Weather Forecasting for Weather Derivatives

Transcription

1 Weather Forecasting for Weather Derivatives Sean D. Campbell and Francis X. Diebold Brown University and University of Pennsylvania

2 The Derivative Instruments Swaps, vanilla and exotic options The Underlying Maximum temperature, minimum temperature Average daily temperature Heating degree days, cooling degree days Precipitation (rainfall, snowfall), humidity, sunshine, etc.

3 Demand Side Estimate the distribution of weather risk and forecast Supply Side Estimate the distribution of the value at maturity and forecast

4 ing and Forecasting Issues Stochastic vs. deterministic Long-horizon vs. short horizon Nonstructural vs. structural Univariate vs. multivariate Top-down vs. bottom-up Dynamic volatility vs. constant volatility Density forecasts vs. point forecasts Forecast evaluation

5 Table 1 Temperature Measuring Stations City State Measuring Station Measuring Station Symbol Atlanta GA Hartsfield Airport ATL Chicago IL O Hare Airport ORD Cincinnati OH Covington, KY CVG Dallas TX Dallas - Fort Worth DFW Des Moines IA Des Moines Int l Airport DSM Las Vegas NV McCarran Int l Airport LAS New York NY La Guardia LGA Philadelphia PA Philadelphia Int l Airport PHL Portland OR Portland Int l Airport PDX Tucson AZ Tucson Airport TUS Table 2 Temperature-Related Variables Variable TMAX TMIN TAVG HDD CDD Definition Maximum Daily Temperature - rounded to the nearest integer Minimum Daily Temperature - rounded to the nearest integer (TMAX+TMIN)/2 max(, 65-TAVG) max(, TAVG-65)

6 Table 3 Missing Data Date Measuring Station Variable Reference Station 3 Year Average Difference 9/15/96 DSM TMAX Omaha, NE /15/96 DSM TMIN Omaha, NE /1/ PHL TMAX Allentown, PA /1/ PHL TMIN Allentown, PA /2/ LGA TMIN New York CP, NY /23/ PDX TMIN Salem, OR /12/ PHL TMAX Allentown, PA /19/ LGA TMAX New York CP, NY /2/ LGA TMAX New York CP, NY /15/ CVG TMAX Columbus, OH /3/ PHL TMIN Allentown, PA /4/ PHL TMIN Allentown, PA Notes: The table contains information regarding the weather stations used when the primary weather station records a missing value. The table lists the date the missing value was recorded, the measuring station recording a missing value, the missing variable and the location of the reference station used in place of the ususal measuring station. The table also reports the thirty-year mean difference in the missing variable as well as the thirty-year standard deviation of the difference.

7 Figure 1 Time Series Plots of Daily Average Temperature Atlanta Chicago C in c in a tti D a lla s D e s M o in e s Las Vegas N e w Y o rk P h ila d e lp h ia P o rtla n d Tucson Notes: Each panel displays a time series plot of daily average temperature,

8 Figure 2 Estimated Unconditional Distributions of Daily Average Temperature Series: Atlanta Sample 1/1/196 11/5/21 Observations Mean Median 63.5 Maximum 92. Minimum 5. Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Chicago Sample 1/1/196 11/5/21 Observations Mean Median 51. Maximum 92.5 Minimum -18. Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Cincinatti Sample 1/1/196 11/5/21 Observations Series: Dallas Sample 1/1/196 11/5/21 Observations Mean Median 55.5 Maximum 9. Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Mean Median 67.5 Maximum 97. Minimum 8.5 Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Des Moines Sample 1/1/196 11/5/21 Observations Mean 5584 Median 52.5 Maximum 92. Minimum Std. Dev Skewness Kurtosis 2944 Jarque-Bera Probability Series: Las Vegas Sample 1/1/196 11/5/21 Observations Mean Median 66.5 Maximum 12. Minimum 19.5 Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: New York Sample 1/1/196 11/5/21 Observations Mean Median 55.5 Maximum 94. Minimum 2.5 Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Philadelphia Sample 1/1/196 11/5/21 Observations Mean Median 55.5 Maximum 92. Minimum.5 Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Portland Sample 1/1/196 11/5/21 Observations Series: Tucson Sample 1/1/196 11/5/21 Observations Mean Median 53. Maximum 86. Minimum 11. Std. Dev Skewness Kurtosis Jarque-Bera Probability Mean Median 68.5 Maximum 99. Minimum 28.5 Std. Dev Skewness Kurtosis Jarque-Bera Probability. Notes: Each panel displays a histogram and associated descriptive statistics of daily average temperature,

9 A Forecasting for Average Temperature

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11 1 Figure 3 Estimated Seasonal Patterns of Daily Average Temperature Fourier Series vs. Daily Dummies A t la n t a C h i c a g o : : : : : : : : 1 F o u r ie r S e a s o n a l D u m m y S e a s o n a l F o u r ie r S e a s o n a l D u m m y S e a s o n a l C in c in a t t i : : : : 1 D a lla s : : : : 1 F o u r ie r S e a s o n a l D u m m y S e a s o n a l F o u r ie r S e a s o n a l D u m m y S e a s o n a l D e s M o i n e s : : : : 1 L a s V e g a s : : : : 1 F o u r ie r S e a s o n a l D u m m y S e a s o n a l F o u r ie r S e a s o n a l D u m m y S e a s o n a l N e w Y o r k : : : : 1 P h ila d e lp h ia : : : : 1 F o u r ie r S e a s o n a l D u m m y S e a s o n a l F o u r ie r S e a s o n a l D u m m y S e a s o n a l P o r t la n d : : : : 1 T u c s o n : : : : 1 F o u r ie r S e a s o n a l D u m m y S e a s o n a l F o u r ie r S e a s o n a l D u m m y S e a s o n a l Notes: We show smooth seasonal patterns estimated from Fourier models,, and rough seasonal patterns estimated from dummy variable models,, as discussed in detail in the text.

12 Figure 4 Inverted Estimated Autoregressive Roots of Daily Average Temperature Atlanta Chicago Cincinatti Dallas Des Moines Las Vegas New York Philadelphia Portland 1. Tucson Notes: Each panel displays the reciprocal roots of the autoregressive lag operator polynomial,, associated with the estimated model, as discussed in detail in the text. The real component of each inverse root appears on the horizontal axis, and the imaginary component appears on the vertical axis.

13 Figure 5 Actual Values, Fitted Values, and Residuals Daily Average Temperature A t la n ta C h ic a g o M o d e l F it R e s id u a l M o d e l F it R e s id u a l C in c in a tti D a l l a s M o d e l F it R e s id u a l M o d e l F it R e s id u a l D e s M o in e s L a s V e g a s M o d e l F it R e s id u a l M o d e l F it R e s id u a l N e w Y o rk P h il a d e l p h ia M o d e l F it R e s id u a l M o d e l F it R e s id u a l P o r t la n d T u c s o n M o d e l F it R e s id u a l M o d e l F it R e s id u a l Notes: Each panel displays actual values, fitted values, and residuals from an unobserved-components model,, as discussed in detail in the text,

14 Figure 6 Estimated Unconditional Distributions of Residuals Daily Average Temperature Series: Atlanta Sample 1/26/196 11/5/21 Observations 1526 Mean -9.78E-15 Median Maximum 2843 Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Chicago Sample 1/26/196 11/5/21 Observations 1526 Mean 8.63E-15 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Cincinatti Sample 1/26/196 11/5/21 Observations 1526 Mean -1.34E-14 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Dallas Sample 1/26/196 11/5/21 Observations 1526 Mean 1.98E-15 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera 3291 Probability Series: Des Moines Sample 1/26/196 11/5/21 Observations 1526 Mean 1.78E-14 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Las Vegas Sample 1/26/196 11/5/21 Observations 1526 Mean -1.9E-14 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: New York Sample 1/26/196 11/5/21 Observations 1526 Mean -1.5E-14 Median Maximum Minimum Std. Dev Skewness Kurtosis Series: Philadelphia Sample 1/26/196 11/5/21 Observations 1526 Mean -1.87E-14 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Jarque-Bera Probability. 5 4 Series: Portland Sample 1/26/196 11/5/21 Observations Series: Tucson Sample 1/26/196 11/5/21 Observations Mean 2.3E-14 Median Maximum Minimum Std. Dev Skewness 287 Kurtosis Jarque-Bera Probability Mean 6.58E-15 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability. Notes: Each panel displays an empirical distribution and related statistics for the residuals from our daily average temperature model,, as discussed in detail in the text.

15 Figure 7 Correlograms of Residuals Daily Average Temperature A tla n ta C hicago Cincinatti D a lla s D e s M o in e s Las Vegas N e w Y o rk Philadelphia Portland Tucson Notes: Each panel displays sample autocorrelations of the residuals from our daily average temperature model,, together with Bartlett s approximate ninety-five percent confidence intervals under the null hypothesis of white noise, as discussed in detail in the text.

16 Figure 8 Correlograms of Squared Residuals Daily Average Temperature Atlanta C h ic a g o C in c in a tti D a lla s D e s M o in e s Las Vegas N e w Y o r k P h ila d e lp h ia P o r tla n d Tucson Notes: Each panel displays sample autocorrelations of the squared residuals from our daily average temperature model,, together with Bartlett s approximate ninety-five percent confidence intervals under the null hypothesis of white noise, as discussed in detail in the text.

17 Figure 9 Estimated Conditional Standard Deviations Daily Average Temperature A t la n t a C h ic a g o Cincinatti D a lla s D es M oines Las Vegas N e w Y o r k Philadelphia Portland Tucson Notes: Each panel displays a time series of estimated conditional standard deviations of daily average temperature obtained from the model,.

18 Figure 1 Estimated Unconditional Distributions of Standardized Residuals Daily Average Temperature Series: Atlanta Sample 1/27/196 11/5/21 Observations Mean.1361 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Chicago Sample 1/27/196 11/5/21 Observations Mean 186 Median.338 Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Series: Cincinatti Sample 1/27/196 11/5/21 Observations Mean.862 Median.6877 Maximum Minimum Std. Dev. 1 Skewness Kurtosis Series: Dallas Sample 1/27/196 11/5/21 Observations Mean.6889 Median Maximum Minimum Std. Dev. 194 Skewness Kurtosis Jarque-Bera Probability Jarque-Bera Probability Series: Des Moines Sample 1/27/196 11/5/21 Observations Mean 136 Median Maximum Minimum Std. Dev Skewness Kurtosis Series: Las Vegas Sample 1/27/196 11/5/21 Observations Mean 145 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Jarque-Bera Probability Series: New York Sample 1/27/196 11/5/21 Observations Mean 5 Median 7442 Maximum Minimum Std. Dev Skewness Kurtosis Series: Philadelphia Sample 1/27/196 11/5/21 Observations Mean.1739 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Jarque-Bera Probability Series: Portland Sample 1/27/196 11/5/21 Observations Mean.1578 Median -642 Maximum Minimum Std. Dev Skewness Kurtosis Series: Tucson Sample 1/27/196 11/5/21 Observations Mean -24 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Jarque-Bera Probability. Notes: Each panel displays an empirical distribution of standardized residuals from our average temperature model,.

19 Figure 11 Correlograms of Standardized Residuals Daily Average Temperature A tla n ta C h ic a g o C incinatti D a lla s D es M oines Las Vegas N e w Y o rk Philadelphia Portland Tucson Notes: Each panel displays sample autocorrelations of the standardized residuals from our daily average temperature model,, together with Bartlett s approximate ninety-five percent confidence intervals under the null hypothesis of white noise, as discussed in detail in the text.

20 Figure 12 Correlograms of Squared Standardized Residuals Daily Average Temperature A tla n ta C h ic a g o C incinatti D a lla s D es M oines Las Vegas N e w Y o rk Philadelphia P o rtla n d Tucson Notes: Each panel displays sample autocorrelations of the squared standardized residuals from our daily average temperature model,, together with Bartlett s approximate ninety-five percent confidence intervals under the null hypothesis of white noise, as discussed in detail in the text.

21 Table 4 Point Forecast Accuracy Comparisons Daily Average Temperature Persistence Climatological Autoregressive EarthSat Atlanta 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Chicago 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Cincinnati 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Dallas 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Des Moines 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Notes: We show each forecast s root mean squared error, measured in degrees Fahrenheit.

22 Table 4 (Continued) Point Forecast Accuracy Comparisons Daily Average Temperature Persistence Climatological Autoregressive EarthSat Las Vegas 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day New York 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Philadelphia 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Portland 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Tucson 1-Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day-Ahead Day Notes: We show each forecast s root mean squared error, measured in degrees Fahrenheit.

23 Figure 13 Forecast Skill Relative to Persistence Forecast Daily Average Temperature Point Forecasts Atlanta Chicago EarthSat EarthSat Cincinatti Dallas EarthSat EarthSat Des Moines Las Vegas EarthSat EarthSat New York Philadelphia EarthSat EarthSat Portland Tucson EarthSat EarthSat Notes: Each panel displays the ratio of a forecast s RMSPE to that of a persistence forecast, for 1-day-ahead through 11-dayahead horizons. The solid line refers to the EarthSat forecast, and the dashed line refers to the autoregressive forecast. The forecast evaluation period is 1/11/99-1/22/1, as discussed in the text.

24 Figure 14 Forecast Skill Relative to Climatological Forecast Daily Average Temperature Point Forecasts Atlanta Chicago Earth Sat Earth Sat Cincinatti Dallas Earth Sat Earth Sat Des Moines Las Vegas Earth Sat Earth Sat New York Philadelphia Earth Sat Earth Sat Portland Tucson Earth Sat Earth Sat Notes: Each panel displays the ratio of a forecast s RMSPE to that of a climatological forecast, for 1-day-ahead through 11-dayahead horizons. The solid line refers to the EarthSat forecast, and the dashed line refers to the autoregressive forecast. The forecast evaluation period is 1/11/99-1/22/1, as discussed in the text.

25 Figure 15 z-statistics, Distributions and Dynamics Daily Average Temperature Distributional Forecasts Atlanta Chicago Atlanta Chicago Earth Sat - - Earth Sat Cincinnatti Dallas 1.1 Correlogram of Z Correlogram of Z^2 1.1 Correlogram of Z^3 Correlogram of Z^ Earth Sat Earth Sat Cincinnati Correlogram of Z Des Moines Correlogram of Z^ Correlogram of Z^3 Las Vegas Correlogram of Z^ Dallas Des Moines Earth Sat Earth Sat Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^4 New York Portland Correlogram of Z Correlogram of Z^2 Correlogram of Z^ Earth Sat Earth Sat Correlogram of Z^4 Philadelphia Tucson Correlogram 2 3of Z Correlogram 1 11of Z^ Correlogram 6 7of Z^ Correlogram of Z^4 Earth Sat Earth Sat Notes: Each row displays a histogram for z, as well as correlograms for the first four powers of z, where z is the probability integral transform of realized November-March HDDs, Dashed lines indicate approximate ninety-five percent confidence intervals in the iid U(,1) case corresponding to correct conditional calibration. See text for details.

26 Figure 15 (Continued) z-statistics, Distributions and Dynamics Daily Average Temperature Distributional Forecasts.6 Las Vegas New York Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^ Philadelphia Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^ Portland Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^ Tucson Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^ Correlogram of Z Correlogram of Z^2 Correlogram of Z^3 Correlogram of Z^4 Notes: Each row displays a histogram for z, as well as correlograms for the first four powers of z, where z is the probability integral transform of realized November-March HDDs, Dashed lines indicate approximate ninety-five percent confidence intervals in the iid U(,1) case corresponding to correct conditional calibration. See text for details.

27 New Directions Richer Climatological Dynamics Formal Unobserved Components ing Forecasting Under the Relevant Loss Function Direct ing of Daily Max and Min Temperatures Multivariate Analysis and Cross-Hedging Weather, Earnings and Share Prices