Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal value, your decson on the null hypothess and ts economc nterpretaton for each hypothess. Conduct all tests at 5% sgnfcance level. In your numercal calculatons: Show all your computatons, complete the calculatons to receve full ponts. Carry your calculatons up to a mnmum of 3 dgts after the decmal pont. All the dervatons and steps n calculatons should be clearly wrtten n the answers. Answers that appear from nowhere wll receve no ponts. You can use your own calculator. PLEASE, ANSWER INDIVIDUAL SECTIONS OF EACH QUESTION IN THE ORDER THEY ARE ASKED. PLEASE DO NOT WRITE IN THE MARGINS. YOU MAY USE THE BACK OF THE PAGE, DO NOT TEAR OF ANY PAGES. 1. (4 ponts) Please menton how one can obtan the measures below usng the lettered dstances n the graph below. If you need sum of all dstances of the same characterstc as a, you can wrte notaton used n the below graph) a. (All your answers should be n the letter Y ˆ + ˆ 0 1X + X 0 1 % $ Y # X a) Error term: b) Resdual: c) SST: d) SSE: e) SSR: f) g) : h) R : ˆ :
) ( ponts) a. For the model Y ı = 1 + X + u, the OLS estmator of the slope coeffcent 1 s ˆ = (X X)(Y Y ) (X X).. Fnd the expresson for ˆ n terms of u.. What are the assumptons about u ; (whch are also called Gauss Markov assumptons). What s the condton for unbasedness? Show (usng a step by step dervaton) that ths estmator s UNBIASED. b. (8 ponts) For the same model let us assume that s an alternatve estmator for defned as = (1/ ) Y 1 + Y $ n # &. Examne whether ths alternatve estmator s unbased or X 1 X n % not? Follow the same steps as n part (a). c. (5 ponts) If Gauss Markov assumptons hold, whch of the followng enttes wll be smaller and why? Var( ˆ ) versus Var( )
3. (5 ponts each part) Accordng to the theory of demand, the quantty demanded of chcken s a functon of ncome and prce of chcken. Q = 1 + I + 3 P 3 + u The followng EVIEWS output reports the results of the estmaton conducted wth a annual sample for the perod 1960 and 198, where Q s the quantty of chcken, n lbs., I s the real dsposable ncome and P 3 s the average wholesale prce of chcken, $/lb.: Dependent Varable: Q Method: Least Squares Date: 10/31/15 Tme: 08:0 Sample: 1960 198 Included observatons: 3 Varable Coeffcent Std. Error t-statstc Prob. C 34.51561 3.855779 8.951655 0.0000 I 0.014884 0.00193 6.785304 0.0000 P -0.1359 0.11905-1.75115 0.0951 R-squared 0.91081 Mean dependent var 39.66957 Adjusted R-squared 0.901903 S.D. dependent var 7.37950 S.E. of regresson.30939 Akake nfo crteron 4.6381 Sum squared resd 106.6517 Schwarz crteron 4.78099 Log lkelhood -50.7744 F-statstc 10.1340 Durbn-Watson stat 0.43741 Prob(F-statstc) 0.000000 a. Before the estmaton what s the sgn you expect for the slope coeffcents and why? Gve an economc reason. b. What are the estmated values of the slope coeffcents and ther estmated standard errors? 3
c. Wrte the ftted equaton for the demand for chcken. d. What are the coeffcent of determnaton and the Sum of Squared Resduals? Compute the Explaned Sum of Squares. e. What s the value for ˆ σ? Compute. f. Construct a 95% confdence nterval for β 1. Accordng to ths result s t lkely that the regresson lne passes through the orgn ( β 1 = 0) 4
g. Test the followng null hypothess: H o : 3 = 0, aganst the alternatve H : β 0 3. Is the slope coeffcent statstcally sgnfcant? A [HINT: COMPUTE THE TEST STATISTIC, THE CRITICAL REGION AND GIVE YOUR DECISION AND ECONOMICS INTERPRETATION] h. Repeat the test of the above null hypothess for the followng alternatve H : β < 0 3. Does your concluson about the null hypothess change? A. For the followng null hypothess: H o : = 3 = 0, aganst the alternatve H A : at least one s non zero, wrte the restrcted equaton. Test the null hypothess usng the nformaton on the EVIEWS results. 5
j. If the followng equaton s estmated Q = 1 + 3 P 3 + u and Sum of Squared Resduals ( û ) s equal to 94., wrte the restrcton mpled by ths new equaton and the valdty of ths restrcton. 4. (10 ponts) Gven the followng regresson equatons; whch pars equatons can be compared usng R statstcs to choose the best model and why? Explan.. Y t = 1 + X t + u t estmated wth 50 observatons. Y t = 1 + X t +Z + u t estmated wth 5 observatons.. log(y t ) = 1 + log(x t ) + u t estmated wth 50 observatons v. Y t = 1 + X t +(1/ Z ) + u t estmated wth 5 observatons. 6
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4. (30 ponts) FORMULA SHEET ˆ β = n X Y X Y n X ( X ) (X = X )(Y Y ) (X X ) where = 1.n 1 ˆβ = Y X = Var( ˆ β ) 1 Var( ˆ β ) Cov( ˆ β ˆ β ) 1 X X ˆ ο n ( X X ) Y X X Y where = 1.n n X ( X ) = ˆ ο ( X 1 X ) = ˆ = û n K X ˆ ο ( X X ) = t stat ˆ β j β j = seˆ( ˆ β ) j R SSE = = 1 SST SSR SST SSE s sum of squared explaned, SSR s sum of squared resduals R SSR /( n K) = 1 TSS /( n 1) Var ( x y) = Var ( x) + Var ( y) Cov( xy) F stat = SSR R SSR SSR U U /( J) /( n K) e k AIC = ln + n n 8
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