Anon

Quantile Regressions 155

market cap increases the debt in the capital structure declines. This finding
may be unique to the petroleum industry. These findings across the distribu-
tion could not have been observed with a classical multivariate regression
analysis.
As explained earlier, in order to make meaningful inferences across
quantiles, it is important to verify that coefficients estimated at each quan-
tile are statistically different. The null hypothesis that the slope coefficients
are all the same is given by

β 1 τ 1 = β 1 τ 2 = β 1 τ 10

Ho: β 2 τ 1 = β 2 τ 2 = β 2 τ 10

β 3 τ 1 = β 3 τ 2 = β 3 τ 10

where β 1 , β 2 , and β 3 are coefficients associated with free cash flow, fixed
asset ratio, and the market cap. There are eight restrictions for each coef-
ficient and with three coefficients there are a total of 24 (3 × 8) restrictions.
The calculated Wald test statistic is 60.49. The critical χ^2 value with 24
degrees of freedom at a 5% significance level is 36.41. Since the test statistic
is greater than the critical value, we reject the null hypothesis that the slope
coefficients are the same across the quantiles. Thus, conclusions drawn for
each quantile are statistically valid.

Key Points

■ (^) In the presence of outliers and a skewed distribution, inferences made
with classical regression analysis may not fully describe the data.
■ (^) The regression tool applied in the presence of non-normal distributions
is the quantile regression.
■ (^) Quantile regressions find parameters that would minimize the weighted
sum of absolute deviations between the dependent and the independent
variables at each quantile.
■ (^) Quantile regressions describe relationships between dependent and
independent variables within the context of time series and cross-
sectional data.
■ (^) In order to make meaningful inferences across quantiles, it is important
to verify that coefficients are different across the quantiles.
■ (^) Quantile regressions are useful tools for risk managers to manage the
tail risk.
■ (^) Applications of quantile regressions to asset management include gener-
ating forecasted returns that can be used in constructing portfolios and
determining the investment style of a portfolio manager.