Nonlinearities in Financial Data

Nonlinearities in Financial Data - Economics 245B...

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Economics 245B Nonlinearities in Financial Data In time-series analysis, one is often interested in forecasting the future value of a random variable. Consider the °rst-order autoregression Y t = °Y t ° 1 + U t ; where U t is a white-noise error with EU 2 t = ± 2 . The unconditional expectation of Y t is zero while E ( Y t j Y t ° 1 ) = °Y t ° 1 , which clearly reveals the great gain in accu- racy achieved with conditional forecasts. For such a model both the unconditional and conditional variances are constant V ar ( Y t ) = ± 2 1 ° ° and V ar ( Y t j Y t ° 1 ) = ± 2 : One might expect better forecast intervals if additional information from the past were allowed to a/ect the forecast variance. The improved forecast intervals would arise under the idea that the mean and variance of the conditional distribution jointly evolve over time. Why might we need models of nonlinearities in °nance? In analysis of asset returns, it is often the case that the conditional mean is unpredictable (at least the excess return above the riskless rate of interest) and so the interesting feature of the conditional distribution to model is the conditional variance. Accurate modeling of the conditional variance of an asset price (or return) is needed to value options on the asset. A related application is to correctly model a conditional covariance matrix for multiple assets to allow one to construct optimal portfolio weights. Let Y t be an asset return, such as the daily return on IBM. It is most often the case that we specify only that Y t is a function of latent shocks f " s g s ± t . Because the model is nonlinear, all structure is captured by the model and we assume that the shocks are iid. A general representation is Y t = f ( " t ; " t ° 1 ; : : : ) : The general representation is often too vague to inform empirical work, so much empirical work is based on Y t = g ( " t ° 1 ; " t ° 2 ; : : : ) + h ( " t ° 1 ; " t ° 2 ; : : : ) " t ;
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often with the assumption that the shocks have mean zero and variance one to identify the conditional location and scale parameters as mean and variance. Such a model might arise if the volatility in the stock price of IBM re±ects the changing structure of the computer market or the changing perception of risk in equities over time. Observe that E t ° 1 Y t = g ( " t ° 1 ; " t ° 2 ; : : : ) and so the conditional variance is E t ° 1 ( Y t ° E t ° 1 Y t ) 2 = h ( " t ° 1 ; " t ° 2 ; : : : ) ± h t ° 1 : The simpler representation imposes substantial mathematical restrictions that one must consider in practice. While the °rst two conditional moments are allowed to vary, the time variation in all higher order conditional moments is in±exibly linked to the time variation in the conditional variance E t ° 1 ( Y t ° E t ° 1 Y t ) M = h M t ° 1 E ° " M t ± : Models of Time-Varying Volatility The quantity h t ° 1 captures time variation in the conditional distribution through the scale parameter. Because the concept of scale is less familiar to °nance econo- mists, it is much more common to refer to h t ° 1 (or h 2 t ° 1 ) as the volatility. To link
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