Define white noise and explain how its independent form and its normal (Gaussian) form differ.
The idea
White noise is the basic building block of time series models. It is a sequence ε(t) with mean zero, constant finite variance σ², and zero autocorrelation at every nonzero lag. That is all the definition demands, so uncorrelated is the key word, not independent. Independent white noise strengthens it. The draws are statistically independent, so any function of them, such as their squares, is also unpredictable, but no particular distribution is imposed. Gaussian white noise is the case where each draw is also normally distributed. For normal variables, zero correlation does imply independence, so Gaussian white noise is automatically independent white noise, while the reverse need not hold. A series whose squares are autocorrelated, such as returns with volatility clustering, can be white noise yet not independent. Because the terms are uncorrelated, the variance of a sum of white noise terms is the sum of their variances.
A practice question
How does independent white noise differ from Gaussian white noise?
- Independent white noise is correlated across time, while Gaussian white noise is notWhy not: Both forms have zero autocorrelation; independence is a stronger statement than zero correlation, not a weaker one.
- Gaussian white noise additionally requires each draw to be normally distributed - correct
- Gaussian white noise has a nonzero mean, while independent white noise has a zero meanWhy not: Both forms have a zero mean; the distinction is about the distribution and dependence, not the mean.
- Independent white noise must be normal, while Gaussian white noise may take any distributionWhy not: Reverses the two: normality is what Gaussian adds; independent white noise may follow any distribution with finite variance.
Independent white noise needs only independent draws with a common variance and mean zero, whatever the distribution. Gaussian white noise adds normality, and for normal variables zero correlation also implies independence.
A harder one
A fund's residuals have zero mean, constant variance and no autocorrelation, but the squared residuals are significantly autocorrelated. Which statement is correct?
- The residuals are not white noise because the variance of the squares variesWhy not: White noise concerns the residuals' own mean, variance and autocorrelation; predictable squares do not violate those.
- The residuals are independent white noise because they are uncorrelatedWhy not: Uncorrelated does not mean independent: autocorrelated squares show that magnitudes are predictable from the past.
- The residuals are white noise but not independent white noise - correct
- The residuals are Gaussian white noise because their variance is constantWhy not: A constant variance says nothing about the distribution; a Gaussian form also needs normality and would have independent squares.
Zero autocorrelation, zero mean and constant variance make the series white noise. Autocorrelated squares reveal dependence, so the series is not independent, and therefore not Gaussian white noise either.
The hardest tier
Daily shocks are Gaussian white noise with standard deviation 0.02. The probability that the two-day sum e_t + e_(t-1) exceeds 0.04 is closest to:
- 7.9% - correct
- 15.9%Why not: Takes the standard deviation of the sum as 0.04 (adding standard deviations, as if perfectly correlated), giving z = 1.
- 2.3%Why not: Uses the one-day standard deviation 0.02 for the two-day sum, giving z = 2.
- 0.2%Why not: Divides the standard deviation by the square root of 2 (an average) instead of multiplying, giving z = 2.83.
Independence of Gaussian white noise makes the variance of the sum 2 x 0.0004 = 0.0008, so sd = 0.0283. z = 0.04 / 0.0283 = 1.414 and P(sum > 0.04) = 7.86%.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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