2026-09-12
By the end of this lecture, you will be able to:
Macroeconometricians have four main tasks:
Describe and summarize macroeconomic data
Make forecasts about future economic conditions
Quantify relationships in the macroeconomy
Advise policymakers on economic decisions
Note
After the economic problems of the 1970s, traditional structural models failed. VARs emerged as a practical alternative.
The Problem in the 1970s:
The Solution: Vector Autoregressions (VARs)
Primary Source: Stock & Watson (2001), Journal of Economic Perspectives
Think of a VAR as a system where:
Simple Example: Unemployment ↔︎ Inflation ↔︎ Interest Rates
Each affects the others over time!
VARs come in three varieties, each with different assumptions:
Let’s explore each in detail…
A reduced-form VAR expresses each variable as a linear function of:
Key Feature
The error terms can be correlated across equations because we haven’t imposed any structure yet.
Think of it as: A descriptive model that captures co-movements without making strong causal claims.
Let’s consider three variables:
Each is modeled with 2 lags (past values).
Note
In practice, lag length is chosen using AIC or BIC criteria.
\[\begin{aligned} \text{UNEM}_t = \beta_{10} &+ \beta_{11}\text{UNEM}_{t-1} + \beta_{12}\text{UNEM}_{t-2}\\ &+ \gamma_{11}\text{INFL}_{t-1} + \gamma_{12}\text{INFL}_{t-2} \\ &+ \phi_{11}\text{R}_{t-1} + \phi_{12}\text{R}_{t-2}+ \mu_{1t} \end{aligned}\]
Reading this equation:
\[\begin{aligned} \text{INFL}_t = \beta_{20} &+ \beta_{21}\text{UNEM}_{t-1} + \beta_{22}\text{UNEM}_{t-2}\\ &+ \gamma_{21}\text{INFL}_{t-1} + \gamma_{22}\text{INFL}_{t-2} \\ &+ \phi_{21}\text{R}_{t-1} + \phi_{22}\text{R}_{t-2}+ \mu_{2t} \end{aligned}\]
Same pattern: Inflation today depends on past values of all three variables.
\[\begin{aligned} \text{R}_t = \beta_{30} &+ \beta_{31}\text{UNEM}_{t-1} + \beta_{32}\text{UNEM}_{t-2}\\ &+ \gamma_{31}\text{INFL}_{t-1} + \gamma_{32}\text{INFL}_{t-2}\\ &+ \phi_{31}\text{R}_{t-1} + \phi_{32}\text{R}_{t-2}+ \mu_{3t} \end{aligned}\]
All three equations together form the VAR system.
Error terms (\(\mu_{1t}\), \(\mu_{2t}\), \(\mu_{3t}\)) represent “surprises”:
The Challenge
Correlated errors make it hard to identify specific shocks. Which shock caused what?
This motivates recursive and structural VARs.
Goal: Make error terms uncorrelated across equations.
How? Add carefully selected contemporaneous values as regressors.
Key Idea: Impose an ordering on variables: 1. UNEM affects INFL contemporaneously 2. UNEM and INFL affect R contemporaneously 3. But R doesn’t affect UNEM or INFL within the same period
Unemployment (first in order): \[\begin{aligned} \text{UNEM}_t = \beta_{10} &+ \beta_{11}\text{UNEM}_{t-1} + \beta_{12}\text{UNEM}_{t-2}\\ &+ \gamma_{11}\text{INFL}_{t-1} + \gamma_{12}\text{INFL}_{t-2} \\ &+ \phi_{11}\text{R}_{t-1} + \phi_{12}\text{R}_{t-2}+ \mu_{1t} \end{aligned}\]
Inflation (second): Now includes \(\text{UNEM}_t\) \[\begin{aligned} \text{INFL}_t = \beta_{20} &+ \delta_{21}\text{UNEM}_{t} + \beta_{21}\text{UNEM}_{t-1} + \beta_{22}\text{UNEM}_{t-2} \\ &+ \gamma_{21}\text{INFL}_{t-1} + \gamma_{22}\text{INFL}_{t-2}\\ &+ \phi_{21}\text{R}_{t-1} + \phi_{22}\text{R}_{t-2}+ \mu_{2t} \end{aligned}\]
Interest Rate (third): Includes both \(\text{UNEM}_t\) and \(\text{INFL}_t\)
\[\begin{aligned} \text{R}_t = \beta_{30} &+ \delta_{31}\text{UNEM}_{t} + \delta_{32}\text{INFL}_{t}\\ &+ \beta_{31}\text{UNEM}_{t-1} + \beta_{32}\text{UNEM}_{t-2}\\ &+ \gamma_{31}\text{INFL}_{t-1} + \gamma_{32}\text{INFL}_{t-2}\\ &+ \phi_{31}\text{R}_{t-1} + \phi_{32}\text{R}_{t-2}+ \mu_{3t} \end{aligned}\]
Result: Error terms are now uncorrelated!
Major Limitation
Results depend critically on the ordering of variables!
The Issue:
Example: With 3 variables, there are 3! = 6 possible orderings.
Which one is “correct”? This requires economic theory…
A structural VAR uses economic theory to:
Key Advantage
Makes explicit economic assumptions rather than arbitrary ordering choices.
Requires: “Identifying assumptions” based on economic theory.
Stock and Watson use this Taylor Rule:
\[R_t = r^* + 1.5(\bar{\pi}_t - \pi^*) + 1.25(\bar{u}_t - u^*)\] \[+ \text{lagged values of } R_t, \pi_t, u_t + \epsilon_t\]
Interpretation:
This equation becomes the interest rate equation in the structural VAR.
Step 1: Estimate Models
Step 2: Forecasting
Step 3: Policy Analysis
Instead of reporting individual coefficients, we focus on:
Does past X help predict Y?
How does a shock to X affect Y over time?
How much of Y’s forecast error is due to shocks in X?
Note
These are much more informative than VAR regression coefficients or R² statistics!
Question: Does knowing the past of X help predict Y?
Not true causality! It’s really “predictive causality.”
Remember
“X Granger-causes Y” means: “Past values of X help forecast Y, beyond what Y’s own past tells us.”
Test: F-test for whether lagged X terms are jointly significant in Y equation.
| Regressor | Inflation (π) | Unemployment (u) | Interest Rate (R) |
|---|---|---|---|
| π | 0.00 | 0.31 | 0.00 |
| u | 0.02 | 0.00 | 0.00 |
| R | 0.27 | 0.01 | 0.00 |
These are p-values. Lower = stronger evidence of Granger causality.
Interpretation:
Question: When we forecast variable Y, what fraction of the forecast error variance is due to shocks in each variable?
Intuition:
Example: How much of interest rate forecast errors are due to:
Variance Decomposition of Interest Rate (R) (Recursive VAR: π, u, R ordering; 1960-2000 quarterly data)
| Horizon (Qtrs) | Forecast SE | π Shocks | u Shocks | R Shocks |
|---|---|---|---|---|
| 1 | 0.85 | 2% | 19% | 79% |
| 4 | 1.84 | 9% | 50% | 41% |
| 8 | 2.44 | 12% | 60% | 28% |
| 12 | 2.63 | 16% | 59% | 25% |
Key Insights:
Question: What happens to all variables when we “shock” one variable?
Setup:
Visual Thinking
Like dropping a stone in a pond and watching the ripples spread.
Most meaningful in models with uncorrelated errors (recursive or structural VAR).
What you’ll see in plots:
Example Questions:
Two Types of Policy Questions:
“What if the Fed unexpectedly raised rates by 0.5%?”
“What if we switched from a Taylor rule to strict inflation targeting?”
What VARs Do Well
✓ Capture co-movements of multiple time series
✓ Standard, widely-accepted diagnostic tools
✓ Good forecasting performance for short horizons
✓ Small VARs are benchmarks for new methods
✓ Don’t require strong theoretical assumptions
Practical Success: Extensively used by central banks and policy institutions.
Time-Varying Parameters (Sims, 1993)
Bayesian VARs
Note
Modern approaches use restrictions and priors to make large VARs tractable.
From Spencer (JMCB, 1989):
Warning
Researchers must be transparent about these modeling choices!
When Using VARs:
Tip
VARs are powerful tools, but their results depend on modeling choices. Transparency and robustness checks are essential!
VARs capture dynamic relationships among multiple time series
Three types serve different purposes:
Focus on interpretable outputs: Granger causality, impulse responses, variance decompositions
Strengths: Flexible, data-driven, good for forecasting
Limitations: Identification challenges, ordering sensitivity, Lucas critique
Essential Papers:
For Applied Work:
Think about:
Thank you!