Resit Examinations 2009/2010: Econometrics Exam at National University of Ireland, Galway, Exams of Econometrics and Mathematical Economics

Download Resit Examinations 2009/2010: Econometrics Exam at National University of Ireland, Galway and more Exams Econometrics and Mathematical Economics in PDF only on Docsity!

Ollscoil na hÉireann, Gaillimh GX_____

National University of Ireland, Galway

Resit Examinations 2009/

Exam Code(s) 3BA1, 3BA5, 3BA6, 4BA4, 4BA8, 1EM1, 1OA1, 3BC1, 4BC2, 4BC3, 4BC4, 4BC5, 1EK3, 1EK2, 1EK3, 3FM Exam(s) B.A., B.A. (ESS), B.A. (PSP), B.A. (Int’l), Erasmus, Occasional, B.Comm., B.Comm. (Language), H.Dip.Econ.Sc. 3rd^ B.Sc.(Fin. Maths & Economics)

Module Code(s) EC Module(s) Econometrics

Paper No. Repeat Paper 1

External Examiner(s) Professor Robert Wright Internal Examiner(s) Professor John McHale Professor Ciaran O’Neill

Instructions: Answer any four questions

Duration 2 hours No. of Pages Department(s) Economics Course Co-ordinator(s) Ciaran O’Neill

Requirements :

Statistical Tables Graph Paper

EC3 6 3 Econometrics Autumn 2010

All questions carry equal marks.

Students answer four questions in 2 hours

1.

a. Briefly explain the principles underlying least squares regression analysis ( 7 marks) b. For a two variable regression model prove that the OLS estimators for the intercept and slope are unbiased and efficient ( 18 marks) 2 a. With examples detail two sources of specification bias in regression analysis and discuss how these might be addressed (7 marks) b. What impact might violation of the normality assumption regarding the error term have on a regression model? Outline two approaches with which you are familiar by which you could examine the normality or otherwise of errors in a regression model.(12 marks) c. In what sense is the confidence interval for an estimated parameter stochastic (6 marks)

3 You are given the following data based on 15 observations:

---

Y = 7. 1913 ; X 1 = 5. 616 ; X 2 = 7. 162 ; Σyi^2 = 4.

Σx1i^2 = 3.9616 Σx2i^2 = 9.1714; Σx1iyi = 4.

Σx2iyi = 5.6391; Σx1ix2i = 4. 9312 ;

(Note, lower case letters denote deviations about the mean)

a. Estimate the intercept and partial slope coefficients (12 marks) b. Test the statistical significance of each slope coefficient using α = 0.05 ( marks) c. Comment on the regression relationship and methods by which it might be improved (5 marks)

4

a. Consider the following regression model: Yi = β 0 + β 1 Xi1 + β 2 Xi2 + β 3 Xi3 + Ui, where Yi is individual income, X 1 an individual’s years in fulltime education, X 2 a dummy variable taking the value one if the individual is female zero otherwise, and X 3 a dummy variable taking the value one if the individual has taken a career break and zero otherwise. Briefly outline the expected signs of each coefficient and the bases for your expectations. How might we test the restriction that gender and having taken career break have no effect on income? (7 marks)

b. Discuss the role of the conventional F test in econometric models. ( marks)

c. What is the difference between a one tailed and two tailed t test on a slope coefficient in an econometric model. What impact might changing from one to the other have on Type 1 and Type 2 errors? (10 marks)

a. Explain what is meant by the term heteroscedasticity as it relates to regression analysis and how its presence may impact on OLS estimators. (8 marks) b. Explain why the visual examination of residuals may be inappropriate for detection of heteroscedasticity.and explain the use of the Park Test in the detection of heteroscedasticity. (10 marks) c. What remedial action might you take in the presence of heteroscedasticity? ( marks)

a. Outline the effects of autocorrelation on least squares estimators. Briefly describe the Durbin-Watson test for autocorrelation and some limitations of the d-statistic. (8 marks) b. In the presence of AR(1) autocorrelation briefly outline the method of estimation that is BLUE. Note you will have to transform the first observation. (10 marks) c. What is meant by multicollinearity? (7 marks)

Formulae Sheet EC363 Econometrics

Two variable model ^ _^ ^ _ β 0 = Y – β 1 X

^ β 1 = Σxiyi Σ xi^2 _ _ xi = (Xi – X) and lower case y = (Yi – Y)

^ ^ Variance of β 0 = Var(β 0 ) = (ΣXi^2 / n Σ xi^2 ). σ^2 (note this involves upper and lower case “x” ^ ^ ^ Standard error β 0 = SE (β 0 ) = √ Var(β 0 ) ^ ^ Variance of β 1 = Var(β 1 ) = σ^2 / Σxi^2 {as before lower case “x” is used to denote deviations} ^ ^ ^ Standard error β 1 = SE (β 1 ) = √ Var(β 1 )

^ σ^2 is estimated by σ^2 = (Σ ei^2 ) / n-

^ Σ ei^2 = Σ(Yi – Yi)^2

r^2 = 1 - Σei^2 / Σyi^2 ^ Σyi^2 = β 1 Σxi^2 + Σei^2

Jarque-Berra test JB = n/6 [S^2 + (K – 3)^2 /4] Where S is skewness and K kurtosis

Forecasting Mean = E(Y│X 0 ) = β 0 + β 1 X 0 _ Var = σ^2 [1/n + (X 0 – X)^2 /Σxi^2 ] ^ Where σ^2 is the variance of Ui (unknown) approximated by σ^2 Confidence interval on forecast

^ ^ ^ β 0 + β 1 X 0 + or - tα/2 SE(Y 0 )

Three variable model ^ ^ ^ Σ ei^2 = Σ(Yi - β 0 - β 1 X1i - β 2 X2i)^2

^ ^ Σ ei^2 = Σyi^2 – β 1 Σx1iyi – β 2 Σ x2iyi

^ _ ^ _ ^ _ β 0 = Y - β 1 X 1 - β 2 X 2

^ β 1 = (Σx1iyi)(Σx2i^2 ) – (Σx2iyi)(Σx1ix2i) (Σx1i^2 )(Σx2i^2 ) - (Σx1ix2i)^2 ^ β 2 = (Σx2iyi)(Σx1i^2 ) – (Σx1iyi)(Σx1ix2i) (Σx1i^2 )(Σx2i^2 ) - (Σx1ix2i)^2 ^ _ _ _ _ Var (β 0 ) = 1/n + X1i^2 (Σx2i^2 ) + X 22 (Σx1i^2 ) – 2 X 1 X 2 (Σx1ix2i). σ^2 (Σx1i^2 )(Σx2i^2 ) - (Σx1ix2i)^2 ^ ^ SE (β 0 ) = √ Var(β 0 ) ^ Var (β 1 ) = Σx2i^2. σ^2 (Σx1i^2 )(Σx2i^2 ) - (Σx1ix2i)^2 ^ ^ SE (β 1 ) = √ Var(β 1 )

^ Var (β 2 ) = Σx1i^2. σ^2 (Σx1i^2 )(Σx2i^2 ) - (Σx1ix2i)^2 ^ ^ SE (β 2 ) = √ Var(β 2 )

^ And we use σ^2 = Σei^2 / n-3 to estimate σ^2

R^2 = ESS/TSS where

TSS = Σyi^2 ^ ^ ESS = β 1 Σx1i yi+ β 2 Σ x2i yi ^ ^ RSS = Σyi^2 - β 1 Σx1i yi - β 2 Σ x2i yi

F test for joint significance

F = (ESS/2) / (RSS/n-k)

_

Adjusted R^2 R^2 = (1-R^2 ) (n-1/n-k)

k = number of estimated parameters including intercept

WALD (Fm,n-k) test of restrictions

F = ( Rur^2 – Rr^2 )/ m ~ Fm, n-k (1 – Rur^2 ) / n-k

Rur^2 = unrestricted model; Rr^2 = restricted model m number of restrictions

For proofs in two variable model _ _ Let wi = xi/Σxi^2 = (X - X) / Σ(X - X)^2