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References—785

 

ˆ

 

 

 

 

T – q

ˆ

ˆ ˆ

 

 

 

 

 

 

 

 

 

 

L1

= L1 + ------------------------

D L1

D

 

 

 

 

 

 

 

 

T – q – K

 

 

 

 

(E.25)

 

ˆ

 

 

 

 

T – q

ˆ

ˆ

ˆ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L0

= L0 + ------------------------

D L1

D

 

 

 

 

 

 

 

 

T – q – K

 

 

 

 

 

Kernel Function Properties

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

q

 

ck

 

 

rB

rn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Truncated uniform

 

0

 

0.6611

 

1 § 5

---

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bartlett

 

1

 

1.1447

 

1 § 3

2 § 9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bohman

 

2

 

2.4201

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Daniell

 

2

 

0.4462

 

1 § 5

---

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parzen

 

2

 

2.6614

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parzen-Riesz

 

2

 

1.1340

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parzen-Geometric

 

1

 

1.0000

 

1 § 3

2 § 9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parzen-Cauchy

 

2

 

1.0924

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Quadratic Spectral

 

2

 

1.3221

 

1 § 5

2 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tukey-Hamming

 

2

 

1.6694

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tukey-Hanning

 

2

 

1.7462

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tukey-Parzen

 

2

 

1.8576

 

1 § 5

4 § 25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes: rB = 1 § (2q + 1) is the optimal rate of increase for the LRCOV kernel bandwidth. rn is the optimal rate of increase for the lag selection parameter in the Newey-West (1987) automatic bandwidth selection procedure. The Truncated kernel does not have theoretically proscribed values for ck and rB , but Andrews (1991) reports Monte Carlo simulations that suggest that these values work well. The Daniell kernel value for rB does not follow from the theory since the kernel does not satisfy the conditions of the optimal bandwidth theorems.

References

Andrews, Donald W. K, and J. Christopher Monahan (1992). “An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator,” Econometrica, 60, 953-966.

786—Appendix E. Long-run Covariance Estimation

Andrews, Donald W. K. (1991). “Heteroskedaticity and Autocorrelation Consistent Covariance Matrix Estimation,” Econometrica, 59, 817-858.

den Haan, Wouter J. and Andrew Levin (1997). “A Practitioner’s Guide to Robust Covariance Matrix Estimation,” Chapter 12 in Maddala, G. S. and C. R. Rao (eds.), Handbook of Statistics Vol. 15, Robust Inference, North-Holland: Amsterdam, 291-341.

Gallant, A. Ronald (1987). Nonlinear Statistical Models. New York: John Wiley & Sons.

Hamilton, James D. (1994). Time Series Analysis, Princeton University Press.

Hansen, Bruce E. (1992a). “Consistent Covariance Matrix Estimation for Dependent Heterogeneous Processes,” Econometrica, 60, 967-972.

Hansen, Bruce E. (1992b). “Tests for Parameter Instability in Regressions with I(1) Processes,” Journal of Business and Economic Statistics, 10, 321-335.

Hansen, Lars Peter (1982). “Large Sample Properties of Generalized Method of Moments Estimators,”

Econometrica, 50, 1029-1054.

Kiefer, Nicholas M., and Timothy J. Vogelsang (2002). “Heteroskedasticity-Autocorrelation Robust Standard Errors Using the Bartlett Kernel Without Truncation,” Econometrica, 70, 2093-2095.

Neave, Henry R. (1972). “A Comparison of Lag Window Generators,” Journal of the American Statistical Association, 67, 152-158.

Newey, Whitney K. and Kenneth D. West (1987). “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55, 703-708.

Newey, Whitney K. and Kenneth D. West (1994). “Automatic Lag Length Selection in Covariance Matrix Estimation,” Review of Economic Studies, 61, 631-653.

Park, Joon Y. and Masao Ogaki (1991). “Inferences in Cointegrated Models Using VAR Prewhitening to Estimate Shortrun Dynamics,” Rochester Center for Economic Research Working Paper No. 281.

Parzen, Emanual (1957). “Consistent Estimates of the Spectrum of a Stationary Time Series,” The Annals of Mathematical Statistics, 28, 329-348.

Parzen, Emanuel (1958). “On Asymptotically Efficient Consistent Estimates of the Spectral Density Function of a Stationary Time Series,” Journal of the Royal Statistical Society, B, 20, 303-322.

Parzen, Emanual (1961). “Mathematical Considerations in the Estimation of Spectra,” Technometrics, 3, 167-190.

Parzen, Emanual (1967). “On Empirical Multiple Time Series Analysis,” in Lucien M. Le Cam and Jerzy Neyman (eds.), Proceedings of the Fifth Berkely Symposium on Mathematical Statistics and Probability, 1, 305-340.

White, Halbert (1984). Asymptotic Theory for Econometricians. Orlando: Academic Press.

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