程序代写代做代考 Alastair Hall ECON61001: Semester 1, 2020-21 Econometric Methods

Alastair Hall ECON61001: Semester 1, 2020-21 Econometric Methods
Problem Set for Tutorial 7
In this question, you investigate the properties the first order autocorrelation of a time series under certain assumptions about how it is generated. This analysis involves the large sample behaviour of terms involving sums that start at t = 2 rather than t = 1. In the cases below, this difference is negligible in large samples and so the large sample behaviour of terms involving sums that start at t = 2 is the same as that of the analogous terms with the sums starting at t = 1. You may find it useful to refer back to Tutorial 5.
1. Define ρˆT to be the sample first-order autocorrelation of the weakly stationary process ut
with mean zero that is,1
􏰔Tt=2 utut−1 ρˆT = 􏰔T u2 .
t=1 t
Let {εt}Tt=1 denote a sequence of i.i.d. random variables with mean zero and variance σε2.
(a) Assume that ut = εt. Show that T1/2ρˆT →d N(0,1).
(b) Assume
where |θ| < 1. Show that ρˆT →p θ. (c) Assume Show that ρˆT →p φ/(1 + φ2). (d) Assume S h o w t h a t ρˆ T →p 0 . ut =θut−1 +εt, ut = εt + φεt−1. ut = εt + φεt−2. 1We use the knowledge that the process has mean zero to simplify the formula by not explicitly stating the formula in terms of ut −E[ut]. 1 In this question you must use the results from Question 1 to evaluate a modeling strategy for esti- mation of a time series regression model 2. Consider the time series regression model yt = x′tβ0 + ut where xt does not contain any lagged values of yt. Suppose a researcher estimates this model via OLS and calculates the OLS residuals {et}Tt=1. The researcher is concerned that the errors may be serially correlated and hence that the OLS estimators are inefficient. To this end, she adopts the following strategy. She assumes that the errors satisfy ut =ρut−1 +εt, (1) where εt is defined as in Question 1 and can be assumed independent of xt, and tests H0 : ρ = 0 versus HA : ρ ̸= 0 using the following decision rule: Reject H0 at the 5% significance level if |T 1/2 􏰔Tt=2 etet−1 rT | > 1.96 where rT = 􏰔Tt=1 e2t
If the test does not reject then she will base inference on the OLS estimators using the stan- dard formula for variance estimator, “σˆT2 (X′X)−1”. If the test does reject then she will base inference on a feasible GLS estimator based on the assumption that the errors are generated by (1), and so the variance is estimator is “(X′Σˆ−1X)−1”.
In light of your answers to Question 1, do you think this is a good strategy? (You may assume that if the errors are generated via (1) then T 1/2(βˆF GLS − β0) has the same large sample distribution as is asymptotically equivalent to T1/2(βˆGLS − β0), and that T−1X′Σˆ−1X − T − 1 X ′ Σ − 1 X →p 0 . )
Hint: it can be shown under certain conditions (which you may assume here) that (i) under the conditions of Question 1(a)& (d), T 1/2rT − T 1/2ρˆT →p 0; (ii) under the conditions on ut in parts (b)-(c) the large sample behaviour of T 1/2rT is determined by T 1/2plimρˆT .
3. Consider now the regression model:
yt = β0,1 + β0,2yt−1 + ut, (2)
where
ut = εt + φεt−1, (3)
where φ ̸= 0 and εt is white noise. Let βˆT be the OLS estimator of β0 = (β0,1, β0,2)′ based on (2). Is βˆT consistent for β0?
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4. Consider the time series regression model
yt = x′tβ0 + ut (4)
where xt = (1, zt)′ and zt is a scalar time series variable. Let βˆT be the OLS estimator of β0 based on (4). Assume that: (i) yt is generated by (4); (ii) (yt, zt) is a weakly stationary and w e a k l y d e p e n d e n t t i m e s e r i e s ; ( i i i ) E [ x t x ′ t ] = Q , a p o s i t i v e d e fi n i t e m a t r 􏰔i x ; ( i v ) E [ x t u t ] = 0 ;
(v) T1/2(βˆT − β0) →d N(0,Vsc) where Vsc = Q−1ΩQ−1, Ω = Γ0 + ∞i=1(Γi + Γ′i) and Γi = Cov[xtut, xt−iut−i]. Propose a decision rule to test H0 : β0,2 = 0 vs HA : β0,2 ̸= 0, being sure to define your test statistic in terms of the data.
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