Econometrics assignment

Problem Set 1

November, 14 2017

Due: 21st of November 2017 In groups (at most 4 persons), use the software STATA when needed to answer

concisely the questions below. Please return a printed version of your results by November 30, in class. Your document should include your answers to each specific question and in any case remember to write name, surname and id number (matricola) of each student in the document.

Question 1

Professor R.C. Fair (Cowles Foundation at Yale University), has for a number of years built and updated models that explain and predict the U.S. presidential elections. The basic premise of the model is that the incumbent party’s share of the two-party (Democratic and Republican) popular vote (incumbent means the party in power at the time of the election) is affected by number of factors relating to the economy and variables relating to the politics such as how long the incumbent party has been in power and whether the President is running for reelection. Fair’s data collect observations for the elections years from 1880 to 2000. The dependent variable is vote, that is, the percentage share of the popular vote won by the incumbent party whereas the regressor is growth, i.e., the growth rate in real per capita GDP in the first three quarters of the election year (annual rate). One would think that if the economy is doing well, and growth is high, the party in power would have a better chance to win the election. For this reason consider the following regression model

vote = α + βgrowth + �.

The regression output is provided in the following Table

1

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Source | SS df MS Number of obs = 33

-------------+------------------------------ F( 1, 31) = 16.51

Model | 407.923492 1 407.923492 Prob > F = 0.0003

Residual | 765.927214 31 24.7073295 R-squared = 0.3475

-------------+------------------------------ Adj R-squared = 0.3265

Total | 1173.85071 32 36.6828346 Root MSE = 4.9706

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vote | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

growth | .6545116 .1610797 4.06 0.000 .3259873 .9830359

_cons | 51.6908 .8711021 59.34 0.000 49.91418 53.46742

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• According to this empirical evidence, is the theory of Professor Fair cor- rect? That is, is there a positive association between these two variables? Define a suitable hypothesis system, build up a testing procedure and decide by using α = 5%.

• Suppose we expect a growth rate of about 0.6 for the next elections. What is the expected percentage share of the popular vote won by the incumbent party? Compare this prediction, by computing the prediction error, to the actual level of vote that is 52.3 percent.

• Do you think a variation of the growth rate of 0.10, i.e., ∆growth= 0.10, may have an impact on the variable vote of at least 1 percentage point? Define a suitable hypothesis system and take a decision at a 5% level.

• According to the output of the above table, what is the level of growth that we expect will not lead to a reelection? .

• Consider now the case in which the growth rate reduces of 0.10. Evaluate the expected variation of the dependent variable, i.e., ∆vote and explain why this variation statistically different from 0 at a 5% level?

• Does this simple regression necessarily capture a causal relationship be- tween the two variables? Explain.

Question 2

Consider the so called constant elasticity model. In particular consider the fol- lowing regression.

log wagei = α + βlog experiencei + ui (0.1)

where log wagei is the log wage for worker i and log experiencei is a log of the years of experiences.

1. Interpret the coefficients of the model. 2. Regarding the linearity concept. The relation between the dependent vari-

able and the regression is no more linear, even though it is still considered a linear regression model. What does linearity refers to?

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Question 3

Using data from 1978 (kiel 1978.dta) for houses sold in Andover, MA, from Kiel and McClain (1995) , the following equation relates housing real price (rprice) to the distance from a recently built garbage incinerator (dist) ex- pressed in miles:

log(rprice) = α + β log(dist)

1. Interpret the coefficient on log(dist). Is the sign of this estimate what you expect it to be?

2. Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist? (Think about the city’s decision on where to put the incinerator.)

3. What other factors about a house affect its price? Might these be corre- lated with distance from the incinerator?

4. Estimate the model

rprice = α + β log(dist)

and interpret the coefficient on log(dist). Evaluate if a variation on the distance of 5% has an impact on the prices of 1,000 dollars (define a suitable t test and decide using α = 5%).

Question 4

Consider the following model to estimate the return to education (dataset: wage.dta):

wagei = α + βeducationi + ui (0.2)

where wagei is the wage for worker i, educationi is a highest education level completed whereas ui is the error term.

1. Provide a descriptive analysis of the data in the way you think is the most informative for the following econometric analysis using the tools you consider appropriate.

2. Estimate the simple linear regression model stated above 3. Predict the fitted values of wage, wagei (hint: use the command predict ).

Find an observation for which the wage is over-predicted and one for which it is under-predicted.

4. Show empirically that the sample average of Ŷi equal to the sample average of Yi and that the sample average of the OLS residuals is equal to zero. (hint: use the command ttest )

5. Show empirically that the predicted value of income at values equal to the sample averages of the explanatory variables belongs to the estimated regression function.

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6. Now estimate the model log(wagei) = γ+θeducationi +εi and explain the meaning of θ contrasting it with the meaning of β. (hint: use the function log() to generate the variable lwage from wage)