25  System of Equations

We have discussed how to estimate the effect of one economic variable on another, and the assumptions on which the estimate would have a (dynamic) causal interpretation. But one equation is often inadequate to characterize the economy, as it does not take into account the feedback between economic variables. For example, an oil price shock would have impact on the price levels, which would trigger adjustments in the monetary policy, which would further exert impact on price levels, real output and so on. To capture the intertwined relationships, it would require a system of equations.

Consider an example of a backward-looking Keynesian system:

\[ \begin{aligned} y_t &= \phi y_{t-1} -\psi(r_{t-1}-\phi_{t-1}) + \epsilon_t^{IS}\\ \pi_t &= \delta\pi_{t-1} + \kappa(y_{t-1} - y_{t-1}^n) + \epsilon_t^{S}\\ r_t &= \beta\pi_t + \gamma(y_t - y_t^n) + \epsilon_t^{MP} \end{aligned} \]

The first equation is the IS curve, which states the negative relationship between output and real interest rate. \(\epsilon_t^{IS}\) is a structural shock of investment-saving decisions that moves the IS curve. We call it structural shock, because it is associated with a structural meaning, not a mere residual from a regression. The second equation describes the Phillips curve, which postulates a positive correlation between inflation and output gap (where \(y_{t}^n\) is the potential output level). \(\epsilon_t^{S}\) is the supply shock, which originates from exogenous supply conditions (such as weather), that could also affect inflation. The third equation is the Taylor’s rule for monetary policy, which sets the interest rate in response to inflation and output gap. \(\epsilon_t^{MP}\) is the monetary policy shock, which is the unpredictable part of the monetary policy decision making.

The set of equations are called structural equations, in a sense that they describe the structure of the economy according to some economic theories (particularly the Keynesian theory). These equations were very popular in 70s and 80s until Sims (1980) questioned their validity. The fact is, these equations impose a lot of restrictions on the relationships between the variables. For example, why output responds to real interest rate but not inflation? Why interest rate does not enter the equation of inflation? Yes, the equations are justified by the theory. But who knows the theory is correct? In reality, economic variables influence each other, often in a way unknown to theorists. So why not model the economy unrestrictively and let the data tell us the relationships between the variables?

\[ \begin{aligned} y_t &= \phi_{11} y_{t-1} + \phi_{12}\pi_{t-1} + \phi_{13} r_{t-1} +\cdots \\ \pi_t &= \phi_{21} y_{t-1} + \phi_{22}\pi_{t-1} + \phi_{23} r_{t-1} +\cdots \\ r_t &= \phi_{31} y_{t-1} + \phi_{32}\pi_{t-1} + \phi_{33} r_{t-1} +\cdots \\ \end{aligned} \]

This gives rise to a vector autoregressive system:

\[ \begin{bmatrix} y_t \\ \pi_t \\ r_t \end{bmatrix} = \sum_{j=1}^{p} \begin{bmatrix} \phi_{j,11} & \phi_{j,12} & \phi_{j,13} \\ \phi_{j,21} & \phi_{j,22} & \phi_{j,23} \\ \phi_{j,31} & \phi_{j,32} & \phi_{j,33} \\ \end{bmatrix} \begin{bmatrix} y_{t-j} \\ \pi_{t-j} \\ r_{t-j} \end{bmatrix} + \begin{bmatrix} u_t^y \\ u_t^\pi \\ u_t^r \end{bmatrix} \]

This is called a vector autoregression (VAR). Ever since being proposed by Sims (1980), VARs have been the Swiss knife for empirical macroeconomists. This chapter offers a thorough introduction of this Nobel prize winning technique. We start by introduce the general general vector processes and the estimation methods. We then explain how VARs map to the structural framework (SVAR). We finish the chapter by a discussion on dimension reduction techniques and the cases when a VAR system is not stationary.