On the Lp-Norm Regression Models for Estimating Value-at-Risk


  • Pranesh Kumar Department of Mathematics and Statistics University of Northern British Columbia Prince George, BC V2N 4Z9, Canada
  • Faramarz Kashanchi Planning and Performance Improvement Northern Health Prince George, BC V2L 5B8, Canada




Value-at-Risk (VaR), Quantile Distributions, Least-Squares estimation, Lp-Norms


Analysis of risk measures associated with price series data
movements and its predictions are of strategic importance in the financial markets
as well as to policy makers in particular for short- and longterm planning for setting up
economic growth targets. For example, oilprice risk-management focuses primarily on
when and how an organization can best prevent the costly exposure to price risk.

Value-at-Risk (VaR) is the commonly practised instrument to measure risk
and is evaluated by analysing the negative/positive tail of the probability distributions of the
returns (profit or loss). In modelling applications, least-squares estimation (LSE)-based
linear regression models are often employed for modeling and analyzing correlated data.

These linear models are optimal and perform relatively well under conditions such as errors
following normal or approximately normal distributions, being free of large size outliers and satisfying
the Gauss-Markov assumptions. However, often in practical situations, the LSE-based linear regression
models fail to provide optimal results, for instance, in non-Gaussian situations especially when the errors
follow distributions with fat tails and error terms possess a finite variance.

This is the situation in case of risk analysis which involves analyzing tail distributions.
Thus, applications of the LSE-based regression models may be questioned for appropriateness
and may have limited applicability. We have carried out the risk analysis of Iranian crude oil price data
based on the Lp-norm regression models and have noted that the LSE-based models do not always
perform the best. We discuss results from the L1, L2 and L∞-norm based linear regression models.

ACM Computing Classification System (1998): B.1.2, F.1.3, F.2.3, G.3, J.2.