TimeSeriesBook.pdf

This test also rejects h2 we can be pretty confident

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this test also rejects H(2), we can be pretty confident that there are three cointegrating relations given as: ˆ β = 1 . 000 0 . 000 0 . 000 0 . 000 1 . 000 0 . 000 0 . 000 0 . 000 1 . 000 - 258 . 948 - 277 . 869 - 337 . 481 . In this form, the cointegrating vectors are economically difficult to in- terpret. We therefore ask whether they are compatible with the following hypotheses: β C = 1 . 0 - 1 . 0 0 . 0 0 . 0 , β I = 1 . 0 0 . 0 - 1 . 0 0 . 0 , β R = 0 . 0 0 . 0 0 . 0 1 . 0 . These hypotheses state that the log-difference (ratio) between consumption and GDP, the log-difference (ratio) between investment and GDP, and the real interest rate are stationary. They can be rationalized in the context of the neoclassical growth model (see King et al., 1991; Neusser, 1991). Each of them can be brought into the form of equation (16.10) where β is replaced by its estimate ˆ β . The corresponding test statistics for each of the three cointegrating relations is distributed as a χ 2 distribution with one degree of freedom, 12 which gives a critical value of 3.84 at the 5 percent significance level. The corresponding values for the test statistic are 12.69, 15.05 and 0.45, 12 The degrees of freedom are computed according to the formula: s ( n - r ) = 1(4 - 3) = 1.
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338 CHAPTER 16. COINTEGRATION respectively. This implies that we must reject the first two hypotheses β C and β I . However, the conjecture that the real interest is stationary, cannot be rejected. Finally, we can investigate the joint hypothesis β 0 = ( β C , β I , β R ) which can be represented in the form (16.9). In this case the value of the test statistic is 41.20 which is clearly above the critical value of 7.81 inferred from the χ 2 3 distribution. 13 . Thus, we must reject this joint hypothesis. 13 The degrees of freedom are computed according to the formula: r ( n - s ) = 3(4 - 3) = 3.
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Chapter 17 State-Space Models and the Kalman Filter The state space representation is a flexible technique originally developed in automatic control engineering to represent, model, and control dynamic systems. Thereby we summarize the unobserved or partially observed state of the system in period t by an m -dimensional vector X t . The evolution of the state is then described by a VAR of order one usually called the state equation. A second equation describes the connection between the state and the observations given by a n -dimensional vector Y t . Despite its simple struc- ture, state space models encompass a large variety of model classes: VARMA, repectively VARIMA models, 1 unobserved-component models, factor mod- els, structural time series models which decompose a given time series into a trend, a seasonal, and a cyclical component, models with measurement errors, etc. From a technical point of view, the main advantage of state space mod- eling is the unified treatment of estimation, forecasting, and smoothing. At the center of the analysis stands the Kalman-filter named after its inven- tor Rudolf Emil K´ alm´ an (Kalman, 1960, 1963). He developed a projection based algorithm which recursively produces a statistically optimal estimate of the state. The versatility and the ease of implementation have made the
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