(_b[c.x1#c.x2]*1.273710907436907
            *2.625666642514989))*(_b[x1]+(_b[c.x1#c.x2]*2.625666642514989))

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _cons |  -.1057673   .0964378    -1.10   0.273    -.2947819    .0832473
------------------------------------------------------------------------------

. 
. *USING NLCOM 
. nlcom exp(_b[_cons]+ (_b[x1]*`mx1') +(_b[c.x1#c.x2]*`mx1' *`mx2'))*(_b[x1]+(_b[c.x1#c.x2]*`mx2'))

       _nl_1: exp(_b[_cons]+ (_b[x1]*1.273710907436907) +(_b[c.x1#c.x2]*1.273710907436907 *2.625666642514989))*(_b[x1]+(_b[c.x1#c.x2]*2.
> 625666642514989))

------------------------------------------------------------------------------
           y | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _nl_1 |  -.1057673   .0964378    -1.10   0.273    -.2947819    .0832473
------------------------------------------------------------------------------

[Luis Espinoza]

Thank you for sharing this example, Andrew! I was not aware of the expression() option! I was able to solve my problem using it!