Question 1

One of the shortest but most important sections in the book is that on the Gauss-Markov theorem.
(a) What is the Gauss-Markov theorem?
(b) Carefully explain what the precise properties specified by the Gauss-Markov theorem mean and why they are desireable for an equation.
(c) What could cause the Gauss-Markov theorem to no longer hold? What should we do in such a situation?

Accepted Answer

(a) The Gauss-Markov theorem is a fundamental result in the field of statistics and econometrics. It states that in a linear regression model, if the errors are uncorrelated, have constant variance, and are on average equal to zero, then the ordinary least squares (OLS) estimator is the best linear unbiased estimator (BLUE) of the true coefficients.

(b) The precise properties specified by the Gauss-Markov theorem are:
- Uncorrelated errors: This means that the errors in the model are not related to each other, and there is no systematic pattern in their relationship. This property is desirable because it ensures that the OLS estimator is efficient and unbiased.
- Constant variance of errors: This property, also known as homoscedasticity, means that the variance of the errors is the same for all observations. This is desirable because it ensures that the OLS estimator is consistent and has minimum variance.
- Errors on average equal to zero: This property ensures that the OLS estimator is unbiased, meaning that it provides estimates that are, on average, equal to the true values of the coefficients.

These properties are desirable for an equation because they ensure that the OLS estimator is the best linear unbiased estimator, meaning that it is efficient, consistent, and has minimum variance. In other words, it provides the most precise and accurate estimates of the true coefficients in the model.

(c) The Gauss-Markov theorem may no longer hold if the assumptions of uncorrelated errors, constant variance, and errors on average equal to zero are violated. This could happen if there is a systematic pattern in the errors (autocorrelation), if the variance of the errors is not constant across observations (heteroscedasticity), or if the errors have a non-zero mean. In such a situation, it is important to diagnose and address these violations in order to obtain reliable and valid estimates. This may involve using alternative estimation methods, such as generalized least squares (GLS) or robust standard errors, to account for the violations of the Gauss-Markov assumptions.

Question 2

In Chapter 4, we discussed seven explicit assumptions about the properties of a regression equation. These assumptions (or, more accurately, the first six of them) are usually referred to as the Classical Assumptions.
(a) Carefully and accurately state the seven assumptions.
(b) State what each assumption actually means in real-world terms (be extremely brief).

Accepted Answer

(a) The seven assumptions about the properties of a regression equation are:

1. Linearity: The relationship between the independent and dependent variables is linear.
2. Independence: The errors in the regression model are independent of each other.
3. Homoscedasticity: The variance of the errors is constant across all levels of the independent variables.
4. Normality: The errors in the regression model are normally distributed.
5. No perfect multicollinearity: There is no perfect linear relationship between the independent variables.
6. Zero conditional mean: The expected value of the error term is zero given the values of the independent variables.
7. No autocorrelation: The errors in the regression model are not correlated with each other.

(b) The real-world meanings of these assumptions are:

1. Linearity: The relationship between the independent and dependent variables can be represented by a straight line.
2. Independence: The errors in the regression model are not influenced by each other.
3. Homoscedasticity: The variability of the errors is consistent across different levels of the independent variables.
4. Normality: The errors in the regression model follow a normal distribution.
5. No perfect multicollinearity: There are no exact linear relationships between the independent variables.
6. Zero conditional mean: The average value of the errors is zero given the values of the independent variables.
7. No autocorrelation: The errors in the regression model do not exhibit a pattern of correlation over time.

Question 3

Briefly identify the following in words or equations as appropriate:
(a) p-value
(b) Sampling distribution of
 $\hat { \beta }$ (c) Type I Error
(d) Level of significance
(e) Given $\hat { \beta }$ = 4.0 and SE ( $\hat { \beta }$ ) = 2.0, test the null hypothesis that $\beta$= 0 (versus the alternative hypothesis that it does not equal zero) with 13 degrees of freedom at the 1% level of significance.

Accepted Answer

(a) See page 135.
(b) See Section 4.2.
(c) See Section 5.1.
(d) See Section 5.2.
(e) t = 2.0, and the critical two-sided, 1%, 13 degree of freedom critical t-value is 3.012, so we can not reject the null hypothesis.

Question 4

Consider the following regression equation for the United States (standard errors in parentheses): $$\begin{array} { c } \hat { \mathrm { P } } _ { \mathrm { t } } = 4.00 - 0.010 \mathrm { PRP } _ { \mathrm { t } } + 0.030 \mathrm { PRB } _ { \mathrm { t } } + 0.20 \mathrm { YD } _ { \mathrm { t } } \ ( 0.005 ) \quad ( 0.020 ) \quad ( 0.04 ) \ \overline { \mathrm { R } } ^ { 2 } = 0.98 \quad \mathrm { n } = 29 \end{array}$$ where: P_t =per capita pounds of pork consumed in time period t PRP_t =the price of pork in time period t PRB_t =the price of beef in time period t YD_t =per capita disposable income in time period t (a) Hypothesize signs and specify the appropriate null and alternative hypotheses for the coefficients of each of these variables. (b) State your decision rules and then test your hypotheses on the above results using the t-test at a 5% level of significance. (c) If you could add one variable to the regression, what variable would you add? Why?

Accepted Answer

The answer of Consider the following regression equation for the...

Quiz 2: Sample Exam for Chapters 4-5