### Introduction

In today’s blog, we put the structural vector autoregressive (SVAR) model to work in a practical application. We use one of the most common applications of SVAR models, monetary policy analysis, to see the SVAR in action.

1. Introduction to the Fundamentals of Vector Autoregressive Models.

2. The Intuition Behind Impulse Response Functions and Forecast Error Variance Decomposition.

3. Understanding and Solving the Structural Vector Autoregressive Identification Problem.

4. Introduction to Granger Causality.

## The Research Question

One of the fundamental concepts of macroeconomics is that the Federal Funds Rate is one of the monetary policy levers that can be used to influence unemployment and inflation.

Students in introductory macroeconomics learn that decreasing the Federal Funds Rate is:

- Expansionary monetary policy.
- Expected to lead to decreases in unemployment and increases in inflation.

Conversely, increasing the Federal Funds Rate is:

- Contractionary monetary policy.
- Expected to lead to increases in unemployment and decreases in inflation.

Today we will see if the data supports these expectations following the analysis by Stock & Watson (2001).

## The Data

The original Stock & Watson (2001) time series data is quarterly and includes three variables:

- The rate of price inflation.
- The unemployment rate
- The Federal Funds interest rate.

## Granger Causality Testing

Granger causality is a helpful diagnostic tool for developing our structural VAR model. It allows us to verify the usefulness of one variable to forecast another.

We'll use the `granger`

procedure from the `TSPDLIB`

library to test the null hypothesis of non-causality.

Standard Granger Causality Test --------------------------------------------------------------------------- Direction Wald Bootstrap Lag p-val Inflation => Unemployment 4.195 0.144 2 FFR => Unemployment 14.387 0.000 2 Unemployment => Inflation 4.788 0.087 2 FFR => Inflation 2.374 0.311 2 Unemployment => FFR 19.886 0.000 2 Inflation => FFR 17.242 0.000 2

There are some helpful insights from these results:

**Unemployment**

- We cannot reject the null hypothesis of the non-Granger causality of inflation on unemployment.
- We can reject the null hypothesis of non-Granger causality of the Federal Funds Rate on unemployment at the 1% level.

**Inflation**

- We can reject the null hypothesis of non-Granger causality of unemployment on inflation at the 10% level.
- We cannot reject the null hypothesis of non-Granger causality of the Federal Funds Rate on inflation.

**Federal Funds Rate**

- We can reject the null hypothesis of non-Granger causality of unemployment on the Federal Funds Rate.
- We can reject the null hypothesis of non-Granger causality of inflation on the Federal Funds Rate.

These insights may help us determine our model, particularly our sign restrictions. For example, it seems reasonable to implement restrictions that:

- Do not allow unemployment nor the Federal Funds Rate to impact inflation. (Note that the 10% level doesn't very strongly indicate that we can reject the null hypothesis).
- Allow only Federal Funds Rate to impact unemployment.
- Allow both unemployment and inflation to impact the Federal Funds Rate.

## Reduced Form VAR

Next, let's estimate the reduced form var model.

For our estimation, we will follow the Stock and Watson (2001) model which allows for a constant and four lags:

$$\begin{aligned} UNEM = \alpha_{UNEM} &+ \phi^{(1)}_{11} UNEM_{t-1} + \phi^{(1)}_{12} INFL_{t-1} + \phi^{(1)}_{13} FFR_{t-1} \\ &+ \phi^{(2)}_{11} UNEM_{t-2} + \phi^{(2)}_{12} INFL_{t-2} + \phi^{(2)}_{13} FFR_{t-2} \\ &+ \phi^{(3)}_{11} UNEM_{t-3} + \phi^{(3)}_{12} INFL_{t-3} + \phi^{(3)}_{13} FFR_{t-3} \\ &+ \phi^{(4)}_{11} UNEM_{t-4} + \phi^{(4)}_{12} INFL_{t-4} + \phi^{(4)}_{13} FFR_{t-4} \\ &+ u_{UNEM, t} \\ INFL = \alpha_{INFL} &+ \phi^{(1)}_{21} UNEM_{t-1} + \phi^{(1)}_{22} INFL_{t-1} + \phi^{(1)}_{23} FFR_{t-1} \\ &+ \phi^{(2)}_{21} UNEM_{t-2} + \phi^{(2)}_{22} INFL_{t-2} + \phi^{(2)}_{23} FFR_{t-2} \\ &+ \phi^{(3)}_{21} UNEM_{t-3} + \phi^{(3)}_{22} INFL_{t-3} + \phi^{(3)}_{23} FFR_{t-3} \\ &+ \phi^{(4)}_{21} UNEM_{t-4} + \phi^{(4)}_{22} INFL_{t-4} + \phi^{(4)}_{23} FFR_{t-4} \\ &+ u_{INFL, t} \\ FFR = \alpha_{FFR} &+ \phi^{(1)}_{31} UNEM_{t-1} + \phi^{(1)}_{32} INFL_{t-1} + \phi^{(1)}_{33} FFR_{t-1} \\ &+ \phi^{(2)}_{31} UNEM_{t-2} + \phi^{(2)}_{32} INFL_{t-2} + \phi^{(2)}_{33} FFR_{t-2} \\ &+ \phi^{(3)}_{31} UNEM_{t-3} + \phi^{(3)}_{32} INFL_{t-3} + \phi^{(3)}_{33} FFR_{t-3} \\ &+ \phi^{(4)}_{31} UNEM_{t-4} + \phi^{(4)}_{32} INFL_{t-4} + \phi^{(4)}_{33} FFR_{t-4} \\ &+ u_{FFR, t} \end{aligned}$$

Our reduced form model estimates 39 total coefficients, 14 for each equation

Variable |
Unemployment Equation |
Inflation Equation |
Federal Funds Equation |
---|---|---|---|

Constant |
0.059 (0.086) |
0.947 (0.372) |
0.541 (0.334) |

$$UNEM_{t-1}$$ |
1.493(0.089) |
-1.145(0.387) |
-1.643(0.348) |

$$INFL_{t-1}$$ |
0.038(0.019) |
0.633(0.081) |
0.068 (0.073) |

$$FFR_{t-1}$$ | 0.005 (0.023) |
0.158 (0.102) |
0.946(0.091) |

$$UNEM_{t-2}$$ |
0.536(0.153) |
1.970(0.665) |
1.824(0.597) |

$$INFL_{t-2}$$ | -0.027 (0.022) |
0.053 (0.095) |
0.226(0.085) |

$$FFR_{t-2}$$ | 0.044 (0.031) |
-0.134 (0.135) |
-0.367(0.121) |

$$UNEM_{t-3}$$ | 1.493 (0.089) |
-1.145(0.387) |
-1.643 (0.348) |

$$INFL_{t-3}$$ | 0.038 (0.019) |
0.633 (0.081) |
0.068 (0.073) |

$$FFR_{t-3}$$ | 0.005 (0.023) |
0.158 (0.102) |
0.946(0.091) |

$$UNEM_{t-4}$$ | 0.536 (0.153) |
1.970 (0.665) |
1.824(0.597) |

$$INFL_{t-4}$$ | -0.027 (0.022) |
0.053(0.095) |
0.226 (0.085) |

$$FFR_{t-4}$$ | 0.044 (0.031) |
-0.134 (0.135) |
-0.367 (0.121) |

Bold values are statistically significant at the 10% level. |

The reduced form model doesn't provide much insight into our research question. It serves primarily as an intermediate step to setting up our structural analysis. The coefficients can be hard to interpret.

## Structural Analysis

To truly look deeper into the relationship between unemployment, inflation, and the Federal Funds Rate in the VAR environment we need to move from the reduced form model to the structural VAR model. To do this we will:

- Implement short-run restrictions that are consistent with our Granger causality testing.
- Compute the impulse response functions.
- Evaluate the forecast error decomposition.

### Implement Identifying Restrictions

For this model, we will implement short-run restrictions which align with the Granger causality findings that:

- Neither unemployment nor the Federal Funds Rate impacts inflation.
- Only the Federal Funds Rate impacts unemployment.
- Both unemployment and inflation to impact the Federal Funds Rate.

Short-run restrictions are implemented following two steps:

- Determining the ordering of our variables such that the short-run impact matrix is lower diagonal.
- Using Cholesky decomposition to find the lower-diagonal short-run impact matrix.

We can determine the correct ordering of our variables by considering the short-run impact matrix in our trivariate structural VAR:

$$\begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}$$

If we implement short-run restrictions using Cholesky decomposition, this matrix becomes:

$$\begin{bmatrix} b_{11} & 0 & 0 \\ b_{21} & b_{22} & 0 \\ b_{31} & b_{32} & b_{33} \end{bmatrix}$$

Note that this implies that:

- $Y_1$ is affected in the short-run only by its own shocks.
- $Y_2$ is affected in the short-run by its own shocks and shock to $Y_1$.
- $Y_3$ is affected in the short-run by its own shocks, shocks to $Y_1$, and shocks to $Y_2$.

Based on this, combined with our findings from the Granger causality testing, we should order our variables $\{ \text{INFL, UNEM, FFR} \}$.

### Impulse Response Functions

After implementing the short-run restrictions, we will compute our impulse response functions. Impulse response functions show us how instant shocks to one variable impact the variables in the model.

The impulse response functions:

- Are computed using short-run restrictions and a unit shock.
- Are computed across a horizon of 24 quarters.
- Are computed with 95% confidence intervals.

### Interpreting Impulse Responses

Looking at the graph above:

- Each individual row reflects the impacts of a specific shock. For example, the first row demonstrates the impacts of an inflation shock.
- Each individual column contains the impacts of shocks to each variable on a specific variable. For example, the first column demonstrates the impacts of shocks to inflation, unemployment, and the Federal Funds Rate to inflation.

There are some helpful insights we can gain from our impulse response functions.

**Inflation**

- If there is a shock to unemployment or the Federal Funds Rate at time period 0, there is no instantaneous impact on inflation. (Consistent with our Granger causality testing and short-run restrictions).
- After an
**initial shock to unemployment**, inflation bounces sharply in the first five quarters before landing on a long-run path of increasing. After 24 quarters it is slightly higher than it started. - After an
**initial shock to the Federal Funds Rate**, inflation bounces sharply in the first five quarters before landing on a long-run path of decreasing. After 24 quarters it is lower than it started.

**Unemployment**

- If there is a shock to the Federal Funds Rate at time period 0, there is no instantaneous impact on unemployment. (Consistent with our Granger causality testing and short-run restrictions).
- After an
**initial shock to inflation**, unemployment instantaneously decreases a very small amount before being pushed to a gradual, smooth path of increasing for roughly the first 20 quarter quarters. After 24 quarters it is higher than it started. - After an
**initial shock to the Federal Funds Rate**, unemployment smoothly increases for roughly 7 periods and smoothly decreases for the remaining periods. After 24 quarters it is slightly lower than it started.

**Unemployment**

- The FFR responds instantaneously to shocks in inflation and unemployment.
- After an
**initial shock to inflation**, the Federal Funds Rate instantaneously increases a very small amount followed by a sharp increase in the first four periods. It eventually returns to a longer-run path of smooth decline. After 24 quarters it is higher than it started. - After an
**initial shock to unemployment**, the Federal Funds rate sharply decreases for the first two periods. It eventually returns to a longer-run path of smooth increase. After 24 quarters it is slightly lower than it started.

### Forecast Variance Error Decomposition

Forecast error variance decomposition (FEVD) "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.

The FEVD for inflation shows that the majority of the forecast error variance results from inflation. This changes very little over time.

Additionally, we can note that the Federal Funds Rate contributes the least to the forecast error variance of inflation.

The graph above shows that all of the initial FEVD of unemployment results from unemployment. Over time, this decreases and the impacts from both inflation and unemployment grow.

The FEVD for the Federal Funds Rate shows that initially forecast error variance is most largely impacted by the Federal Funds Rate changes.

However, we can see as we move forward, the Federal Funds Rate is most largely impacted by unemployment changes.

### Conclusion

Today's blog provides a practical application of the VAR framework. Through the lens of monetary policy and macroeconomic outcomes, we see the SVAR in action. After this blog, you should have a stronger understanding of:

- How to use Granger causality testing to inform model selection.
- How to implement short-run identification restrictions.
- How to conduct and interpret structural VAR analysis.

### Further Reading

- Introduction to the Fundamentals of Time Series Data and Analysis
- Introduction to the Fundamentals of Vector Autoregressive Models
- The Intuition Behind Impulse Response Functions and Forecast Error Variance Decomposition
- Introduction to Granger Causality
- Understanding and Solving the Structural Vector Autoregressive Identification Problem

Erica has been working to build, distribute, and strengthen the GAUSS universe since 2012. She is an economist skilled in data analysis and software development. She has earned a B.A. and MSc in economics and engineering and has over 15 years combined industry and academic experience in data analysis and research.