Help with Gravity Model: unidentifiable interaction term and intra-national trade (ppmlhdfe)

[YH jordaan]

Dear Stata users, Joao Santos Silva, and fellow gravity modelers,


I have encountered some problems in my empirical research related to carbon leakage and trade.

Context

My dataset comprises bilateral trade data at the sectoral level, with the primary dependent variable being emissions embodied in imports. Specifically, I estimate the following equation:

C_ijs,t = exp[α₂ d(EPS) × INTL_ij + γ_it + δ_jt + v_st + μ_ijs + μ_bt + α₁ log(1+ D_ijt)] + ϵ_ijs,t (1)

• C_ijs,t denotes the CO2 content of imports of country i from country j in sector s, including both international and intra-national (i.e. domestic) trade (in CO2).
• d(EPS) × INTL_ij is the interaction between the difference in environmental policy stringency (d(EPS) = EPS_it - EPS_jt) and a dummy variable INTL_ij that equals 1 for international trade. I expect α₂ to be positive.
• γ_{it and} δ_jt represent exporter-time and importer-time fixed effects, respectively. v_st captures sector-time fixed effects. μ_ijs denotes country-pair-sector fixed effects. μ_bt accounts for border-year fixed effects, which control for changes in international trade costs relative to intra-national trade costs.
• D_ij is a time-varying bilateral trade cost variable from the ESCAP-WB Trade Cost Database, expressed in ad valorem equivalent form.
• Eq. 1 is estimated using the PPML estimator, i.e. the ppmlhdfe command in Stata.

Questions:

1. Identification Issue: According to Heid et al. (2021), including intra-national trade allows for the identification of any non-discriminatory trade policies as well as non-trade policies even when exporter- and importer-time fixed effects are included. Despite this, STATA drops the interaction term d(EPS) × INTL_ij when estimating eq. 1. Could anyone explain why this might be happening and suggest potential solutions? See the construction of the interaction below, as well as the regression output.

 gen intl = 1 if importer!=exporter
(13,566 missing values generated)

. replace intl=0 if importer==exporter
(13,566 real changes made)

. gen intl_dif_ij_EPS = intl * (EPS21_i - EPS21_j)
(266,266 missing values generated)

ppmlhdfe co2_content intl_dif_ij_EPS log_trade_cost, absorb(j#i#s s#year i#year j#year inter#year) cluster(i#j#s)

(dropped 1796 observations that are either singletons or separated by a fixed effect)
warning: dependent variable takes very low values after standardizing (7.9168e-12)
note: 1 variable omitted because of collinearity: inter_dif_ij

HDFE PPML regression                              No. of obs      =    366,454
Absorbing 5 HDFE groups                           Residual df     =     21,966
Statistics robust to heteroskedasticity           Wald chi2(1)    =      86.73
Deviance             =  1.34680e+10               Prob > chi2     =     0.0000
Log pseudolikelihood =  -6735979874               Pseudo R2       =     0.9965

Number of clusters (i#j#s)  =     21,967
                               (Std. err. adjusted for 21,967 clusters in i#j#s)
--------------------------------------------------------------------------------
               |               Robust
   co2_content | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
---------------+----------------------------------------------------------------
  intl_dif_ij_EPS |          0  (omitted)
log_trade_cost |  -.6371896   .0684216    -9.31   0.000    -.7712935   -.5030856
         _cons |     19.199   .0476588   402.84   0.000     19.10559     19.2924
--------------------------------------------------------------------------------

2. Symmetrical Fixed Effects: As an alternative, I have estimated the model with symmetrical importer-time and exporter-time fixed effects (i.e., γ_it is constrained to be equal to δ_jt). In this setup, the interaction term is identifiable. However, I am wondering to which extent these symmetrical fixed effects account for fluctuations in the economy of the importing country. What are your thoughts on this?
3. Trade Cost Variable: Unrelated to the above issues, I'd like to check whether it is correct to include the time-varying trade cost variable D_ij in logs, since it is specified in ad-volorem form?

Snippet of my dataset:

* Example generated by -dataex-. For more info, type help dataex
clear
input str9(importer exporter) int year str6 IND float(co2_content intl_dif_ij_EPS log_trade_cost)
"ARG" "ARG" 2000 "D10T12"   6844510 0 0
"ARG" "ARG" 2001 "D10T12"   6636712 0 0
"ARG" "ARG" 2002 "D10T12"   7670751 0 0
"ARG" "ARG" 2003 "D10T12"         . 0 0
"ARG" "ARG" 2004 "D10T12"  14759698 0 0
"ARG" "ARG" 2005 "D10T12"  14213412 0 0
"ARG" "ARG" 2006 "D10T12"  14847242 0 0
"ARG" "ARG" 2007 "D10T12"  14501948 0 0
"ARG" "ARG" 2008 "D10T12"  15278275 0 0
"ARG" "ARG" 2009 "D10T12"  16977000 0 0
"ARG" "ARG" 2010 "D10T12"  16744057 0 0
"ARG" "ARG" 2011 "D10T12"  18253434 0 0
"ARG" "ARG" 2012 "D10T12"  19692390 0 0
"ARG" "ARG" 2013 "D10T12"  20047822 0 0
"ARG" "ARG" 2014 "D10T12"  19017780 0 0
"ARG" "ARG" 2015 "D10T12"  22300582 0 0
"ARG" "ARG" 2016 "D10T12"  20025090 0 0
"ARG" "ARG" 2017 "D10T12"  20574622 0 0
"ARG" "ARG" 2018 "D10T12"  18297452 0 0
"ARG" "ARG" 2000 "D13T15" 2203199.8 0 0
"ARG" "ARG" 2001 "D13T15" 2000815.5 0 0
"ARG" "ARG" 2002 "D13T15" 1695733.8 0 0
"ARG" "ARG" 2003 "D13T15"         . 0 0
"ARG" "ARG" 2004 "D13T15"   5339110 0 0
"ARG" "ARG" 2005 "D13T15"   5409009 0 0
"ARG" "ARG" 2006 "D13T15"   5589320 0 0
"ARG" "ARG" 2007 "D13T15"   5600412 0 0
"ARG" "ARG" 2008 "D13T15"   5364749 0 0
"ARG" "ARG" 2009 "D13T15"   5094218 0 0
"ARG" "ARG" 2010 "D13T15"   5585709 0 0
"ARG" "ARG" 2011 "D13T15"   6007067 0 0
"ARG" "ARG" 2012 "D13T15"   5912168 0 0
"ARG" "ARG" 2013 "D13T15"   5945553 0 0
"ARG" "ARG" 2014 "D13T15"   5762494 0 0
"ARG" "ARG" 2015 "D13T15"   5971323 0 0
"ARG" "ARG" 2016 "D13T15"   5253716 0 0
"ARG" "ARG" 2017 "D13T15"   4744749 0 0
"ARG" "ARG" 2018 "D13T15"   4328012 0 0
"ARG" "ARG" 2000 "D16"    154349.52 0 0
"ARG" "ARG" 2001 "D16"    122990.27 0 0
"ARG" "ARG" 2002 "D16"    127674.02 0 0
"ARG" "ARG" 2003 "D16"            . 0 0
"ARG" "ARG" 2004 "D16"     923668.8 0 0
"ARG" "ARG" 2005 "D16"     807155.3 0 0
"ARG" "ARG" 2006 "D16"     885259.3 0 0
"ARG" "ARG" 2007 "D16"     910833.1 0 0
"ARG" "ARG" 2008 "D16"       991047 0 0
"ARG" "ARG" 2009 "D16"     820032.3 0 0
"ARG" "ARG" 2010 "D16"     855373.7 0 0
"ARG" "ARG" 2011 "D16"     927323.4 0 0
"ARG" "ARG" 2012 "D16"       917962 0 0
"ARG" "ARG" 2013 "D16"     927639.4 0 0
"ARG" "ARG" 2014 "D16"     819333.5 0 0
"ARG" "ARG" 2015 "D16"     862390.1 0 0
"ARG" "ARG" 2016 "D16"     737336.1 0 0
"ARG" "ARG" 2017 "D16"     704515.9 0 0
"ARG" "ARG" 2018 "D16"     630378.1 0 0
"ARG" "ARG" 2000 "D17T18"   1695272 0 0
"ARG" "ARG" 2001 "D17T18"   1516552 0 0
"ARG" "ARG" 2002 "D17T18"   1433771 0 0
"ARG" "ARG" 2003 "D17T18"         . 0 0
"ARG" "ARG" 2004 "D17T18"   2964082 0 0
"ARG" "ARG" 2005 "D17T18"   2804171 0 0
"ARG" "ARG" 2006 "D17T18"   3010510 0 0
"ARG" "ARG" 2007 "D17T18"   3075446 0 0
"ARG" "ARG" 2008 "D17T18"   3185869 0 0
"ARG" "ARG" 2009 "D17T18"   2949060 0 0
"ARG" "ARG" 2010 "D17T18"   3004854 0 0
"ARG" "ARG" 2011 "D17T18"   3290745 0 0
"ARG" "ARG" 2012 "D17T18" 3235521.5 0 0
"ARG" "ARG" 2013 "D17T18" 3329973.5 0 0
"ARG" "ARG" 2014 "D17T18"   3192178 0 0
"ARG" "ARG" 2015 "D17T18"   3385247 0 0
"ARG" "ARG" 2016 "D17T18"   3151508 0 0
"ARG" "ARG" 2017 "D17T18"   3081584 0 0
"ARG" "ARG" 2018 "D17T18"   2902027 0 0
"ARG" "ARG" 2000 "D19"      5685618 0 0
"ARG" "ARG" 2001 "D19"      5809087 0 0
"ARG" "ARG" 2002 "D19"      8098276 0 0
"ARG" "ARG" 2003 "D19"            . 0 0
"ARG" "ARG" 2004 "D19"      8438462 0 0
"ARG" "ARG" 2005 "D19"      6761695 0 0
"ARG" "ARG" 2006 "D19"      5338450 0 0
"ARG" "ARG" 2007 "D19"      5825263 0 0
"ARG" "ARG" 2008 "D19"      7250279 0 0
"ARG" "ARG" 2009 "D19"      9712234 0 0
"ARG" "ARG" 2010 "D19"     10521705 0 0
"ARG" "ARG" 2011 "D19"     11448484 0 0
"ARG" "ARG" 2012 "D19"     13411457 0 0
"ARG" "ARG" 2013 "D19"     13968090 0 0
"ARG" "ARG" 2014 "D19"     14092115 0 0
"ARG" "ARG" 2015 "D19"     14753895 0 0
"ARG" "ARG" 2016 "D19"     14871400 0 0
"ARG" "ARG" 2017 "D19"     14944739 0 0
"ARG" "ARG" 2018 "D19"     14026114 0 0
"ARG" "ARG" 2000 "D20T21"   3231112 0 0
"ARG" "ARG" 2001 "D20T21" 3504779.5 0 0
"ARG" "ARG" 2002 "D20T21" 3108923.5 0 0
"ARG" "ARG" 2003 "D20T21"         . 0 0
"ARG" "ARG" 2004 "D20T21"   6660840 0 0
end

Thank you all in advance for helping me out on this.

Best wishes,

Yannick

Sources:

Heid, B., Larch, M., & Yotov, Y. V. (2021). Estimating the effects of non‐discriminatory trade policies within structural gravity models. Canadian Journal of Economics/Revue canadienne d'économique, 54(1), 376-409.

[Joao Santos Silva]

Dear YH jordaan,

I am afraid I cannot help you much. On 1, you are apparently including a fixed effect that absorbs that interaction, so all I can suggest is that you check carefully that you are including the fixed effects you intend to. In any case, I do not think that imposing symmetrical fixed effects is a good idea.

Best wishes,

Joao