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Macroeconometrics 1: The Basic Building Blocks

(Graded Homework Problems)^*

宏观计量经济学作业代写 This problem continues the one with the same background as in this Lesson‘s practice problems with solutions.

1.Let x_s,tbe the 1 × k vector of explanatory variables for observation s such that  宏观计量经济学作业代写

so that for ‹ = 1, …, n

x_s,t = [ x_s1,t x_s2,t … x_sk,t ],

and

where ‹ = 1, 2, …, n.  Show that the OLS estimator β^  can be written as:

2. This problem continues the one with the same background as in this Lesson‘s practice problems with solutions.  Let β^  be the OLS estimate from the regression of ¢_t  on y_t. Let X_t be a k × k non-singular matrix and define:  宏观计量经济学作业代写

z_{s,t≡ xs,t} At

for ‹ = 1, …, n. Therefore, x_s,t is 1 × k and is a nonsingular linear combination of x_s,t. Let  Zt  be  the  n × k  matrix  with  rows  x_s,t.   Let  β˜  denote  the  OLS  estimate

These lecture notes closely and sometimes literally follow sections from: Dowling, Edward T. Intvo− ductson to Mathematsca1 Economscs. Shaum‘s Outlines, 3rd ed., McGraw Hill, 2001 Nicholson, Walter. Mscvoeconomsc fheov4: Bassc Pvsncsp1es and Estenssons. South-Western College Pub, 9th ed., £OO4¡ Simon, Carl P., and Lawrence Blume. Mathematscs fov Economssts. New York: Norton, fi994¡ Sydsaeter, Knut, and Peter J. Hammond. Mathematscs fov Economsc Ana14sss. Prentice Hall, 1995.

from  a  regression  of  y_t  on  Z_t. Show  that  the  estimated  variance  matrix  for  β˜  is

3.Thisproblem continues the preceding one as well as the one with the same background as in this Lesson‘s practice problems with solutions.  Let β^  be the OLS estimate from the regression of ¢_t on X_t. Let A_t be a h × h non-singular matrix and define:

z_s,t≡ x_s,tA_t

for s = 1, …, n. Therefore, x_s,t is fi × h and is a nonsingular linear combination of x_s,t.

Let  Z_t  be  the  n × h  matrix  with  rows  x_s,t.   Let  β˜  denote  the  OLS  estimate  from  a regression of y_t on Z_t.  Let βˆj  be the OLS estimates from a regression of y_t on:

1, x_2,t, …, x_k,t

and, let β˜j  be the OLS estimates from a regression of y_t on:

1, a₂x_2,t, …, a_kx_k,t,

where the as are constants and a_j  ƒ= 0  for j  = X, …, h.  State the mathematical rela- tionship between β˜j  and βˆj .

4.Consider the following lagged income determination model:  宏观计量经济学作业代写

In equilibrium, Y_t = C_t ‡ I_t akin to the development in this Lesson‘s practice problems with solutions.

(a)Find the time path of national incomeY_t.

(b)Characterize the stability of the time path obtained in part(a).

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