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In this paper, we propose an extreme conditional quantile estimator. Derivation of the estimator is based on extreme quantile autoregression. A noncrossing restriction is added during estimation to avert possible quantile crossing. Consistency of the estimator is derived, and simulation results to support its validity are also presented. Using Average Root Mean Squared Error (ARMSE), we compare the performance of our estimator with the performances of two existing extreme conditional quantile estimators. Backtest results of the one-day-ahead conditional Value at Risk forecasts are also given.

Correct specification of a loss/returns distribution is key to the accuracy of a risk measure such as Value at Risk. As noted in [

The challenge of quantile crossing has been addressed by smoothing suggestions in [

The lack of monotonicity in estimation of conditional quantiles is addressed in [

Parametric quantile regression is used in [

We seek to improve the extreme conditional quantile estimator in [

Let

(Risk Measure). A risk measure is a function

We assume that

The corresponding

Let

Note that equation (

There exists

From the sample analog of equation (

Given that Assumptions

To still maintain dependence and ensure positivity of the scale function, this study incorporated a scale function in the form of a quantile autoregressive (QAR) function on the absolute of the nonstandardized residuals. This was achieved by replacing

Given that the QAR process defined in equation (

Most of the financial datasets are heavy-tailed [

Consequently,

This study applied Points Over Threshold (POT) method because it uses more data leading to better estimates compared to the Block Maxima method. POT models the distribution

We can find (positive-measurable function)

A major challenge in POT is accuracy in choosing a threshold to separate extreme observations from the center of the distribution [

For convenience in making inferences on variability of the estimated quantiles, a recommendation in [

The overall distribution of the standardized residuals was obtained by splicing the GPD with the empirical bulk distribution at the threshold using the approach in [

Let

From equation (

Let

Using equation (

Given that the QAR processes defined satisfy Assumptions

To evaluate the accuracy of our estimators, a sample of size

Sample path superimposed with median.

Clearly, from the sample path, there is some level of volatility clustering, which is common in most financial data. An ACF plot of the resulting standardized residuals in Figure

Autocorrelation plot of resulting residuals.

Mean excess plot.

The corresponding shape and scale parameter estimates from the GPD fit were 0.1156182 and 1.272937, respectively. Table

Sample statistics.

0 | 0.50 | 0.75 | 0.95 | 0.99 | |
---|---|---|---|---|---|

Minimum | −15.81 | −3.58 | −2.80 | −0.73 | 1.76 |

Median | 0.67 (0.01, 1.38) | 0.65 (0.48, 0.83) | 1.15 (0.99, 1.32) | 2.47 (2.34, 2.61) | 4.07 (3.97, 4.17) |

Mean | 0.69 | 0.66 | 1.16 | 2.48 | 4.07 |

Maximum | 10.69 | 3.22 | 3.55 | 4.42 | 5.47 |

Figure

Quantile estimates at various

The accuracy of our extreme quantile estimator was evaluated using the Average Root Mean Squared Error (ARMSE). The RMSE seeks to return the Mean Squared Error (MSE) to the original scale of the sample. For

ARMSE for

Sample size ( | |||
---|---|---|---|

250 | 0.96844 | 0.74493 | 0.62015 |

500 | 0.79632 | 0.60788 | 0.56179 |

1000 | 0.71974 | 0.54056 | 0.53254 |

2000 | 0.71416 | 0.49585 | 0.50037 |

3000 | 0.69751 | 0.47862 | 0.49713 |

4000 | 0.69608 | 0.46439 | 0.48776 |

We note that, for a large enough sample (2000 observations and above), ARMSE is lowest when

ARMSE for

Sample size ( | Without noncrossing constraint | With noncrossing restriction | ||||
---|---|---|---|---|---|---|

RRQ | ECQ | AECQ | RRQ | ECQ | AECQ | |

250 | 0.58236 | 0.64552 | 0.73307 | 0.58200 | 0.64230 | 0.72728 |

500 | 0.56989 | 0.61336 | 0.60766 | 0.56816 | 0.61083 | 0.60341 |

1000 | 0.53386 | 0.58654 | 0.53386 | 0.55780 | 0.58529 | 0.53186 |

2000 | 0.56051 | 0.58204 | 0.49562 | 0.55800 | 0.57880 | 0.49360 |

3000 | 0.55478 | 0.57663 | 0.48235 | 0.55235 | 0.57369 | 0.48073 |

4000 | 0.55271 | 0.57619 | 0.47032 | 0.55013 | 0.57295 | 0.46832 |

RRQ, restricted regression quantiles in [

Based on ARMSE, both RRQ and ECQ perform better than AECQ for small samples. However, as the sample size increases, AECQ outperforms both RRQ and ECQ. The decreasing ARMSE with increase in the sample size for AECQ and ECQ confirms that both are consistent estimators of the extreme conditional quantile. Also, for sample size above 2000, the rate of convergence of the AECQ estimator is higher than that of the ECQ estimator. It was not possible to comment on the consistency of the RRQ estimator, since its ARMSE fluctuated with increase in sample size. The consistent reduction in ARMSE when noncrossing constraint is added during estimation, confirming that indeed this constraint increases accuracy of resulting estimators.

In Section

Therefore, the conditional coverage hypothesis can be assessed using the likelihood ratio:

Table

Sample size ( | RRQ model | ECQ model | AECQ model | ||||||
---|---|---|---|---|---|---|---|---|---|

250 | 0.000 | 0.001 | 0.000 | 0.014 | 0.013 | 0.002 | 0.001 | 0.006 | 0.000 |

500 | 0.000 | 0.001 | 0.000 | 0.036 | 0.010 | 0.003 | 0.009 | 0.005 | 0.000 |

1000 | 0.000 | 0.000 | 0.000 | 0.042 | 0.008 | 0.003 | 0.019 | 0.004 | 0.000 |

2000 | 0.000 | 0.001 | 0.000 | 0.010 | 0.005 | 0.005 | 0.002 | ||

3000 | 0.000 | 0.001 | 0.000 | 0.012 | 0.006 | 0.005 | 0.003 | ||

4000 | 0.000 | 0.000 | 0.000 | 0.009 | 0.005 | 0.004 | 0.002 |

Observe that, as a consequence of consistency, the accuracy of ECQ and AECQ forecasts improves with increase in sample size. It can also be seen that all the three models perform poorly under

We have derived the extreme conditional quantile estimator and used it to obtain the one-step-ahead conditional Value at Risk forecast for a simulated financial distribution. Consistency of our estimators has been proved and illustrated through Monte Carlo simulations. We noticed that adding the noncrossing restriction during estimation improves accuracy of the resulting extreme conditional quantile estimator. Backtesting results from the one-step-ahead conditional Value at Risk forecasts indicate that independence and conditional coverage tests in [

Let

The functional form of

Considering the possible range

For

Since

and so

Similarly, when

Combining equations (

Thus,

To prove Condition 3(i), let

To verify Condition 4, let

Thus,

The proof proceeds in a similar way to the proof of Lemma

Observe that

Note that

Let

Combining Lemma

The data used in the article were simulated, and the data generating process (DGP) is included within the article.

The authors declare that they have no conflicts of interest.

The authors acknowledge the Pan-African University of Basic Sciences, Technology and Innovation (PAUSTI) for funding this research.