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Back Matter

Nir Klein
Published Date:
August 2011
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    Appendix 1. Unobserved Component Models using Kalman Filter

    Model 1

    The state space form of the univariate filter can be presented as follows:

    The variables ytc and gt represent the cyclical component of yt (the output gap) and the trend growth, respectively. and ϵtc and ϵtg are residual terms of mean 0 and variances σc2 and σg2, respectively. The cyclical component of output follows an autoregressive process, and θ is lower than 1 to ensure a stationary process. The smoothness of the trend component is controlled by constraining the relative variance (σc2/σg2) to be equal to 1,600, as in the HP filter. The system can be estimated by Kalman filter, using equation (2) as a signal equation and equations (3) to (5) as the transitional equations.

    Model 2

    In this model, we add a backward-looking Phillips curve as a second signal equation in the system presented previously, which implies that inflation path is affected by past inflation rates as well as current and past output gaps, as follows:

    Where πt is the inflation rate and is a white noise process of mean 0 and variance σπ2. The parameters p and q refer to the lags of inflation and output gap, respectively.

    Model 3

    In this third model, we add the following standard backward-looking IS curve to the second model, such that the system includes three signal equations:

    Where rt is the real short-term rate and ϵty is the white process of mean 0 and variance σy2. The parameter r^t reflects the unobserved natural real rate, which is affected by the trend growth, as follows:

    The smoothness of r^t is controlled by constraining the relative variance of ϵty and ϵtr(σy2/σr2) to λ.

    Figure A1.The estimated total factor productivity (in natural logarithm), 1985-2009

    Figure A2.The cyclical components of energy production and employment, 1985-2010

    Figure A3.South Africa’s output gap by main sectors

    Figure A4.South Africa: Participation rate and discouraged work-seekers

    Source: Quarterly Labor Force Survey, SASTAT.

    Figure A5.REER change in selected EMs, 2008Q2-2010Q4

    Source: INS database.

    Figure A6.The share of South Africa’s exports of goods by regions

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