# Addressing Conditional Heteroscedasticity in SVAR Models

## Introduction

Structural VAR models are powerful tools in macroeconomic time series modeling. However, given their vast applications, it is important that they are properly implemented to address the characteristics of their underlying data.

In today’s blog, we build on our previous discussions of SVAR models to examine the use of SVAR in the special case of conditional heteroscedasticity.

We will look more closely at:

• Conditional heteroscedasticity.
• The impacts of conditional heteroscedasticity on SVAR models.
• Estimating structural impulse response functions (SIRF) in the presence of conditional heteroscedasticity.
• An application to the global oil market.

## What is Conditional Heteroscedasticity?

Simply put, heteroscedasticity is defined as a non-constant variance for the residuals of a model. As noted by Mignon (2022, chapter 4), when the errors are non-spherical, heteroscedasticity may come from:

• Heterogeneity of the sample.
• Missing variables.
• Asymmetric distributions of the variables (distributions of income and wealth, for example).
• Improper transformation of the variables (linear instead of log-linear).
• Nature of the data (averaged observations coming from different samples).

Conditional heteroscedasticity is a particular form of heteroscedasticity. Many macroeconomic and finance applications, such as daily financial time series or monthly growth rates, exhibit periods of low volatility and other periods of high volatility. These clusters of volatility are also known as conditional heteroscedasticity because one period’s volatility is related to previous periods' volatility.

## What are the Implications of Conditional Heteroscedasticity?

Linear regression models assume that there is no heteroscedasticity, and the presence of any type of heteroscedasticity can impact the validity of regression results if not accounted for properly.

In particular, while the OLS coefficient estimates will be unbiased:

• OLS estimators are no longer efficient.
• Covariance matrix of the coefficients will be inconsistent.
• Standard inferences will be incorrect.

## What are the Consequences for Inference on Structural Impulse Response Functions (SIRFs)?

Recursive VAR systems are highly common in empirical work in macroeconomics and finance. However, as previously noted, macroeconomic and financial data can often suffer from conditional heteroscedasticity. For this reason, it is important to consider the impacts of conditional heteroscedasticity in the context of structural VAR.

In SVAR models, impulse response functions are often a more important result than the estimated parameters. They help us to better understand the dynamic relationship between the variables in our model.

Because SIRFs are widely used for analysis, it’s imperative to have a clear understanding of the uncertainty in their estimations. This is widely done using some type of bootstrapped confidence intervals which provide insight into the statistical likelihood of a SIRF.

MethodDescription
Pairwise bootstrapDraws samples from the joint distribution of the dependent and independent variables.
Wild bootstrapA new sample is generated by multiplying the prediction residual by a random variable and adding it to the prediction.
Moving block bootstrapData is split into overlapping blocks of length 𝝀 and a specified number of blocks are drawn at random, with replacement from the portioned data.

In the context of bootstrapped SIRFs, when not properly accounted for, conditional heteroscedasticity can lead to distorted inferences. However, not all bootstrap methods perform equally when facing conditional heteroscedasticity.

Brüggemann et al. (2016) have shown that in the presence of conditional heteroscedasticity:

• The wild and pairwise bootstraps underestimate the true asymptotic variances.
• The corresponding confidence intervals with pairwise and wild bootstraps are typically too narrow and miss the true value of the SIRF.

The main message, here, is that the pairwise bootstrap and the wild bootstrap underestimate the actual estimation uncertainty in the presence of conditional heteroscedasticity. If you detect conditional heteroscedasticity in your data, the MBB is better than the pairwise and wild bootstrap to estimate the true degree of uncertainty in the SIRF.

## What is Moving Block Bootstrap?

The basic idea of the bootstrap rests on the assumption that the simulated data sample that we use does a good job representing the actual population data. The moving block bootstrap does this by:

1. Generating consecutive, overlapping “blocks” of the original sample.
2. Drawing a specified number of “blocks” at random, with replacement.
3. Combine the sampled blocks to create a complete bootstrapped sample.

This method of sampling is particularly relevant for time series data because it maintains the relationship between neighboring observations.

## An Application to the Effects of US-China Political Tensions on the Oil Market

As an example application of MBB in the SVAR framework, we’ll look at the newly released paper of Cai, Mignon, and Saadaoui (2022).

### Background

This motivation behind this paper is built on the intuition that causal interactions between political events and economic developments exist. It specifically considers the idea that increasing political tensions between the US and China will impact the world economy and the oil market. This is a reasonable concern, given that the US and the Chinese economies are the largest consumers of oil (around 20% and 16% of the world's consumption, respectively), according to the BP Statistical Review of World Energy 2021.

### Data

The paper focuses on data over the period spanning from January 1971 to December 2019.

Relying on an extended period allows them to:

• Highlight the growing influence of the Chinese economy on the international scene
• Have a complete picture of the evolution of the political relationships between China and the US through time.

#### Measuring political tensions

In order to capture the effects of political tensions on the oil market, the newly released paper of Cai, Mignon and Saadaoui (2022) relies on a quantitative measure of political relationships developed in the Institute of International Relation of Tsinghua University.

This index:

• Fluctuates between -9 and 9 according to the occurrence of “bad” or “good” political events, using a scale similar to the Goldstein scale (Goldstein, 1992).
• Shows improved relationships between US and China at the end of the 1970s and at the end of the 1990s, when positive diplomatic developments have occurred.
• Indicates the relationship deteriorated considerably during the Tiananmen Square Event in 1989, after the bombing of the Chinese embassy in Belgrade in 1999, and during Trump’s administration.

#### Measuring the oil market

This paper uses three oil market indicators:

• Global oil supply.
• World oil demand.
• Real price of price, measured by the WTI spot price deflated with the US consumer price index.

### Model

#### The SVAR setup

Today, we will be considering an SVAR model that examines the effects of US-China political tensions on the oil market. This model is based on the newly released paper from Cai, Mignon, and Saadaoui (2022).

This model considers four endogenous variables, included in logarithmic terms:

• US-China political tension index
• Oil supply
• Oil demand
• Oil price

#### Estimation

We use the recursive identification scheme by using Cholesky decomposition to obtain a lower-triangular matrix.

The identification scheme:

• Reflects the hypothesis in the literature that political tensions influence contemporaneous market developments, but the reverse causality takes some time.
• Captures our primary interest, the short-run effects of political tensions on the oil market.

The baseline SVAR model:

• Is estimated from January 1971 to December 2019.
• Includes 24 lags.

#### Impulse response functions

The confidence intervals for the structural IRFs are computed using MBB to address the issue of conditional heteroscedasticity. Estimation follows the methodology in Brüggemann, R., Jentsch, C., & Trenkler, C. (2016).

The impulse response functions:

• Have an 80-month horizon.
• Reflect the impact of a 1% decrease in the US-China PRI measure.
• Include 68% and 90% confidence intervals.

From these impulse response functions, we can see that a 1% decrease in the US-China relationships leads to:

• Insignificant short-run impacts.
• A long-run increase in oil production.
• A long-run decrease in oil demand.
• A long-run increase in real oil prices.

## Conclusion

Today's blog looks more closely at the issue of inference in the structural VAR model. In particular, we consider the impact of conditional heteroscedasticity on inferences or structural impulse response functions.

### References

Brüggemann, R., Jentsch, C., & Trenkler, C. (2016). Inference in VARs with conditional heteroskedasticity of unknown form. Journal of Econometrics, 191(1), 69-85. https://doi.org/10.1016/j.jeconom.2015.10.004

Cai, Y., Mignon, V., & Saadaoui, J. (2022). Not All Political Relation Shocks are Alike: Assessing the Impacts of US-China Tensions on the Oil Market. Energy Economics, 106199. https://doi.org/10.1016/j.eneco.2022.106199

Mignon, V. (2022). Econométrie: Théorie et applications. Economica. 2ème edition.

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