Chapter 4

Question 1

We adopt the least-squares criterion

Answer 1

THE METHOD OF ORDINARY LEAST
SQUARES

Question 2

We want to minimize the sum of the squared residuals

Answer 2

THE METHOD OF ORDINARY LEAST
SQUARES

Question 3

Three Statistical Properties of
OLS Estimators

Answer 3

I. The OLS estimators are expressed solely in
terms of the observable quantities (i.e. X and Y).
Therefore they can easily be computed.

II. They are point estimators (not interval
estimators). Given the sample, each estimator
provide only a single (point) value of the
relevant population parameter.

III. Once the OLS estimates are obtained from the
sample data, the sample regression line can be
easily obtained.

Question 4

The properties of the
regression line

Answer 4

1. It passes through the sample means of Y and X
2. The mean value of the estimated Y = Y; is equal to the mean value of the actual Y
3. The mean value of the residuals is zero
4. The residuals û; are uncorrelated with the predicted Y
5. The residuals û are uncorrelated with X₁; that is, sum hat u_{i}*X_{i} = 0 .

Question 5

The Classical Linear
Regression Model: The Assumptions Underlying the Method of Least Squares

Answer 5

Assumption 1: Linear regression model.
Assumption 2: X values are fixed in repeated sampling
Assumption 3: Zero mean value of disturbance u.
Assumption 4: Homoscedasticity or equal variance of u,.
Assumption 5: No autocorrelation between the disturbances.
Assumption 6: Zero covariance between u; and Xi, or E(uiXi) = 0 Formally,
Assumption 7: The number of observations n must be greater than the number of parameters to be estimated.
Assumption 8: Variability in X values.
Assumption 9: The regression model is correctly specified.
Assumption 10: There is no perfect multicollinearity.

Question 6

The regression model is linear in the parame- ters

Answer 6

Assumption 1: Linear regression model.

Question 7

. Values taken by the regressor X are considered fixed in repeated samples. More technically, Xis assumed to be nonstochastic.

Answer 7

Assumption 2: X values are fixed in repeated sampling

Question 8

Given the value of X, the mean, or expected, value of the random disturbance term u; is zero. Technically, the conditional mean value of u; is zero. Symbolically, we have E(u; X) = 0

Answer 8

Assumption 3: Zero mean value of disturbance u.

Question 9

Given the value of X, the vari- ance of u, is the same for all observations. That is, the conditional variances of u; are identi- cal. Symbolically, we have
var (u; Xi) = E[u; - E(u; | Xi)]2 = E(ut | Xi) because of Assumption 3
where var stands for variance.

Answer 9

Assumption 4: Homoscedasticity or equal variance of u,.

Question 10

Given any two X values, X, and Xj (ij), the correlation between any two u, and u; (ij) is zero.

Answer 10

Assumption 5: No autocorrelation between the disturbances.

Question 11

cov (ui, Xi) = E[u-E(u)] [X-E(X))] = E[u(X-E(X))] since E(u_{i}) = 0 = E(uX) - E(X)E(u) since E(X) is nonstochastic = E(u,X) since E(u_{i}) = 0 = 0 by assumption

Answer 11

Assumption 6: Zero covariance between u; and Xi, or E(u_{i}*X_{i}) = 0 Formally,

Question 12

Alternatively, the number of observations n must be greater than the number of explanatory variables.

Answer 12

Assumption 7: The number of observations n must be greater than the number of parameters to be estimated.

Question 13

The X values in a given sample must not all be the same. Technically, var (X) must be a finite positive number. 13

Answer 13

Assumption 8: Variability in X values.

Question 14

Alternatively, there is no specification bias or error in the model used in empirical analysis

Answer 14

Assumption 9: The regression model is correctly specified.

Question 15

That is, there are no perfect linear relationships among the explanatory variables

Answer 15

Assumption 10: There is no perfect multicollinearity.

Question 16

Given the assumptions of the classical linear regression model, the least-squares estimators, in the class of unbiased linear estimators, have minimum variance, that is, they are BLUE.

Answer 16

Gauss-Markov Theorem

Question 17

An estimator, say the OLS estimator , is said to be a

Answer 17

best linear unbiased estimator (BLUE)

Question 18

An estimator, say the OLS estimator , is said to be a best linear unbiased estimator (BLUE) of β2 if the following hold:

Answer 18

1. It is linear, that is, a linear function of a random variable, such as the dependent variable Y in the regression model.
2. It is unbiased, that is, its average or expected value, E(ẞ2), is equal to the true value, ẞ2.
3. It has minimum variance in the class of all such linear unbiased estimators; an unbiased estimator with the least variance is known as an efficient estimator.

Question 19

that is, a linear function of a random variable, such as the dependent variable Y in the regression model.

Answer 19

1. It is linear,

Question 20

that is, its average or expected value, E(ẞ2), is equal to the true value, ẞ2.

Answer 20

It is unbiased,

Question 21

an unbiased estimator with the least variance is known as an efficient estimator.

Answer 21

It has minimum variance in the class of all such linear unbiased estimators;

Question 22

• TSS:
• ESS:
• RSS:

Answer 22

total sum of squares
explained sum of squares
residual sum of squares

Question 23

is the sample correlation coeffient

Answer 23

r

Question 24

Some of the properties of r

Answer 24

1. It can be positive or negative, the sign depending on the sign of the
   term in the numerator of (3.5.13), which measures the sample covariation of
   two variables.
2. It lies between the limits of -1 and +1; that is, - 1 <= r <= 1
3. It is symmetrical in nature; that is, the coefficient of correlation be- tween X and Y(r XY ) is the same as that between Y and X(r YX ).
4. It is independent of the origin and scale; that is, if we define X_{i} ^ * = aX_{i} + C and Y i ^ * = bY_{i} + d , where a > 0 b > 0 and c and d are constants
   then r between X and Y is the same as that between the original variables X and Y.
5. If X and Y are statistically independent (see Appendix A for the defi- nition), the correlation coefficient between them is zero; but if r = 0, it does not mean that two variables are independent. In other words, zero correla- tion does not necessarily imply independence. [See Figure 3.11(h).]
6. It is a measure of linear association or linear dependence only; it has no meaning for describing nonlinear relations. Thus in Figure 3.11(h), Y = X2 is an exact relationship yet r is zero. (Why?)
7. Although it is a measure of linear association between two variables, it does not necessarily imply any cause-and-effect relationship, as noted in
   Chapter 1.

Question 25

an unbiased estimator with the least variance is known as

Answer 25

an efficient estimator.

Question 26

var:
se:
σ²:
σ:
6:

Answer 26

• variance
• standard error
• the constant homoscedastic variance of ui
• the standard error of the estimate
• OLS estimator of σ

Question 27

How to minimize the sum of the residual

Answer 27

Through a good model or theory

Question 28

The opposite of homescedasticity

Answer 28

heteroscedasticity

Question 29

The coefficient of determination

Answer 29

r²

Question 30

How much of the variation of variable Y explained by variable X

Answer 30

r²