##### Document information

*Ollscoil na hÉireann, Gaillimh GX_____
National University of Ireland, Galway
*

**Examinations 2010/2011
**

**Exam Code(s) **3BA1, 3BA5, 3BA6, 4BA4, 4BA8, 1EM1, 1OA1, 3BC1,
4BC2, 4BC3, 4BC4, 4BC5, 1EK3, 1EK2, 1EK3, 3FM1

**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) **EC422
**Module(s) **Applied Econometrics

Paper No. Repeat Paper 1

External Examiner(s) Dr Pat McGregor 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

**EC422 Applied Econometrics Resit 2011
All questions carry equal marks.
Students answer four questions in 2 hours
**

1.

a. Briefly detail the steps involved in the conduct of an econometric study (8 marks)

b. Outline the principles underlying ordinary least squares regression analysis (9 marks)

c. Distinguish between the coefficient of multiple determination and the adjusted coefficient of multiple determination. Which would use when assessing a regression function and why? (8 marks)

2 a. Give an account of the desirable properties of an estimator (7 marks) b. Construct the 95% confidence intervals for the predicted value of Y in the

following regression function when X = 262.5 and when X = 345. (10 marks)

^ Y = 7.6182 + 0.0814X1i

_ ^ Where n = 10, X = 262.5, σ2 = 6.4864 and Σxi2 = 51562

c. Interpret and comment on the confidence interval (8 marks)

3

You are given the following data based on 15 observations: _ _ _ Y = 0.2033; X1 = 1.2873; X2 = 8.0; Σyi2 = 0.016353

Σx1i2 = 0.359609; Σx2i2 = 280; Σx1iyi = 0.066196

Σx2iyi = 1.60400; Σx1ix2i = 9.82000

(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 (8

marks) c. Comment on the regression relationship (5 marks)

4 a. What is multi-collinearity and what are its consequences for OLS

estimators (12 marks)

b. Detail how you ascertain whether a model suffered from multi-collinearity (7 marks)

c. With use of examples briefly detail common sources of multi-collinearity (6marks)

5 a. Briefly discuss what is meant by “under” and “over” estimation in

regression analysis and outline the impact of each on OLS estimates (12 marks)

b. Detail the steps involved in the conduct of the WALD test (6 marks) c. What, if any, impact will errors in measurement with respect to the

dependent or independent variable have on OLS estimators. (7 marks)

6 a. What is heteroscedasticity and what are its consequences for OLS

estimators (8 marks) b. Outline the method of weighted least squares as a means of addressing

heteroscedasticity (8 marks) c. What is meant by autocorrelation, and how would you test for it? (9

marks)

**Formulae Sheet EC422 Econometrics
**

**Two variable model
**^ _ ^ _
β0 = Y – β1 X

^

β1 = Σxiyi Σ xi2

_ _ xi = (Xi – X) and lower case y = (Yi – Y)

^ ^

Variance of β0 = Var(β0) = (ΣXi2 / n Σ xi2) . σ2 (note this involves upper and lower case “x”

^ ^ ^

Standard error β0 = SE (β0) = √ Var(β0) ^ ^

Variance of β1 = Var(β1) = σ2/ Σxi2 {as before lower case “x” is used to denote deviations}

^ ^ ^

Standard error β1 = SE (β1) = √ Var(β1)

^

σ2 is estimated by σ2 = (Σ ei2) / n-2

^

Σ ei2 = Σ(Yi – Yi)2

r2 = 1 - Σei2 / Σyi2

^

Σyi2 = β1Σxi2 + Σei2

**Jarque-Berra test
**JB = n/6 [S2 + (K – 3)2 /4]
Where S is skewness and K kurtosis

**Forecasting
**Mean = E(Y│X0) = β0 + β1X0

_ Var = σ2 [1/n + (X0 – X)2/Σxi2]

^

Where σ2 is the variance of Ui (unknown) approximated by σ2

Confidence interval on forecast ^ ^ ^

β0 + β1X0 + or - tα/2 SE(Y0)

**Three variable model
**^ ^ ^

Σ ei2 = Σ(Yi - β0 - β1 X1i - β2X2i)2

^ ^

Σ ei2 = Σyi2 – β1Σx1iyi – β2Σ x2iyi

^ ˉ_ ^ ˉ_ ^ ˉ _

β0 = Y - β1X1 - β2 X2

^

β1 = (Σx1iyi)(Σx2i2 ) – (Σx2iyi)(Σx1ix2i) (Σx1i2 )(Σx2i2 ) - (Σx1ix2i)2

^

β2 = (Σx2iyi)(Σx1i2 ) – (Σx1iyi)(Σx1ix2i) (Σx1i2 )(Σx2i2 ) - (Σx1ix2i)2

^ _ _ _ _

Var (β0) = 1/n + X1i2 (Σx2i2 ) + X22(Σx1i2 ) – 2 X1X2 (Σx1ix2i) . σ2

(Σx1i2 )(Σx2i2 ) - (Σx1ix2i)2

^ ^

SE (β0) = √ Var(β0) ^

Var (β1) = Σx2i2 . σ2

(Σx1i2 )(Σx2i2 ) - (Σx1ix2i)2

^ ^

SE (β1) = √ Var(β1)

^

Var (β2) = Σx1i2 . σ2

(Σx1i2 )(Σx2i2 ) - (Σx1ix2i)2

^ ^

SE (β2) = √ Var(β2)

^

And we use σ2 = Σei2/ n-3 to estimate σ2

R2 = ESS/TSS where

TSS = Σyi2

^ ^

ESS = β1Σx1i yi+ β2Σ x2i yi ^ ^

RSS = Σyi2 - β1Σx1i yi - β2Σ x2i yi

F test for joint significance

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

_ Adjusted R2 R2 = (1-R2) (n-1/n-k)

k = number of estimated parameters including intercept

**WALD (Fm,n-k) test of restrictions
**

F = ( Rur2 – Rr2 )/ m ~ Fm, n-k (1 – Rur2 ) / n-k

Rur2 = unrestricted model; Rr2 = restricted model m number of restrictions

**Chow Test
**F = (RSSR – RSSUR)/K

RSSUR/n1 + n2 – 2K

Where RSSR is the residual sum of squares from the restricted model, RSSUR the residual sum of squares from the unrestricted model, K the number of estimated parameters

**Ramsey reset test
**FM;N,K = (RSSR – RSSUR)/M

RSSUR/ N-K Alternatively

FM,N-K = ( Rur2 – Rr2 )/ m ~ Fm, n-k (1 – Rur2 ) / n-k

Where M is the number of restrictions, k the number of estimated parameters

**For proofs in two variable model
**_ _

Let wi = xi/Σxi2 = (X - X) / Σ(X - X)2