Bootstrapping Impulse-Response Functions of Structural VAR Models

Author and guest post by Eren Ocakverdi

This blog piece intends to introduce a new add-in (i.e. BOOTSVAR) that extends the current capability of EViews’ available features for obtaining bootstrapped draws of impulse response functions from structural VAR estimations.

Table of Contents

1. Introduction
2. Residual Based Bootstrapping
3. Application to Estimation of Sacrifice Ratio for Turkish Economy
4. Files
5. References

Introduction

Impulse-response analysis is the most preferred way to reveal the interaction between the variables of interest modelled through regular or structural VAR analyses. The uncertainty in the estimation process is depicted by the confidence intervals, which is usually determined by bootstrap methods as they maintain the contemporaneous relationships among residuals.

Residual Based Bootstrapping

Suppose that we have the following underlying data generating process (DGP): \begin{align*} A_0 y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + u_t \end{align*} Here, $y_t = [y_{1t}, y_{2t}, \ldots, y_{mt}]'$, $A_i$ are $(m \times m)$ coefficient matrices and $u_t = [u_{1t}, u_{2t}, \dots, u_{mt}]'$ white noise disturbance matrices.

The bootstrap algorithm is then as follows:

1. Estimate the model to obtain model parameters ($\hat{A}_0, \hat{A}_1, \ldots, \hat{A}_p$) and store the residuals ($\hat{u}_t$).
2. Compute the centered residuals: $\hat{u}_t - \frac{1}{T} \sum_{t=1}^{T} \hat{u}_t$.
3. Start the bootstrap loop:
   • 3a) Randomly draw with replacement from centered residuals to generate new bootstrap residuals ($u_t^* = [u_{1t}^*, u_{2t}^*, \dots, u_{mt}^*]' $).
   • 3b) Generate new bootstrap time series: $y_t^* = \hat{A}_0^{-1} \left(\hat{A}_1 y_{t-1}^* + \dots + \hat{A}_p y_{t-p}^* + u_t^* \right)$.
   • 3c) Re-estimate the coefficient matrices based on the new bootstrap time series and obtain the impulse response coefficients.
   • 3d) Repeat the steps a to c for a large number of times.

Application to Estimation of Sacrifice Ratio for Turkish Economy

The sacrifice ratio is a macroeconomic concept that addresses the cost of reducing inflation in terms of lost output. It is usually calculated under the assumption that monetary policy is implemented to permanently bring the inflation rate down (i.e. by 1 percentage point) and the tradeoff will be a cumulative loss in the output (i.e. % of GDP).

Example here is based on the two-variable structural VAR model of Cecchetti and Rich (2001), where the system includes only inflation and output. Turkish CPI and GDP figures over the 2005q1-2025q3 period are used in the analysis (see Figure 1).

Figure 1: GDP and CPI (annual % growth rates, seasonally adjusted data)

A two-variable VAR model with GDP growth and the change in the inflation rate is estimated (see bootsvar_example.prg). Authors imposed a long-run restriction (à la Blanchard and Quah, 1989) for their model, which ensures aggregate demand shocks to have no permanent effect on the level of output. This corresponds to “recursive long-run impulse response (F triangular)” in EViews’ terms and can also be defined at the outset. Monetary shock is expected to be disinflationary, so we need to make an adjustment to its sign in the factorization matrix. To run the procedure on the estimated base VAR model, we can use the add-in (see Figure 2).

Figure 2: GUI of the add-in (see the help file in the add-in’s folder for details)

Figures 3a and 3b plot the accumulated responses of GDP growth and the change in inflation rate to a monetary policy shock, respectively. They denote the responses of GDP and inflation in levels over the next five years. Responses display expected patterns where GDP growth declines after the shock, but eventually returns to its initial level. Inflation, however, displays a permanent decline on the order of 1 percent.

Figure 3a: Accumulated response of GDP growth to a monetary shock

Figure 3b: Accumulated response of change in the inflation rate to a monetary shock

Since sacrifice ratio is the relative impact of monetary policy on output and inflation over a certain time horizon (e.g. five years), we need to compute the size of the cumulative output loss after a permanent disinflation is achieved. Summing the accumulated response of GDP divided by the response of inflation will yield an estimate of this measure (see Figure 4).

Figure 4: Distribution of bootstrapped sacrifice ratios

Bootstrap estimates of sacrifice ratio demonstrate that nearly 0.4 percent of GDP is forgone to permanently lower inflation by 1 percentage point over the next five years. However, authors’ primary goal was to characterize the precision of sacrifice ratio estimates, since the identification of economic relationships and their magnitudes are usually subject to uncertainty. Wide confidence interval around the point estimate indicates a high degree of imprecision and casts a shadow to its reliability.

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Files

• BOOTSVAR_EXAMPLE_DATA.WF1
• BOOTSVAR_EXAMPLE.PRG

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References

1. Cecchetti, S. G., and Rich R. W. (2001), "Structural Estimates of the U.S. Sacrifice Ratio", JBES, vol. 19, no 4, 416-427.
2. Lütkepohl, H., (2000), "Bootstrapping impulse responses in VAR analyses", SFB 373 Discussion Papers 2000,22, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.