SECTION A

1.(a) Let A denote a m  m symmetric matrix. If A is positive deinite then what does this imply about any quadratic form involving A?          [1 marks]

1.(b)  If A is a m m positive deinite matrix then what is rank(A), the rank of A. Justify your answer. [3 marks]

1.(c) Consider

A1   =    ]

and

A2   =   [  5(7)

3(5)  ] .

Verify whether or not A1  and A2  are positive deinite, being sure to justify your answer. [4 marks]

2. Let {(yi , x )}i(/)   be a sequence of independently and identically distributed (i.i.d.) random vectors. Suppose that yi is a dummy variable and so has a sample space of {0, 1} with P (yi   = 1|xi ) = ^(x βi(/) 0 ) where ^( · ) is the cumulative distribution function of the logistic distribution. Derive:

(a) Derive E[yi |xi], E[yi] and Var[yi |xi].                                 [4 marks]

(b) Now suppose that a researcherestimates a logit model based on {(yi , x )} and let xi,j  be the jth  element of xi  and a continuous random variable.

(i) Derive the marginal response of P (yi  = 1|xi ) to a change in xi,j  that is, ∂^(x βi(/) 0 )/∂xi,j .                [2 marks]

(ii) What is the limit of the marginal response function in your answer to part(i) as the index x βi(/) 0 tends to ininity? Provide an intuitive explanation for why this must be the case.                          [2 marks]

3. Let aˆT  be an estimator of the unknown parameter a0 .  A researcher claims that “if aˆT  is an unbiased estimator of a0  then it must also be a consistent estimator of a0”. Evaluate this claim, being sure to justify your argument briely and deine any statistical concepts to which you refer.                                           [8 marks]

4. Consider the following model:

yi   =  xi β0  + ui ,       i = 1, 2, . . . , n                               (1)

where xi  is a scalar observable variable, ui  is the unobserved error.  Let zi  be a q  1 vector of observable variables.  Assume {(xi , ui , z )i(/) / } is a sequence of independently and identically distributed random vectors.

(a) State the conditions under which zi said to be a valid instrument for xi in this model. Can these conditions be tested? If so then explain how? If not then

[2 marks]

(b) Suppose now that

xi   =  zi(/)V0  + vi ,

where E[zi z ] =i(/) Mzz , a positive deinite matrix, and for wi  = (ui , vi )/ , E[wi |zi] = 0 and Var[wi |zi] =   0 . Provide a set of conditions involving the parameters  of the model, β0 ,V0 , and   0  under which xi  is an endogenous regressor and  zi  is a valid instrument for xi  in this model, being sure to justify your answer  carefully.                                [6 marks].

5. Let {et } be a univariate white noise process.

(a) Assess which of the following three series are covariance stationary provid- ing a justiication for your answer in each case:

(i) ut  = et ;

(ii) vt  = ( — 1)t + et ;

(iii) wt  = ( — 1)t et .

[5 marks]

(b) Assess which of the series in part (a) are strictly stationary providing a justi- ication for your answer in each case.         [3 marks]

SECTION B

6. Consider the linear regression model

y  =  Xβ0  + u                                              (2)

where y is T  1 with tth  element yt , X is T  k with tth  row xt(/)  =  [1, x2(/),t], u is T  1 with tth  element ut , β0  is a k  1 vector of unknown parameters. Assume that (2) is the true model for y , X is ixed in repeated samples, rank(X) = k , u ~ N(0, a0(2)IT ) for some unknown scalar constant a0(2) . Let Z be a T  k matrix that is ixed in repeated samples with rank(Z) = k and assume Z/ X is nonsingular. Deine T  = (Z X)/ — 1 Z y, and let/ T,i  be the ith element of T .

(a) Show that T  is an unbiased estimator of β0 .

(b) Show that Var[T] = a0(2)(X/ Z(Z/ Z) — 1 Z/X) — 1 .

[4 marks]

[9 marks]

(c) State the formula for Cov[T,i , T,j ] in terms of a0(2)  and (X/ Z(Z/ Z) — 1 Z/ X) — 1 . [2 marks]

(d) Show that T  ~ N ( β0 , Var[T] ).                                                [3 marks]

(e) A researcher argues that given the results in parts (a), (b) and (d) there is no reason to prefer inferences based on the OLS estimator of β0  over inferences based on T . Do you agree? Justify your answer.      [4 marks]

(f) Suppose now that X and Z are stochastic with E[u|Z] = 0.  Is T  an unbi- ased and/or a consistent estimator of β0 ? Justify your answer but there is no need to provide a formal analysis of the probability limit of T .     [8 marks]

7.(a) Consider the model

ut   =  wt  + wt —2 ,       t = 1, 2, . . . , T,                             (3)

where   0, and {wt }—∞  is a sequence of independently and identically dis- tributed random variables with E[wt] = 0 and Var[wt] = a2 . Let u denote the T  1 vector with tth element ut . Derive Var[u] =    in terms of  and a2 .    [14 marks]

7.(b) Consider the times series regression model

yt   =  xt(/)β0  + ut , t = 1, 2, . . . , T                                  (6)

where xt  = (1, yt — 1 )/ , β0  = (β0,1 , β0,2 , )/  and {ut } is generated as in part (a). You may assume that yt  has the following MA(∞) representation:

∞

工

i=0

where uy  is a constant and ψ0,1  0.

(i) Evaluate whether yt — 1  is contemporaneously exogenous in (6).  [10 marks] (ii) Evaluate whether yt — 1  is strictly exogenous in (6).                        [6 marks]

8.(a) Consider the linear regression model

yi   =  xi(/)β0  + ui                                                                           (4)

where xi  = (1, x2(/),i )/  and β0  are k  1 vectors. Assume {(ui , x2(/),i )/ , i = 1, 2, . . . N} are independentlyand identically distributed with: (i) E[xix ] =i(/) Q, a inite, positive deinite k k matrix of constants; (ii) E[ui |xi] = 0; (iii) Var[ui |xi] = h(xi ) > 0. The OLS estimator of β0  is βˆN   =  (X/ X) — 1 X/ y where y is a N  1 vector with ith element yi , X is a N  k matrix with ith  row xi(/) .

Show that

N1/2(βˆN   — β0 )    N (0, Vh ) ,

where Vh  = Q — 1    h Q — 1 ,   h  = E[h(xi )xix ]i(/) .                                            [8 marks]

Hint:  you may quote the generic form of the  Weak Law of Large Numbers,

N — 1 对 zi     uz , but must verify uz   for the speciic choices of zt  relevant to

your answer. Also you may quote the generic form of the Central Limit Theorem,

N — 1/2 对(zi — uz )  N(0,   ), but must verify uz  and    for the speciic choices of zi  relevant to your answer.

8.(b) Consider the following model

yt   =  β0,1  + β0,2x2,t  + β0,3x3,t  + ut , for t = 1, 2, . . . , T.

Let u be the T  1 vector with tth  element ut  and X be the T  3 matrix with tth row (1, x2,t, x3,t). Suppose that a researcher estimates the model via OLS based on sample of size T = 100, and obtains the itted equation:

t  = 0.389 + 0.336x2,t + 1.896x3,t           R2  = 0.455                       (5)

(1 .216)        (0 .234)                 (0 .927)

[1.362]        [0.286]                 [1.602]

{1.321}       {0.321}              {1.421}

where OLS Standard errors in parentheses ( ), White Standard Errors in square brackets [ ] and Newey-West Standard Errors in { }. In parts (i) - (iii) speciied on the next page:

.  Discuss how to test the hypothesis stated with correct asymptotic size α

providing the relevant test statistic and its distribution.

.  If more than one test may be formed based on the information in equation

(5) state so, providing details on how to perform each test.

.  Perform the test given the information in equation (5), discussing the choice

of test in the case more than one option is available.

8.(b)    (i)

H0  : β0,1 = 0;    HA  : β0,1   0

for α = 0.05 where Var(ut |X) = a2 |x2,t| and E[utut —j |X] = 0 for all t and all

j  0.                                                                                            [7 marks]

(ii)

H0  : β0,2    0;    HA  : β0,2  > 0

for α  =  0.01 if ut   =  et x1,t  where Corr(et , et —j )  =  exp(   |j|) for all t and

j .                                                                                                  [7 marks] (iii)

H0  : β0,2  = β0,3 = 0;    HA  : β0,2   0 and/or β0,3  0

for α = 0.1 where u|X ~ N(0, a2IT ).                                             [8 marks]

9. Let {(yi , x )}i(/)   be a sequence of independently and identically distributed (i.i.d.) random vectors. Suppose that yi is a dummy variable and so has a sample space of {0, 1} with P (yi   = 1|xi ) = (x βi(/) 0 ) where ( · ) is the cumulative distribution function of the standard normal distribution.

(a) Assume that xi  = 1.  Show that the maximum likelihood estimator of β0  is βˆ =  — 1 () where  is the sample mean of y and  — 1 ( . ) denotes the inverse of the cumulative distribution function of the standard normal distributionthat

is, if (z) = p then z =  — 1 (p) for any z ∈ ( — ∞ , ∞).                   [12 marks]

(b) A researcher is interested in modeling the probability that a citizen of a US town votes in favour of an increase in the local tax rate to provide additional funding for public schools as a function of certain family and household char- acteristics. Let yesvm be a dummy variable that takes the value one if the citizen votes in favour of the tax increase and the explanatory variables are: loginc, the log of annual household income; ptcon, the log of property taxes paid in the year the vote took place; years the number of years the voter has been living in the community; school, a dummy variable that takes the value one if the voter works in the public school system; and four other control variables denoted x1, x2, x3 and x4 below.  Using the Stata output on the next page answer the following questions.

(i) Test whether the amount of property taxes paid by a citizen affects the  probability that they vote for the tax increase. Be sure to specify the null  and alternative hypothesis, and the decision rule.                  [4 marks] (ii) What do the results reveal about how household income affects the probability of voting in favour of the tax increase? Be sure to justify your  answer.                                                                                  [4 marks] (iii) Fifty nine out the ninety ive citizens in the sample voted for the increase.  Use the Likelihood Ratio statistic to test whether the explanatory vari-  ables in the model collectively help to explain the probability that a citi-  zen votes in favour of the tax increase. Be sure to specify the null and alternative hypotheses, and the decision rule, and to explain how you calculate the test statistic.                                                    [10 marks]

9.(b) contd The Stata output for the model is as follows in which certain portions have been deleted for the purpose of this question:

Probit regression                                  Number  of  obs     =             95

Log  likelihood =  -52 .844