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Question 1 In this question we study the price of a cup of coffee at various cafes in Sydney (city==1) and Melbourne (city==0). We ran a two-sample t test, and the results are as follows: . ttest lnprice, by(city) Two-sample t test with equal variances --------------------------------------------------------------------Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+----------------------------------------------------------0 | 19 1.291262 .0346825 .1511775 1.218397 1.364127 1 | 27 1.478113 .0366481 .1904291 1.402782 1.553444 ---------+----------------------------------------------------------combined | 46 1.400935 .0290226 .196841 1.342481 1.45939 ---------+----------------------------------------------------------diff | -.1868511 .0525339 -.2927262 -.0809759 --------------------------------------------------------------------diff = mean(0) - mean(1) t = -3.5568 Ho: diff = 0 degrees of freedom = 44 (a)

As you can see, we chose to work with logarithms of prices, rather than with the prices themselves. What could be a reason for that choice?

(b)

What is the economic interpretation of the number –0.1868511 in the bottom left?

(c)

Test whether the means are equal in both cities.

(d)

Write a regression model that you could use to test the same hypothesis as in part (c), and describe which test needs to be performed in your model.

(e)

Now suppose that we also collect data from Brisbane (city==2) and we wish to test the hypothesis that the means are equal in all three cities. How would you do that?

Question 2 The time-to-maturity of a loan is the time between the moment the loan is taken out and the moment it is due to be repaid. Higher annual interest rates tend to be charged for longer times-tomaturity, since there is a higher chance that the borrower might turn out to be incapable of repayment – more random accidents happen in thirty years than in three months. In the following Stata output, intrate is the interest rate (in %) that the Greek government paid on its loans during the crisis in early 2012, and lnmatur is the natural logarithm of the corresponding time-to-maturity (in months). . regress intrate lnmatur Source | SS df MS ---------+---------------------------Model | 950.056558 1 950.056558 Residual | 40.1330494 17 2.36076761 ---------+---------------------------Total | 990.189607 18 55.0105337

Number of obs F( 1, 17) Prob > F R-squared Adj R-squared Root MSE

= = = = = =

19 402.44 0.0000 0.9595 0.9571 1.5365

--------------------------------------------------------------------intrate | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+----------------------------------------------------------lnmatur | 5.575181 .2779143 20.06 0.000 4.988833 6.161529 _cons | 15.65086 1.122539 13.94 0.000 13.28251 18.01921 --------------------------------------------------------------------. summarize lnmatur Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------------lnmatur | 19 3.83485 1.303105 1.098612 5.886104 (a)

Why might we have chosen this linear–log form for the model?

(b)

Give economic interpretations for the coefficient estimates b1 and b2 .

(c)

If the Greek government were planning to take out a loan that would mature in forty years, and they asked you to predict which interest rate they should expect to pay, would you give them a conditional mean forecast or an actual value forecast? Why?

(d)

Compute the prediction interval that you selected in part (c).

(e)

Why did you not need to take retransformation bias into account in part (d)?

Question 3 We have collected data on the annual number of cars of twenty different brands sold in Australia (sales, in number of cars), as well as each brand’s average retail price (price, in dollars), their annual marketing expenditure (mark, also in dollars), and whether or not they assemble some of their cars domestically (domestic, dummy variable). We wish to investigate how all of these factors influence sales, and we settle on the following regression model: . regress lnsales lnprice lnmark domestic Source | SS df MS ---------+---------------------------Model | 15.2271807 3 5.0757269 Residual | 4.73289348 16 .295805843 ---------+---------------------------Total | 19.9600742 19 1.05053022

Number of obs F(XXXX, XXXX) Prob > F R-squared Adj R-squared Root MSE

= 20 = 17.16 = 0.0000 = XXXX = XXXX = .54388

--------------------------------------------------------------------lnsales | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+----------------------------------------------------------lnprice | -1.389525 .2392136 -5.81 0.000 -1.896635 -.8824151 lnmark | .1775161 .125631 1.41 0.177 -.0888098 .443842 domestic | .6156965 .389287 1.58 0.133 -.209555 1.440948 _cons | 21.35649 3.31581 6.44 0.000 14.32728 28.38569 --------------------------------------------------------------------(a)

I have removed four numbers from this table, indicated by “XXXX”. Compute them.

(b)

Describe what the coefficient estimate –1.389525 means, in economic terms.

(c)

Suppose we wish to test the claim that the Australian car market is completely pricedriven, so that marketing and whether production is done domestically are irrelevant. Regressing lnsales only on lnprice gave an RSS of 10.69, whereas regressing lnsales only on lnmark and domestic gave an RSS of 14.72. Which of these two numbers is useful for testing our claim, and why?

(d)

Test the claim described in part (c).

Question 4 Let yi be the number of hours per week a person spends on household chores, and xi the number of rooms in this person’s house. Assume that the following relationship exists between these variables:   α1 + α2 xi + ui if person i is unemployed, yi = β + β2 xi + ui if person i is self-employed,  1 γ1 + γ2 xi + ui if person i is an employee,

with the usual assumptions on ui . You may assume that every person in the sample is in exactly one of these three categories. As discussed during the course, we may estimate this model by defining dummy variables   1 if person i is an employee, 1 if person i is self-employed, and d emp,i = d self,i = 0 otherwise 0 otherwise and then running the regression yi = α1 + δ1 d self,i + ζ1 d emp,i + α2 xi + δ2 d self,i xi + ζ2 d emp,i xi + ui .

(a)

Why isn’t there also an unemployment dummy d unemp,i in the regression model?

(b)

Consider the hypothesis “α1 = β1 = γ1 and α2 = β2 = γ2 ”. Explain what this hypothesis means in plain English, and describe how you would use the regression model to test it.

(c)

Consider the hypothesis “α2 = β2 = γ2 ”. Explain what this hypothesis means in plain English, and describe how you would use the regression model to test it.

(d)

Consider the hypothesis “β1 = γ1 and β2 = γ2 ”. Explain what this hypothesis means in plain English, and describe how you would use the regression model to test it.

Question 5 In this question, we discuss two different Cobb–Douglas-type production models, which relate a firm’s production yi (in units) in a year to its use of capital ki (in dollars) and labour ℓi (in man-hours) in that year. Other factors that may influence production are d i , a dummy that equals one for high-tech firms and zero for others, and xi , GDP growth in the state that the firm is based in. The two models that we will consider are ln yi = β1 + β2 ln ki + β3 ln ℓi + ui ,

and

(1)

ln yi = β1 + β2 ln ki + β3 ln ℓi + β4 d i + β5 xi + ui .

(2)

(a)

If the true population model is given by equation (1) but we estimate model (2), which problem do our estimators b1 , b2 , and b3 have?

(b)

If the true population model is given by equation (2) but we estimate model (1), which problem do our estimators b1 , b2 , and b3 have?

From this point on, only consider model (2). (c)

Economists often assume that production technologies have constant returns to scale, which means that if a firm would double its usage of both capital and labour, its production would also be expected to double, all else being equal. Show that in our model the constant returns to scale assumption is equivalent to the hypothesis β2 + β3 = 1. (Feel free to ignore retransformation bias for this question; it cancels out anyway.)

(d)

The hypothesis in part (c) cannot be tested directly using the Stata output from estimating the regression model (2). Describe which other regression(s) you would need to run, and how you would use its/their output to test this hypothesis.

Some Stata commands and output follow below and on the next page:

. regress lnprod lncap lnlabour d x Source | SS df MS ---------+---------------------------Model | 175.977585 4 43.9943962 Residual | 30.2371483 25 1.20948593 ---------+---------------------------Total | 206.214733 29 7.11085286

Number of obs F( 4, 25) Prob > F R-squared Adj R-squared Root MSE

= = = = = =

30 36.37 0.0000 0.8534 0.8299 1.0998

--------------------------------------------------------------------lnprod | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+----------------------------------------------------------lncap | .2987886 .1979317 1.51 0.144 -.1088593 .7064366 lnlabour | .7374641 .2159732 3.41 0.002 .292659 1.182269 d | 3.986826 .4456447 8.95 0.000 3.069004 4.904649 x | .472238 .179805 2.63 0.015 .1019227 .8425532 _cons | -5.460113 4.722212 -1.16 0.259 -15.18569 4.265465 ---------------------------------------------------------------------

. predict lnyhat (option xb assumed; fitted values) . generate lnyhat2=lnyhatˆ2 . generate lnyhat3=lnyhatˆ3 . regress lnprod lncap lnlabour d x lnyhat2 lnyhat3 Source | SS df MS ---------+---------------------------Model | 177.168375 6 29.5280625 Residual | 29.0463582 23 1.26288514 ---------+---------------------------Total | 206.214733 29 7.11085286

Number of obs F( 6, 23) Prob > F R-squared Adj R-squared Root MSE

= = = = = =

30 23.38 0.0000 0.8591 0.8224 1.1238

--------------------------------------------------------------------lnprod | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+----------------------------------------------------------lncap | .1592639 4.164654 0.04 0.970 -8.455979 8.774506 lnlabour | .6067685 10.30999 0.06 0.954 -20.72107 21.93461 d | 2.918479 57.0932 0.05 0.960 -115.1878 121.0248 x | .3585001 6.624572 0.05 0.957 -13.34547 14.06247 lnyhat2 | -.0304219 1.064497 -0.03 0.977 -2.232502 2.171658 lnyhat3 | .0019188 .0261343 0.07 0.942 -.052144 .0559817 _cons | -.0038604 135.6833 -0.00 1.000 -280.6861 280.6784 --------------------------------------------------------------------(e)

The F test that compares the fit of the two models in this Stata output has a specific name. Give that name, and describe what it is useful for. Finally, perform the test....