How to perform quantile-on-quantile regression in this paper Oil prices, US stock return, and the dependence between their quantiles?

The paper "Oil prices, US stock return, and the dependence between their quantiles" by Hoang et al. (2018) investigates the relationship between oil prices and US stock returns using quantile-on-quantile regression. This method allows for analyzing the relationship between two variables across different quantiles of their distributions.

To perform quantile-on-quantile regression, follow these general steps:

Collect data: Gather data on the two variables of interest, in this case, oil prices and US stock returns, over a specific time period.

Divide the data into quantiles: Divide the data for each variable into quantiles (e.g., deciles, quartiles, or quintiles) based on their distributions. This means that each variable will have multiple quantiles, each containing a different range of values.

Choose the quantile levels: Select the quantile levels that you want to analyze. For example, you could choose to compare the 10th percentile of oil prices with the 90th percentile of US stock returns.

Estimate the conditional quantile: For each combination of quantile levels, estimate the conditional quantile of one variable given the other variable using quantile regression. This involves estimating the slope and intercept coefficients for the quantile regression model.

Analyze the results: Examine the estimated coefficients to determine the strength and direction of the relationship between the two variables at each quantile level.

In the case of the Hoang et al. (2018) paper, the authors used the following specific steps:

Data: They collected daily data on West Texas Intermediate (WTI) crude oil prices and the S&P 500 index returns from January 1990 to May 2017.

Quantiles: They divided the data for each variable into quintiles based on their distributions.

Quantile levels: They chose to compare the 10th and 90th quantiles of each variable.

Quantile regression: They estimated the conditional quantile of stock returns given oil prices and vice versa using quantile regression.

Results: They found evidence of a positive relationship between oil prices and US stock returns at the higher quantiles, indicating that when oil prices are high, US stock returns are also high. They also found evidence of a negative relationship at the lower quantiles, indicating that when oil prices are low, US stock returns are lower.

Overall, to perform quantile-on-quantile regression, you will need to select the appropriate quantile levels to analyze, estimate the conditional quantiles using quantile regression, and analyze the results to gain insights into the relationship between the two variables at different parts of their distributions.

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