Question

Question 3

A valuable property of the ln (natural logarithm) function is that:

Question 3 options:

	ln(x+∆x) – ln(x) is approximately equal to ∆x/x when ∆x/x is small.
	ln(x+∆x) – ln(x) is approximately equal to the percentage change in x when ∆x/x is small.
	Both (a) and (b)
	None of the above.

Question 4

Suppose you estimate the following regression model using OLS: Y_i = β₀ + β₁X_i + β₂X_i² + β₃X_i³+ u_i. You estimate that β₃has a value of 0.6 with a standard error of 0.7. This implies:

Question 4 options:

	You can reject the null hypothesis that the regression function is linear.
	You cannot reject the null hypothesis that the regression function is quadratic.
	You can reject the null hypothesis that the regression function is quadratic.
	None of the listed options.

Answer 1

3. The correct option would be

• Both (a) and (b)

The reason being that, for a function y=f(x), we have . For y=ln(x), we would have . Now, we know that or . Hence, we have . This means that tends to , as decreases (tends to zero). Hence, as decreases, since tends to , we have tends to . Now, as decreases, so would , and tends to . Also, is the percentage change in x too.

4. The correct option would be

• You cannot reject the null hypothesis that the regression function is quadratic.

For the OLS equation be , the restriction that would make the regression as , which is quadratic.

The null hypothesis of the test of individual significance for X-cube would be , which is equivalent to the statement that the regression is quadratic. The alternative hypothesis would be (which is equivalent to the statement that the regression is cubic). The test statistic would be or or . The degree of freedoms are unknown, but we may say that the given t-statistic is quite low. Also, for df greater than 10, and the p-value of the calculated t is lesser than 0.20 for df greater than 100. Hence, we fail to reject the null that the regression is quadratic.