- Nir Klein
- Published Date:
- August 2011
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The state space form of the univariate filter can be presented as follows:
The variables and gt represent the cyclical component of yt (the output gap) and the trend growth, respectively. and and are residual terms of mean 0 and variances and , 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 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.
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 . The parameters p and q refer to the lags of inflation and output gap, respectively.
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 is the white process of mean 0 and variance . The parameter reflects the unobserved natural real rate, which is affected by the trend growth, as follows:
The smoothness of is controlled by constraining the relative variance of and 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.