Multiplicative versus additive IV Poisson

[Zuhura Anne]

I am using the IV Poisson to address the endogeneity issues in my gravity model. However, when I use the additive version of the IV Poisson, I do not get convergence. When I use the multiplicative version of the IV Poisson, I get convergence.

What could be the reason behind this phenomenon?

[Jeff Wooldridge]

The additive model really doesn't make much sense. You have a nonnegative y, and the model is y = exp(x*b) + u, and so u cannot range freely from x: we need u >= -exp(x*b). Since some of the elements of x are treated as exogenous, this is a problem. Frankly, I was surprised Stata allowed that option.

The multiplicative model makes more sense when y >= 0.

I know this doesn't really answer your question -- it's more like my saying to forget the additive model.

P.S. I should note that we like to see the commands you typed and what Stata returned. It's easier to provide advice, as a general rule.

[Zuhura Anne]

Thank you Prof. Wooldridge for the feedback.
These are the commands I used:

ivpoisson gmm exports covariate1 (logged) covariate2 covariate3 covariate4 covariate 5 (variable to be instrumented(logged)= instrumental variable(logged)

ivpoisson gmm exports covariate1 (logged) covariate2 covariate3 covariate4 covariate 5 (variable to be instrumented (logged)= instrumental variable(logged)), multiplicative

Also, are the coefficients in the multiplicative model interpreted in the same way as in the gravity model or it changes. For instance, will the coefficient of logged variables still be interpreted as elasticities?

[Joao Santos Silva]

Dear Jeff Wooldridge,

I must be missing something because your argument appears to imply that standard Poisson regression really doesn't make much sense, and I am sure you do not mean that. Can you please elaborate?

Best wishes and many thanks,

Joao
PS: Zuhura Anne try estimating

  
 ppml exports instrumental variable(logged) covariate1 (logged) covariate2 covariate3 covariate4 covariate5

and show us the results.

[Zuhura Anne]

Dear Joao,
These are the results:

ppml R_exports IVLogged logdistw contig comlang_off comcol RTA

note: checking the existence of the estimates

Number of regressors excluded to ensure that the estimates exist: 0
Number of observations excluded: 0

note: starting ppml estimation
note: R_exports has noninteger values

Iteration 1:   deviance =   1247263
Iteration 2:   deviance =  986796.1
Iteration 3:   deviance =  955333.6
Iteration 4:   deviance =  954239.9
Iteration 5:   deviance =  954237.5
Iteration 6:   deviance =  954237.5
Iteration 7:   deviance =  954237.5

Number of parameters: 7
Number of observations: 2941
Pseudo log-likelihood: -480617.68
R-squared: .13137755
Option strict is: off
------------------------------------------------------------------------------
             |               Robust
   R_exports |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    IVLogged |   .1351713   .0434426     3.11   0.002     .0500254    .2203173
    logdistw |  -.0994044   .1848547    -0.54   0.591    -.4617128    .2629041
      contig |   1.285097   .2834192     4.53   0.000     .7296058    1.840589
 comlang_off |   .9650616   .1908264     5.06   0.000     .5910488    1.339074
      comcol |   -1.88645   .2257126    -8.36   0.000    -2.328839   -1.444062
         RTA |   1.195954   .1569171     7.62   0.000     .8884025    1.503506
       _cons |   3.631157   1.578206     2.30   0.021     .5379311    6.724384
------------------------------------------------------------------------------

I have rescaled exports by dividing by 1 million

[Joao Santos Silva]

Thank you, Zuhura Anne; I do not see any problem with these results. Do you still have convergence problems if you use the additive ivpoisson with the rescaled data?

Best wishes,

Joao

[Zuhura Anne]

I still have the convergence problem with the additive ivpoisson when I use the rescaled data.

However, there is convergence when I include the lag of the endogenous variable to the instrumental variable that I already have:

 
 ivpoisson gmm exports covariate1 (logged) covariate2 covariate3 covariate4 covariate 5 (endogenous variable(logged)= L.endogenous  variable instrumental variable(logged)

But I did not want to use the lag of the endogenous variable as an IV due to the criticisms around its use in literature and I can't test for its orthogonality in to be sure that its a valid instrument.

By the way, can one include the country pair fixed effects and importer and exporter fixed effects in the IVPoisson?

[Joao Santos Silva]

Dear Zuhura Anne,

Thanks for this. If you ware willing to send me your data by email, I'll be happy to look into this for you.

On your other question, I do not think the estimator is valid country pair fixed effects and importer and exporter fixed effects.

Best wishes,

Joao

[Zuhura Anne]

Dear @Joao, I have sent the data file.

[Joao Santos Silva]

Dear Zuhura Anne,

Thank you for sending the data. If you use the option tech(nr) the estimator converges.

Best wishes,

Joao

[Zuhura Anne]

Thank you so much Joao for the assistance. Are the coefficients interpreted the same way as in the ppml, whereby if a covariate is logged, its coefficient is an elasticity?

[Jeff Wooldridge]

Since I just threw that coment out there, I should provide a better explanation. It's true that when all x's are treated as exogenous, the moment conditions lead to the Poisson quasi-MLE. But in that case the additive error is definitional; it is just a way of getting the moment conditions of the Poisson QMLE. In fact, in my 1992 IER paper on alternatives to the Box-Cox transformation, where I suggested modeling E(y|x) directly when y >= 0, I made the point that the error can be defined to be additive or multiplicative: one gets the same conditional mean no matter what.

Once an endogenous variable is introduced, that changes everything. Now exp(x*b) is no longer a conditional mean. It behooves us to explain what it is, and how the parameters relate to something "structural." In the leading case where an element of x is endogenous we have in mind omitted variables. But then in the formulation y = exp(x*b) + u, why does the omitted variable have an additive effect when all other variables have multiplicative effects? And now the constraint u >= -exp(x*b) has real meaning. To me, it makes more sense to think of the structural model as

E(y|x,r) = exp(x*b + r)

where r is the omitted variable. This omitted variable is correlated with one or more elements of x. This leads to the Mullahy multiplicative approach, or the control function approach that I discuss in my MIT Press book. Note how r is treated just like the elements of x: all are inside the mean function and E(y|x,r) > 0 for all x, r, and b. If endogeneity is due to measurement error in an x, then the error term is still multiplicative. I can't really think of why one would ever have all observed variables have multiplicative effects but the omitted variable an additive effect. If part of u is an omitted variable, then the additive model implies

E(y|x,r) = exp(x*b) + c*r,

which seems weird to me.

Take a related case, where y is binary or fractional and we use a logit model, G(x*b), when x is exogenous. Like the Poisson, the moment conditions based on y = G(x*b) + u, E(u|x) = 0 using x as IVs delivers the logit MLE or QMLE (fractional case). Should we now use the model y = G(x*b) + u when x includes an endogenous variable? I wouldn't, and I haven't seen it suggested. An approach that is popular is to start with

E(y|x,r) = G(x*b + r)

and then deal with the correlation between r and elements of x using a control function approach. Or, one can use a joint MLE or QMLE. To me, this is very similar to the case where y >= 0, whether it is a count, continuous, or has a corner at zero.

The bottom line is that taken a set of moment conditions that makes sense when all x are exogenous and errors are definitional does not necessarily make sense when x inclcudes endogenous variable.

[Joao Santos Silva]

Dear Jeff Wooldridge,

Thank you very much for clarifying your views on this. If we start from your omitted variables formulation, we can write

E(y|x,r) = exp(x*b + r) = exp(x*b)exp(r) = exp(x*b) + exp(x*b)(exp(r) - 1)

which has the form of an exponential model with an additive "error". For the GMM estimation, we need an instrument z that is orthogonal to an error term. If, as usual, we let y = E(y|x,r) + e, the multiplicative approach assumes that

E{[e + exp(x*b)(exp(r) - 1)]/exp(x*b)| x,z } = 0,

while the additive formulation assumes

E{e + exp(x*b)(exp(r) - 1)| x,z } = 0.

So, the only difference between the two sets of moment conditions is that the multiplicative model down-weights the errors with large exp(x*b). I do not have a view on which of the two moment conditions is more likely to hold, but I guess it will depend on the application and on the relative importance of e and of the error caused by the omission of r. So, I would not dismiss the additive model.

Best wishes and many thanks again,

Joao

[Jeff Wooldridge]

I'll go through your argument more carefully, but here is my initial reaction. In the multiplicative case, it is sufficient for the exogenous variables to be independent of r. That is the sensible notion of exogeneity here. So if x2 is the exogenous elements of x and it suffices that (x2,z) isindependent of r. This is the analog of the assumption we make in the model y = x1*b1 + x2*b2 + u, except we don't need full independence. For the error you have constructed, exp(x*b)(exp(r) - 1), it seems heroic to think (x2,z) would be uncorrelated with exp(x*b)(exp(r) - 1), given in depends on the endogenous variables x1 in a nonlinear way.

Okay, I just went through your derivation in the multiplicative case. It is misleading in a couple of respects. First, I don't think you mean to condition on all of x in your expectations, as those include endogenous variables and those cannot be used as IVs. Second, we would always assume the standard exclusion restriction

E(y|x,r,z) = E(y|x,r)

which means that, in your notation, E(e|x,r,z) = 0. That means E(e|x2,z) = 0 and we do not have to worry about e in the orthogonality conditions. The same is true in the additive or multiplicative case. But in the multiplicative case, the moment condition reduces to

E[exp(r)|x2,z] = 1,

with the "1" on the RHS simply being a normalization that is WLOG if x includes an intercept. So, sufficient is that r is independent of (x2,z), which is what Mullahy's original paper showed. This is a "structural" and standard assumption.

To me, the moment conditions in the additive case are ad hoc, while in the multiplicative case they follow from the standard assumption of what we mean by exogeneity of control variables and external instruments.

[Jeff Wooldridge]

Concerning interpretation of the coefficients, there is a subtle difference in the multiplicative and additive cases. If we agree that quantities should be defined in terms of conditional means (as I proposed in my 1992 paper), then in the omitted variable case that is E(y|x,r). So if

E(y|x,r) = exp(x*b + r)

then the bj have the same interpretation as in the the exogenous case because the partial of log E(y|x,r) with respect to xj is simply bj. If xj = log(wj) then bj is an elasticity.

In the additive case,

E(y|x,r) = exp(x*b) + r and so the expression becomes bj*exp(x*b)/[exp(x*b) + r], and this depends on the unobserved r. Of course, r is taking to have zero mean, so bj is the semi-elasticity (or elasticity) at E(r) = 0. This is different from the multiplicative model, where the semi-elasticity (or elasticity) does not depend on r or any of the x.