Orthogonal GARCH and covariance matrix forecasting: The Nordic stock markets during the Asian financial crisis 1997–1998

In risk management, modelling large numbers of assets and their variances and covariances in a unified framework is often important. In such multivariate frameworks, it is difficult to incorporate GARCH models and thus a new member of the ARCH-family, Orthogonal GARCH, has been suggested as a remedy to inherent estimation problems in multivariate ARCH modelling. Orthogonal GARCH creates positive definite covariance matrices of any size but builds on assumptions that partly break down during stress scenarios. This article therefore assesses the stress performance of the model by looking at four Nordic stock indices and covariance matrix forecasts during the highly volatile years of 1997 and 1998. Overall, Orthogonal GARCH is found to perform significantly better than traditional historical variance and moving average methods. Out-of-sample evaluation measures include symmetric loss functions (RMSE), asymmetric loss functions, operational methods suggested by the Basle Committee on Banking Supervision, as well as a forecast evaluation methodology based on pricing of simulated ‘rainbow options’.     

A nested factor model for non-linear dependencies in stock returns

The aim of our work is to propose a natural framework to account for all the empirically known properties of the multivariate distribution of stock returns. We define and study a ‘nested factor model’, where the linear factors part is standard, but where the log-volatility of the linear factors and of the residuals are themselves endowed with a factor structure and residuals. We propose a calibration procedure to estimate these log-vol factors and the residuals. We find that whereas the number of relevant linear factors is relatively large (10 or more), only two or three log-vol factors emerge in our analysis of the data. In fact, a minimal model where only one log-vol factor is considered is already very satisfactory, as it accurately reproduces the properties of bivariate copulas, in particular, the dependence of the medial point on the linear correlation coefficient, as reported in Chicheportiche and Bouchaud [Int. J. Theor. Appl. Finance, 2012, 15]. We have tested the ability of the model to predict out-of-sample the risk of non-linear portfolios, and found that it performs significantly better than other schemes.     

Some Results in the Theory of Arbitrage Pricing

This paper derives a stronger version of Huberman's recent “preference free” pricing theorem. This pricing result relates the expected return on an asset to its factor responses and the covariance structure of the residuals from a linear factor model. It must characterize any infinite asset economy in which no arbitrage opportunities are present whether or not the factor model has uncorrelated residuals. This result provides the intuition for the role of residual risk in the pricing model and eliminates some classes of arbitrage opportunities still present under Huberman's bound. Some applications to empirical tests and performance measurement are also discussed.     

Bias of a Value-at-Risk estimator

We develop the analytical second-order bias of a Value-at-Risk estimator based on an ARCH(1) volatility specification when the parameters are estimated by the method of quasi maximum likelihood. We show that the bias results from two sources: assumption on the distribution of the standardized residuals and the parameter estimation error.     

Generalized runs tests to detect randomness in hedge funds returns

The major contribution of this paper is to make use of generalized runs tests (Cho and White, 2011) to analyze the randomness, i.e. the lack of persistence, in both absolute and relative returns of hedge funds. We find that about 42% of the HFR universe exhibit iid absolute returns over the period spanning 2000 to 2012. These funds are mainly found in proportions within the Macro and Equity Hedge strategies. A similar result holds for relative returns. We also find that funds having non-iid returns often exhibit ARCH effects and structural breaks, with largest breaks located within financial crises. Also, only a small percentage displays persistence in their relative performance, 8.2% to 16.7% of the universe, mainly found in proportions within the Relative Value and Event-Driven strategies. The robustness of results is challenged by implementing the tests on a crisis-free period. We find similar results for absolute returns. For relative ones, differences appear across strategies and benchmarks, but still both ARCH and breaks are present. Our work contributes to the hedge fund literature in terms of methodology, portfolio allocation, and performance measurement.     

Multivariate fractionally integrated APARCH modeling of stock market volatility: A multi-country study

Tse (1998) proposes a model which combines the fractionally integrated GARCH formulation of Baillie, Bollerslev and Mikkelsen (1996) with the asymmetric power ARCH specification of Ding, Granger and Engle (1993). This paper analyzes the applicability of a multivariate constant conditional correlation version of the model to national stock market returns for eight countries. We find this multivariate specification to be generally applicable once power, leverage and long-memory effects are taken into consideration. In addition, we find that both the optimal fractional differencing parameter and power transformation are remarkably similar across countries. Out-of-sample evidence for the superior forecasting ability of the multivariate FIAPARCH framework is provided in terms of forecast error statistics and tests for equal forecast accuracy of the various models.     

Spatial Autocorrelations and Urban Housing Market Segmentation

This paper seeks to let data define urban housing market segments, replacing the conventional administrative or any pre-defined boundaries used in the previous housing submarket literature. We model housing transaction data using a conventional hedonic function. The hedonic residuals are used to estimate an isotropic semi-variogram, from which residual variance–covariance matrix is constructed. The correlations between hedonic residuals are used as identifier to assign housing units into clusters. Standard submarket identification tests are applied to each cluster to examine the segmentation of housing market. The results are compared with the prevailing structure of market segments. Weighted mean square test shows that the defined submarket structure can improve the precision of price prediction by 17.5%. This paper is experimental in the sense that it represents one of the first attempts at investigating market segmentation through house price spatial autocorrelations.     

The quality of market volatility forecasts implied by S&P 100 index option prices

This study examines the performance of the S&P 100 implied volatility as a forecast of future stock market volatility. The results indicate that the implied volatility is an upward biased forecast, but also that it contains relevant information regarding future volatility. The implied volatility dominates the historical volatility rate in terms of ex ante forecasting power, and its forecast error is orthogonal to parameters frequently linked to conditional volatility, including those employed in various ARCH specifications. These findings suggest that a linear model which corrects for the implied volatility's bias can provide a useful market-based estimator of conditional volatility.     

Financial prediction with constrained tail risk

A new class of asymmetric loss functions derived from the least absolute deviations or least squares loss with a constraint on the mean of one tail of the residual error distribution, is introduced for analyzing financial data. Motivated by risk management principles, the primary intent is to provide “cautious” forecasts under uncertainty. The net effect on fitted models is to shape the residuals so that on average only a prespecified proportion of predictions tend to fall above or below a desired threshold. The loss functions are reformulated as objective functions in the context of parameter estimation for linear regression models, and it is demonstrated how optimization can be implemented via linear programming. The method is a competitor of quantile regression, but is more flexible and broader in scope. An application is illustrated on prediction of NDX and SPX index returns data, while controlling the magnitude of a fraction of worst losses.     

Stochastic Mortality Modeling: Key Drivers and Dependent Residuals

This article proposes an alternative framework for modeling the stochastic dynamics of mortality rates. A simple age basis combined with two stochastic period factors is used to explain the key mortality drivers, while the remaining structure is modeled via a multivariate autoregressive residuals model. The latter captures the stationary mortality dynamics and introduces dependencies between adjacent age-period cells of the mortality matrix that, among other things, can be structured to capture cohort effects in a transparent manner and incorporate across ages correlations in a natural way. Our approach is compared with models with and without a univariate cohort process. The age- and period-related latent states of the mortality basis are more robust when the residuals surface is modeled via the multivariate time-series model, implying that the process indeed acts independently of the assumed mortality basis. Under the Bayesian paradigm, the posterior distribution of the models is considered to explore coherently the extent of parameter uncertainty. Samples from the posterior predictive distribution are used to project mortality, and an in-depth sensitivity analysis is conducted. The methodology is easily extendable in multiple ways that give a different form and degree of significance to the different components of mortality dynamics.     

The selection of multinational equity portfolios: forecasting models and estimation risk

The problem considered is the selection of a portfolio of international assets, particularly the forecasting of the inputs to a selection algorithm. Four models of the asset return generating process are considered, two of which ignore the international nature of the universe of assets, two which exploit it in different ways. Several estimation methods are considered for each component: expected return, variance and covariance of returns. The combinations of model and estimation method are first evaluated in terms of their forecasting performance for the components mentioned for the individual assets. The universe used is the components of the Financial Times Eurotrack 100 Index. Significant differences were found between the forecasting accuracy of the methods considered for each component. In the final stage of the analysis, a comparison of the returns on portfolios chosen using each combination showed a significant difference. The analysis suggests that the choice of estimation method is more critical than the choice of pricing model.     

Volatility forecasts and at-the-money implied volatility: a multi-component ARCH approach and its relation to market models

This article explores the relationships between several forecasts for the volatility built from multi-scale linear ARCH processes, and linear market models for the forward variance. This shows that the structures of the forecast equations are identical, but with different dependencies on the forecast horizon. The process equations for the forward variance are induced by the process equations for an ARCH model, but postulated in a market model. In the ARCH case, they are different from the usual diffusive type. The conceptual differences between both approaches and their implication for volatility forecasts are analysed. The volatility forecast is compared with the realized volatility (the volatility that will occur between date t and t + ΔT), and the implied volatility (corresponding to an at-the-money option with expiry at t + ΔT). For the ARCH forecasts, the parameters are set a priori. An empirical analysis across multiple time horizons ΔT shows that a forecast provided by an I-GARCH(1) process (one time scale) does not capture correctly the dynamics of the realized volatility. An I-GARCH(2) process (two time scales, similar to GARCH(1,1)) is better, while a long-memory LM-ARCH process (multiple time scales) replicates correctly the dynamics of the implied and realized volatilities and delivers consistently good forecasts for the realized volatility.     

Analysis of Spatial Autocorrelation in House Prices

This article examines spatial autocorrelation in transaction prices of single-family properties in Dallas, Texas. The empirical analysis is conducted using a semilog hedonic house price equation and a spherical autocorrelation function with data for over 5000 transactions of homes sold between 1991:4 and 1993:1. Properties are geocoded and assigned to separate housing submarkets within metropolitan Dallas. Hedonic and spherical autocorrelation parameters are estimated separately for each submarket using estimated generalized least squares (EGLS). We find strong evidence of spatial autocorrelation in transaction prices within submarkets. Results for spatially autocorrelated residuals are mixed. In four of eight submarkets, there is evidence of spatial autocorrelation in the hedonic residuals for single-family properties located within a 1200 meter radius. In two submarkets, the hedonic residuals are spatially autocorrelated throughout the submarket, while the hedonic residuals are spatially uncorrelated in the remaining two submarkets. Finally, we compare OLS and kriged EGLS predicted values for properties sold during 1993:1. Kriged EGLS predictions are more accurate than OLS in six of eight submarkets, while OLS has smaller prediction errors in submarkets where the residuals are spatially uncorrelated and the estimated semivariogram has a large variance.     

VOLATILITY FORECASTS,TRADING VOLUME,AND THE ARCH VERSUS OPTION‐IMPLIED VOLATILITY TRADE‐OFF

We investigate empirically the role of trading volume (1) in predicting the relative informativeness of volatility forecasts produced by autoregressive conditional heteroskedasticity (ARCH) models versus the volatility forecasts derived from option prices, and (2) in improving volatility forecasts produced by ARCH and option models and combinations of models. Daily and monthly data are explored. We find that if trading volume was low during period t?1 relative to the recent past, ARCH is at least as important as options for forecasting future stock market volatility. Conversely, if volume was high during period t?1 relative to the recent past, option‐implied volatility is much more important than ARCH for forecasting future volatility. Considering relative trading volume as a proxy for changes in the set of information available to investors, our findings reveal an important switching role for trading volume between a volatility forecast that reflects relatively stale information (the historical ARCH estimate) and the option‐implied forward‐looking estimate.     

What Drives Housing Markets: Fundamentals or Bubbles?

This study applies the dynamic Gordon growth model which is in the circumstance of rational bubbles to decompose log price-rent ratio into three parts, i.e., rational bubbles, discounted expected future rent growth rates and discounted expected future returns. The latter two terms represent housing fundamentals. The magnitudes of the components of price-rent ratio’s variance are estimated to distinguish the relative impact of the three parts on housing prices. Using time series data from the housing markets in the four largest cities in China (1991:Q1–2011:Q1 for Shanghai, Guangzhou and Shenzhen; 1993:Q2–2011:Q1 for Beijing), this paper presents a number of empirical findings: (a) the variance of rational bubbles is much larger than the variance of price-rent ratio, and rational bubbles contribute more fluctuations directly to price-rent ratio than the expected returns or the expected rent growth rates do; (b) the covariance between rational bubbles and expected returns or expected rent growth rates is also large; (c) the positive covariance of rational bubbles and expected returns implies that high expected returns coexist with bubbles, which differs from previous findings that lower expected returns drive asset prices; (d) the negative covariance of rational bubbles and expected rent growth rates indicates that the larger the bubbles are, the lower the expected rent growth rates are; (e) the positive covariance of expected returns and expected rent growth rates reveals under-reaction of the housing markets to rents.     

Exchange volatility and trade performance in Morocco and Tunisia: what have we learned so far?

This paper attempts to assess two interesting issues for two small open economies (Morocco and Tunisia). First, it analyses the historical behaviour of nominal exchange rate, differential price and real exchange rate uncertainties. Second, it investigates the stability of the interaction between exchange volatility and exports in nominal and real terms. Our main results reveal that the effect of differential price volatility on exports exceeds that of nominal exchange rate by a large margin in terms of duration of persistence, ARCH and GARCH effects and intensity of shock. The relationship appears complex. In Morocco, it is negative and significant in 75.82% (as average) of cases in nominal terms and in 77.22% in real terms. This link is stronger in Tunisia with averages, respectively, equal to 85.88% and 89.99%. We associate the apparently mixed results to the differential price uncertainty itself sensitive to ups and down oil price movements, switching regime and leverage effects.     

Jackknife empirical likelihood for parametric copulas

For fitting a parametric copula to multivariate data, a popular way is to employ the so-called pseudo maximum likelihood estimation proposed by Genest, Ghoudi, and Rivest. Although interval estimation can be obtained via estimating the asymptotic covariance of the pseudo maximum likelihood estimation, we propose a jackknife empirical likelihood method to construct confidence regions for the parameters without estimating any additional quantities such as the asymptotic covariance. A simulation study shows the advantages of the new method in case of strong dependence or having more than one parameter involved.     

Multivariate Lévy processes with dependent jump intensity

In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence.     

GARCH model selection criteria

Abstract

The autoregressive conditional heteroscedasticity (ARCH) family of models has grown to encompass a wide range of specifications, each of which is designed to enhance the ability of the model to capture the characteristics of the data. In this paper, the ability of a number of model selection criteria to correctly identify the data generating process in simulated data is established. The results of this study suggest that the Hannan-Quinn and stochastic complexity criteria provide a superior level of performance for ARCH and generalized ARCH (GARCH) processes compared to the more commonly used criteria. Where leverage and/or power effects are present, however, none of the procedures considered perform well. A new LM based test for the presence of nonlinearity and power effects is introduced and tested.     

Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects

This paper provides empirical support for the notion that Autoregressive Conditional Heteroskedasticity (ARCH) in daily stock return data reflects time dependence in the process generating information flow to the market. Daily trading volume, used as a proxy for information arrival time, is shown to have significant explanatory power regarding the variance of daily returns, which is an implication of the assumption that daily returns are subordinated to intraday equilibrium returns. Furthermore, ARCH effects tend to disappear when volume is included in the variance equation.