n a previous post, the impulse response functions for the German macroeconomic variables were estimated and graphically depicted using STATA. The dialogue focused on the interpretation of the impulse response graphs. While that entry was concerned with the practical estimation of a German economy model, this post will focus on the statistical definition of impulse response functions. Once the theory explained, a model will be estimated, and impulse responses calculated to provide context. Still, again the ambitious aim of this post will be to answer the following questions:

**What are some of the assumptions behind impulse response functions and the underlying Vector Autoregressive (VAR) macroeconomic model?****How are impulse response functions derived from a VAR?****What do impulse responses tell us about the U.S. economy and where do they fall short in describing it?**

**Statistical Theory**

The Vector Moving Average (VMA) description of a stationary VAR system derives the Impulse Response Functions (IRF) of a model, using the VMA representation of a stationary VAR model as the starting point.

The equation above is the VMA model with the structural error terms, but it is useful to write the expression in terms of the reduced form residuals.

To simplify notation the matrix of coefficients within the summation sign will be written in compact form using this definition:

The moving average representation now can be written more compactly in terms of the structural error terms.

The impact multiplier represents the instantaneous reaction of an external shock in one variable to another is written as:

Plots of this function on the y-axis with time on the x-axis would yield an impulse response graph. The summations of all impulse response functions as the forecast horizon approaches infinity are finite because the series is assumed to be stationary:

The summation above is referred to as the long-run multiplier.

**U.S. Economic Model**

Using U.S. quarterly data on inflation, unemployment, and interest rates, I replicated the analysis of Stock and Watson that appeared in the Journal of Economic Perspectives (Volume 14, Number 4, Fall 2001). Consistent with their results, I found that there are significant long-term effects on the economy when there are one-standard deviation shocks to these variables.

Using a Choleski decomposition on a VAR model with ordering 1) inflation, 2) unemployment, and 3) interest rates, I calculate the following impulse response functions for the U.S. unemployment rate:

A one standard deviation shock to the inflation rate increases the unemployment rate. The effect becomes statistically significant seven quarters after the excitement, and unemployment decreases to its previous value of about 24 years later or six years.

A one standard deviation shock to unemployment causes unemployment to peak about 2-3 quarters; then it begins to decrease eventually overshooting, leading to a decrease in unemployment about 12-16 quarters.

A one standard deviation shock to interest rates increases unemployment. Unemployment reaches a maximum of about nine quarters after the initial interest rate shock to the economy.

**About the Author**

JJ Espinoza is Senior Full Stack Data Scientist, Macroeconomist, and Real Estate Investor. He has over ten years of experience working in the world’s most admired technology and entertainment companies. JJ is highly skilled in data science, computer programming, marketing, and leading teams of data scientists. He double-majored in math and economics at UCLA before going on to earn his master’s in economics, focusing on macro econometrics and international finance.

You can follow JJ on Medium, Linkedin, and Twitter.