The moving average MA representation of the I 2 model is expressed as x t C 2 t

The moving average ma representation of the i 2 model

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The moving average (MA) representation of the I ( 2 ) model is expressed as: x t = C 2 t i = 1 i s = 1 ( ε s + Φ D s + μ 0 + μ 1 s ) | {z } ε s + C 1 t i = 1 ( ε i + Φ D i + μ 0 + μ 1 i ) | {z } ε i + C * ( L ) ( ε t + Φ D t + μ 0 + μ 1 t ) + A + B t (23) where C 2 = β 2 ( α 0 2 Θ β 2 ) - 1 α 0 2 , β 0 C 1 = α 0 Γ C 2 , β 0 1 C 1 = α 1 0 ( I p - Θ C 2 ) , and Θ = Γ β α 0 Γ + I p - k - 2 i = 1 Λ i . A and B are functions of both the initial values and the model parameters ( Johansen 1992 ). 13 Matrix C 2 can be expressed as C 2 = ˘ β 2 α 0 2 , where ˘ β 2 = β 2 ( α 0 2 Θ β 2 ) - 1 , so that α 0 2 t i = 1 i s = 1 ε s can be interpreted as the measure of the s 2 trends which load into the variables in x t with the weights ˘ β 2 ( Juselius 2006 ). The likelihood ratio test for the joint hypothesis of r cointegrating relationships and s 1 and s 2 trends, labeled H ( r , s 1 , s 2 ) , versus H ( p ) is given by: - 2log Q ( H ( r , s 1 , s 2 ) | H ( p )) = - T log ˜ Ω - 1 ˆ Ω (24) where ˜ Ω and ˆ Ω are, respectively, the covariance matrices estimated under H ( r , s 1 , s 2 ) and H ( p ) . 14 4. Stylized Facts Figure 1 a shows the evolution of the natural logarithm (log) of the nominal exchange rate, measured as Chilean pesos (CLP) per US dollar (USD) and the log of the relative prices, measured as the ratio between the Chilean consumer price index (CPI) and the US CPI. Relative prices exhibit a positive but decreasing slope, reflecting the fact that from 1986 until 1999, Chilean prices were growing faster than US prices, but after 1999 the growth in relative prices decreased. This might be associated with the partial implementation of inflation targeting in Chile in 1990, which reduced annual inflation from 26% in 1990 to 3% in 1997. In the same panel, the nominal exchange rate undergoes long and persistent swings around relative prices, suggesting that PPP may hold only as a very long-run condition. 13 From the MA representation ( 23 ), it follows that the unrestricted constant, μ 0 , cumulates once to a linear trend and twice to a quadratic trend. In addition, the unrestricted trend, μ 1 , cumulates once to a quadratic trend and twice to a cubic trend. To avoid the latter, quadratic and cubic trends have been restricted to zero in the subsequent analysis. For further information, see Chapter 17 in Juselius ( 2006 ). 14 The distribution of the this is found in Johansen ( 1995 ) provided that model ( 22 ) does not restrict deterministic components; otherwise see Rahbek et al. ( 1999 ).
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Econometrics 2017 , 5 , 29 8 of 21 Figure 1 b shows the PPP gap, defined as the difference between the log of relative prices and the log of the nominal exchange rate. The deviations exhibit long persistent swings, but it seems that the upward trend in relative prices is canceled by the upward trend in nominal exchange rate. Figure 1. ( a ) Nominal exchange rate (CLP/USD) and relative prices (Chilean CPI/US CPI); ( b ) Deviations from PPP. Monthly data 1986:1–2013:04. CLP: Chilean peso, USD: U.S. dollar.
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