Home assignment 2. Multiple Linear Regression Model. Transformations of variables.

The International College of Economics and Finance

Econometrics-2012.

To be submitted by October, 12

Home assignment 2. Multiple Linear Regression Model. Transformations of variables.

1. A student has data on annual household expenditure on clothing, Y, annual household wage income, W, and annual household non-wage income, NW, all measured in U.S. dollars, for 100 households for 2004. He is considering fitting the model

Y_i = a + b₁ W_i + b₂ NW_i + u_i

The correlation between W and NW is 0.95. Explain what is correct, mistaken, confused or incomplete in the following statement:

“The high correlation between W and NW will definitely give rise to the problem known as multicollinearity. If ordinary least squares (OLS) is used to fit the model, the coefficients will be biased in small samples but consistent in large ones. The standard errors will also be biased, probably downwards, and the t tests and F test will be invalid. The problem will be even worse if there is a high correlation between Y and W or Y and NW. One way of dealing with the problem is to run two separate regressions, one with Y regressed on W, the other with Y regressed on NW. Multicollinearity is not usually a serious problem in models with only one explanatory variable”.

1. The teacher of econometrics is interested what factors are essential if determining the midterm exam results of 21 students attending her course, in per cent. The teacher uses the data on attending lectures and number of home assignments submitted by the students and the grades of the midterm examination to estimate a linear regression model.

where is the exam results of the students, – a proportion of lectures attended, – a proportion of home assignments submitted.

(a) Give the interpretation to the coefficients of the model. Is it possible to give reasonable interpretation to the constant term of the equation? Try to state the general conditions when the interpretation of the constant term is meaningful? Answering questions a)-c) ignore the issues of statistical validity of the model as they are discussed in d).

(b) The coefficient by the variable is three times as much as the coefficient by the variable . Does it follow from here that attending lectures is three times more useful for the student (as measured in midterm exam grade) than submitting home assignment? Explain.

(c) Suppose there are 20 hours of lectures and 10 hours of self-study for preparing home assignment before midterm exam. The student who had important meeting in the morning got a call that the meeting is cancelled and so she has an additional hour for her study. What would you advice her from the analysis of equation: to attend lecture in econometrics or to go home to prepare her home assignment? How your conclusion changes if home assignments are twice less time consuming and require not 10 but 5 hours of self-study.

(d) Discuss the problem of significance of the coefficients. Pay special attention to the choosing of the appropriate significance level for each coefficient. Under what conditions is it possible to use one-tailed test here?

(e) Comment the value of determination coefficient. Is it possible to use and interpret equation with such a small level of determination coefficient? Is the equation as a whole significant (do the test)? What a null hypotheses and alternative hypotheses are used in this test?

(f) The teacher decided to include into equation an additional variable ‑ the grade of the student at final Statistics exam (all students took Statistics course last year). The result was as following

Comment and discuss this equation using different tools (t -statistic, F -statistic, adjusted R²) and compare the conclusions. The teacher noticed that in both equation the sum of all coefficients is not equal to 100: comment.

1. For your data set EAEF, estimate multiple linear regression model of LOG(EARN) on HGC, HGCF, HGCM and the composite indicator ASVABC.

(a) Give the interpretation to the coefficients of equation. Analyze the significance of variables and the equation as a whole and explain the results.

(b) Analyse the significance of the group of all HGCs. Why there any signs of the presence of multicollinearity? What is multicollinearity, what are its consequences? How the problem of multicollinearity could be solved in your case?

(c) introduce linear restriction(s) for HGCs (try the cases of one and two restrictions) and test them;

(d) estimate the final specification and comment on it. Discuss the procedure of testing linear restrictions using t-tests. Implement the procedure and discuss the results. Are they fully the same as for the F-tests?).

(e) How the interpretation of your final equation would change if you decide to use as a dependent variable EARN instead of LOG(EARN). How linear and logarithmic regressions could be compared?

(f) Try to use different indicators of abilities (ASVAB2-ASVAB6). Propose the abilities indicator (likely a combination of the listed above) the most appropriate for your data set.

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