Define dummy variable and give two examples analyze the

Question

Practice Questions for the Final Exam:

Theoretical Part-

1. Define dummy variable and give two examples.

2. Analyze the three different types of data (cross-sectional, time series, panel data).

3. Define R² and R^-2. What is their important property? Show the relation between them and their differences. Analyze as much as you can.

4. (a) Analyze fully the assumption of "homoscedasticity" of a CLRM. Moreover, which are the differences with heteroscedasticity? (b) Analyze fully the assumption of specification bias and use an example to show your intuition.

5. State the CLRM (Classical linear Regression Model) Assumptions.

6. State the Gauss-Markov Theorem and provide full definitions of the characteristics of a BLUE estimator (unbiasedness, linearity, efficiency).

7. Analyze the procedure of the Maximum Likelihood (ML) Estimator for the following bivariate regression model: Y_i = β₁ + β₂X_i + u_i. Specify any advantages or disadvantages the ML estimator has over the OLS estimator. The formula for Log Likelihood is given as:

inL = -nlnσ² - n/2 ln(2π) - 1/2 ∑ (Y_i - β₁ - β₂X_i)²/σ²

8. You are given the following non-linear regression model

Y = β₁X^-β_2e^u, where, Y: dependent variable, X: independent variable, e: exponent, u: error term, betas: coefficients. Make the necessary transformation/s so that the model can be estimated by using OLS method.

9. You are given the following two models:

(GPDI)^{^} =  -1026.5 + 0.30GDP

       se =  (257.58)          (0.04)

where GDPI and GDP are measured in billions of dollars

(GPDI*)^{^} = 0.94GDP*

        se  = (0.115)

where GDPI* and GDP* are the standardized versions of the variables GDPI and GDP.

Interpret the coefficient of GDP and GDP* with economic reasoning.

10. Analyze briefly the different reasons of doing hypothesis testing in a multiple regression model.

11. Discuss two types of specifications errors that we may have in a classical linear regression model. Use an example for each case.

11. Choose the correct answer.

i. The α in confidence interval given by Pr (β_i^{^} - δ ≤ β_i ≤ β_i^{^} + δ) = 1 - α is known as:

a. Confidence coefficient

b. Level of confidence

c. Level of significance

d. Significance coefficient

ii. The 1 - α in confidence interval given by Pr (β_i^{^} - δ ≤ β_i ≤ β_i^{^} + δ) = 1 - α is known as:

a. Confidence coefficient

b. Level of confidence

c. level of significance

d. Significance coefficient

iii. Standard error of an estimator is a measure of

a. Population estimator

b. Precision of the estimator

c. Power of the estimator

d. Confidence interval of the estimator

iv. For a regression through the origin, the intercept is equal to

a. 1

b. 2

c. 0

d. -1

v. Which of the following statements is correct?

a. Multicollinearity arises when explanatory variables are highly correlated with each other.

b. Multicollinearity can be identified by examining the pattern of correlations among explanatory variables

c. A high correlation between the dependent variables and a given independent variable is a sign of mulicollinearity.

d. All of the above are correct

e. Only (a) and (b) are correct

f. Only (b) and (c) are correct

Empirical Part-

Problem 1 - The table contains the ACT scores and the GPA for eight college students. GPA is based on a 4- point scale and has been rounded to one digit after the decimal.

Student	GPA	ACT
1	2.8	21
2	3.4	24
3	3.0	26
4	3.5	27
5	3.6	29
6	3.0	25
7	2.7	25
8	3.7	30

Obtain the estimates α₀^{^} and α₁^{^} in the linear regression model: (GPA_i)^{^} = α₀^{^} + α₁^{^}ACT

Problem 2 -

(I) The following equation is part of a nutrition-based efficiency wage model: Total calories Cals were regressed on the number of meals MG given to guests at ceremonies, the number of meals ME given to employees and the number of meals MO given to guests on other occasions:

Cals = β₀ + β₁MG + β₂ME + β₃MO +_e         (Model A)

The expected sign of the coefficients are β₁ > 0, β₂ > 0, β₃ > 0

You ran the regression in Eviews, by using OLS, for the period 1960-1999 and you obtained the following output:

Dependent Variable: CALS

Method: Least Squares

Included observations: 40 after adjustments

Variable	Coefficient	Std. Error         t-Statistic
C	27.59394	17.41539
MG	0.607160	0.157120
ME	0.092188	2.311452
MO	0.244860	0.011095
R-squared		Mean dependent var
Adjusted R-squared	0.989590	S.D. dependent var	19.53879
S.E. of regression		Akaike info criterion
Sum squared resid	143.0726	Schwarz criterion
Log likelihood	-82.24700	Hannan-Quinn criter.
F-statistic		Durbin-Watson stat	0.897776

a. Some of the standard errors and t-statistics of the coefficients are missing from the output. Calculate them (be careful about the signs). Remember that the Null Hypothesis is:

H₀: β_i = 0. Test in α = 5% significance level. Your t-critical is given as 2.03. Do you reject or fail to reject the Null?

b. Interpret the effect of MG and ME variables with economic reasoning.

c. The value of R² is missing from the output. Calculate it.

d. The standard error of regression is not visible in the above output. Calculate it by using the relevant formula: σ^{^2} = ∑u^{^}_i²/n-k, where n is the number of observations and k is the number of parameters.

e. The F value (for overall significance) you obtained is missing. Calculate it and test it for 5% significance level. The F_critical(5%,3,36) is given as 2.87. Note that we are testing jointly for all the coefficients, excluding the intercept.

 f. Calculate the 95% Confidence Intervals for the coefficients of MG and MO.

(II) You decide to re-specify Model A, by dropping variable MO. Your model becomes:

Cals = β'₀ + β'₁ MG + β'₂ ME + ε  (Model B)

You run the model and you obtain the following output:

Dependent Variable: CALS

Method: Least Squares

Included observations: 40 after adjustments

Variable	Coefficient	Std. Error         t-Statistic	.
C	35.45670
MG	2.567990
ME	0.892676
R-squared	0.860390	Mean dependent var
Adjusted R-squared	0.852844	S.D. dependent var	19.53879
S.E. of regression	7.495264	Akaike info criterion
Sum squared resid	2078.622	Schwarz criterion
Log likelihood	-135.7692	Hannan-Quinn criter.
F-statistic	114.0122	Durbin-Watson stat	0.678483

a. We do not know which model is better performed between model A and model B. Which model is better in terms of model building? Make use of the Information Criteria to find out which model is better.

b. The output is problematic. The standard errors and the t-statistics are missing. Use the Wald Test to do Restriction Testing between models A and B. The Null Hypothesis is: H₀: β₃ = 0. The Fcritical is given as 4.11 and the X₁² critical value is given as 3.841 (both for a 5% significance level). What can you conclude regarding the dropped variable MO?

Problem 3 - lnip_t+1 = β₀ + β₁lnol_t + β₂lnrsr_t + β_3emplt + u_t

The above linear regression model states that the industrial productivity (lnip) is positively affected by the stock market returns (lnrsr) and the employment ratio (empl) and negatively affected by the oil prices (lnlo). We are examining the USA market growth during the period 1960-1999.

In 1980 we have the introduction of the personal computer and we want to test whether there is any structural change after the year of 1980:

H₀: There was no structural break (or change) after 1980

H₁: There was a structural break (or change) after 1980

You get the following three outputs by doing the Chow Test.

Dependent Variable: LNIP

Method: Least Squares

Sample: 1960 1999

Included observations: 40

Variable	Coefficient	Std. Error         t-Statistic	Prob.
C	27.59394	1.584458          17.41539	0.0000
LNOL	-0.607160	0.157120        -3.864300	0.0004
LNRSR	0.092188	0.039883          2.311452	0.0266
EMPL	0.244860	0.011095          22.06862	0.0000
R-squared	0.990391	Mean dependent var	50.56725
Adjusted R-squared	0.989590	S.D. dependent var	19.53879
S.E. of regression	1.993549	Akaike info criterion	4.312350
Sum squared resid	143.0726	Schwarz criterion	4.481238
Log likelihood	-82.24700	Hannan-Quinn criter.	4.373414
F-statistic	1236.776	Durbin-Watson stat	0.897776
Prob(F-statistic)	0.000000

Dependent Variable: LNLIP

Method: Least Squares

Sample: 1960 1979

Included observations: 20

Variable	Coefficient	Std. Error         t-Statistic	Prob.
C	27.59882	2.433883          11.33942	0.0000
LNOL	-0.899693	0.297873        -3.020394	0.0081
LNRSR	0.181932	0.098121          1.854171	0.0822
EMPL	0.265328	0.058970          4.499342	0.0004
R-squared	0.913357	Mean dependent var	34.28700
Adjusted R-squared	0.897112	S.D. dependent var	6.199594
S.E. of regression	1.988596	Akaike info criterion	4.389592
Sum squared resid	63.27225	Schwarz criterion	4.588738
Log likelihood	-39.89592	Hannan-Quinn criter.	4.428467
F-statistic	56.22198	Durbin-Watson stat	1.116410
Prob(F-statistic)	0.000000

Dependent Variable: LNLIP

Method: Least Squares

Sample: 1980 1999

Included observations: 20

Variable	Coefficient	Std. Error         t-Statistic	Prob.
C	16.18376	3.874379          4.177124	0.0007
LNOL	-0.345689	0.136746        -2.527964	0.0224
LNRSR	0.151866	0.046860          3.240840	0.0051
EMPL	0.272712	0.008611          31.67100	0.0000
R-squared	0.993168	Mean dependent var	66.84750
Adjusted R-squared	0.991887	S.D. dependent var	13.68186
S.E. of regression	1.232329	Akaike info criterion	3.432546
Sum squared resid	24.29817	Schwarz criterion	3.631692
Log likelihood	-30.32546	Hannan-Quinn criter.	3.471421
F-statistic	775.3400	Durbin-Watson stat	1.665783
Prob(F-statistic)	0.000000

a. Explain the procedure of the Chow Tests, i.e the steps you have to take in order to make conclusions about the structural stability.

b. Calculate the Chow F-Statistic. According to the F statistic you just calculated do you reject or fail to reject the Null Hypothesis? What that means for your data? You are given that Fcritical(0.05, 4, 32) = 2.69.