Modeling the extremes of temperature time series Debbie J. Dupuis Department of Decision Sciences HEC Montréal
Outline
Fig. 1: S&P 500. Daily negative returns (losses), Realized Variance (RV) and Jump component (J) from January 1, 2000 December 31, 2014. Red dots indicate days for which a threshold set at the 98th quantile of the loss distribution is exceeded.
Following Davison and Smith (1990), the tail of the conditional distribution of Y t can be decomposed as Pr(Y t > y F t 1 ) = Pr(Y t > u F t 1 ) Pr(Y t u > y Y t > u, F t 1 ) where F t 1 is the information set of the process up to time t 1. φ t = Pr(Y t > u F t 1 ) a dynamic model such as a generalized linear model [or a non-homogeneous Poisson process for the counts N u (t)] Pr(Y t u > y Y t > u, F t 1 ) a GP distribution with parameters depending on covariates
Dynamic EVT Modeling in Finance Requires finding an economically sound source of information that can be used to explain the time-varying behaviour of the extremes. Current approaches exploit information in past daily exceedances. Chavez-Demoulin et al (2005) suggest : using a self-exciting process to model the probability of exceeding a high threshold of the loss distribution, and use a time-varying GP distribution with the past exceedances as covariates to model the size of the excesses. Alternatively, Chavez-Demoulin et al (2014) model the intensity parameter of the non-homogeneous Poisson process describing the exceedance rate and the time-varying scale parameter of the GP with non-parametric Bayesian smoothers.
F t : information set generated by the daily price path high-frequency (HF) data, i.e. intra-daily returns augment the available information set with HF data new information set H t where F t H t Realized Peaks-over-threshold [Bee, Dupuis and Trapin (2016)]
Realized measures are non-parametric estimators of the variation of the price path of an asset. They ignore the variation of prices overnight and sometimes the variation in the first few minutes of the trading day when recorded prices may contain large errors. Good background for realized measures : Barndorff-Nielsen and Shephard (2007) and Andersen et al. (2009). Let r t, = p t p t be the discretely sampled -period return on day t of the log-price. The most common realized measure is the realized variance (RV), RV t = N rt 1+j, 2, j=1 where N = 1/. It can be shown that, if prices are observed without noise then, as 0, this measure consistently estimates the quadratic variation of the price process on day t.
When the price process includes a jump component, the quadratic variation contains the contributions of both continuous and discontinuous sample paths. As these two components may contain different sources of information, define the bipower variation (BV) BV t = π 2 N r t 1+j, r t 1+(j 1), j=2 which consistently estimates the variation due to the continuous sample path. The jump variation J t can then be estimated by taking the difference between RV and BV, J t = max(0, RV t BV t ).
Fig. 2: S&P 500. Daily negative returns (losses), Realized Variance (RV) and Jump component (J) from January 1, 2000 December 31, 2014. Red dots indicate days for which a threshold set at the 98th quantile of the loss distribution is exceeded.
Fitted non-homogeneous Poisson models for specifications : I. log λ t = ζ 0 + ζ 1 log RV t 1 II. log λ t = ζ 0 + ζ 2 log BV t 1 + ζ 3 log(1 + J t 1 ) Tab. 1: S&P 500 returns over 2005-2009. *,**,*** indicate significance at the 5%, 1% and 0.1% levels, respectively. Threshold set to 90th quantile of the loss distribution. ζ 0 ζ 1 ζ 2 ζ 3 I. 3.34*** 0.62*** (0.52) (0.06) II. 3.49*** 0.62*** -0.02 (0.59) (0.07) (0.05)
Fitted dynamic GP models with scale specifications : I. log ν t = κ 0 + κ 1 log RV t 1 II. log ν t = κ 0 + κ 2 log BV t 1 + κ 3 log(1 + J t 1 ) Tab. 2: S&P 500 returns over 2005-2009. ξ is constant. *,**,*** indicate significance at the 5%, 1% and 0.1% levels, respectively. Threshold set to 90th quantile of the loss distribution. κ 0 κ 1 κ 2 κ 3 ξ I. -0.95 0.42*** 0.00 (0.61) (0.07) (0.07) II. -0.83 0.42*** -0.01-0.02 (0.67) (0.08) (0.05) (0.07)
Fitted non-homogeneous Poisson with HF and LF covariates LF. log λ t = ζ + ζ LF LM t 1 HF. log λ t = ζ + ζ LF LM t 1 + ζ HF log RV t 1 Tab. 3: S&P 500 returns over 2005-2009. *,**,*** indicate significance at the 5%, 1% and 0.1% levels, respectively. LM t 1 Model ζ ζ LF ζ HF I t 1 LF. -2.36*** 0.52* (0.10) (0.24) HF. 3.70*** -0.42 0.65*** (0.55) (0.25) (0.06) log DR t 1 LF. 1.76*** 0.48*** (0.46) (0.06) HF. 3.64*** -0.45** 1.07*** (0.52) (0.16) (0.17) log R2 t 1 LF. -0.48 0.17*** (0.40) (0.24) HF. 3.27*** -0.04 0.65*** (0.51) (0.03) (0.06)
Fitted dynamic GP models with HF and LF covariates. LF. log ν t = κ + κ LF LM t 1 HF. log ν t = κ + κ LF LM t 1 + κ HF log RV t 1 Tab. 4: S&P 500 returns over 2005-2009. ξ is constant. *,**,*** indicate significance at the 5%, 1% and 0.1% levels, respectively. LM t 1 Model κ κ LF κ HF W t 1 LF. -4.75*** 26.39*** (0.10) (6.27) HF. -1.87*** 15.19** 0.32*** (0.59) (5.24) (0.06) log DR t 1 LF. -1.57** 0.36*** (0.56) (0.07) HF. -0.76-0.17 0.59** (0.61) (0.19) (0.20) log R2 t 1 LF. -3.66*** 0.08** (0.41) (0.04) HF. -1.04** -0.04 0.46*** (0.61) (0.05) (0.08)
We know that temperature does not follow the assumptions made for the price process. We assumed price process P t = exp (p t ) is given by the stochastic integral Z t p t = p 0 + σ s db s, 0 where B s is a standard Brownian motion, and σ s is some (possibly random) process for which the integrated variance, Z t IV = σ 2 s ds, 0 is well defined. It has been show that, as N, RV converges to IV in probability. Moreover, the RV also converges in distribution in the sense that RV (N) IV q2t R t 0 σ 4 s ds, is approximately distributed as N(0,1) when N is large. Does the spirit of high frequency dynamics being informative on low frequency extremes carry over to extremes of temperature time series?
Fig. 3: For temperatures in March from January 1, 1973 to December 31, 2015. Red dots indicate days for which a threshold set at the 97th quantile of the March distribution is exceeded. Max daily temp 15 20 25 30 35 1973 1987 2001 2015 LAX 03 Realized Vol 0 20 40 60 80 100 1973 1987 2001 2015
Fig. 4: For temperatures in July from January 1, 1973 to December 31, 2015. Red dots indicate days for which a threshold set at the 97th quantile of the July distribution is exceeded. Max daily temp 20 25 30 35 1973 1987 2001 2015 Ely 07 Realized Vol 0 50 150 250 1973 1987 2001 2015
Fig. 5: For temperatures in May from January 1, 1973 to December 31, 2015. Red dots indicate days for which a threshold set at the 97th quantile of the May distribution is exceeded. Max daily temp 10 15 20 25 30 35 1973 1987 2001 2015 JFK 05 Realized Vol 0 20 40 60 80 120 1973 1987 2001 2015
Fig. 6: For temperatures from January 1, 1973 to December 31, 2015. J F M A M J J A S O N D 0 10 20 30 40 0 10 20 30 40 Daily Max Temp sqrt(rv) LAX
Fig. 7: For temperatures from January 1, 1973 to December 31, 2015. J F M A M J J A S O N D 10 0 10 20 30 40 10 0 10 20 30 40 Daily Max Temp sqrt(rv) Ely
Fig. 8: For temperatures from January 1, 1973 to December 31, 2015. J F M A M J J A S O N D 10 0 10 20 30 40 10 0 10 20 30 40 Daily Max Temp sqrt(rv) JFK
Fig. 9: Square root RV. For temperatures from January 1, 1973 to December 31, 2015. 19.01 19.02 19.03 19.04 19.05 19.06 19.07 19.08 19.09 19.10 19.11 19.12 0 5 10 15 LAX sqrt(rv) 19.01 19.02 19.03 19.04 19.05 19.06 19.07 19.08 19.09 19.10 19.11 19.12 0 5 10 15 20 25 Ely sqrt(rv) 19.01 19.02 19.03 19.04 19.05 19.06 19.07 19.08 19.09 19.10 19.11 19.12 2 4 6 8 10 12 JFK sqrt(rv)
Fitted non-homogeneous Poisson with HF and LF covariates LF. log λ t = ζ + ζ TD t + ζ LF LM t 1 HF. log λ t = ζ + ζ TD t + ζ LF LM t 1 + ζ HF log RV t 1 Tab. 5: Max Daily Temp 1973-2015. *,**,*** indicate significance at the 5%, 1% and 0.1% levels, respectively. Threshold set to 97th quantile of daily max temp distribution. LM t : 3-day rolling mean of daily max temp. Model ζ ζ TD ζ LF ζ HF LAX March LF. -11.30*** 0.71 0.36*** (0.82) (0.49) (0.03) HF. -9.98*** 1.57* 0.09 1.08* (1.06) (0.64) (0.08) (0.55) Ely July LF. -35.00*** 1.24 0.91*** (4.08) (1.07) (0.11) HF. -62.56*** 1.87 0.80*** 5.83*** (9.46) (1.44) (0.13) (1.63) JFK May LF. -10.27*** -0.45 0.34*** (1.09) (0.59) (0.04) HF. -9.06*** -0.47 0.40*** -0.96* (1.29) (0.58) (0.05) (0.44)
Fig. 10: Ely July - Fitted Intensity. Months of July 1973 to July 2015. intensity 0 2 4 6 8 Fitted intensity Upper CI Transf Lower CI Transf 0 200 400 600 800 1000 1200 time
There are many unresolved issues...
Not really RV as we do not have a fixed -period. Fig. 11: Number of HF per month. from January 1, 1973 to December 31, 2015. LAX Ely JFK Frequency 0 20 40 60 80 100 120 140 Frequency 0 50 100 150 200 Frequency 0 50 100 150 200 700 800 900 1100 600 800 1000 1200 700 900 1100
Is there a better way to consolidate intra-day temperature dynamics so as to yield a more informative covariate? Unresolved issues, continued... No interesting results for dynamic GP yet. Is there a better way to deal with the seasonality?
Thank you for your attention!
References Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2009), Parametric and nonparametric volatility measurement, Handbook of Financial Econometrics, 67 138. Barndorff-Nielsen, O. E. and Shephard, N. (2007), Variation, jumps and high frequency data in financial econometrics, Econometric Society Monograph. Cambridge University Press : Cambridge, UK. Bee, M., Dupuis, D.J., Trapin, L. (2016), Realized Peaks over Threshold : a High-Frequency Extreme Value Approach for Financial Time, submitted. Chavez-Demoulin, V., Davison, A., and McNeil, A. J. (2005), Estimating Value-at-Risk : A point process approach, Quantitative Finance, 5, 227 234. Chavez-Demoulin, V., Embrechts, P., and Sardy, S. (2014), Extreme-quantile tracking for financial time series, Journal of Econometrics, 181, 44 52. Davison, A. C. and Smith, R. L. (1990), Models for exceedances over high thresholds, Journal of the Royal Statistical Society : B, 52, 393 442.