by Eric · Published May 6, 2021 · Updated June 16, 2022
If you aren't familiar with vector autoregressive models, you may want to start with our previous blog, Introduction to The Fundamentals of Vector Autoregressive Models.
VAR models are widely used in finance and econometrics because they offer a framework for understanding the intertwined relationships of multivariate time series data in a systematic manner.
Reduced-form VAR estimates can be complex, difficult to understand, and generally aren’t insightful on their own. More valuable insights come from structural analysis. In structural analysis, we apply the VAR relationship to understand the dynamic relationship between the variables in our model.
Structural analysis begins with the structural vector autoregression (SVAR). SVAR applies restrictions that allow us to identify the impacts that exogenous shocks have on the variables in the system.
Once the SVAR model is estimated, impulse response functions and forecast error variance decomposition are two of the most important structural analysis tools for examining those impacts.
| Applications of structural analysis after VAR |
|---|
| What impact does monetary policy have on real GDP? |
| How does a shock to income impact consumption paths? |
| Do exchange rate shocks pass through to international currencies? |
Impulse response functions trace the dynamic impact to a system of a “shock” or change to an input. While impulse response functions are used in many fields, they are particularly useful in economics and finance for a number of reasons:
In stationary systems, we expect that the shocks to the system are not persistent and over time the system converges. When the system converges, it may or may not converge to the original state, depending on the restrictions imposed on our structural VAR model.
For example, Blanchard and Quah(1989) famously demonstrated the use of long-run restrictions in a structural VAR to trace the impact of aggregate supply and aggregate demand shocks on output and unemployment. In their model:
As a result, when a positive aggregate supply shock occurs, output converges to a higher level than before the shock.
There is a clear modeling procedure to obtaining the impulse response functions:
Impulse responses are most often interpreted through grid graphs of the individual responses of each variable to an implemented shock over a specified time horizon.
Let’s look at an example to see how we can interpret these graphs.
The graph above shows the impulse response functions for a VAR(2) of income, consumption, and investment. These IRFs show the impact of a one standard deviation shock to income.
In order to estimate the structural VAR, short-run restrictions on the model were employed. These restrictions are such that:
Let’s look first at the IRF tracing the impact of the shock to income on income itself. In this graph, we see:
The consumption graph shows:
In the investment response to the income shock, we note that there:
Forecast error variance decomposition (FEVD) is a part of structural analysis which "decomposes" the variance of the forecast error into the contributions from specific exogenous shocks.
Intuitively this is useful because it:
As an example, FEVD may be used to explain how much various shocks, like supply and demand shocks, technology shocks, or monetary policy shocks, contribute to business cycle variations or long-term economic growth.
Like impulse response functions, forecast error variance decompositions are generally presented graphically, as either a bar graph or an area graph. At each time period, the graph plots the composition of the error variance across shocks to all the variables.
To understand how we interpret FEVD let's look at an example VAR(4) model (with a time trend and constant) of inflation, per-capita output, and the Federal Funds rate.
The plot above graphs the FEVD of the Federal Funds rate. This plot, like all FEVD plots:
From this we can tell that:
Today we've provided an intuitive look at impulse response functions and forecast error variance decompositions. These two multivariate time series tools are fundamental applications of the structural VAR model.
After today, you should have a better understanding of what these tools are and how to apply them.
Eric has been working to build, distribute, and strengthen the GAUSS universe since 2012. He is an economist skilled in data analysis and software development. He has earned a B.A. and MSc in economics and engineering and has over 18 years of combined industry and academic experience in data analysis and research.
Eric Post author May 15, 2023 at 7:26 am Hello, Thank you for your comment. I am happy you found the blog helpful! In regard to your question about the graphs, the graphs are labeled such that the first variable listed is the response variable and the second is the shock variable. For example, the graph titled "Consumption to Income" reflects the response of consumption to the income shock. Hope this helps! Best,
Eric
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