the estimator β^is an unbiased estimator of β with var(β^|X) = σ2(XTX)−1. Moreover, β is the Best Linear Unbiased Estimator (BLUE}

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

 Section 1, we discussed properties of LSE β and residual e = (In − X(XTX)−1XT)y.

Based on the notation defined in Section 2.1, please show that

• the estimator β^is an unbiased estimator of β with var(β^|X) = σ2(XTX)−1. Moreover,

β is the Best Linear Unbiased Estimator (BLUE}

Suppose that {(Yi, Xi) : i = 1, · · , n} is a sequence of independently and identically dis- tributed (i.i.d.) random variables with Yi, Xi ∈ R. Assume that var(Xi) = σ2 . We consider

the simple linear regression model

 

where si i.∼i.d. N (0, σ2) and si is independent of Xi.

• In some applications (e.g., when collecting data), we are not able to precisely measure

Xi,  but  instead,  we  can  only  observe  Xi∗.   It  is  called  mismeasurement.   As  a  result,

we  usually  have  model  (1)  with  Xi  replaced  by  Xi∗

in this situation. Please find the

estimators of βx based on Xi and Xi∗, and denote them by βx and βx, respectively.

• We usually build up the relationship between Xi and Xi∗by the following model:

Xi∗ = Xi + δi,                                                       

 

βx        ω1βx and βx         ω2βx for some non-negative values ω1 and ω2 as n       , where

p

“−→” represents convergence in probability.  Also, you should specify the exact values of

ω1 and ω2.

(2). Suppose that we run 1000 repetitions. Based on your “artificial” data, calculate numerical results for βx, βx, var(βx) and var(βx). Summarize your numerical results as the following table and compare with (a), (b), and (c).

 

• Summarize your findings in (a) - (d).

3. Suppose f (y) is a probability density function (pdf). Let

 where f (r)(y) is the rth derivative of f (y).

 When f is pdf of N (µ, σ2), please find R(f (2)) so that we are able to obtain the bandwidth based on normal scal

• Show that under some conditions,

Which kinds of conditions do we need here? Does standard normal distribution satisfy these conditions?

 Consider the wool prices data set (txt) that reports the wool prices at weekly markets. The response of interest is the log price difference between the price of a particular wool 19

µm (cents per kilogram clean) and the floor wool price (cents per kilogram clean) at markets:

 Fit the data by a simple linear regression model and a polynomial model of order

Give scatterplot of the data and add the two  fitted lines,  one  for simple linear  model  and one for polynomial model. Put clear and proper legends on it.

Fit the data by local constant kernel estimator and local linear kernel estimator. Choose the bandwidths in these two estimators by the CV method. Give scatterplot of the data and add the two fitted lines. Put clear and proper legends on it.

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