Essay: Importance of volatility forecasts for commodities

Several reasons for the importance of volatility forecasts for commodities are given by Kroner et al. (1995). Commodity prices are one of the most volatile of all major asset classes. Many developing countries have economies that are tied to the success of commodity exports. Second, commodity price volatility has implications for institutions and developed countries giving loans, investments, or aid to developed countries since a drop in prices can lead to default in developing countries. Third, future volatility of asset prices is important in option pricing as a higher forecast of future volatility leads to a higher price of options on those assets.
The ARCH class of models was developed beginning with Engel (1982) to allow the variance to depend on past information. This is important because many financial and economic time series data show conditional heteroscedasticity. The ARCH model permits using a conventional regression specification for the mean equation with a variance that is able to change over time.
The GARCH, or generalized autoregressive conditional heteroscedasticity model, was developed by Bollerslev (1986) partly due to the fact that the ARCH model does not fully capture the degree of persistence observed during financial crises. In GARCH models, the forecast of the next period’s variance is based on the current period variance and the squared return.
Awartani & Corradi (2005) note that a negative correlation exists between stock returns and volatility is common in financial data. This phenomenon is known as asymmetric volatility. One possible reason for this is the leverage effect (Black 1976) in which it is theorized that a drop in value of a company increases their leverage making the stock riskier. Another theory is the effect of volatility feedback, or time-varying risk premiums (Bekaert & Wu 2000). If volatility is priced into the asset, then an anticipated volatility increase raises the required return.
The standard ARCH and GARCH models capture volatility clustering, where large shocks are likely to be followed by large shocks, but do not allow the sign of returns to affect volatility. Bad news tends to increase volatility and good news tends to decrease it. Asymmetric GARCH, where the variance can be affected differently by positive or negative shocks, can fit the data better than standard GARCH models. Two of the most widely used asymmetric GARCH models are the GJR-GARCH model of Glosten et al. (1993) and the EGARCH model of Nelson (1991). The GJR-GARCH, the simplest model of the two, adds an extra term in the coefficient for the squared return with an indicator variable that is equal to 1 if last period’s return is equal or greater than the mean and 0 otherwise. The EGARCH model uses log-variance instead of variance to avoid the possibility of a negative variance.
Andersen & Bollerslev (1998) note that squared returns are very noisy but are an unbiased measure of volatility. Lopez (2001) shows that squared returns have an asymmetric distribution and are 50% lower or 50% greater than volatility approximately 75% of the time. The squared returns are often used as a proxy for volatility since the ‘true’ volatility cannot be observed. The use of squared returns ensures a correct ranking of models using a quadratic loss function (Awartani & Corradi 2005).
Awartani & Corradi (2005) evaluate out of sample forecasts of several GARCH models emphasizing the predictive ability of the asymmetric component in the conditional variance equation. They compare the asymmetric models to GARCH (1,1) as a benchmark. The data used is the S&P 500 daily return. They use squared returns as a proxy for volatility. They use the Diebold and Mariano (1995) tests to compare the non-nested models, the Clark and McCracken (2001) tests for the nested models, and the White (2000) reality check for a joint comparison of all models. They conclude that the GARCH (1,1) is beaten by asymmetric GARCH models. Their results were robust to changes in the forecasting horizon.
Brownless et al. (2011) document the effect estimation choices have on forecast performance. They use daily log returns of the S&P 500 adjusted for splits and dividends. The sample used covers the period from 1990 to 2008. They find that the threshold GARCH, or GJR-GARCH, model performed the best in forecasts across several asset classes and volatility regimes. Their estimates of parameters showed some slow-varying drift over time, but that using the longest estimation period possible produced the best forecasts. Re-estimating the models at least weekly did help to reduce the effects of parameter drift. They also found that using a student’s t distribution did not improve forecasting performance despite the presence of heavy tails in financial data.
Poon and Granger (2003) conduct a survey of forecasting research focusing on two questions: Is forecastable? And if so, which methods produce the best forecasts. They survey four categories of volatility forecasts: historical volatility models, implied volatility models, the GARCH class of models, and stochastic volatility models.
One example of a historical volatility model is the random walk where last period’s standard deviation is the forecast for this period. An improvement is the moving average model where old estimates are discarded. The exponential smoothing and exponential weighted moving average are further refinements which place greater weight on the more recent estimates. Stochastic volatility models (SV) produce the properties observed in financial return data such as heavy tails, persistence, and asymmetry. A survey of stochastic volatility models was done by Ghysels et al. (1996). One major drawback of SV models is that the volatility noise term leads to having no closed form solution so they cannot be estimated by maximum likelihood (Poon & Granger 2003). Implied volatility is the level of volatility that solves the option pricing model for the market price given the other variables such as time to expiration and interest rate. The VIX, a volatility index, is based on implied volatility of options on the S&P 500. This is seen as an indicator of market expectations over the next 30 days. This information can be utilized in forecasting models.
Poon and Granger (2003) produce pairwise comparisons of forecasts from 66 studies. The overall ranking suggests that implied volatility produces the best forecast, and GARCH and historical volatility are roughly equal. There were 17 studies that compared different versions of GARCH models. GARCH was found to be superior to ARCH models. Some studies were conducted just to demonstrate a certain method of interest could be useful. The authors noted possible publication bias: some studies might not have been published if the desired result was not achieved. Their conclusion was that volatility in financial markets is forecastable over short periods of time. They concluded this result does not violate market efficiency since forecasts of volatility are not inconsistent with current asset price models.
Kroner et al. (1995) develop a method of forecasting commodity price volatility over long horizons up to 225 days. They combine information from implied volatility with time series forecasts to produce forecasts of long term volatility that are more accurate than short term methods. They use mean square error to evaluate forecasts. Except for one commodity, GARCH and the combined forecasts dominated the implied standard deviation and historical volatility forecasts. Their results are robust to out of sample forecasts of horizons of different lengths across many different commodities.
Kroner et. Al (1995) also note that long term forecasts using time series methods are generally not useful since the forecast is typically just the unconditional mean and variance if the volatility is mean- reverting. Incorporating investors’ expectations through implied volatility introduces some information about expected future price movement. The forecasting performance of the time series models suggest that the time series history contains some useful information not incorporated in the future expectations through implied volatility. This suggests that either option markets are inefficient or that the option pricing models used were incorrect (Kroner et al. 1995).
An important aspect of forecasting is the evaluation of forecasting models. Some evaluation methods used in the literature include mean error (ME), mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE). One less common method is Theil’s U, which compares one forecast to a benchmark. Another is LINEX, which weighs positive and negative errors differently but depends on a parameter that is chosen subjectively. If one model happens to have smaller error as measured by every evaluation method to another model, then a comparison is easy (Granger 1999). However, this is rarely the result. Comparisons based on some average of the evaluation methods are used in practice (Poon & Granger 2003).
Hansen and Lunde (2005) compare 330 GARCH class models in tests of out of sample forecasting of conditional variance. They use Deutsche Mark to U.S. Dollar exchange rate data and IBM stock return data. They investigate the performance of many of the more sophisticated GARCH models compared to a benchmark GARCH (1,1) model. They find no evidence that the benchmark was outperformed with exchange rate data. The t-distributed specification did perform better than the Gaussian specification for the exchange rate data. For the IBM return data, the benchmark model was outperformed by the asymmetric GARCH models. The t-distributed specification was outperformed by the Gaussian specification for the IBM data.
Hansen and Lunde (2005) use realized variance as a proxy for volatility. The daily realized variance is calculated using intraday data. They show realized variance to be approximately unbiased and the measure becomes more precise as the time interval decreases. One complication with using intraday data is that many assets are not traded 24 hours per day. The authors use a scaling estimator to yield a measure of volatility for the whole day. Another issue with intraday data is its limited availability.
Patton (2011) compares forecasts of volatility that use proxies for volatility. Proxies have imperfections but must be used since the true volatility is an unobserved, or latent, variable. He notes that even conditionally unbiased estimators of volatility can be very noisy. The noise in proxy variables cause errors in forecast ranking, and affect the asymptotic efficiency of statistical tests commonly used. Patton then derives necessary and sufficient conditions for a loss function to be robust to proxy noise. Loss functions that are robust to proxy noise are important not just for volatility forecasting. Patton also notes that they would be important for measuring the true rate of GDP growth, default probabilities, and conditional correlations between assets in portfolios.
Poon and Granger (2005) discuss some of the problems that outliers pose for forecasting. They note that outliers, or abnormally large observations, are generated by different processes than the normal daily observations. Therefore, outliers have a large impact on modeling volatility and forecasts generated by those models. They conclude that forecasting models do not necessarily have to be able to predict the timing and size of extreme shocks to be useful. It is not necessarily fair to penalize models for this. They propose that outliers can be modeled by extreme value theories or crisis models. They note that 50 to 58 percent of stock index volatility is forecastable over a horizon of 1 to 20 days. They also note that one area for future research is to determine whether exogenous variables, such as macroeconomic data, interest rates, or trading volume, can improve forecasting power.