Granular Instrumental Variables

I propose to conduct methodological research on Granular Instrumental Variables (GIV). My
research will focus on expanding the theoretical foundations of this fledgling area of inquiry. I will apply my results to analyze suitable empirical applications.

The GIV methodology constructs instruments from panel data (many individuals (N) observed at many different time periods (T)) to estimate structural time series models with endogenous regressors (Gabaix and Koijen, 2021). The construction of instruments depends on the factor structure of the error and the 'granularity' of the setting. That is, the unobserved error has latent factors and an idiosyncratic component, with the effect of the latent factor varying across individuals. Granular settings are characterized by a few large agents whose impact is not negligible, in addition to infinitesimal agents, as in the case of perfect competition (Gabaix, 2009).

Panels with endogenous regressors require external instruments for consistent estimation (Ahn et al., 2013; Bai and Ng, 2010; Bai, 2009). But such instruments are very difficult to find and may often require a deep understanding of the individual setting. GIV is promising as we construct economically intuitive and valid instruments from within the system. GIV can be constructed when N is small, T is large (limited panel) and when both N and T are large (large panels). This leads to interesting applications in macroeconomics and empirical finance respectively.

The core idea of the GIV methodology is that a linear transformation of the idiosyncratic errors is a valid instrument for the estimation of the endogenous variable. The key methodological challenge is that we do not separately observe these idiosyncratic error terms.

But as a new area of research, the methodological work on GIV is very limited. Gabaix and Koijen (2021) lay out the idea and the asymptotic theory for fixed N and large T. Banafti and Lee (2022) extend this to the large N and T setting.

This paves the way for methodological research about the estimation and inference of the GIV methodology with a focus on consistent factor estimation, instrument weakness, optimal instruments, over-identification, and so on. I propose to answer questions in this research setting under two streams: 1. Large panels: N and T large, and 2. Limited panels: N fixed, and T large.

Research Stream 1: Large Panels

Under this research stream, I will undertake advanced inferential issues like weakness of instrument, optimal instrument, and over-estimation. I will also compare GIV with other possible estimators along these angles.

Weakness of the instrument is one of the primary concerns in any IV setting (Andrews et al., 2019; Stock and Wright, 2000; Antoine and Renault, 2021). Banafti and Lee (2022) show that when the distribution of market shares does not have sufficiently fat tails, the first stage relationship between the instrument and the exogenous variable becomes very weak. Similarly, the presence of weak factors/factor loadings can affect the quality of the instruments Bai and Ng (2023). I propose to develop the theory for the identification and inference of GIV under weak instruments and factors.

There is not yet a unified theory of optimal GIV. I will be exploring the question of optimal GIV from the point of view of asymptotic efficiency. I will also be working on Generalized Method of Moments (GMM) estimation using over-identified estimators.

Research Stream 2: Limited Panels

In limited panels, we have a small number of individuals who are observed many times. An example is the monthly yields on the sovereign debt of Eurozone countries (N = 12) observed for 30 years (T = 12 x 30 = 360). The construction of the GIV from the panel in (1) depends on our ability to isolate the idiosyncratic shocks.