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In macroeconomics and macroeconomic models, shocks, innovations, and disturbances are very prominent and often mentioned in the literature.
In general, is there a difference between macroeconomic shocks, disturbances, and innovations, or can these terms be used interchangeably?
I don't think there is really a consensus and the definition depend on the field and context. Within applied macroeconomics, I prefer the definitions used by Blanchard and Watson (1986) and Bernanke (1986) and others. Valerie Ramey gives a good discussion of the definition in "Macroeconomic shocks and their propagation" in the Handbook of macroeconomics, vol. 2. You can see a discussion in a YouTube video here.
In short, a shock is an economically meaningful primitive exogenous force. We can think of these as the "empirical counterparts to the shocks we discuss in our theories." An innovation, on the other hand, is the residual from a reduced form VAR or similar model. An identifying assumption may establish a link between the innovation and the shock, but they are distinct concepts.
Within this context, the definition of a disturbance is less clear. Ramey mentions in the Handbook chapter that "shocks are most closely related to the structural disturbances in a simultaneous equation system."
In Greene "Econometric Analysis" the disturbance of linear regression, $$\epsilon_t = y_t - \mathbf{x_t}\boldsymbol{\beta}$$
is estimated with the residual, $$e_t = y_t - \mathbf{x_t b}$$
In the text, disturbances $\epsilon_t$ are also called innovations. Qualitative events in an economy that impact a process or model's variables or error term are shocks.
Here's an example of why your question is difficult and depends on the framework.
Case 1: Take the classical OLS case $Y = \beta X + \epsilon$.
In this case, the $\epsilon$ can thought of as an error term which represents noise around the response. So, the mean of the response is $\beta X$ but on any particular response is not going to be exactly equal to that mean because of $\epsilon$ which represents the variation around that mean.
Case 2: Take a simple dynamic model in econometrics such as the ADL(1,0) so that $Y_t = \rho Y_{t-1} + \beta X_{t} + \epsilon_t$.
In this case, $\epsilon_t$ is not really noise because it's going to stay in $Y_{t}$ after that period is finished so it's actually part of the model. It's almost like an exogenous variable rather than noise so, to me, the best term in this case for $\epsilon$ would probably be innovation. It has what economists refer to as a "permanent" effect on the response.
So, my point is that each case can be different. This paper by Qin explains all of this in much more detail and more clearly.
