Week 11 Common Errors and Feedback Summary

1. Overview

This week’s assignment was to estimate a 3-Variable Vector Autoregression (VAR) for the US economy (GDP Growth, Inflation, Interest Rate) and identify monetary policy shocks using the Cholesky decomposition. The submissions were generally excellent, with most of you correctly implementing the estimation and identification strategy.

2. Technical Remarks

A. Data Transformations and Interest Rates

  • Standard Practice: In monetary VARs (like the classic Christiano, Eichenbaum, Evans models), we typically use:
    • GDP: Log-difference (Growth rate) or Log-level (if cointegration is handled).
    • Prices: Log-difference (Inflation).
    • Interest Rate: Levels (e.g., 5.0 for 5%).
  • Observation: Most of you did this correctly. One submission differenced the interest rate ($\Delta R_t$). While valid econometrically to ensure stationarity, it changes the interpretation of the shock (a shock to the change in rate vs. the rate itself).

B. Cholesky Identification

  • Ordering Matters: The Cholesky decomposition $P$ of the covariance matrix $\Sigma$ is lower triangular. This implies a recursive structure:
    • Variable 1 (GDP) responds only to Shock 1 on impact.
    • Variable 2 (Inflation) responds to Shock 1 and Shock 2 on impact.
    • Variable 3 (Interest Rate) responds to all three shocks on impact.
  • Monetary Policy Assumption: The assumption that the central bank observes GDP and Inflation before setting the rate (but the economy responds to the rate with a lag) implies $R_t$ should be ordered last. You all followed this correctly.

C. The “Price Puzzle”

  • Observation: Many of you found that in response to a contractionary monetary policy shock (Interest Rate $\uparrow$), inflation initially rises before falling.
  • Interpretation: This is a famous result in empirical macroeconomics known as the Price Puzzle. It is often attributed to the Fed having information about future inflation that the VAR model lacks (e.g., commodity prices). The Fed raises rates because it sees inflation coming, so we see rates rise and then inflation rise, mistakenly attributing the inflation to the rate hike. Adding commodity prices to the VAR often solves this.

3. Best Practices Checklist

  • Check Matrix Dimensions: Ensure your LHS ($Y_t$) and RHS ($Y_{t-1}$) matrices have the same number of rows after lags are created.
  • Verify Cholesky: In MATLAB, chol(Sigma) produces an upper triangular matrix. For the standard recursive form $Y_t = P \epsilon_t$, we need a lower triangular matrix. Always use chol(Sigma, 'lower') or transpose the result.
  • Label Shocks: Be careful which column of the Cholesky matrix corresponds to which variable. If $R_t$ is the 3rd variable, the monetary policy shock is the 3rd column.

4. Conclusion

Great work on this econometric assignment. Understanding VARs and identification is crucial for empirical macroeconomics.