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CALIFORNIA STATE UNIVERSITY, SACRAMENTO Department of Economics
Economics 141/241 Prof. Yang
Answers to Assay #1
Compare and adverse amid the afterward concepts:
Estimator and appraisal An estimator is artlessly a aphorism or blueprint that tells us how to go about ciphering a citizenry quantity, such as citizenry mean. An estimate is artlessly the after bulk taken by an estimator.
Correlation and causation: Alternation measures the bulk of beeline association of two
Variables, i.e. how carefully the two are accompanying to anniversary other. The causation implies a causal accord amid the abased capricious and absolute variable. Two credibility amid the two agreement are: 1) the alternation does not necessarily betoken the causation, and 2) account is adumbrated from a approach or antecedent assay results.
PRF and SRF: PRF specifies the accord (in the population) amid the mean (average) bulk of the depedent capricious (Y) agnate to anniversary bulk of the absolute capricious (X). This accord may be authentic algebraically as E(Y Xi ) = B1 B2Xi. Graphically, the citizenry corruption band is a band that passes through the conditional agency of Y. SRF specifies the accord (in the sample) amid the Yi , estimator of E(Y Xi ), or the estimator of the citizenry codicillary mean, and its primary account X. Algebraically, it may be accounting as YI = b1 b2Xi.
A. Anniversary access represents a collective probability-its gives the anticipation that the accidental capricious
X takes a accustomed value, like 1, and Y takes a accustomed value, like 8.5.
The bordering PDF represents the anticipation that X assumes a accustomed value, like 2,
behindhand of the ethics taken by Y. f(X=2) = 0.40 f(Y=17.5) = 0.28
The codicillary PDF is authentic as f(Y X) = P(Y=y X=x ). It gives the anticipation that
Y takes the bulk of y codicillary on the ability that X has affected the value of x.
One simple adjustment of artful the codicillary PDF is: f(Y X) = f(X,Y) / f(X).
F(Y=11.5 X=2) = 0.28 / 0.40 = 0.70.
E(Y X=2) = S Y f(Y X=2) = 8.5 x 0.10/0.40 11.5 x 0.28/0.40 17.5 x 0.02/0.40
= 8.5 x 0.25 11.5 x 0.7 17.5 x 0.05 = 2.125 8.05 0.875 = 11.05
A. Z = (X – m )/ (s / basis of n) and Z ~ N(0,1)
t = (X – m ) / (S/ sq. basis of n), area S = sample standard deviation. The consistent t capricious authentic aloft follows Students t administration with (n-1) degrees of freedom.
Difference? The agency of the two distributions are zero, so they are same. The only aberration amid them lies in the variance: The about-face of the accepted normal administration is 1, while the about-face of the t-distribution is k/k-2. As the degrees of abandon increases, the t-distribution resembles or approximates the accustomed distribution.
State whether the afterward statements are true, false, or uncertain.
An estimator of a constant is a accidental variable, but the constant is nonrandom.
True. An estimator is a aphorism or blueprint and is a accidental variable; the constant is an appraisal and it is a anchored cardinal or nonrandom.
An aloof estimator of a parameter, say m X, agency that it will consistently be according to
False. An aloof estimator may be or may not be according to the accurate parameter. The
mean or accepted bulk of aloof estimator is according to the accurate parameter. E(m X ) = m X.
C. An able estimator agency an estimator with minimum variance.
Uncertain. To be an able estimator, the estimator needs to accept both the
unbiasedness and minimum variance. Therefore, an able estimator agency more than
estimator with minimum variance.
Refer to Figure 5-4 on folio 132, Essentials of Econometrics.
State with affidavit whether the afterward statements are true, false, or uncertain. Be precise.
The academic absurdity appellation ui and the balance appellation ei beggarly the aforementioned thing.
False. The academic absurdity appellation ui is the absurdity appellation in the PRF and the balance absurdity appellation ei is the absurdity appellation in the SRF and it is the appraisal of ui with a accustomed sample. e = absolute Yi estimated or predicted Y.
A beeline corruption archetypal agency a archetypal beeline in the variables.
False. It is beeline in the parameter.
In the citizenry corruption function, the corruption coefficients Bs are random variables. On the added hand, the corruption coefficients, b1 and b2 in the SRF are papameters.
False. Bs are parameters, while the corruption coefficients in the SRF are accidental variables.
A. Since X ~ N(70,9), Z=(X-m X) / aboveboard root of 9. Z = (75-70) / 3 @ 1.67. Therefore, we appetite to acquisition P(Z > 1.67). For a graphical illustration, accredit to Figure 3-3 (A) on folio 70, Essentials of Econometrics.
B. Since X ~ N(m , s 2), sample beggarly of X ~ N(m X, S2X /n). Accustomed the administration of sample mean, again (sample beggarly of X – m ) / (S / aboveboard basis of N) will chase t- distribution.
For help, see the altercation on t administration in the textbook.
H0: m = 7.5; HA: m 7.5
t = (6.5 7.5) / (2/square basis of 20) = 1/ [2/4.472] = 1/0.447 @ 2.37. Since the degrees of abandon is N-1 = 19, the analytical t bulk for 5% akin of aplomb for the two appendage assay [Note: HA: m 7.5 with two possibilities and appropriately we charge a two-tail test] is 2.093 from the t-table. Since the computed t statistics is beyond than the analytical t value, we adios the absent hypothesis.
Alternatively, one can set up a aplomb breach and see if the accepted value is aural or alfresco the aplomb breach in testing, as is illustrated in Example 4.3 in the Essetnials of Ecoonometrics, p. 108. The arbiter archetype uses a = 1% and this assay catechism asks you to use a =5%. From the Appendix t-table, the analytical t-value for the 19 degrees of abandon is 2.093.
The aplomb breach is: sample beggarly of X tc (S/square basis of n), area S = sample accepted deviation. By bung in after ethics to the aloft aplomb interval, we obtain: 6.5 2.093 (2/sq. basis of 20) = 6.5 0.936. So the aplomb breach is:
5.564 m X 7.436. Since the accepted bulk of accurate beggarly is 7.5, we note that it is alfresco the aplomb interval. Therefore, we adios the hypothesis, the same aftereffect as we acquired from the t-test, which is simpler and accessible to use.
A. Sketch the corruption line.
Interpretation of the intercept. A austere estimation of the ambush is: when retail bulk is according to zero, the boilerplate bulk of coffee burning is 2.691 cups per being per day. But this estimation is absurd. What does the ambush term again represents? It about represents the influences of afar variables in the regression.
C. Slope coefficient: When the retail bulk of coffee increases by one dollar a pound, the coffee burning decreases by 0.4795 pounds on the average. Remember: In corruption analysis, we are analytical the accord amid the codicillary mean of Y and its determinants, say X.
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