Econ 360 - Fall 2012

Problem Set 1 Answers

True/False

(30 points)

1. FALSE If (ai , bi ) : i = 1, 2, . . . , n and (xi , yi ) : i = 1, 2, · · · , n are sets of n pairs of numbers, then: n n n

(ai xi + bi yi ) =

i=1 i=1

ai x i +

i=1

bi yi

2. FALSE If xi : i = 1, 2, . . . , n is a set of n numbers, then: n n n n n

(xi − x) = ¯

i=1 n i=1

2

x2 i

− 2¯ x

i=1

xi +

i=1

x = ¯

i=1

2

x2 − n¯2 x i

where x = ¯

1 n i=1

xi

3. TRUE If xi : i = 1, 2, . . . , n is a set of n numbers and a is a constant, then: n n

a xi = a

i=1 n i=1

xi = a n x ¯

where x = ¯

1 n i=1

xi

4. FALSE If X and Y are independent random variables then: E (Y |X) = E (Y )

1

5. TRUE If {a1 , a2 , . . . , an } are constants and {X1 , X2 , . . . , Xn } are random variables then: n n

E

i=1

ai X i

=

i=1

ai E (Xi )

6. FALSE For a random variable X, let µ = E (X). The variance of X can be expressed as: V ar(X) = E X 2 − µ2

7. TRUE For random variables Y and X, the variance of Y conditional on X = x is given by: V ar(Y |X = x) = E Y 2 |x − [E (Y |x)]2

8. TRUE An estimator, W , of θ is an unbiased estimator if E (W ) = θ for all possible values of θ. 9. FALSE The central limit theorem states that the average from a random sample for any population (with ﬁnite variance) when it is standardized, by subtracting the mean and then dividing by the standard deviation, has an asymptotic standard normal distribution. 10. TRUE The law of large numbers states that if X1 , X2 , . . . , Xn are independent, identically distributed random variables with mean µ, then ¯ plim Xn = µ

2

Multiple Choice Questions

(a) ceteris paribus (b) correlation (c) causal eﬀect (d) independence

(20 points)

11. The idea of holding “all else equal” is known as

12. If our dataset has one observation for every state for the year 2000, then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) time series data (d) panel data 13. If our dataset has one observation for every state for the year 2000 and another observation for each state in 2005, then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) time series data (d) panel data 14. If our dataset has one observation for the state of Indiana each year from 1950-2005 then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) time series data (d) panel data 15. Consider the function f (X, Y ) = (aX + bY )2 . What is (a) 2aX (b) a(aX + bY ) (c) 2a(aX + bY ) (d) a2 X ∂f (X,Y ) ∂X

3

Long Answer Questions

(50 points)

16. The sum of squared deviations (subtracting the average value of x from each observation on x) is the sum of the squared xi minus n times the square of x. There are ¯ several ways to show this, here is one: n n

xi (xi − x) ¯

i=1

=

i=1 n

(xi − x + x) (xi − x) ¯ ¯ ¯

n

=

i=1 n

(xi − x) (xi − x) + ¯ ¯

i=1 n

x (xi − x) ¯ ¯

=

i=1

(xi − x)2 + x ¯ ¯

i=1 n

(xi − x) ¯ (xi − x) = 0, so ¯

and we know that

i=1 n

=

i=1

(xi − x)2 ¯

17. There are several ways to show that this expression equals the sample covariance between x and y, here is one: n n

xi (yi − y ) ¯

i=1

=

i=1 n

(xi − x + x) (yi − y ) ¯ ¯ ¯

n

=

i=1 n

(xi − x) (yi − y ) + x ¯ ¯ ¯

i=1

(yi − y ) ¯

=

i=1

(xi − x) (yi − y ) ¯ ¯

18. Correlation and causation are not always the same thing. (a) A negative correlation means that larger class size is associated with lower test performance. This could be because the relationship is causal meaning that having a larger class size actually hurts student performance. However, there are other reasons we might ﬁnd a negative relationship. For example, children from more aﬄuent families might be more likely to attend schools with smaller class sizes, and aﬄuent children generally score better on standardized tests. Another possibility is that...