The study examines 54 variables8 divided into 5 factors namely: interest rate factor, the money supply factor, the inflation factor, the output factor and the crypto factor. These factors are studied in a Structural Factor Augmented VARx framework. The factors were extracted using Principal component analysis (PCA) and selected on the basis of the variables that maximum explained by it. The variables in each factor are close alternatives of each other. To mention, say for example, the money supply factor includes all narrow, broad and broader measures of money supply. The rationale for finding linkages among all these variables dates back to the 16th and 17th century whereby French philosopher Jean Bodin attributed the increase in rising prices in Western Europe to the monetary metals imported. This further got famous known as classical quantity theory of money in the 19th century and so on Humphrey (1974).
Several studies (Friedman & Schwartz, 1963a; Friedman & Schwartz, 1982; King & Plosser, 1983; Chow & Shen, 2004; Favara & Gordani, 2006; Ćorić, Perović, & Šimić, 2012; and Sharma & Nurudeen, 2020) have been conducted that analyse the relationship between these variables. However, ours is one of the first few FAVAR approaches that have been conducted in the Indian context which is structural and offers economic interpretation to the results.
The 54 variables are monthly frequency spanning from M05 2013 to M10 2021. The data sources include the Reserve Bank of India and crypto data from statista.com. For details of the variables see Appendix 5.2.
The objective of the study is to analyse the linkages among the macroeconomic variables considering crypto currency as an exogenous variable in a ‘datarich environment’. Several studies, conducted with smaller SVARs, show the existence of a price puzzle which is generally attributed to the lack of information in the model.
Further, the individual series are subjected to Augmented Dickey Fuller unit root test. The level of integration of each series is mentioned in the Appendix on data information. Mostly, the series are first difference stationary or integrated of order 1, that is, I (1). There are three methods in which the factors can be extracted: (i) the principal component analysis (PCA), (ii) the Bayesian method, and (iii) spectral analysis. We have used PCA to generate our factors, pioneered by Pearson (1901) and suggested by Ng & Perron (2001). The factors are derived from the standardised9 and stationary series and based on the maximum absolute loadings value. Each variable is assigned to a broader macro factor. Four factors are derived with help of PCA and these factors are taken as given for estimation purpose of the model parameters
In this modelling approach (Bernanke, Boivin, & Eliasz, 2005) observables and nonobservables series are introduced. The observables are those series which can directly be measured; similarly, non observables are those series which cannot directly be measured, for example, output gap. Since output gap cannot be observed, factors extracted can be used to represent the series. In either case, both observables and nonobservables are allowed to follow a VAR process.
After the application of Factor Augmentation to VAR framework by Bernanke, Boivin, & Eliasz (2005) many researchers followed suit and applied the methodology in various research works. For example, Stock & Watson (2005a); Stock & Watson (2005b); Eickmeier & Breitung (2005); Eickmeier (2007); Stock & Watson (2002); Negro & Otrok (2008); Bagliano & Morana (2009); Ahmadi & Ritschl (2009); Stock & Watson (2010); and Huh, Kim, Kim & Park (2014). Fransesco & Fabio (2006) also mention the PCA factor extraction method and further a Structural FAVAR analysis.
To understand the application of a FAVAR methodology, first, we demonstrate the factor extraction process. Thus, we consider the equation below in which Zt is a vector with large macroeconomic variables. Now, to disentangle Zt from observed and unobserved elements reported by Y, X and F respectively, we have the following relation.
$${Z}_{t} = {Г}_{1}{F}_{t} + {Г}_{j}{Y}_{t} +{Г}_{k}{X}_{t}+ {e}_{t}$$
1
where, Zt is a N*1 vector of macroeconomic variables, Г1 and (Гj & Гk) are Vectors of N*M factor loadings and structural coefficients (Yt is endogenous and Xt is exogenous) respectively. et is the random disturbance term with zero mean and constant volatility. Eq. (1) could be augmented in VAR framework and could be rewritten as follows:
$$\left[\begin{array}{c}{F}_{t}\\ \begin{array}{c}{Y}_{t}\\ {X}_{t}\end{array}\end{array}\right]=Q1+Q2\left(L\right)\left[\begin{array}{c}{F}_{t}\\ \begin{array}{c}{Y}_{t}\\ {X}_{t}\end{array}\end{array}\right]+ vt$$
2
where, Q1 is the vector of intercepts, Q2 is the Matrix of factor loadings and is the Kx1 vector of reduced from errors, which are also assumed to have zero mean and constant variance.
In order take care of the criticism that the estimates from Eq. (2) lack economic meaning, the FAVAR of Eq. (2), could be extended to include structural identification of the factors, thus the model becomes FASVAR, and could be specified as below:
AYt = Ѳ1 + Ѳ2 (L) Yt + Ѳ3 (L) Xt+ Bet (3)
here, A is an N*N contemporaneous impact matrix which measures the simultaneous response of the variables. B is an N*N matrix, and it represents the instantaneous impact of the structural shocks. Yt is N*1 vector of endogenous variables. Xt is n*1 vector of exogenous factor, however, It only acts as a constraint and impulse response is calculated for the endogenous variables considering the exogenous factor only as a constraint.
The term, Ѳ2(L)Yt represent the dynamics component of the explanatory variables and et is N*1 vector of structural shocks. When we divide both sides by A, we obtain a reduced form equation as follows:
where, A is M*N matrix of contemporaneous response, B is the M*K variance covariance matrix, et is the vector of structural shocks.
There are different ways in which restriction can be imposed to identify Eq. (4) based on A model, B model, and or AB model. Thus, the restrictions could be recursive as suggested by Sims (1980). Wold restriction as suggested by Wold (1969), restriction based on heteroskedasticity and signs as in Kilian (2011) and restriction based on economic theory. For the purpose of our study, a recursive restriction is used as suggested by Sims (1980). Thus, the model needs k *(k1/n) restrictions to identify the model as exactly or over identified. The restrictions on the model are as follows:
A = \(\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}Ui\\ Uii\\ Uiii\\ Uiѵ\end{array}\right]\) B= \(\left[\begin{array}{cccc}b11& 0& 0& 0\\ b21& b22& 0& 0\\ b31& b32& b33& 0\\ b41& b42& b43& b44\end{array}\right]\left[\begin{array}{c}ei\\ eii\\ eiii\\ eiѵ\end{array}\right]\) (5)
where, A is a diagonal matrix, and measures the contemporaneous response of the macroeconomic variables in the system. The B matrix is a lower triangular matrix and it measures the impact of contemporaneous shock. The exogenous factor only acts as a constraint in the model and the impulse responses are generated of the endogenous variables only, however considering crypto as a constraint in the entire process. Comparison of the SFAVAR and SFAVARx is made to check the impact of crypto currency on the behaviour of key macroeconomic variables.
Structural Factors
We partition the vector of economic variables Zt so that each variable is explained by one of the following structural factors:

Output factor: The output factor includes both quarterly real GDP data and Index of Industrial Production (IIP) data both use based and sectoral.

Inflation factor: The inflation factor includes all the data of Consumer Price Index (CPI) and Wholesale Price Index (WPI). Since WPI was the focus of RBI prior to the FIT Framework period and CPI subsequently.

Money supply factor: All the broad and broader measures of money supply are included.

Interest Factor: This factor includes the measures of interest rates used by both RBI and market determined rates.

Crypto factor: It includes five most traded crypto currencies in India.