Econ4261_Hw4_key

Econ4261_Hw4_key - ECON 4261: Introduction to Econometrics...

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Unformatted text preview: ECON 4261: Introduction to Econometrics Fall 2009 Problem Set 4 Answer Key Exercise 1 Data on a three variable regression Y i = β 1 + β 2 X 2 i + β 3 X 3 i + u i yield the following results: X X = 33 40 20 20 60 X Y = 132 24 92 X ( Y- ¯ Y ) 2 = 150 (i) What is the sample size? (ii) Estimate the regression. (iii) Estimate the standard error of ˆ β 2 and test the hypothesis that β 2 is zero. (iv) Test the same hypothesis by running the restricted regression and exam- ining the difference in the residual sum of squares. Answer: [i] In the regression model we have a constant term and two ex- planatory variables therefore X X matrix is 3 x 3 and given by X X = n ∑ X 2 i ∑ X 3 i ∑ X 2 i ∑ X 2 2 i ∑ X 2 i X 3 i ∑ X 3 i ∑ X 2 i X 3 i ∑ X 2 3 i Therefore (1 , 1) element of X X gives the sample size n = 33. [ii] Writing the regression equation in the deviation form y i = β 2 x 2 i + β 3 x 3 i + u i From the first column of X X matrix, observe that the regressors X 2 and X 3 have zero mean. Therefore, the lower right 2 x 2 sub-matrix of X X is in the deviation form. Since ∑ ( X- ¯ X )( Y- ¯ Y ) = ∑ ( X- ¯ X )( Y ) = ∑ XY , the lower 2 x 1 sub-vector of X Y is also in deviation form. ˆ β = ˆ β 2 ˆ β 3 = 40 20 20 60- 1 24 92 =- . 2 1 . 6 The intercept coefficient is estimated as ˆ β 1 = ¯ Y- ˆ β 2 ¯ X 2- ˆ β 3 ¯ X 3 = ¯ Y = 132 / 33 = 4 1 [iii] In order to estimate the variance of ˆ β 2 first obtain RSS from the model by: ˆ u ˆ u = Y Y- ˆ βX Y = 150-- . 2 1 . 6 40 20 20 60- 1 = 7 . 6 Notice that ∑ ( Y i- ¯ Y ) 2 = Y Y = 150. The estimated var-cov matrix is: var ˆ β 2 ˆ β 3 = ˆ u ˆ u /n- k 40 20 20 60- 1 = . 0076- . 0025- . 0025 . 0051 Then the test-statistic for the null hypothesis β 2 = 0 is given by: t = ˆ β 2 q var ( ˆ β 2 ) =- . 2 √ . 0076 =- 2 . 294 The table values for the critical values for t are t = 2 . 042 at α = 0 . 05 and t = 2 . 750 at α = 0 . 01. Thus, we reject the null hypothesis at 0 . 05 significance level but fail to reject at 0 . 01 significance level....
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This note was uploaded on 02/23/2011 for the course ECON 4261 taught by Professor Staff during the Fall '08 term at Minnesota.

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Econ4261_Hw4_key - ECON 4261: Introduction to Econometrics...

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