Second, we employ the Kubinec (2019) method of adjusting for missing data. Missing data rates in firm surveys can be substantial, and in this case, we have reason to be concerned about non-ignorable missingness. Some firms may not want to report political activities they undertake, either because these are quasi-legal or illegal in their economy or because they do not trust the survey enumerators to keep their data private. As a result, missingness may be correlated with the latent trait, where either high or low influence firms may under- or over-report political activities and connections.

To adjust for this, we implement a two-stage IRT model to account for selection into item response. The first stage is a separate IRT equation with shared influence parameters Θ _i_ but different item parameters _γj′_<sup>and</sup><sup>_β_</sup> _j′_<sup>.Ifaresponse</sup><sup>_yij_ismissing</sup><sup>_yijm_=1,theposteriordistributionisequaltothefollowing:</sup> And if the survey response is observed _yijm_ =0, we have the following posterior distribution: As such, the missingness mechanism allows for the observed responses to be either inflated or deflated depending on whether higher or lower values of the latent trait are correlated with missingness for a given item _j_ , which is determined by the sign and level of the missingness discrimination parameter _γj′_<sup>.14While</sup> relatively straightforward, this missingness adjustment will permit us to handle the most likely scenario in which highly influential firms are less likely to report political activities (or vice versa). Importantly, the missingness adjustment also means that our scores are defined over the whole sample regardless of how many responses each firm _i_ answered. If missingness is ignorable for a given item _j_ (a case in which _γj′_<sup>=0),themodelwillappropriatelyinflatetheposteriorvarianceofΘ</sup><sup>_i_toaccountfordatathatis</sup> missing completely at random.

On its own, the model will have multiple possible solutions to the values of the parameters, resulting in a multi-modal posterior (Bafumi et al., 2005). A common solution to this issue is to pin a specific item discrimination to an assumed value.<sup>15</sup> We choose to fix the discrimination parameter at 1 for a measure widely believed to give political influence: whether a firm has a direct political connection. Such connections have been extensively studied and shown to garner firms political favors and access, a point that was suggested above and elaborated more fully in Section 3. The discrimination parameters for the remaining items, _γj_ , in turn can be interpreted relative to the discrimination given to that political connection. We do not constrain the sign of any _γj_ , meaning there is a possibility for both a positive relationship—indicating complements—across items or a negative one—indicating substitutes.<sup>16</sup>

> 14This parameterization is essentially a hurdle model defined over a latent variable, where that variable represents item missingness in the data.

> 15Note that a pinned value of 1 is not the only possibility, nor is it limited to items. Individual units ( _i_ ) may also be fixed. In the political science literature on ideal points and polarization, for example, legislators on either end of the political spectrum may be given fixed values (See Bafumi et al. (2005) and Clinton et al. (2004) for discussions. These approaches have also been extended to analyze the political leanings of judges as well (See Bonica and Sen (2021)).

> 16In the use of IRT models in standardized testing, for instance, discrimination parameters are lower-bounded by zero,

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