## Introduction

This blog provides a non-technical look at impulse response functions and forecast error variance decomposition, both integral parts of vector autoregressive models.

If you're looking to gain a better understanding of these important multivariate time series techniques, you're in the right place. We cover the basics, including:

- What is structural analysis?
- What are impulse response functions?
- How do we interpret impulse response functions?
- What is forecast error variance decomposition?
- How do we interpret forecast error variance decomposition?

## What is structural analysis?

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 valuables 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 |
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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? |

## What are impulse response functions?

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:

- They are consistent with how we use theoretical economic and finance models. Theoretical economists develop a model, then ask how outcomes change in the face of exogenous changes.
- They can be used to predict the implications of policy changes in a macroeconomic framework.
- They employ structural restrictions which allow us to model our believed theoretical relationships in the economy.

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:

- They allow aggregate supply shocks to have lasting effects on output.
- Assume that aggregate demand shocks do not have lasting effects on long-run output.

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:

- Determine appropriate restrictions based on theory and/or previous empirical models.
- Estimate the structural VAR model.
- Predict the impulse response functions for a specified time horizon along with their confidence bands.
- Plot the predicted IRF and their confidence bands.

## How do we interpret 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.

### Example: VAR(2) Model of Consumption, Investment, and Income

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:

- Income shocks cannot contemporaneously (i.e. immediately) impact investment.
- Consumption shocks cannot contemporaneously impact income or investment.

Let’s look first at the IRF tracing the impact of the shock to income on income itself. In this graph, we see:

- The initial shock to in income in the first period.
- This shock quickly dies as the impact returns to almost zero in the second period.
- A slight increase in income in periods 2-4, with a post-shock peak in period 4.
- The impact converges back to zero after period 4.

The consumption graph shows:

- A quick jump in consumption at the time of the income increase -- this is consistent with the economic theory that consumption is a normal good (it increases with increases in income).
- A second spike in consumption occurs around the third period -- this is likely a lagging response to the increase in investment.

In the investment response to the income shock, we note that there:

- Is no first-period impact of the income shock on investment. This is by design and results directly from the restrictions implemented in order to estimate the SVAR.
- A short period of positive impact periods 2-4 which converges back to zero.

## What is forecast error variance decomposition?

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:

- Demonstrates how important a shock is in explaining the variations of the variables in the model.
- Shows how that importance changes over time. For example, some shocks may not be responsible for variations in the short-run but may cause longer-term fluctuations.

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.

## How do we interpret forecast variance error decompositions (FEVD)?

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:

- Has a Y-range from 0 to 100%.
- Shows the contributions from each individual shock as a portion of the total area (or bar) at any time period.

From this we can tell that:

- In the initial period, approximately 90% of the variation in the Federal Funds rate is from shocks to the Federal Funds rate itself and most of the remaining 10% is from inflation.
- The contribution of per-capita output to the variation in the Federal Funds rate changes fairly rapidly over the first 5 periods and eventually seems to converge at around 40%.
- The contribution of inflation to variations in the Federal Fund rate rises more slowly than that of per-capita output and levels off at roughly 20%.
- The system becomes stable after 15-20 periods.

### Conclusion

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.

### Further Reading

- Introduction to the Fundamentals of Time Series Data and Analysis
- Introduction to the Fundamentals of Vector Autoregressive Models
- Introduction to Granger Causality
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 15 years of combined industry and academic experience in data analysis and research.