(a) Set up the least squares estimation procedure for this problem, which will consist in minimizing the sum of squared residuals. 2 t.

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1 Insrucions: The goal of he problem se is o undersand wha you are doing raher han jus geing he correc resul. Please show your work clearly and nealy. No credi will be given o lae homework, regardless of he excuse. Please wrie your answers in he space provided. Analyical Problems (100 poins) 1. [20 poins] Suppose you have a sample of T observaions on he variable Y and ha you wan o calculae he sample mean. Unforunaely, a memory lapse has caused you o forge he formula for he sample mean. Being an asue and resourceful economerician, you realize ha his calculaion can be inerpreed as esimaing he following regression Y = µ + ε (a) Se up he leas squares esimaion procedure for his problem, which will consis in minimizing he sum of squared residuals min and compue he firs order condiion. Express he formula for he esimaor. µ T = 1 ε 2 (b) Alernaively, calculae by he mehod of momens, he esimaor for µ by realizing ha you only need one momen condiion, namely E(ε ) = 0. 1

2 2. [30 poins] An economic model relaing food expendiures (Y) and income level (X) of households posulaes ha his relaionship can be modeled as, Y = log( βx ) In order o calculae his empirical relaionship for he U.S. economy, you collec a sample of T individual s food expendiures and income levels. (a) Based on he above heoreical model, se up he corresponding saisical model ha would allow you o compue β from he sample of daa you colleced. Hin: ake ino consideraion ha he relaionship will no hold exacly for he daa due o sampling errors, approximaions errors, or oher sources. (b) Using he Leas Squares principle, derive he esimaor of β from your saisical model. Hin: Depending on he way you se up he saisical model, you may have a nonlinear regression problem. Worry no because his does no make he compuaions any harder. (c) Use he Mehod of Momens o compue he esimaor for β. (d) Being a clever economerician, you suddenly realize ha his economic relaionship could be modified o look like a linear regression model. Wrie down he linear regression ha would allow you o compue an esimae of β wih he convenional, leas squares esimaor. 2

3 (e) Commen on he economic appropriaeness of he economic model posulaed above by discussing he properies of he elasiciy of food expendiures o income. Hin:The elasiciy in quesion is defined as: d[logy ] e = = d[log X ] dy dx X Y Hins: For any W, and Z, log(wz) = log(w) + log(z), and d(log[ f ( s)] = ds f '( s) f ( s) 3. [25 poins] Suppose a sample of daa of size T was generaed by he model: Y = α + βx + ε wih he usual assumpions. Now, suppose ha a researcher execues he following ransformaions of he daa: z = y y w = x x where y and x denoe he sample means of Y and X respecively. Wih hese variables, he researcher esimaes he following regression Z = γ + δw + u (a) Use he normal equaions o show ha ˆ γ = 0 and ˆ δ = ˆ β 3

4 (b) Suppose X = {1,2,3,4,5} and Y = { 5,5,9,9,12}. Use wo graphs o represen he regression lines corresponding o he model ha relaes Y and X, and he model ha relaes Z and W. Observing wha he wo graphs look like and he resuls in (a), explain in a few words wha is going on and wheher i makes a difference or no o esimae he model wih he consruced variables Z and W. (c) Do you expec ha he fi achieved by he regression of Z on W will be beer/he same/worse han he fi achieved in he regression of Y on X? Explain why. 4. [25 poins] The following quesions are based on he following oupu from EViews Regression 1 Dependen Variable: CRIME_RATE Mehod: Leas Squares Dae: 02/01/01 Time: 13:59 Sample(adjused): 1 90 Included observaions: 87 Excluded observaions: 3 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C UR R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression F-saisic Sum squared resid Prob(F-saisic)

5 Regression 2 Dependen Variable: CRIME_RATE Mehod: Leas Squares Dae: 02/01/01 Time: 14:01 Sample(adjused): 1 90 Included observaions: 87 Excluded observaions: 3 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C HSPERCENT R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression F-saisic Sum squared resid Prob(F-saisic) The daa used in hese regressions consiss on couny-level daa for 87 counies sampled randomly across all saes in The variable CRIME_RATE is he oal number of crimes divided by he oal populaion in ha couny and convered o a percenage; he variable UR is he couny s unemploymen rae in percen, he variable HSPERCENT is he percenage of he populaion ha finished high-school. (a) Fill he missing cells and compue he mean of he independen variables. (b) Using regression 1, inerpre he effecs ha an increase of 1% in he unemploymen rae has on he crime rae. Similarly, using regression 2, explain wha would happen o he crime rae if he percen of people ha finished high school wen up 5 percenage poins. (c) From a policy poin of view, wha acions would mos effecively reduce he crime rae? (d) In one paragraph, prepare an execuive summary wih your recommendaions o figh crime based on hese regressions. Adjuss your commens o he evidence conained in he regression oupu only. 5

6 EViews Exercise (50 poins) Use EViews o answer he following quesion. I is ypically argued ha high levels of unemploymen benefis help explain why unemploymen raes in Europe are higher han in he U.S. Below are he average unemploymen raes of OECD counries for he period ; he benefi replacemen raes (i.e., percenage of your salary paid by unemploymen benefis); and he benefi duraion (number of years of paid benefis). Counry Unemploymen Rae Benefis (%) Duraion Ausria Belgium Denmark Finland France Germany Ireland Ialy Neherlands Norway Porugal Spain Sweden Swizerland U.K Canada U.S Japan Ausralia New Zealand These daa are conained in he file ps3.xls, which is in EXCEL forma. Answer he following quesions: (a) Tes he asserion ha he U.S. unemploymen rae is lower han in Europe. Find ou he average Unemploymen rae in Europe (excluding Canada, U.S., Japan, Ausralia and New Zealand) and es H 0 : Average unemploymen rae in Europe = Unemploymen rae in he U.S. a a 90\% confidence level. (b) Wha are he basic saisics and correlaions beween UR and he Benefi Level BL? UR and Benefi Duraion BR? BL and BD? Wha do you conclude from his exploraory analysis? (c) Plo he daa: Do scaer plos and fied regression lines for: UR and BL;UR and BD; and BL and BD. (d) Regress UR on BD. Wha can you say abou he relaionship beween hem? (e) Regress UR on BL. How do his resuls compare o (d)? (f) Regress BD on BL. Wha is he relaionship? (g) Some praciioners prefer o use he logarihm of UR in heir regressions. Do your resuls for (d) and (e) change when you ransform UR o log(ur)? (h) How do your answers change for counries wih UR<8%? (i) Prepare an execuive summary based on he previous empirical analysis (1/2 page maximum) wih your recommendaions o he Presiden of a counry wih 15% Unemploymen, 90% of Benefi Levels ha can las up o 5 years. The goal is o reach a 10% Unemploymen rae. 6