Richard T. Baillie

Fractionally integrated generalized autoregressive conditional heteroskedasticity

Abstract The new class of Fractionally Integrated Generalized AutoRegressive Conditionally Heteroskedastic (FIGARCH) processes is introduced. The conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations. Unlike (I(d) processes for the mean, Maximum Likelihood Estimates (MLE) of the FIGARCH parameters are argued to be T 1 2 - consistent . The small-sample behavior of an approximate MLE procedure is assessed through a simulation study, which also documents how the estimation of a standard GARCH model tends to produce integrated, or IGARCH, like estimates. An empirical example with daily Deutschmark — U.S. dollar exchange rates illustrates the practical relevance of the new FIGARCH specification.

Long Memory Processes and Fractional Integration in Econometrics

Abstract This paper provides a survey and review of the major econometric work on long memory processes, fractional integration, and their applications in economics and finance. Some of the definitions of long memory are reviewed, together with previous work in other disciplines. Section 3 describes the population characteristics of various long memory processes in the mean, including ARFIMA. Section 4 is concerned with estimation and examines semiparametric procedures in both the frequency and time domain, and also the properties of various regression based and maximum likelihood techniques. Long memory volatility processes are discussed in Section 5, while Section 6 discusses applications in economics and finance. The paper also has a concluding section.

The Message in Daily Exchange Rates: A Conditional-Variance Tale

Formal testing procedures confirm the presence of a unit root in the autoregressive ploynomial of the univariate time series representation of daily exchange-rate data. the first differences of the logarithms of daily spot rates are approximately uncorrelated through time, and a generalized autoregressive conditional heteroscedasticity model with daily dummy variables and conditionally t-distributed errors is found to provide a good representation to the leptokurtosis and time-dependant conditional heteroscedasticity. The parameter estimates and characteristics of the models are found to be very similar for six different currencies. these apparent stylized facts carry over to weekly, fortnightly, and monthly data in which the degree of leptokurtosis and time-dependant heteroscedasticity is reduced as the length of the sampling interval increases.

Stock Returns And Volatility

Most asset pricing models postulate a positive relationship between a stock portfolios expected returns and risk, which is often modeled by the variance of the asset price. This paper uses GARCH in mean models to examine the relationship between mean returns on a stock portfolio and its conditional variance or standard deviation. After estimating a variety of models from daily and monthly portfolio return data, we conclude that any relationship between mean returns and own variance or standard deviation is weak. The results suggest that investors consider some other risk measure to be more important than the variance of portfolio returns.

Intra-Day and Inter-Market Volatility in Foreign Exchange Rates

Four foreign exchange spot rate series, recorded on an hourly basis for a six-month period in 1986 are examined. A seasonal GARCH model is developed to describe the time-dependent volatility apparent in the percentage nominal return of each currency. Hourly patterns in volatility are found to be remarkably similar across currencies and appear to be related to the opening and closing of the worlds major markets. Robust LM tests designed to deal with the extreme leptokurtosis in the data fails to uncover any evidence of misspecification or the presence of volatility spillover effects between the currencies or across markets.

Analysing Inflation by the Fractionally Integrated ARFIMA-GARCH Model

This paper considers the application of long-memory processes to describing inflation for 10 countries. We implement a new procedure to obtain approximate maximum likelihood estimates of an ARFIMA-GARCH process; which is fractionally integrated I(d) with a superimposed stationary ARMA component in its conditional mean. Additionally, this long-memory process is allowed to have GARCH type conditional heteroscedasticity. On analysing monthly post-World War II CPI inflation for 10 different countries, we find strong evidence of long memory with mean reverting behaviour for all countries except Japan, which appears stationary. For three high inflation economies there is evidence that the mean and volatility of inflation interact in a way that is consistent with the Friedman hypothesis. Copyright 1996 by John Wiley & Sons, Ltd.

Price discovery and common factor models

Abstract If a financial asset is traded in more than one market, common factor models may be used to measure the contribution of these markets to the price discovery process. We examine the relationship between the Hasbrouck (J. Finance (50) (1995) 1175) and Gonzalo and Granger (J. Bus. Econ. Stat. 13 (1995) 27) common factor models. These two models complement each other and provide different views of the price discovery process between markets. The Gonzalo and Granger model focuses on the components of the common factor and the error correction process, while the Hasbrouck model considers each markets contribution to the variance of the innovations to the common factor. We show that the two models are directly related and provide similar results if the residuals are uncorrelated between markets. However, if substantive correlation exists, they typically provide different results. We illustrate these differences using analytic examples plus a real world example consisting of electronic communications networks (ECNs) and other Nasdaq market makers.

The forward premium anomaly is not as bad as you think

Abstract The forward premium anomaly refers to the widespread empirical finding that the slope coefficient in the regression of the change in the logarithm of the spot exchange rate on the forward premium is invariably less than unity, and often negative. This “anomaly” implies the apparent predictability of excess returns over uncovered interest rate parity (UIP), and is conventionally viewed as evidence of a biased forward rate and/or of evidence of a time-varying risk premium. This paper presents a stylized model that imposes UIP and allows the daily spot exchange rate to possess very persistent volatility. The model is calibrated around realistic parameter values for daily returns and the slope coefficient estimates in the anomalous regressions with monthly data are found to be centered around unity, but are very widely dispersed, and converge to the true value of unity at a very slow rate. This theoretical evidence is shown to be consistent with the empirical findings for the monthly sample sizes typically employed in the literature. Hence, the celebrated unbiasedness regression does not appear to provide as much evidence as previously supposed concerning the possible bias of the forward rate.

A multivariate generalized ARCH approach to modeling risk premia in forward foreign exchange rate markets

Assuming that daily spot exchange rates follow a martingale process. we derive the implied time series process for the vector of 30-day forward rate forecast errors from using weekly data. The conditional second moment matrix of this vector is modeled as a multivariate generalized ARCH process. The estimated model is used to test the hypothesis that the risk premium is a linear function of the conditional variances and covariances as suggested by the standard asset pricing theory literature. Little support is found for this theory; instead lagged changes in the forward rate appear to be correlated with the ‘risk premium.’ This study examines spot and forward exchange rates at a weekly level for four different currencies. It is shown that the vector of forward market forecast errors can be parameterized as a vector moving average (MA) process where the MA coefficients can be theoretically determined from knowledge of the martingale behavior of exchange rates. The conditional covariance matrix is then estimated by assuming a multivariate GARCH structure which depends on a relatively small number of parameters. A range of LM tests confirms that the model provides an adequate description of the first- and second-order moments of the conditional density of the data. The vector MA process is then used to provide some bounds on the magnitude of the risk premium. Series of tests are also applied to the estimated model to test for the inclusion of terms that would be implied by a time varying risk premium. The results are not consistent with any standard model of asset pricing, but do provide evidence for the existence of this type of effect.

Prediction In Dynamic Models With Time Dependent Conditional Variances

Abstract This paper considers forecasting the conditional mean and variance from a single-equation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with time-dependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum MSE predictor and the conditional MSE are presented. We also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with GARCH(1, 1) innovations. These results are then used in the construction of ex ante prediction confidence intervals by means of the Cornish-Fisher asymptotic expansion. An empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results.