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# 4. A piece of economic theory implies the pair of series X3 and K are generated by X: " 05E. = U131 where "11: = "Lt1 + 51:: X: " 51/: = "2:, where

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4. A piece of economic theory implies the pair of series X3 and K are generated by X: “ 05E. = U131 where ”11: = “Lt—1 + 51:: X: “ 51/: = “2:, where 1132: = Wat—1 + 52:1 with (Eu, Eat)’ vector white noise, Ipl &lt; l and 0: 7E 13. a) Is XE 1(1) or HO)? Is Y; I(1) or NO)? If both are I(1), are there conditions under which they are cointegrated? If yes, state the cointegrating vector. b) Let X00” and Fool; denote the long run optimal linear forecasts of X1: and K; madeat time t. (Formally, X00“ 2 limgHoo Xtht.) Information available at time t includes Xt,}Q,Xt_1,Yt_1, . . .. Find a functional relationship between 200'; andYOOIE'c) [OPTIONAL FOR EXTRA CREDIT] Show that (Xhl’gy follows a 1VAR(1). (Hint: This part is easier to solve using matrix notation.) d) [OPTIONAL FOR EXTRA CREDIT] Find the error correction representation ofthe system. Show that fl &gt; D and o: &lt; D are necessary conditions for the long run equilibrium to be stable. (This part may get a bit tedious. You can start from the system in part a) or b).)