6.Explain the difference between ratio, interval, ordinal, and nominal scales. Give an example of each. (pp. 30–31)

Ratio Scale For a variable X, : X1 and X2, the ratio X1/X2 and the distance (X2 − X1) are meaningful quantities. Also, there is a natural ordering of the values along the scale.Therefore, comparisons such as X2 ≤ X1 or X2 ≥ X1 are meaningful. Most economic variables belong to this category. Thus, it is meaningful to ask how big is this year’s GDP compared with the previous year’s GDP.Interval Scale An interval scale variable satisfies the last two properties of the ratio scale variable but not the first. Thus, the distance between two time periods, is meaningful, but not the ratio of two time periods .Ordinal Scale A variable belongs to this category only if it satisfies the third property of the ratio scale (i.e., natural ordering). Examples ; income class (upper, middle, lower). For these variables the ordering exists but the distances between the categories cannot be quantified. Students of economics will recall the indifference curves between two goods, each higher indifference curve indicating higher level of utility, but one cannot quantify by how much one indifference curve is higher than the others.Nominal Scale Variables in this category have none of the features of the ratio scale variables. Variables such as gender (male, female) and marital status (married, unmarried, divorced, separated) simply denote categories. Question: What is the reason why such variables cannot be expressed on the ratio, interval, or ordinal scales? As we shall see, econometric techniques that may be suitable for ratio scale variables may not be suitable for nominal scale variables. Therefore, it is important to bear in mind the distinctions among the four types of measurement scales discussed above.,

7. Explain how you can use the general F test to test for a structural change (Chow test). Illustrate with an example

The Chow test is a statistical and econometric test of whether the coefficients in two linear regressions on different data sets are equal. The Chow test was invented by economist Gregory Chow. In econometrics, the Chow test is most commonly used in time series analysis to test for the presence of a structural break. In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.

8.Explain the dummy variable trap using an example. (pp. 301–303).

The dummy variable trap is where we have a case of perfect collinearity in the sense that unless one dummy variable is omitted from the regression equation, the sum of all dummy variables add up to 1, giving an inaccurate answer. When there are dummies in all observations, the constant term has to be excluded. If a constant term is included in the regression, it is important to exclude one of the dummy variables from the regression, making this the base category against which the others are assessed. If all the dummy variables are included, their sum is equal to 1 (which stands for the variable X0 to the constant term B0), resulting in perfect multicollinearity. This is referred to as the dummy variable trap.

9. Explain how you can use the general F test to test for a restriction on the parameters (restricted least squares). Illustrate with an example. (pp.267–273).

The F-Test Approach: Restricted Least Squares

The preceding t test is a kind of postmortem examination because we try to

find out whether the linear restriction is satisfied after estimating the “unrestricted’’

regression. A direct approach would be to incorporate the restriction

(8.7.3) into the estimating procedure at the outset. In the present example,

this procedure can be done easily. From (8.7.3) we see that

β2 = 1 − β3 (8.7.5)

or

β3 = 1 − β2 (8.7.6)

Therefore, using either of these equalities, we can eliminate one of the β coefficients

in (8.7.2) and estimate the resulting equation. Thus, if we use

(8.7.5), we can write the Cobb–Douglas production function as

ln Yi = β0 + (1 − β3) ln X2i + β3 ln X3i + ui

= β0 + ln X2i + β3(ln X3i − ln X2i) + ui

or

(ln Yi − ln X2i) = β0 + β3(ln X3i − ln X2i) + ui (8.7.7)

or

ln (Yi/X2i) = β0 + β3 ln (X3i/X2i) + ui (8.7.8)

where (Yi/X2i) = output/labor ratio and (X3i/X2i) = capital labor ratio, quantities

of great economic importance.

Notice how the original equation (8.7.2) is transformed. Once we estimate

β3 from (8.7.7) or (8.7.8), β2 can be easily estimated from the relation

(8.7.5). Needless to say, this procedure will guarantee that the sum of the

estimated coefficients of the two inputs will equal 1. The procedure outlined

in (8.7.7) or (8.7.8) is known as restricted least squares (RLS). This procedure

can be generalized to models containing any number of explanatory

variables and more than one linear equality restriction. The generalization

can be found in Theil.13 (See also general F testing below.)

How do we compare the unrestricted and restricted least-squares regressions?

In other words, how do we know that, say, the restriction (8.7.3) is

valid? This question can be answered by applying the F test as follows.

10.Illustrate in the example of a savings function how you can use dummy variables as an alternative to the Chow test (pp. 306–310). THE DUMMY VARIABLE ALTERNATIVE TO THE CHOW TEST9

In Section 8.8 we discussed the Chow test to examine the structural stability

of a regression model. The example we discussed there related to the

relationship between savings and income in the United States over the

period 1970–1995. We divided the sample period into two, 1970–1981 and

1982–1995, and showed on the basis of the Chow test that there was a difference

in the regression of savings on income between the two periods.

However, we could not tell whether the difference in the two regressions

was because of differences in the intercept terms or the slope coefficients or

both. Very often this knowledge itself is very useful.

Referring to Eqs. (8.8.1) and (8.8.2), we see that there are four possibilities,

which we illustrate in Figure 9.3.

1. Both the intercept and the slope coefficients are the same in the two regressions.

This, the case of coincident regressions, is shown in Figure 9.3a.

2. Only the intercepts in the two regressions are different but the slopes

are the same. This is the case of parallel regressions, which is shown in

Figure 9.3b.

3. The intercepts in the two regressions are the same, but the slopes are

different. This is the situation of concurrent regressions (Figure 9.3c).

4. Both the intercepts and slopes in the two regressions are different.

This is the case of dissimilar regressions, which is shown in Figure 9.3d.

The multistep Chow test procedure discussed in Section 8.8, as noted earlier,

tells us only if two (or more) regressions are different without telling us

what is the source of the difference. The source of difference, if any, can be

pinned down by pooling all the observations (26 in all) and running just one

multiple regression as shown below10:

Yt = α1 + α2Dt + β1Xt + β2(DtXt) + ut (9.5.1)

where Y = savings

X = income

t = time

D = 1 for observations in 1982–1995

= 0, otherwise (i.e., for observations in 1970–1981)

11. Explain the use of a dummy variable in an interactive (or multiplicative) form. Illustrate in an example. (pp. 310–312).

Dummy variables are a flexible tool that can handle a variety of interesting

problems. To see this, consider the following model:

Yi = α1 + α2D2i + α3D3i + βXi + ui (9.6.1)

where Y = hourly wage in dollars

X = education (years of schooling)

D2 = 1 if female, 0 otherwise

D3 = 1 if nonwhite and non-Hispanic, 0 otherwise

In this model gender and race are qualitative regressors and education is

a quantitative regressor.11 Implicit in this model is the assumption that the

differential effect of the gender dummy D2 is constant across the two categories

of race and the differential effect of the race dummy D3 is also constant

across the two sexes. That is to say, if the mean salary is higher for

males than for females, this is so whether they are nonwhite/non-Hispanic

or not. Likewise, if, say, nonwhite/non-Hispanics have lower mean wages,

this is so whether they are females or males.

In many applications such an assumption may be untenable. A female

nonwhite/non-Hispanic may earn lower wages than a male nonwhite/non-

Hispanic. In other words, there may be interaction between the two qualitative

variables D2 and D3. Therefore their effect on mean Y may not be simply

additive as in (9.6.1) but multiplicative as well, as in the following model.

ˆY

i = α1 + α2D2i + α3D3i + α4(D2iD3i) + βXi + ui (9.6.2)

where the variables are as defined for model (9.6.1).

From (9.6.2), we obtain:

E(Yi | D2i = 1, D3i = 1, Xi) = (α1 + α2 + α3 + α4) + βXi

which is the mean hourly wage function for female nonwhite/non-Hispanic

workers. Observe that

α2 = differential effect of being a female

α3 = differential effect of being a nonwhite/non-Hispanic

α4 = differential effect of being a female nonwhite/non-Hispanic

which shows that the mean hourly wages of female nonwhite/non-Hispanics

is different (by α4) from the mean hourly wages of females or nonwhite/non-

Hispanics. If, for instance, all the three differential dummy coefficients are

negative, this would imply that female nonwhite/non-Hispanic workers earn

much lower mean hourly wages than female or nonwhite/non-Hispanic

workers as compared with the base category, which in the present example

is male white or Hispanic.

Now the reader can see how the interaction dummy (i.e., the product of

two qualitative or dummy variables) modifies the effect of the two attributes

considered individually (i.e., additively).