Homework 3

(Due 3/5)

HAC

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  1. Prove and explain the following statement in Newey and West (1987, p. 703): While $\widetilde{S}_{T}$ is consistent, it need not be positive semi-definite in any finite sample when $m$ is not zero.

  2. Explain the following statement in New and West (1987, p. 704): Hansen (1982) suggested the use of spectral methods for the estimation of $S_{T},$ motivated by the fact the in the covariance stationary case the limit of $S_{T}$ is $2\pi $ times the spectral density of MATH at frequency zero, where
    MATH
    where
    MATH

  3. Show that
    MATH
    Then the Newey-West estimator is basically the variance of $m^{th}$ sums, MATH Hence the Newey-West estimator can be constructed as:
    MATH
    where MATH is a Bartlett window.

  4. Explain the statement in New and West (1987, p. 704): That $\widehat{S}_{T}$ is positive semi-definite follows from the positive semi-definiteness of the sample autocovariance function.

  5. Suppose $h_{t}$ is a scalar and follows an AR(1) process, i.e.,
    MATH
    where $\varepsilon _{t}$ is $iid$ with mean zero and a constant variance MATH Show that
    MATH

  6. Explain the Assumption (iii) in Theorem 2 in New and West (1987, p. 705). What is a mixing sequence? Why do we need to assume the mixing condition(s)?

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