If not, formulate the best alternative estimator and comment on why it is BLUE and thus, preferable to the OLS estimator.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

MODERN ECONOMETRICS

INSTRUCTIONS TO CANDIDATES:

Answer FOUR questions

ALL questions carry EQUAL marks

Only University approved calculators are allowed

This paper consists of EIGHT questions on FOUR pages 

1) Assume that the error term assumptions A.1-A.5 hold for an estimator b which is the GLS estimator denoted by bGLS=(X’V-1X)-1X’ V-1y.

a) Explain and prove the properties of unbiased (proposition 1.1) and efficiency that must hold in order for the GLS estimator to be the Best Linear Unbiased Estimator (BLUE).

[Marks 15]

b) Explain and prove the property of consistency such that the vector of regressors X and the error term 𝜀 are not correlated. [Marks 10]

Hint: The assumptions that must be taken into account are the following: A1. 𝑦 = 𝑋𝛽 + 𝜀 (Linearity)

A2. 𝐸(𝜀𝑖|𝛸) = 0 (Strict exogeneity)

A3. The rank of the nxK matrix X is K with probability 1 and n ≥ K (No perfect multicollinearity).

A4. 𝐸(𝜀𝑖2|𝛸) = 𝜎2𝐼𝑛 > 0, 𝑖 = 1, … , 𝑛 and 𝐸(𝜀𝑖𝜀𝑗|𝛸) = 0, for i≠j. A5. 𝜀|𝛸~𝑁(0, 𝜎2𝐼𝑛) (Normality of error term).

[Total Marks 25]

2) Assume that assumption A.4 is violated and the variance of the residual term ε is

Var(ε|X)=σ2V, where V=V(X)≠In is a known positive definite matrix.

a) Discuss whether the OLS estimator is still BLUE. [Marks 5]

b) If not, formulate the best alternative estimator and comment on why it is BLUE and thus, preferable to the OLS estimator. [Tip: You are advised to formulate the new estimator using the same process as the OLS estimator. The mean and variance of the new estimator is needed]. [Marks 20]

[Total Marks 25]

3) The following regression model is to be estimated

𝑌𝑡 = 𝛽0 + 𝛽1𝑋𝑡 + 𝛽2𝑍𝑡 + 𝛽3𝐺𝑡 + 𝑢𝑡

a) Discuss how the Bresuch-Godfrey (LM) test works. How it is different to the Durbin-Watson serial correlation test? [Marks 13]

b) If the Durbin-Watson value of this regression is equal to 2.40 and the Bresuch- Godfrey (LM) test is insignificant when using at least 1 lag in the error term, what would that imply for the regression? What would be the most suitable estimator? [Marks 12]

[Total Marks 25]

4) The following regression model is to be estimated

𝑌𝑡 = 𝛽0 + 𝛽1𝑋𝑡 + 𝛽2𝑍𝑡 + 𝛽3𝐺𝑡 + 𝑢𝑡

a) Discuss how to proceed with estimation when the variance of the residual term is Var(εt)=σ2Χt. What is the most suitable estimator? Provide the formulation of the regression in order to obtain that estimator. [Marks 13]

b) Assume that the Breusch-Pagan-Godfrey test results in insignificant evidence of heteroskedasticity, but White’s test is the only test indicating significant results of heteroskedasticity. What would this imply for this regression? What would be the best way to proceed to estimation? [Tip: It would be useful to mention the hypotheses of those tests.] [Marks 12]

[Total Marks 25]

5) The following regression model is to be estimated

𝑌𝑡 = 𝛽0 + 𝛽1𝑋𝑡 + 𝛽2𝑍𝑡 + 𝛽3𝐺𝑡 + 𝑢𝑡

a) Mention the definition of endogeneity. What is the main difference between the Instrumental Variables (IV) and the Two-Stage Least Squares (2SLS) estimators? (Tip: The formulation of the IV estimator is needed) [Marks 15]

b) Assume that endogeneity emerges for 𝑋𝑡. What actions must be taken into account in order to formulate a consistent estimator? [Marks 5]

c) When is the Two-Stage Least Squares (2SLS) equivalent to the Generalised Method of Moments (GMM) estimator? [Tip: Proof is not needed] [Marks 5]

6) Assume that the following model is estimated by using OLS

𝑌𝑡 = 𝑏0 + 𝑏1𝐶𝑡 + 𝑏2𝐼𝑡 + 𝜀𝑡

Yt= Gross National Income (or Gross Domestic Production)

Ct= Total Consumption

It= Total Investment

a) When is this estimation process considered to produce incorrect estimations (that is, spurious OLS regression)? [Marks 5]

b) What is the definition of a unit root and what is the main problem caused? [Marks 8]

c) Assume that at least one of the above time series is not integrated in levels. How could we identify such outcome? (Tip: Use one of the time series as an example) [Marks 12]

[Total Marks 25]

7) Assume that the following model is given

𝑌𝑡 = 𝑏0 + 𝑏1𝐶𝑡 + 𝜀𝑡

a) Explain fully the concept of co-integration. How is the Engle-Granger approach implied in this case? Make sure to mention the whole process, starting by assuming at least one time series is not stationary. [Marks 13]

b) When is the Johansen approach preferred to Engle-Granger? Discuss how Johansen’s test for co-integration works mentioning the whole process. [Marks 12]

[Total Marks 25]

8) Assume the following model

𝑌𝑡 = 𝛽0 + 𝛽1𝑌𝑡−1 + +𝛽2𝑌𝑡−2 + 𝛾0𝑋𝑡 + 𝛾1 𝑋𝑡−1 + 𝛾2𝑋𝑡−2 + 𝑢𝑡

a) If the time series are I(1) stationary and co-integrated in levels, produce the Error Correction model (ECM) of this equation. Discuss the significance of the coefficients and the error correction mechanism. [Marks 15]

b) When is the model considered to converge to its long run equilibrium? When does a short-run relationship emerge between Y and X? [Marks 5]

c) What happens to the ECM if the there is no I(0) co-integration between Y and X? [Marks 5]

[Total Marks 25]

Attachments:

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