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Section: New Results

Causal inference by independent component analysis with Application of to American macro-economic data

Participants : Jacques Lévy-Vehel, Anne Philippe, Marie-Anne Vibet

The aim of this work is to study the causal relationships existing among macro-economic variables under investigation, and trace out how economically interpreted random shocks affect the system. Structural vector of autoregressive models (SVAR) are usually applied in this kind of study and the causal structure is driven by the data. In this work, independent component analysis (ICA) is implemented in order to guaranty the identifiability of the causal structure. However, the use of ICA can only be done under the hypothesis that the residuals are non-Gaussian, an hypothesis easily verified with economic data.

The vector of autoregressive (VAR) model has the following reduced representation :

where, $Y_t$, is the vector of contemporaneous variables of dimension $K\times 1$, $p$ is the number of autoregressive variables, $A_j$, for $j=1,...,p$, are matrices of dimension $K\times K$ estimated by the model and $u_t$ is the vector of random disturbances of dimension $K\times 1$ and assumed to be a zero-mean white noise process, $u_t\sim N(0_K,{\Sigma }_u)$. Given enough data, both ${\Sigma }_u$ and all matrices $A_j$ can be corretly estimated by the VAR model.

However, the VAR model is not sufficient for policy analysis. Indeed, using the Moving Average representation of a stable VAR :

where ${\Phi }_0=I_K$ and ${\Phi }_j$, $j\ge 1$, are the coefficients matrices representing the impulse responses of the elements of $Y_t$ to the disturbances $u_{t-j}$. This representation is not unique.

The structiral VAR (SVAR) is essentially a VAR equipped with a particular choice of a matrix $P$ so that $Y_t={\sum }_{j=0}^{\infty }{\Phi }_{j}P{P}^{-1}{u}_{t-j}={\sum }_{j=0}^{\infty }{\Psi }_j{ϵ}_{t-j}$

where ${ϵ}_{t-j}$ are independent random shocks economically interpreted. To this aim, the ICA procedure is then used to find the proper matrix $P$ using the hypothesis that the residuals, ${ϵ}_{t-j}$, are non-Gaussian.

We used the VAR-LINGAM procedure developped by Moneta et al [28] and their package written for R software. We started by testing this procedure with a series of simulations study. We tackled the following questions : Are the coefficients of the matrices $B$ and $A$ well estimated by the VARLINGAM procedure ? Is the bootstrap function appropriate, and in particular, does it estimate properly the standard error of the coefficients of matrices $A$ and $B$ ? And how long should the economic data be in order to estimate correctly the coefficients of the matrices $B$ ?

As the conclusion to all these studies were correct enough, we went on analysing our real data that consists of 6 weekly time series US macro-economic data, reported from the first week of January 1996 to April 2016 : The BofA Merrill Lynch US Broad Market Index, The Bofa Merril Lynch US Corporate Index, Equity Indices S&P, 500, Federal Funds Rates,Treasury Bills, Other Factors Draining Reserve Balances.

The conclusions of this work is in discussion with economists and a paper will soon be written.