Econ4261_Hw4_key

# Econ4261_Hw4_key - ECON 4261: Introduction to Econometrics...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 02/23/2011 for the course ECON 4261 taught by Professor Staff during the Fall '08 term at Minnesota.

### Page1 / 6

Econ4261_Hw4_key - ECON 4261: Introduction to Econometrics...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online