15  The Structural Shock Framework

The counterfactual framework introduced in the last section defines the dynamic causal effect of any variable on another. As economists, we are more interested in understanding the causal relationships between important forces that drive the economy. We now introduce the structural shock framework, or the Slutzky-Frisch paradigm. This paradigm is explicitly or implicitly embedded in virtually every mainstream macroeconomic models or econometric models. It is not an essential component of time series analysis. But, as we would like to approach the topic from an economist’s perspective, it is good to have this framework in mind for many of our applications.

The structural shock framework envisions our economy as a complex system driven by a set of fundamental structural forces and coordinated by numerous price signals that automatically balance the demand and supply of all goods and services. The structural forces could be technology progress, climate change, policy changes and so on. These structural shocks are the primitive forces underlying our economy. When a structural shock happens, it triggers a reallocation of economic resources guided by market forces. In theoretical works, we are interested in modelling the system as a whole, particularly how resources are allocated optimally by market forces. In empirical works, we are interested how to recover the underlying structural shocks and estimate their causal effect on other economic variables.

In the language of time series analysis, we can envision our economy as an MA process, in which the observable variables (output, employment, inflation, etc) are the outcomes of accumulated past and current structural shocks:

\[ \boldsymbol y_t = \boldsymbol\Theta(L) \boldsymbol\epsilon_t, \]

or

\[ \begin{bmatrix} y_{1t}\\ y_{2t}\\ \vdots\\ y_{nt} \end{bmatrix} = \sum_{j=0}^{\infty} \begin{bmatrix} \theta_{j,11} & \theta_{j,12} & \cdots & \theta_{j,1m}\\ \theta_{j,21} & \theta_{j,22} & \cdots & \theta_{j,2m}\\ \vdots & \vdots & \ddots & \vdots \\ \theta_{j,n1} & \theta_{j,n2} & \cdots & \theta_{j,nm} \end{bmatrix} \begin{bmatrix} \epsilon_{1,t-j}\\ \epsilon_{2,t-j}\\ \vdots\\ \epsilon_{m,t-j} \end{bmatrix}. \]

\(\boldsymbol y_t\) represents the vector of economic variables of concern. The space of \(\boldsymbol y_t\) are spanned by \(m\) structural shocks (current and past): \(\{\boldsymbol \epsilon_{t-j}\}_{j=0}^{\infty}\).

Structural shocks are conceptual constructions that are primitive, unforeseeable, and uncorrelated underlying forces. Whether structural shocks do exist or not is an open question. But they are useful constructions that enable econometricians to disentangle different driving forces of the outcome variable.

In reality, almost every economic variable is endogenous. For example, monetary policy (interest rate) is set by the monetary authority based on their assessment of the economic conditions. But we can also imagine, there is a “genuine” component of the monetary policy, which may come from the personality of the policymaker and his mental conditions when he make the decision, that is not predictable from other variables. This genuine component is what we deem as the “monetary policy shock”. It is a shock in a sense that it is not predictable. It speaks for its own sake and contribute to the economic outcomes independently.

We do not observe the structural shocks directly. The observable variables are linear combinations of the structural shocks. For example, we may think of the observed interest rate as a linear combination of the monetary policy shock together with supply-side shocks and others.

\[ i_t = \theta_{1}(L)\epsilon_{t}^{\text{MP}} + \theta_{2}(L)\epsilon_{t}^{\text{SS}} + \cdots \]

Therefore, regressing inflation or output on interest rate will not give the causal effect of the monetary policy. Because interest rate does not represent the “genuine” monetary policy shock. It is determined by other economic variables and there are multiple structural forces at work. There are numerous literature that works on methods to isolate the “monetary policy shock” from the observed interest rates. Such way of constructing the structural shocks is not only a conceptual idea, but also a prerequisite for meaningful interpretation of the coefficients of econometric models.