Eonomics 835
Spring 1994
D. Maki
                    Midterm Examination

There are three parts to this examination, each with two
questions.  Answer one question from each part (a total of
three questions).  All questions are equally weighted.
Total time allowed=two hours.

Part I.  Answer one of the following two questions.

1.   a. For the sample regression model Y=a+bX+u, show that
b is a BLUE estimator under the CLR model assumptions.  Make
it clear at what points in your demonstration each of the
assumptions is used.
     b. For the sample regression model Y=bX+e (no intercept
estimated) show whether the ols estimator b is an unbiased
estimator if there is a non-zero intercept in the true
population model.
c.  If your computer program only does simple regression and
does not estimate an intercept, how would you estimate the
slope coefficient if the true population intercept is
nonzero?

2.  Two researchers each estimated the same sample
regression model, Y = a + bX +e, but used different
(independent) samples of size n=8.  Their output is
reproduced below (standard errors of estimated coefficients
in parentheses).

Researcher A:  Y = 1.875 + 0.750X,  R2 = 0.45, S2X = 0.286
                   (1.197) (0.339)

Researcher B:  Y = 1.153 + 0.972X,  R2 = 0.99, S2X = 23.143
                   (0.267) (0.038)

a.  Why do you suppose researcher B obtained such a high R2
     value compared to researcher A?
b.  What slope coefficient would obtain if the two samples  
     were combined into one sample of size n=16?
c.  What t-value obtains for testing the hypothesis that the
     slope coefficients in the two populations are equal?

Part II.  Answer one of the following two questions.

3.  For each of the following statements regarding the
sample multiple regression model with two independent
variables:   Y = a + b1 X1 + b2 X2 + e, note whether it is
always true, always false, or uncertain (true in some cases,
but not others).  Explain why.

a.  The estimator b1 = ä[y(x1 - b12x2)]/ä(x1 - b12x2)2,
     where lower case x and y represent mean deviated       
     variables, and b12 represents the slope coefficient in
     the regression of X1 on X2.
b.  In a given set of data it is observed that r12 = 0.9,
     rY2 = -0.2, and rY1 =0.8 (rY2 is the correlation       
     between Y and X2, etc.).
c.  If X2 = X12 (a parabola is being fitted), then b1 is    
     equal to b in the simple regression Y = a + bX1 + e.
d.  R2 = rY12 + rY22.

4.  For each of the following applications, indicate how you
would estimate the nonlinear function indicated  (in all
cases a, b and c are parameters to be estimated; Y, X and Z
are variables, and e is the natural constant).

a.  Y = a + bXc
b.  Y = a[bX-c + (1-b)Z-c]-1/c, (a constant elasticity of   
     substitution (CES) production function, where Y is     
     output and X and Z are inputs)
c.  Y = a(1 + b)X
d.  Y = abXcZ
e.  Y = a + becX
f.  Y = aXbZc

Part III.  Answer one of the following two questions.

5.  For the sample regression model Y = Xb + e:
a.  Show that the least squares estimator, b=(XtX)-1XtY.
b.  Show that b is unbiased if the CLR assumptions hold.
c.  Derive the variance of the b vector.
d.  Explain how you would calculate a 95% confidence        
     interval for Y when the X variables take on some       
     particular values, say X1t ... XKt.  Be specific.
e.  Explain how you would test the joint hypothesis that the
     intercept is zero while the slope coefficients sum to  
     unity.

6.  Assume Y represents number of years of education
completed, X2 represents father's income, and X3 represents
urban versus rural residence (coded zero for urban, unity
for rural).  The X variables (but not Y) are mean deviated
to produce:

XtX = 8  0  0        and XtY = 109           YtY = 1535
      0 68 -8                   47
      0 -8  2                   -6.5

Calculate test statistics for the following hypotheses:
a.  The coefficient of father's income is zero.
b.  The coefficient of urban versus rural in zero.
c.  The coefficients of both father's income and urban      
     versus rural are simultaneously zero.