Modeling Error Variances Understanding CH, ARCH and GARCH Models J. Stuart McMenamin Itron s Forecasting Brown Bag Seminar March 9, 011
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011 Brown Bag Seminars Modeling Error Variances --Understanding CH, ARCH and GARCH models -March 9, 011 What is a Good Model? -June 1, 011 System Operations Forecasting - September 13, 011 Variables For Each Time Horizon-December 13, 011 All at noon, Pacific Time All are recorded and available for review after the session. 009, Itron Inc. 3
Agenda Overview of error variance assumptions in linear models Tests for constant variance (homoskedasticity) Specifications for CH, ARCH, GARCH Estimation of heteroskedastic models Application to financial data (S&P 500) Application to energy usage data (daily load) Conclusions 009, Itron Inc. 4
The Standard Linear Model Standard linear model Y = βx + e t t t > e values are independent > identically distributed > normally distributed Skedasis is greek for dispersion The skedistic function is the variance function for a model In the standard linear model, the variance function is very simple: Var Cov ( e ) = σ ( variance is constant) t ( e,e ) 0 t t j = When variance is not constant, errors are heteroskedastic Constant variance is called homoskedastic 009, Itron Inc. 5
Depiction of Homoskedasticity In the single variable case (Y=a+bX+e) Y Constant Variance True Model X 009, Itron Inc. 6
Implications of Heteroskedasticity Least squares parameter estimators are unbiased and consistent Least squares parameter estimators are not efficient (the variances of the estimated parameters are not the minimum variance estimates). The problem is that least squares weights all squared errors equally. Observations when variances are large will tend to have large errors and even bigger squared errors. These observations will have too large an influence on the estimates. The standard solution is to build a variance model and use generalized (weighted) least squares > The goal is to weight each squared error by the inverse of its variance. > This can be accomplished in the simple case by dividing Y and the X s by the estimated standard error (square root of the variance) for each observation. 009, Itron Inc. 7
Tests for Heteroskedasticity Estimate Least Squares and get residuals Estimate model of residuals > ei = c0 + c1z1 + cz +... (Breusch-Pagan) > ei = c0 + c1z1 + cz +... (Glejser) ( )... > lnei = c0 + c1z1 + cz + (Harvey- Godfrey) > ei = a0 + a1x1 +... + akxk + b1x1 +... + c1x1x +... (White) Compute Lagrange Multiplier (LM) test statistic = n R Under null hypothesis of homoskedasticerrors, LM has a chi squared distribution with L-1 degrees of freedom (where L is the number of parameters in the residual model). 009, Itron Inc. 8
Have you run any tests for heteroskedasticityin your sales or peak forecasting models? 009, Itron Inc. 9
Forms of Heteroskedasticity Discussed Today Contitional heteroskedasticity Var ( e ) = v = c + cz1 + c Z... t t 0 1 t t + Autoregressive conditional heteroskedasticity (ARCH Engle 198) ( e ) = v = c + c e + c e +... Var = t 0 1 t 1 t t + Generalized autoregressive conditional heteroskedasticity (Bollerslev 1986) ( e ) t + Var = vt = c0 + c1et 1 + cet +... + d1vt 1 + dvt... Innovation Terms Persistence Terms 009, Itron Inc. 10
That s Not All From Glossary to ARCH (GARCH), Bollerslev, 007 Available at: http://faculty.chicagobooth.edu/jeffrey.russell/teaching/finecon/readings/glossary.pdf 009, Itron Inc. 11
Estimation Method for ARCH/GARCH Model for GARCH(1,1) Y t = βxt + et etisn0, ( vt ) v = c + ce d v t 0 1 t 1 + 1 t 1 Maximize the Likelihood Function L = t 1 πv t exp 1 e v Equilavent is to minimize ln(l) ln(l) 1 t ln ( v ) Find the parameters (β,c,d) that minimize this function > Note the parallels to OLS and GLS t + e v t t t t 009, Itron Inc. 1
Data for S&P 500 Picture of constant variance Daily Close Change 003 004 005 006 007 008 009 010 Change 003 004 005 006 007 008 009 010 009, Itron Inc. 13
The Model in MetrixND 009, Itron Inc. 14
Regression Result Delta Regression Confidence 1 σf = σˆ 1+ + Band T t= 1 003 004 005 006 007 008 009 010 ( Xf X) T ( X t X) 009, Itron Inc. 15
White Test for Heteroskedasticity Residual Squared n R χ 95%,8 = = 53.5 =.73 Prob =.0000 Predicted Variance 003 004 005 006 007 008 009 010 009, Itron Inc. 16
ARCH(1) Model Delta ARCH Confidence Band 003 004 005 006 007 008 009 010 009, Itron Inc. 17
GARCH(1,1) Model Delta GARCH Confidence Band 003 004 005 006 007 008 009 010 009, Itron Inc. 18
What do we learn We get different parameters. The regression parameters are the average daily change in each year. The variance model of indicates that volatility is persistent. We get a direct estimate and forecast of volatility. Regression GARCH(1,1) 009, Itron Inc. 19
How does this apply to our typical problem Typical problem is explaining and forecasting: > Customers > Energy usage (sales) > System peaks Example with daily energy data 009, Itron Inc. 0
Daily Energy Scatter Plot Visually the data appear to be relatively homoskedastic once temperature is accounted for Weekdays Saturday Sunday Holiday 009, Itron Inc. 1
Do you use weighted least squares or another method to adjust for heteroskedasticityin your sales and peak forecasting models? 009, Itron Inc.
Regression Model Actual Predicted Residual 009, Itron Inc. 3
Applying the White Test n R χ 95%,40df = 64.6 = 6.5 Probability =.008 009, Itron Inc. 4
GARCH(1,1) Model GARCH(1,1) Regression 009, Itron Inc. 5
Comparison of Regression and GARCH Results Regression Sigma GARCH Sigma Regression and GARCH predicted values are almost identical Correlation =.9996 009, Itron Inc. 6
Comparison of Confidence Bands Regression Confidence Bands GARCH(1,1) Confidence Bands 009, Itron Inc. 7
Conclusion Extreme heteroskedasticityis unlikely for energy consumption data. Variances are relatively stable compared to financial data. Energy processes are not volatile in a GARCH sense, where volatility today generates volatility tomorrow. When it gets hot, the loads increase, and variances may go up, but when it cools off, the load returns quickly to a normal low variance value. There is little or no volatility persistence. Estimated parameters are not likely to change significantly due to heteroskedasticity corrections. Predicted and forecasted values are not likely to be impacted significantly by heteroskedasticity corrections. Modeling of energy market prices is a more likely place to apply the ARCH and GARCH types of techniques, especially if it is valuable to quantify and forecast price volatility. 009, Itron Inc. 8
Should we include more functionality in MetrixND to test and adjust for heteroskedasticity? 009, Itron Inc. 9
011 HANDS-ON WORKSHOPS Energy Forecasting Week - May 16-0, Las Vegas > One-Day SAE Modeling Workshop May 18 > One-Day MetrixIDR Workshop May 18 Fundamentals of MetrixND - June 6-7, Boston Itron UC - September 18-0, Phoenix Fundamentals of Sales & Demand Forecasting September -3, Boston Fundamentals of Short-Term and Hourly Forecasting September 8-30, San Diego Forecasting 101 - October 4-6, San Diego Press *6 to ask a question OTHER FORECASTING MEETINGS Energy Forecasting Week - May 16-0, Las Vegas > Annual ISO/RTO Forecasting Summit May 16-17 > Long-Term Forecasting/EFG Meeting May 19-0 011 Itron Users' Conference - September 18-0, Phoenix Hotel registration deadline: April 18, 011 For more information and registration: www.itron.com/forecastingworkshops Contact us at: 1.800.755.9585, 1.858.74.60 or forecasting@itron.com 009, Itron Inc. 30