Homework 1: The two-variable linear model Answer key to analytical questions

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1 Econ471: Appled Econometrcs Walter Sosa-Escudero Homework 1: The two-varable lnear model Answer key to analytcal questons I) Excercses 1. Consder the data shown n the next table, on consumpton C and ncome Y for countres A and B (measured n dollars): Country A Country B Obs C Y C Y (a) For country A, compute the sample means and standard devatons for both varables. Compute the covarance and the correlaton coeffcent based on the next auxlar table: Obs C Y (C C) (Y Ȳ ) (C Ĉ)(Y Ŷ ) Note: It s suggested that ths exercse be done n Excel or by hand, just to explore the mechancs of the formulae. (b) Construct a scatter dagram for consumpton and ncome (consumpton n the Y axs and ncome n the X axs). (c) Repeat the exercse for country B. Compare the means and standard devatons. (d) Compare the covarances for both countres. Comment ntutvely.. Usng the results found on Exercse 1 for the data on consumpton and ncome for country A, calculate by hand (wthout a computer) the least-squares estmators of α and β of the lnear consumpton functon C t α + βy t + u t. Gve an economc nterpretaton of the results. 3. Suppose that the data on consumpton for country A s altered n the followng way: 1

2 (a) Observatons on consumpton and ncome are measured n cents nstead of dollars. (b) Observatons on consumpton are measured n cents and observatons on ncome n dollars. (c) Observatons on consumpton are measured n dollars, but ncome s measured n Crowns (1 dollar 7.85 Crowns). (d) To all the observatons on ncome and consumpton the number 10 s added (arbtrarly). (e) Only to the observatons on ncome the number 10 s added. For all the cases, compute the covarance, and the least-squares estmatons of α and β of the lnear model. Comment your fndngs ntutvely. 4. Ths excercse wll formalze the prevous results. Consder the lnear model: Y α + βx + u, 1..., n, where ˆα y ˆβ are the least-squares estmators of α and β. Consder the followng changes ntroduced n the varables of the model. (a) All observatons of X and Y are multpled by a constant k. Let Z kz. Then z Z Z k(z Z) kz Let ˆβ be the OLS estmator usng all varables multpled by a constant. Then ˆβ y kx ky k x x y ˆβ Now ˆα Ȳ ˆβ X kȳ ˆβk X k(ȳ ˆβ X) k ˆα (b) Only the observatons of X are multpled by a constant k. Now ˆβ y kx y x y x k x k 1 k ˆβ And ˆα Ȳ ˆβ X Ȳ 1 k ˆβ k X Ȳ ˆβ X α (c) A constant k s added to each observaton of X. Now Z Z + k, then z Z Z (Z + k Z k) z Then And ˆβ y x y ˆβ ˆα Ȳ ˆβ X Ȳ ˆβ( X + k) Ȳ ˆβ X k ˆβ ˆα k ˆβ Show mathematcally how the orgnal estmators can be altered by these changes,.e., calculate the new least-squares estmators and pont out how these estmators can be obtaned from the orgnal estmators. Gve an economc example where these stuatons can appear. 5. True or false: Frst ndcate whether the followng statements are true or false and then justfy your answer.

3 (a) In the two-varable lnear model f the coeffcent of determnaton R s equal to one, then the relaton between the varables s exact and resduals are all zero. True: R s zero f and only f RSS 0. But RSS e s zero f and only f e 0 for every, so the relaton s exact and all errors are zero. (b) In the two-varable lnear model, f V (Y ) V (X) then the slope n a regresson model of Y on X s equal to the slope n a regresson model of X en Y. True: z Let V (Z) n. The slope coeffcent n the model where Y s the dependent varable and X s the ndependent varable s ˆβ Y X x y and when the dependent varable s X and the ndependent varable s Y : ˆβ XY x y y Hence, when V (Y ) V (X), y, so ˆβ Y X ˆβ XY (c) The fact that R s equal to zero ndcates that varables are unrelated. False: t means that varables are lnearly unrelated. They may be non lnearly related (d) Under the classcal assumptons the least-squares estmator s the best lnear estmator. False: t s the best lnear UNBIASED estmator. (e) The fact that the least-square estmator s unbased s a drect consequence of the homoscedastcty assumpton of the error term. False: the homoscedastcty assumpton s NOT used n the proof of unbasedness (f) A crucal assumpton of the lnear model s that the sum of the resduals s zero. False: e 0 s a consecuence of mnmzng the sum of squared resduals (from the frst order condtons). It s not an assumpton. (g) The fact that resduals n the lnear model estmated by least-squares have zero mean s a consequence of assumng that the expected value of the error term s zero. False: t s just a consequence of the frst order condtons of the OLS mnmzaton problem (h) The assumpton that the error term s normally dstrbuted s necessary to demonstrate that the least-squares estmator s unbased. False: normalty does not play any role n the proof of unbasedness. II) Emprcal Problems Note: for ths sesson you must use Stata. See the attached Computer Handout on detals on how and where to use Stata for ths homework. 1. In ths exercse we wll use real data on consumpton and ncome. The data set (datacons.xls) contans nformaton for four seres: year, rgdpc (real gross domestc product per capta n constant dollars of 1985), cons (partcpaton of consumpton on gdp n %) and pop (populaton 3

4 n thousands). We have 41 annual observatons (1950 to 1990) and the source of nformaton s the Penn World Table. A basc lnear consumpton functon wll be estmated: C t α + βy t + u t t 1950,..., 1990 where C, Y and u represent consumpton, ncome and an error term. (a) Based on the nformaton provded, construct consumpton and ncome varables. Descrbe the exact unts of measure. Descrbe brefly the economc and statstcal characterstcs of the model to estmate. Try to predct the results, sgns of the coeffcents, values, etc. (b) Graph the temporal evoluton of both seres (tme n the X axs and the seres n the Y axs). Descrbe the temporal behavor of each varable. Interpret tendences, cycles and any other relevant aspect. (c) Consderng consumpton n the Y axs and ncome n the X axs, construct a scatterplot. Make a comment. (d) Estmate a basc lnear consumpton model. Interpret n economc and statstcal terms the followng results: estmated coeffcents (consder specally the unts of measure), standard errors, t statstcs, p-values, R, F statstc and any other nformaton that s consdered relevant. (e) Test the hypothess that ncome s not relevant to explan consumpton. Test the sgnfcance of the constant term and gve an economc nterpretaton. Test the hypothess that the ncome coeffcent s equal to one. Gve an economc nterpretaton. (f) Graph the resduals over tme and make a comment.. The data set wages.xls presents nformaton about hourly wages n dollars (wage) and years of educaton (yearse) for 1606 men between 4 and 54 years old. For example, a person wth 1 years of educaton ndcates that the person have fnshed secondary school. The purpose of ths exercse conssts n studyng the relaton between educaton and wages. (a) Construct a scatter dagram between educaton and wages. Make a comment. (b) Estmate a smple model between wage (dependent varable) and yearse (educaton). (c) Interpret carefully the estmated ntercept and slope. (d) Test the hypothess that educaton s not relevant to explan wages. (e) Comment on the R. Explan the meanng of a low R and a t statstc of the educaton so hgh. Accordng to ths result, would you be wllng to say that the estmated model s bad? 4

5 Curostes: The Penn World Table s an nternatonal database frequently used n growth studes and economc development. It s avalable n: A database descrpton s presented n Alan Heston and Robert Summers, The Penn World Table (Mark 5): An Expanded Set of Internatonal Comparsons, Quarterly Journal of Economcs, May 1991, pp The nformaton for the educaton problem comes from a household survey, that s, a large survey collected at some perod for many households. Try to fnd out f there s regular household survey n your home country. It mght provde a very valuable source of nformaton for the emprcal project. 5