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ECON217_HW_Cointegration_Key

# ECON217_HW_Cointegration_Key - Homework(cointegration 1 Ans...

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2 Homework (cointegration) 1. What does Y t ~ I(d) mean? Ans. Y t ~ I(d) means (1-L) d Y t is stationary; that is after differencing d times, the series {Y t } is stationary. 2. Let Y t (Y 1t , Y 2t )’. What does “ Y t ~ CI(d,b)” mean? Ans. Y t ~ CI(d,b) means the elements of the vector process are cointegrated of order (d,b). That is, the elements of Y t are individually I(d) and there is a linear combination of the element, say, Z t such that Z t is I(d-b).cointegrated of order (d,b) if there exists Z t = Y t + X t such that Z t is I(d-b). Z t is known as the equilibrium error. The case d=1=b receives the most attention in the literature. 3. Explain how the cointegration concept could be used to study economic equilibrium relationships. Use an economic example to illustrate your argument. 4. Consider the bivariate system Y 1t = Y 2t + 1t and Y 2t = Y 2t-1 + 2t , where 1t = 1t + k 1 , 2t = 2t + k 2 , k 1 and k 2 are constants, and 1t and 2t are independent white noises. a. Show that Y t = (1 - L) 1 t 2 t = 1 L 0 1 k 1 k 2 + 1 L 0 1 1 t 2 t b. Show that E( Y t ) = 1 2 1 L 0 1 k 1 k 2 and the cointegrating vector is orthogonal to E( Y t ). c. Show Y t = 1 2 + 1 L 0 1 1 t 2 t , where u 1 u 2 is defined in (b). Hence, show that Y t = 1 2 + 1 0 Z t-1 + 1 0 1 1 t 2 t where 1 = 1 + k 1 . Ans. The bivariate system can be written as (1 - L) 1 t 2 t = 1 L 0 1 1 t 2 t = 1 L 0 1 k 1 k 2 + 1 L 0 1 1 t 2 t = 1 2 + 1 L 0 1 1 t 2 t

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3 Thus, E( Y t ) = ( 1 2 )’. By direct multiplication, we find (1 - ) is orthogonal to ( 1 2 ). In general the cointegrating vector is orthogonal to the E( Y t ).
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