rt=byyS,t+rt (3.8) where economic growth for region i at time t is yi,t, the size-weighted economic growth across all regions in the union at timetisyS,t, and the short-term interest rate set a timet by the central bank isrt. Economic growth disturbances in regioniare assumed to be the sum of a common shockηt and an idiosyncratic shockνi,t local to that region. Aggregate economic growth disturbances ytS = ηt+νS,t in equation (3.7) are thus the sum of the common shock ηt plus the size-weighted average of idiosyncratic shocksνS,t. Finally, the monetary policy shock at time t is denoted rt. All shocks uit, ηt and rt are i.i.d across dates, with the regional shocksνit also i.i.d acrossi’s.
The parameterbr, which is assumed to be homogeneous across regions, is the sensitivity of economic growth to the short-term interest rate. The parameterby is the central bank’s standard response function to aggregate economic growth. In a typical economy by > 0 and br < 0. When economic activity is above its long run average, the central bank increases the short-term interest rate, which reduces economic growth and prevents an overheating of the economy as the central bank desired.
3.4.3 Traditional monetary policy identification in the simple economy
The objective of the monetary policy identification literature is to estimate, br, the sen-sitivity of economic growth to the short-term interest rate. However, estimating equation (3.7) by itself leads to endogeneity due to simultaneity. Solving forrt results in
rt= by
1−bybrytS+ 1
1−bybrrt (3.9)
and thus a regression ofyS,t on rtproduces a biased estimate of br with the independent variable correlated with the error term
E[rtytS] = br
1−bybrE[ytSytS]6= 0. (3.10) The traditional solution to the endogeneity problem is to extract monetary policy shocks rt directly. These can be used to identify br in equation (3.7) as they are, by definition, not correlated with the error term ytS. There are various techniques in the literature for extracting monetary shocks. For example, using narrative methods with historical case studies (Friedman and Schwartz (1963)), moneraty policy rate innovations estimated through VAR model’s (Christiano et al. (1999)), and policy deviations away from typical (i.e. the Taylor rule) policy responses (Romer and Romer (2004) Coibion (2012)). More
recently, high-frequency data has also been used to extract monetary policy shocks. The idea is that the changes in interest rates in narrow windows surrounding policy announce-ments can be treated as unexpected moveannounce-ments (Gertler and Karadi (2015) Nakamura and Steinsson (2018)).
However, as Ramey (2016) stresses, in recent decades monetary policy changes have been well anticipated by market participants ahead of the policy announcements. The predictability of policy changes means that true monetary policy shocksrt are very small.
This is a challenge for all the existing approaches to monetary policy identification. By attempting to extract and use rt, the econometrician replaces an endogeneity problem with an issue of statistical power.
3.4.4 GIVs monetary policy identification in the simple economy
GIVs provides an alternative monetary policy identification strategy. The approach is to apply an instrumental variables estimation to
yE,t =brrt+ytE (3.11)
whereyE,t for equal-weighted economic growth across regions at timetand the error term ytE =ηt+νE,tis the sum of the common economic growth shockηtand the equal-weighted average of idiosyncratic growth shocksνE,t. The “granular” instrumental variable for the estimation is
zt=yΓ,t (3.12)
whereyΓ,t is the difference between the size-weighted and equal-weighted averages of eco-nomic growth. Forztto be a valid instrument for the estimation of equation (3.11), both the relevance condition E[ztrt]6= 0 and the exclusion restriction condition E[ztytE] = 0 must hold. Below the intuition of the instrument with respect to these conditions is explained. The econometric proofs are provided in Appendix 3.9.1.
For the exclusion restriction condition to hold, the instrument must be uncorrelated with the estimation error termytE =ηt+νE,t. The crucial characteristic of the instrument from this perspective is that it is made up of idiosyncratic shocks only. The factors common across regions {rt, ηt} cancel when calculating the difference between yS,t and yE,t and hence we are left withzt=yΓ,t =νΓ,t. The instrument is therefore uncorrelated with the common shockηtin the estimation error term. Further, given the regional shocks are i.i.d
across i, the gamma-weighted average is uncorrelated with the equal-weighted average, and thus the instrumentνΓ,t is uncorrelated with the νE,t in the estimation error term.7
For the instrument to be relevant, it must also have predictive power for the short-term interest rate. The crucial characteristic of zt from this perspective is the weight Γi,t on the idiosyncratic shocksνi,t. By placing greater weight on the shocks of the larger regions, the instrument correlates with aggregate economic growth. The short-term interest rate is set in response to aggregate economic growth, and it therefore moves when there are shocks to large regions. This relationship is captured in first term of equation (3.9).
3.4.5 GIVs: Key intuition and required characteristics in the data
To summarise so far, a GIVs approach to identification is to harness idiosyncratic shocks in the cross-section of the economy and use them to make causal inference on the effect of monetary policy in the aggregate. Shocks to the economic growth of one region in the currency union leads to changes in monetary policy, with these changes in the short-term interest rate being exogenous from the perspective of the other regions.
GIVs methods therefore requires two features in the data. First, there needs to be sufficiently large idiosyncratic shocks in the cross section. Rich heterogeneity in business cycles and taylor rule residuals across states (documented in section 3.3) help validate the application of GIVs methods in a US monetary policy setting from this perspective.
Second, there needs to be size variation across regions. The larger a region, the more its idiosyncratic shocks effect aggregate monetary policy, the more powerful it’s shock is as a component of the granular instrumental variable.8
As shown by GK, the excess Herfindahl index
h= v u u t−1
N +
N
X
i=1
S2i (3.13)
is a useful measure of expressing the panel’s size-weight variation in the context of a GIVs approach. Figure 3.4 presents the excess Herfindahl index for US states withSi defined as statei’s fraction of the aggregated US population. The size variation has been increasing over our sample and todayh= 16%.
7see appendix for details on the importance of the independence assumption for this result
8Indeed, if all regions are equal sized thenzt=νS,t−νE,t=νE,t−νE,t= 0.
3.4.6 Dealing with heterogeneous loadings on the common factor
One extension of the model that is important in practice is a relaxation of the assumption that all regions load equally on the common factorηt. In this section, I denote the loading of region i on the common shock as λi. In the previous sections, the assumption was λi = 1 for alli. With this adjustment to the model, equation (3.6) becomes
yi,t =brrt+λiηt+νi,t (3.14) and the error terms for equations (3.7) and (3.11) are now ytS = λSηt+νS,t and ytE = λEηt+νE,t respectively. The GIV in this setting is
zt=yΓ,t =λΓηt+νΓ,t (3.15) the sum of a common shock componentλΓηtand the gamma-weighted idiosyncratic shocks νΓ,t. The common shock component, which results from the heterogeneous loadings λi, means the instrument is correlated withytE. The exclusion restriction therefore no longer holds.
To resolve this issue, GK recommend to first compute the difference between regional economic growth and equal-weighted economic growth
yi,t−yE,t = (λi−λE)ηt+ (νi,t−νE,t) (3.16) with this variable in essence removes date fixed effects (including the endogenousrt) from the panel data of yi,t’s. A factor model, such as Principal Component Analysis (PCA), can be run on this newly created variable to estimate the latent factor. The estimated factors are then used as control variable in the instrumental variables estimation.
There are many further extensions possible of the GIVs approach, with the reader referred to GK for details.