Section A Answer ALL SEVEN questions in section A.
1. Each part of the following is either a true or false statement. If you think Part n is True, your answer should be: ”(n) T” and if you think it is false, ”(n) F”.
Consider the linear model with k regressors, including a constant y = XJβ + ε,
where it is known that plim 1 XJε /= 0 (n denotes sample size). Suppose the columns of the matrix Z contain m > k instruments, including a constant, such that
plim 1 ZJε = 0, plim 1 ZJX = QZX, a matrix of rank k, and plim 1 ZJZ = QZZ, a matrix of rank m. Let Xˆ and yˆ denote the fitted values from regressing the columns of X and y
respectively on Z. Which of the following will give consistent estimates of β?
(a) (Xˆ JXˆ )−1Xˆ Jy. (1 mark)
(b) (XJXˆ )−1XJyˆ. (1 mark)
(c) (XJX)−1XJy. (1 mark)
(d) (ZJX)−1ZJy. (1 mark)
(e) (ZJX)−1XJy. (1 mark)
2. Each part of the following is either a true or false statement. If you think Part n is True, your answer should be: ”(n) T” and if you think it is false, ”(n) F”.
A researcher estimates the following dynamic regression using annual data, where yt is the log demand for a certain good and xt is its log price, which we take to be exogenous.
yt = 117.7 + 0.373yt−1 − 0.156xt − 0.062xt−1 + et.(50.284) (0.177) (0.019) (0.029)n = 46(Standard errors in parentheses.)
(a) A t-test that the variable xt should be omitted from the regression would be rejected at the 5% level. (1 mark)
(b) If xt were to fall by one unit we would expect yt to fall by 0.156. (1 mark)
(c) A small coefficient for yt−1 means that it takes longer for the system to adjust to a new equilibrium value. (1 mark)
(d) The estimated long run elasticity of demand is approximately 0.218. (1 mark)
(e) OLS estimates of this regression are inconsistent if the disturbance term is a first-order moving average process. (1 mark)
3. Suppose we are trying to estimate the linear relationship yi = xJiβ + σϵi, where
ϵi ∼ N (0, 1). Our sample has been truncated from below, so that it contains only
individuals who report a positive value for the dependent variable, yi > 0. Explain, using diagrams or otherwise, why ordinary least squares estimates of β will not be consistent. Outline a method that would produce consistent results. (6 marks)
4. In order to begin to model a quarterly univariate time series, the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) are calculated. The following results are obtained.
Lag SACF SPACF
1 0.832 0.832
2 0.694 -0.066
3 0.519 -0.019
4 0.372 -0.083
5 0.236 0.017
6 0.176 -0.134
7 0.112 -0.104
8 0.095 -0.030
Briefly explain how the sample partial autocorrelation function (SPACF) is calculated and discuss what type of time series process would fit the table. (6 marks)
5. A researcher is interested in the relationship between the elements of an m × 1
vector, Yt, using the equivalent expressions
Yt = α + Φ1Yt−1 + Φ2Yt−2 + st,
∆Yt = Π(Yt−1 − µ) + Γ1∆Yt−1 + st, t = 3, . . . , n,
where ∆Yt = Yt − Yt−1 and st ∼ IID(0, Ω). Show how the parameters µ, Π and Γ1
relate to the parameters α, Φ1 and Φ2. (6 marks)
6. Define what it means for a time series process, yt, to be stationary. Under what conditions is an autoregressive moving average (ARMA) process stationary? Show that the following ARMA (2,1) process is stationary: yt = yt−1 − 0.21yt−2 + εt + 0.7εt−1, t ∈ (−∞, +∞) where εt is a white noise process with variance σ2. (6 marks)
7. Suppose that an econometrician is interested in the relationship between two variables, y1t and y2t using a sample t = 1, . . . , n. They want to estimate the following simultaneous regressions model
y1t = γ12y2t + β1z1t + ϵ1t, (1)
y2t = γ21y1t + β2z2t + ϵ2t, (2)
where z1t and z2t are exogenous explanatory variables, γ12, γ21 0 and s = (ϵ1t, ϵ2t)J
contains unobserved disturbances with E(s) = 0, Var(s) = Ω. Explain why ordinary least squares estimates of Equation (1) and Equation (2) will not be consistent.
Propose a method to obtain consistent estimates of γ12 and β1 in Equation (1), taking care to state any conditions and assumptions required. (6 marks)
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