Uj
u gt ' x l w'i Rj fi,,. 5.2.21 As in the prior problem, we form the Lagrangian function ., differentiate it with respect to u gt , set the resulting equations to zero, and then solve for w. For the Lagrangian function we have c L - wflu S jf, - u gt ' i - 1 - v'l Rj . 5.2.22 Differentiating I. with respect to w and setting the result equal to zero, well a ve Combining 5.2.23 with 5.2.21 we have w, - M I y-ir' ,x - 111 . 5.2.24 5. The Capital Asset 'riling .m.iicl Note thai we t an express u gt...
Info Ksw
A consistent estimator Df can easily be constructed using the maximum likelihood estimators of Mm and o'fn. To compute a consistent estimator of S0, an assumption is necessary to reduce the summation in 5.6.8 to a finite number of terms. Section A.3 in the Appendix discusses possible assumptions. Defining Sj as a consistent estimator of So, l 7' D7 S7'Dr -1 is a consistent estimator of the covariance matrix of 0. Noting that a R0 where R 1 0 8 lN, a robust estimator or Var a is l VlRtDVs D,...
Info Okx
to the large sample distribution.'' The large sample distribution o J under the mill hypothesis will be chi-square with the degrees of freedom equal to the number of restrictions imposed by the null hypothesis. 6.2.1 Portfolios as Earlors with a IliskJ'ree Asset We first consider the case where the factors are traded portfolios and there exists a riskfrec asset. The unconstrained model will be a AMaclor model expressed in excess returns. Define Z, as an Nxl vector of excess returns for N assets...
Cj
where denotes convergence in probability. The fact that this ratio exceeded one for many historical slock returns series led Cowlcs and Jones 1937 lo conclude lhal this represents conclusive evidence of .structure in slock prices.1 However, the assumption of a zero drift is critical in determining the value off . In particular, C will exceed one for an 11D random walk with drill, since a drift either positive or negative clearly makes sequences more likely than reversals. To see this, suppose...
V
This measure of risk aversion rises as the the surplus consumption ratio 5', declines, that is, as consumption declines toward habit.17 The requirement that consumption always be above habit is satisfied automatically in microeconomic models with exogenous asset returns and endogenous consumption, as in Constantinides 1990 and Sundarcsan 1989 . It presents a more serious problem in models with exogenous consumption processes. To handle this problem Campbell and Cochrane 1995 specify a nonlinear...
The Predictability of Asset Returns
one ok hie earliest and most enduring questions of financial econometrics is whether financial asset prices are forecastable. Perhaps because of the obvious analogy between financial investments and games of chance, mathematical models of asset prices have an unusually rich history that predates virtually every other aspect of economic analysis. The fact that many prominent mathematicians and scientists have applied their considerable skills to forecasting financial securities prices is a...
Info Fya
Where lt e Depending on whether d is negative or positive, the spectral density of 2.6.1 at frequency zero will either be zero or infinite. i 2.6.2 The Hurst-Mandelbrut Retailed Range Statistic The importance of loug-raugc dependence in asset markets vvas first studied by Mandelbrot 1971 , who proposed using the range over standard deviation, or R S, statistic, also called the rescaled range, to detect long-range dependence in economic time series. The R S statistic was originally developed by...