CCER 计量经济学 第三次作业和答案

Intermediate Econometrics Class 1

Problem Set 3 with Answers

Handout Date: Dec. 4th, 2011

Due Date: Dec. 9th, 2011 (Hand in BEFORE class)

1.An estimated equation is

with

, and SSR = 1.5

Use F-statistic to test the following

(1).

(2).

(3).

(Hint: The tests involve only the sub-matrix in the lower-right corner of .

Refer to TA materials for the formula of inverses of partitioned matrices.)

(1).

equals the single element at the lower right-hand corner of

,which is 2.5.

Then the F-statistic is calculated as

It falls well short of any usually critical value for . So we cannot reject .

(2).

only involves the elements in the sub-matrix in the lower

right-hand corner of

The F-statistic equals

From the tables of F distribution, , so we cannot reject the null at

5% significance level.

(3).

Thus the test statistic becomes

Again, the test statistic falls well short of any usually critical value for . So we

cannot reject .

2. A four-variable regression using quarter data from 1958 to 1976 inclusive gave an estimated

equation

The explained sum of squares was 109.6, and the residual sum of squares, 18.48.

(1).When the equation was re-estimated with three seasonal dummies added to the

specification, the explained sum of squares rose to 114.8. Test for the presence of seasonality.

To test for the presence of seasonality we test the joint significance of the three seasonal dummy variables. The restricted is 18.48, while the unrestricted is

The rule-of-thumb F-statistic is calculated as

The 5% critical value is (is usually not given in statistic tables,

so here we use the instead). We can reject the hypothesis of no seasonality at 5%

significance level.

(2).Two further regressions based on the original specification were run for the sub-periods

1958.1 to 1968.4 and 1969.1 to 1976.4, yielding residual sums of squares of 9.32 and 7.46, respectively. Test for the constancy of the relationship over the two sub-periods.

To test the parameter consistency over the two sub-samples, consider the Chow test,

The 5% critical value is . Hence we cannot reject the hypothesis of

parameter constancy at 5% significance level.

3.Survey records for a large sample of families show the following weekly consumption

expenditure (Y) and weekly income (X):

Y 70 76 91 …… 120 146 135 X 80 95 105 …… 155 165 175

* * *

Families with an asterisk (*) reported that their income is higher than in the previous year.

(1).To examine the impact of weekly income on weekly consumptions, one sets up the

following model

He is concerned that the error terms may have heterogeneous variance. Derive the robust standard error of .

Under HSK, the large sample distribution of is

The sample estimate of is

where

The robust standard error of is the 2nd diagonal element of the estimated covariance

matrix of

(2).If he wants to estimate directly the elasticity of consumption with respect of income, how

should he modify the model in (1).

(3).If he wants to test whether the event of an increase in income, holding the level of income

unchanged, helps to explain the consumption behavior, how should he extend the model in (1)?

(4).If he wants to test whether the marginal propensity to consume (the slope coefficient) of

families experiencing an increase in income is different from that of families who did not experience an increase, how should he extend the model in (3)?

4.Consider the equation

where is the cumulative college grade point average, is size of high school

graduating class, in hundreds, is academic percentile in graduating class, is

combined SAT score, is a dummy gender variable, and is a dummy variable

which is one for student-athletes.

(1).What are your expected signs for the coefficients in this equation? Explain.

Holding all other variables constant, the expected sign for high school size should be negative, but at a diminishing rate, because larger high schools tend to have lower teacher-to-student ratios, and the effect becomes less important as the size increase. The higher sat should be positively related to GPA. So should hsperc and female (why should this be the case might be controversial; either because female students tend to study harder to overcome gender discrimination in society, or they tend to take classes where they excel more). I suppose the coefficient for athlete might be negative. However, this might just be my own prejudice.

(This answer is provided by the solution manual of Introductory Econometrics: A Modern Approach. It is only for your reference. You’ll receive full credit so long your arguments make sense.)

(2).To allow the effect of being an athlete to differ by gender, how should you extend the

model? Write out the null hypothesis if you want to test whether there is no ceteris paribus difference between women athletes and women nonalthletes.

Adding to the model we have:

In this setup, the intercepts for 4 different categories are:

Male non-athlete

Male athlete

Female non-athlete

Female athlete

So the test between female athletes and female non-athletes is the test of

5.One application of ADL models is the Adaptive Expectation Model:

?(5.1)

?(5.2)

where

demand for money

interest rate (observables)

equilibrium, optimum, or expected long-run interest rate (unobservable)

the coefficient of expectation (,)

Rewrite Eq.(5.2)

?(5.3)

Substitute Eq. (5.3) into Eq. (5.2)

?(5.4)

(1).Lag Eq. (5.1) one period, then substitute it into Eq. (5.4). You should be able to show that

the short-run demand is in essence an ADL process of the observables. Write out

the model, and calculate the long-run impact multiplier of .

The ADL model is

Use lag operator to rewrite the model

The long-run impact multiplier of is

Note the long-run impact multiplier of in the short-rum model is essentially the coefficient

of in Eq. (5.1), the equilibrium/long-run model.

(2).Now consider another application that incorporates the partial adjustment of into the

Adoptive Expectations Model:

where are defined as in (1), and:

actual capital stock (observable)

desired level of capital (unobservable)

the coefficient of adjustment

Show that the observed short-run demand is in essence an ADL process. (Hint: If

you derive the model correctly, you will find the error terms are serially correlated.)

The ADL model is