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Additional info for Hayek's Conservative Liberalism (Political Theory and Political Philosophy)
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50) negative definite. Hence all requirements of the Theorem are met which proves stability of x (Y), Y. Since this is a single point, it is also the smallest attractor of the true system. Hence the stable solution x = x (Y), y = Y of the true system is unique. Remark 1. 51) 14 Cf. Bellman pp. 243 - 245. 42). Assume A to be nonsingular. 52) and this implies that the eigenvalues of J are approximately the eigenvalues of exA and (D - CA- 1 B), respectively, if ex is sufficiently large. e. if small changes in the system leave the qualitative behaviour of the system unchanged 15.
52 Consider the regression model YI = a~ XI + UI, t = 1,2, ... 6) with YI E JR, XI E JR n observations, al E 1R n coefficients to be estimated, and UI E JR normal disturbances for the time periods t = 1,2, ... , T. The coefficients are assumed to be generated by a random walk with normal disturbances v I E JR n: al = al_\ + VI, t = 2, 3, ... , T; VI'" JV (0, E). 7) The estimation problem can be discussed in two steps. First, the time path a\, a2, ... , aT of the coefficients is estimated under the assumption that the variances a 2 and E are known.
Remark 2. In the example of Sect. 1 ("The Marshallian Market") the moving equilibrium method leads to the correct equilibrium as well as to the correct stability conclusion independently of the speeds of adjustments. This is not, however, always the case. g. 54) with the associated moving equilibrium equation Y=-Y. 52) is unstable, its trace being positive. Remark 3. The Theorem requires continuously differentiable Ljapunov functions. This seems no severe restriction in view of the fact that stability implies in any case existence of continuous Ljapunov functions and is usually proved, in nonlinear applications, by means of Ljapunov functions which are typically piecewise continuous.