Question

Consider the model yi = β0 +β1X1i +β2X2i +ui . We fail to reject the null hypothesis H0 : β1 = 0 and β2 = 0 at 5% when: a) A F test of H0 : β1 = 0 and β2 = 0 give us a p value of 0.001 b) A t test of H0 : β1 = 0 give us a p value of 0.06 and a t test of H0 : β2 = 0 a p value of 0.06 c) A F test of H0 : β1 = 0 and β2 = 0 give us a p value of 0.1 d) A t test of H0 : β1 = 0 and H0 : β2 = 0 give us a t equal to 1.5

Suppose that a researcher, using wage data by county on 300 randomly selected counties in the North and 240 counties in the South, estimates the following OLS regression: W age ˆ = 9.92 + 1.02xNorth R2 = 0.12 (.34) (.29) where Wage is measured in dollars per hour and North is a binary variable that is equal to 1 if the county is in the North and 0 if it is in the South. Define the regional gap as the difference in mean earnings between North and South. a) What is the regional gap estimated in our model? b) Is it significantly different from zero? c) Construct a 95% confidence interval for the regional gap. d) In the sample, what is the mean wage of counties in the South? in the North? e) Another researcher uses these same data but regresses Wages on South, a variable that is equal to 1 if the county is located in the South and 0 if it is not. What are the regression estimates calculated from this regression? W age ˆ = ... + ...xSouth, R2 = ...

Answer 1

The model is:

yi = β0 +β1X1i + β2X2i +ui

The hypotheses are:

H0 : β1 = 0 and β2 = 0

H1: Not all slope coefficients are simultaneously zero

For testing the joint significance of coefficients ,we conduct the F-test.

The decision rule is as follows:

At α = 0.05, if the p-value of F-test < 0.05 => Reject H0

If p-value > 0.05 => Do not reject H0

In option a) The p-value for F-test = 0.001 i.e. p-value < 0.05, which implies that we need to reject the null hypothesis.

In option (b) and option (d) , t-test is used which is used to test individual significance of coefficients and is not valid for testing joint significance of coefficients.

Hence option c) is the correct option because p-value of F-test = 0.1 i.e. p-value > 0.05, which implies that we fail to reject the null hypothesis, H0.

Given: n= 300 + 240 = 540

W age ˆ = 9.92 + 1.02xNorth        , R2 = 0.12

               (.34) (.29)

a) The regional gap is given by the slope coefficient β1^ in the regression, which in this case is 1.02.

b)

To test the hypothesis:

H0: β1 = 0

H1: β1 ≠ 0

We do the t-test:

t = (β1^ - β10) / se(β1^)

= (1.02 - 0) / 0.29

= 3.517

For this sample, df = n-2 = 540-2 = 538

At α = 0.05 we have α/2 = 0.025, the critical value of t0.025, 298 = 1.96.

Since the calculated t-value > critical t-value, we reject the null hypothesis. Hence, β1 i.e., the regional gap is significantly different from zero.

c)

We know that 100(1 - α) % confidence interval for β1 is:

              [ β1^ ± tα/2 xse(β1^) ]

So, a 95% confidence interval for β1 is:

( 1.02 ± 1.96 x 0.29)

= ( 1.02 - 1.96 x 0.29, 1.02 + 1.96 x 0.29)

= ( 0.4516, 1.5884)

d)

In the dummy regression, W age ˆ = 9.92 + 1.02xNorth

Since the dummy variable is for North, the reference category is South.

So, the mean wages in South is $ 9.92 per hour.

Since, the slope coefficient is the differential coefficient or the regional gap, we obtain the mean wages in North by adding this value to the mean wages in the South.

The mean wages in North is $9.92 + $1.02 = $10.94 per hour.

e)

If wages are regressed on a dummy variable South that takes the value 1 for countries in South, the regression will be:

W age ˆ = 10.94 - 1.02xSouth

Since, the reference category is North in this case, the intercept gives the mean wages in North.

Also, the regression is exactly same as before apart from the change in reference category and dummy variable, so the R2 value will be the same i.e. 0.12.