In a previous post the impulse response functions for the German macroeconomic variables where 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 model of the German economy, this post will focus on the statistical definition of impulse response functions. Once the theory is carefully laid out, a model will be estimated and impulse responses calculated to provide context, but 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 can be used to derive 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

in order 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 and can be written as:

Plots of this function on y-axis with time on the x-axis would yield an impulse response graph. The summations of all of the impulse response functions as the forecast horizon approaches infinity are finite because the series are 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 affects to 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 for the U.S. unemployment rate:

A one standard deviation shock to the inflation rate increases the unemployment rate, the effect becomes statistically significant 7 quarters after the shock, and unemployment returns decreases to its previous value about 24 years later or 6 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 later.

A one standard deviation shock to interest rates increases unemployment. Unemployment reaches a maximum about 9 quarters after the the initial interest rate shock to the economy.