Following Canova and Ciccarelli (2013)5, we can rewrite Equation (3) in a simultaneous equation form by stacking the N N n equations6: (4) y t = X t β + ε t ,   where   ε t ~ I I D ( 0 , Σ )

y t ≡ [ y 1 t ; y 2 t ; … ; y N t ] is stacked endogenous variables for all countries;

The length of the coefficient vector β is N N n ( 2 + p N N n + ( q + 1 ) N x ) , which can easily exceeds the sample size available. Noting that β varies with countries and with variables among others, it is therefore possible to model the variations of β by K parsimonious factors7: (5) β = F 1 φ 1 + F 2 φ 2 + … + F K φ K + ξ

This is essentially a linear model of the coefficient vector β . The “regressor” or factor F j has known inputs which can be determined with respect to the corresponding element in β , while φ j captures the “factor loading” effect. For example, φ 1 is usually the common factor across countries and variables, so it is a scalar and the corresponding F 1 is a vector of 1s. φ 2 could capture the factors common within countries, so its length is N and the corresponding F 2 is an indicator matrix with columns identifying countries. φ 3 could capture the factors common within variables, so its length is N n and the corresponding F 3 is an indicator matrix with rows identifying variables. One can flexibly add or omit more factors in the model to suit the scenario at hand.

Combine Equations (4) and (5), we then end up with a multivariate regression model: (6) y t = ∑ j = 1 K X ˜ j t φ j + ε ˜ t ,   where   X ˜ j t = X t F j   and   ε ˜ t = ε t + X t ξ

This is a much more parsimonious form compared to Equation (3), since the parameters to be estimated are φ j , which can then be used to derive β via Equation (5). Classical OLS can provide the consistent estimates.

Following Pesaran et al. (2004) and Dees et al. (2007), the exogenous state variable z t in the general PVAR Equation (3) can be divided into two parts: the common global exogenous variables, such as oil price, and the country-specific exogenous variables. In GVAR models, the country-specific exogenous variables are defined as weighted averages of corresponding foreign variables. Since the foreign variables are included in the exogenous state variables, the endogenous state variable only includes domestic endogenous variables. Equation (3) can therefore be written as a VARX( p , q ) model: (7) y i t = A i ( t ) + B i ( L ) y t − 1 + C 1 i ( L ) y i t * + C 2 i ( L ) z t + ε i t ,   where   ε i t ~ I I D ( 0 , Σ i i )

y i t is all the N n endogenous variables at t for country i ;

The exogenous state variable y i t * is defined as: y i t * ≡ ∑ j = 1 N w i j y j t ,   with   w i i = 0

The weights w i j can be derived from bilateral trades or capital flows between country i and the rest of the world. Intuitively, the relative importance of country j ’s GDP on that of country i can be described by the trade share of country j to country i relative to the rest of the world; the relative importance of country j ’s investment on that of country i can be described by the capital flow share of country j to country i relative to the rest of the world.

Note that the dimensionality of the PVAR is greatly reduced, since the “intertemporal interdependencies” are now captured by a single factor y i t * . Hence, other things equal, there will be N N n 2 ( p N − p − q ) less parameters to be estimated in the VARX( p , q ).

To estimate GVAR, the first step is to conduct country-by-country estimation of Equation (7) or a VECMX form of it to take into account cointegration: (8) Δ y i t = A i ( t ) + α i ( β i y ′ y i , t − p + β i y * ′ y i , t − p * + β i z ′ z i , t − p ) + C 1 i ( L ) Δ y i t * + C 2 i ( L ) Δ z t + ε i t

The second step is then to combine the country-specific VARX into a global specification—the GVAR. Note that in this step there is no estimation involved, and it is purely algebraic derivation. Define a “selection matrix”, which is an indicator matrix picking up country-specific endogenous variables from the global vector of endogenous variables: y i t = S i y t . Also define a “link matrix”, which consists of the weights of country-specific exogenous variables from the global vector of endogenous variables: y i t * = W i y t . Rewrite the estimated Equation (7) using the two matrices for country i : (9) S i y t = A ^ i ( t ) + B ^ i ( L ) y t − 1 + C 1 ^ i ( L ) W i y t + C 2 ^ i ( L ) z t + ε i t

Rearrange Equation (9), resulting in a country-specific model: (10) D ^ i y t = A ^ i ( t ) + E ^ i ( L ) y t − 1 + C 2 ^ i ( L ) z t + ε i t

Stack all country-specific models to finally get the GVAR: (11) y t = F ^ ( t ) + P ^ ( L ) y t − 1 + Q ^ ( L ) z t + ε t ,   or   y t = X t β ^ + D ^ − 1 ε t

y t ≡ [ y 1 t ; y 2 t ; … ; y N t ] is stacked endogenous variables for all countries;

The estimated GVAR in Equation (11) can be used for further inferences such as impulse responses and simulation. A remarkable advantage of GVAR is that the spillover effects of shocks across countries can be explicitly described. For example, one can easily derive the effect of a productivity shock in the US on the output in the UK.

The DSGE model component can be based on Harada (2017) NK model with innovation sector, Dees et al. (2014) MCNK model, Romer (1990) model, Grossman and Helpman (1991) model, Aghion and Howitt (1992) model, or any other model with a separate R&D sector. Since our focus is innovation—a real economic activity, we could put aside the nominal/monetary aspects without losing generality8 and employ an RBC-type DSGE model. In this scenario, endogenous variables can include output ( Y t ), consumption ( C t ), sector-specific employment ( L s t ) and capital ( K s t ) with s = 1 , 2 , human capital ( H t ) and knowledge capital ( X t ), whose variations are explained by exogenous variables including goods sector productivity ( A 1 t ), R&D sector productivity ( A 2 t ), capital flows and trade flows.

The multi-sector DSGE model for each country specifies the following optimization problems under rational expectations:

Household: max ∑ t = 0 ∞ β t u ( C t , l e i s u r e t ) , subject to budget and time constraints9;

Each optimization problem will result in a set of first order conditions describing the equilibrium relationships between the endogenous variables and exogenous variables.

In essence, the link equations are used to associate variables across countries to determine the variables which are treated as exogenous from each specific country’s perspective but endogenous from the global perspective.

Firstly, the production function in R&D sector is assumed to be dependent of both domestic and foreign knowledge capital stocks. The “foreign” concept here is the same as that in GVAR, i.e., each country is exposed to a different subset of global knowledge capital. It is reasonable to assume that this knowledge capital X t * is a weighted average of all its trade partners.

Moreover, note that currently the trade flows and capital flows, along with exchange rates, are treated as exogenous to keep things simple. To fully endogenize these variables, we need to redefine C t as a composite consumption including both domestic goods ( C H t ) and foreign goods ( C F t ), and also set up a new sector—trading firms as in Monacelli (2003). In this extended case, we need to specify how the exchange rate is determined such as PPP or UIP.

After the two structural components are specified, we will have N equations for the N endogenous variables. There are well-established techniques to analyse this dynamic stochastic equation system which is nonstationary and nonlinear. First, to deal with nonstationarity, we can either apply HP filtering or (preferably) divide the nonstationary variables by a basis variable to convert them into stationary ratios or rates. Second, to deal with nonlinearity, a standard procedure is to log-linearize the stationarized nonlinear system around the steady state and use perturbation method to solve the linearized equation system. Note that the structural parameters need to be calibrated using some initial values θ 0 , so that the equation system can be solved numerically. After obtaining the linearized solution of the endogenous variables in terms of the state variables, we can then estimate the best θ ^ following ML procedure or Bayesian procedure. Alternatively, we can also use the Indirect Inference technique as in Chen et al. (2017). Indirect inference is a simulation-based empirical test for DSGE models using reduced-form econometric regressions as auxiliary models. In our case, we can use the GVAR developed in the previous section as the auxiliary model for our purpose.

Based on the estimated value of the structural parameters, we can then draw inferences such as impulse response functions to be compared with those in the econometric models. However, to make meaningful comparison, we need to impose proper restrictions to identify structural shocks from PVAR.