DOFIN - Bazele econometriei - 2010NR.2

I. (1,5p) Argumentali scurt gi la obiect validitatea urmdtoarelor afirmalii:1. Dacd P(A):0,1 , P(B):0,5 Si A(\ B =Q atunci P(lw B)= 0,0 .

2. DacdP(B):0,1qi AnB=/ atunci f(l1n)=ft.3. Dacd P(B):0,0s , p(.ql B)= 0,5 ei P(,tlE):0,01 atunci r(n1 t)= 0,8080.

4. Dac6, P(A):0,7 Si P(8):0,6 atunci f(l o B)>0,5 .

5. Dacd P(A):0,5 Si P(8):0,6 atunci este posibil ca Ao B = Q.

II. (1,5p) Argumentali scurt qi la obiect rispunsul la urmdtoarele intrebbri:l. Care sunt caracteristicile unui estimator ,,bun" in large sample?2. Care sunt elementele necesare pentru a efectua un test valid de tip Wald?3. Care sunt ipotezele necesare gi suficiente pentru a asigura consistenla estimatorului OLS

pentru un model liniar?Se considerd un model liniar caracterizat prin variabile explicative exogene qi inovaliiautocorelate. Estimatorul OLS este BLUE?Se considerd un model liniar caracterizat prin variabile explicative exogene qi inovaliiheteroskedastice. Este valid testul / construit pebaza erorilor standard clasice OLS?

III. (1,5p) Se considera un vector aleator bidimensional (X.)') care are funclia de densitate de

reparli{ie data de .f (", y)= (2 - x * y) .lro,,l(x)'1r0.,10).

1. Sa se determine funclia de densitate marginala pentru {'2. Sd se determine P(X < 0.5, Y 30.75), precum si P(f <0.75 | X < 0,5) .

IV. (1,5p) Pentru a analiza timpul scurs dintre doi cumpdrdtori succesivi dintr-un magazin se

urllizeazd. modelul statistic (Tr,Tr,....,f,) t.t.a unde Z este o variabil6 aleatoare avAnd funclia de

densitate de repartilie datd de f (x)= j exp(- i)' 1ro,-r("). Se qtie cd Elrl= e qi VARlrf= 0' .

1. Sa se determine estimatorul MLE pentru parametrul d;2. Sa se determine distribulia asimptotici a estimatorului 0rr, .

V. (1,5p)InurmaestimiriiprinOLSamodeluluiliniarderegresie Z,=fo+frX,+FrY,*q,undea[x]=0.0s,vARfxl=0.t , nlrl=0.r,runfrf=o.z , cov(x.r)=0, cov(Y.t):0 au fost

oblinutele rezultatele prezentate in Anexu.l. Comentali rezultatele;

2. Si se determine ElZ,l si VARlrl.

VI. (1,5p) Se considera urmdtorul model macroeconomic simplificat exprimat in formd structurald:

lY, = C, + NX, (t)

lq = Fo + f,.y, + €, (z)

unde I reprezintd devialia PIB-ul fatl de trend, C devialia consumului fala de trend. 1/X

devialia exporlului net fa![ de trend, E[]']= E[C']= f[,VX] :0. t, sunt inovalii sferice cu medie

0 gi variantd 6) .iar C)V(NX,,a,)= 0.

1. Sd se determine forma redusd a modelului:2. Sd se argumenteze o modalitate de estimare consistentd a parametrilor fi.mcjiei de consum

(i.e. ecuatia 2).

4.

Anexd

Deoendent Variable: Z

Method: Least SquaresSample (adjusted): 3 400

Included observations: 398 after adjustments

Variable Coefficient Std' Error t-Statistic Prob'

c -0.00202 0.001333 -1 515556 0 1304

x -0.320708 0.047913 -6.693574 0 0000

Y 0.197357 0.047658 4 141131 0 0000

R-squared 0.'121839 Mean dependent var -0 00178

Adjusted R-squared 0117392 S'D. dependent var 0'028244

S.E. of regression 0.026497 Akaike info criterion -4 416091

Sum squared resid 0.277318 Schwarz criterion -4'386042

Log likelihood 881 .8021 Hannan-Quinn criter' -4 404189

f-statistic 27.40171 Durbin-Watson stat 1 943491

Prob(F-statistic) 0

Wald Test:Equation: OLS

Test Statistic Value df Probability

F-statistic 0.100899 (2' 395) 0.904

Chi-square 0.201798 2 0'904

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std Err'

0.3 + C(2) -0.020708 0.047913

-0.2 + C(3) -0.002643 0 047658

Restrictions are linear in coefficients

Breusch-Godfrey Serial Correlation LM Test

F-statisticObs*R-squared

Coefficient Std

c -0 000524

x -0.053761Y -0.254188

RESID(-I) o.27 5916

RESID(-2) 0.105687

R-squared 0.006937

AdjustedR-squared -0.003171

S.E. of regression 0.026472Sum squared resid 0.275394

Log likelihood 883.1874

F-statistic 0.686311

Prob(F-statistic) 0.601773

1 372622 Prob. F(2,393)2.760884 Prob.Chi-Square(2)

0.25470 2515

Test Equation:Dependent Variable: RESID

Method: Least SquaresDafe. O2lO4l10 Time. 12:27

Sample: 3 400lncluded observationsi 398

Presample missing value lagged residuals set to zero

Vanable Error t-Statlstic Prob.

0.001383 -0.378855 0.7050.135257 -0.397474 0.6912

0.160651 -1.582243 0.11440.166996 1.652232 0.0993

0.'153445 0.688764 0.4914

Mean dependent var '1.17E-18

S.D. dependent var 0.02643

Akaike info criterion -4.413002Schwarz criterion -4.362921

Hannan-Quinn criter. -4.393165

Durbin-Walsonstat 1.992666

Heteroskedasticity Tesl: White

F-statistic 0.760469

Obs-R-squared 3.823456

Scaled explained SS 3.35825

Test Equation:Dependent Variable: RESID^2Method: Least SquaresDale: 02lo4l1o Time: 12:32

Sample: 3 400lncluded observations: 398

Variable Coefficient

aXx2

YY^2

0.00071 I0.000246-0.054677-0.065591-0.0009790.034725

Prob. F(5,392)Prob. Chi-Square(5)Prob. Chi-Square(5)

Std. Error lstatistic Prob.

0.57880.57510.6449

10.65468 0

o 144791 0.885-1.226356 0.2208-1.067318 0.2865-0.579163 0.56280.775832 0.4383

6.75E-05o.0017020.0445850.061454

0.001690.0447 59

R-squared 0.009607

AdjustedR-squared -0.003026

S.E. of regression 0.000933

Sum squared resid 0.000341

Log likelihood 2215.132F-statistic 0.760469

-statrstic) 0.578781

Mean dependent var 0.000697

S.D.dependentvar 0.000932

Akaike info criterion -11.10117

Schwaz criterion -11 04107

Hannan-Quinncriler. -11.07736

Durbin-Watson stat 2.039288

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Series: ResidualsSample 3 400Observations 398

Mean 1.17e-18Median 0.000413Maximum 0.076770Minimum -0.073472

Std. Dev. 0 026430Skewness 0.041635Kudosis 2.783442

Jarque-Bera O.892702Probability 0.639959

o bzs -o bso -o bzs 0.ooo o 025 0 oso 0 o7s

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