Autocorrelation: Properties, Estimation, and Time-Series Applications

All flashcards

Wiener–Khinchin theorem

The autocorrelation function and power spectral density are Fourier-transform pairs.

Hermitian symmetry

For a complex-valued WSS process, autocorrelation at a lag is the complex conjugate of its value at the opposite lag.

Hassani −1/2 identity

For a nonconstant sample centered by its sample mean, the positive-lag ACFs sum to −1/2 under the common lag-zero denominator convention.

Circular correlation

Correlation computed by an unpadded FFT treats the sequence as wrapping around at its ends.

Newey–West standard errors

Heteroskedasticity-and-autocorrelation-consistent standard errors used for regression inference.

Quiz questions

1. What symmetry does the autocorrelation of a real-valued WSS process satisfy?Answer: It is even, with equal values at opposite lags.
2. What relationship does the Wiener–Khinchin theorem establish?Answer: Autocorrelation and power spectral density are Fourier-transform pairs.
3. For a finite-order moving-average process of order $q$, what is true of its theoretical ACF at lags greater than $q$?Answer: It is zero at all lags greater than $q$.
4. Which test can assess higher-order residual autocorrelation and accommodate lagged dependent variables among regressors?Answer: The Breusch–Godfrey test.
5. For a nonconstant sample centered by its sample mean, what is the sum of all positive-lag sample ACFs under the common lag-zero denominator convention?Answer: It equals $-1/2$.