Elements of Econometrics
SECTION A
Answer all questions from this section
1. We are interested in investigating the factors governing the precision of regression coefficients. Consider the model Yi = β1 + β2X2i + β3X3i + oi with OLS parameter estimates β^1, β^2 and β^3. Under the Gauss Markov assumptions, where σ2 is the variance of o and rE E is the sample correlation between X2 and X3.
(a) (4 marks) Provide four factors that help with obtaining more precise parameter estimates for, say, β^2.
(b) (4 marks) In light of your answer to (a), discuss the concept of near multicollinearity. What consequences does its presence have when considering single and joint significance testing of our slope parameters?
2. Consider the linear regression model
Yt = βO + β1Xt + β2Yt—1 + ut, t = 1, ..., T
where the errors ut are distributed independently of the regressors Xt and |β2| < 1. You suspect that the, mean zero, errors exhibit autocorrelation.
(a) (2 marks) Explain what we mean by the concept of autocorrelation.
(b) Assume that ut follows an AR(1) process.
i.(3 marks) Discuss, for the given model, the consequences for the ordinary least squares estimator. Support your answers with suitable arguments.
ii.(3 marks) Discuss how you would detect the presence of autocorrelation in the errors in this model. Clearly indicate the null and alternative hypothesis, the test statistic, and the rejection rule.
3. For the population of men who grew up with disadvantaged backgrouns, let poverty be a dummy variable equal to one if a man is currently living below the poverty line, and zero otherwise. The variable age is age and educ is total years of schooling. Let vocat be an indicator equal to unity if a man’s high school offered vocational training. Using a random sample of 850 men, you obtain
Pr (poverty=1^|educ,age,vocat ) = Ω (0.453 — 0.016 age — 0.087 educ — 0.049 vocat)
where Ω (z) = exp (z) /(1 + exp (z)) is the logit function.
(a) (5 marks) It is argued that using the logit regression model is better than using the linear probability model when explaining the binary variable poverty. Discuss the benefits/drawbacks of using the logit regression model when trying to explain a binary variable.
(b) (3 marks) For a 40-year old man, with 12 years of education, what is the estimated effect of having vocational training available in high school on the probability of cur- rently living in poverty?
Hint: Clarity of computations required is enough, no need to give an exact number.
4. The following model jointly determines monthly child support payments and monthly visita- tion rights for divorced couples with children:
support = 1 + 2 visits + 3 finc + 4 fremarr + 5 dist + o1
visits = β1 + β2 support + β3 mremarr + β4 dist + o2
We assume that children live with their mothers, so that fathers pay child support. Thus, the first equation is the father’s "reaction function": it describes the amount of child support paid for any given level of visitation rights and the other exogenous variables finc (father’s income), fremarr (binary indicator if father remarried), and dist (miles currently between the mother and father’s residence). Similarly the second equation is the mother’s reaction function: it describes visitation rights for a given amount of child support; mremarr is a binary indicator for whether the woman is remarried.
(a) (3 marks) Examine the identification of each structural equation.
(b) (5 marks) Your friend suggests you should implement the IV estimator to estimate the β parameters consistently. He tells you to use finc as instrument for support. Provide a critical discussion of this suggestion.
5. Consider the simple linear regression model
Yi = βO + β1Xi + ui
under the classical linear regression model assumptions, where Xi is fixed under repeated sampling. The usual OLS estimators β^O and β^1 are unbiased for their respective population parameters. Let β˜1 be the estimator of β1 obtained by assuming the intercept is zero.
(a) (4 marks) Show that the restricted least squares estimator of β1 is given by
(b) (4 marks) Find E β˜1 in terms of the Xi, βO and β1. Verify that β˜1 is unbiased for β1
when the population intercept is zero. Are there other cases where β˜1 is unbiased?
SECTION B
Answer three questions from this section.
6. Let us consider the estimation of a hedonic price function for houses. The hedonic price refers to the implicit price of a house given certain attributes (e.g., the number of bed- rooms). The data contains the sale price of 546 houses sold in the summer of 1987 in Canada along with their important features. The following characteristics are available: the lot size of the property in square feet (lotsize), the numbers of bedrooms (bedrooms), the number of full bathrooms (bathrooms), and a dummy indicating the presence of aircondi- tioning (airco).
Consider the following ordinary least squares results
log^(price)i=7.094 + 0.400 log(lotsize)i + 0.078 bedroomsi +(6.1)
(.232)
[.233]
(.O28)
[.O28]
(.O15)
[.O17]
0.216 bathroomsi + 0.212 aircoi, n = 546, RSS = 32.622
(.O23)
[.O24]
(.O24)
[.O23]
The usual standard errors are in parentheses, the heteroskedasticity robust standard errors are in square brackets, and RSS measures the residual sum of squares.
(a) (5 marks) Interpret the parameter estimates on log (lotsize) , bedrooms, and airco. Briefly discuss the statistical significance of the results.
(b) (5 marks) Suppose that lot size was measured in square metres rather than square feet. How would this affect the parameter estimates of the slopes and intercept? How would this affect the fitted values? Note: the conversion (approximate) 1m2 = 10ƒt2.
(c) (5 marks) We are interested in testing the hypothesis HO : βbedrooms = βbathrooms against the alternative HA : βbedrooms βbathrooms . Discuss a test for this hypothesis
that makes use of the following restricted regression result
log^(price)i = 6.994 + 0.408 log(lotsize)i + 0.127 bbroomsi + 0.215 aircoi, (6.2)
(.234)
(.282)
(.O11)
(.O24)
n = 546, RSS = 33.758
where bbrooms = bedrooms + bathrooms. Clearly indicate the assumptions you are making for this test to be valid.
(d) (5 marks) You are interested in testing for the presence of heteroskedasticity. Say you are told that the variance is increasing with log (lotsize) . Discuss how you would test for the presence of heteroskedasticity. What is the name of the test you are proposing?
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