QFC Quantitative Finance Statistical & Mathematical Analysis 2 — Questions and Answers
Question 1: Which test is used to assess whether a time series is stationary by testing for a unit root?
- Jarque-Bera test
- Augmented Dickey-Fuller test (Correct answer)
- Ljung-Box test
- Kolmogorov-Smirnov test
Correct answer: Augmented Dickey-Fuller test
The Augmented Dickey-Fuller (ADF) test checks for the presence of a unit root, with rejection of the null indicating stationarity.
Question 2: In a multivariate normal distribution, if the covariance between two variables is zero, what can be concluded?
- The variables are uncorrelated but may be dependent
- The variables are independent (Correct answer)
- The variables have equal variances
- The correlation coefficient equals one
Correct answer: The variables are independent
For multivariate normal distributions, zero covariance implies statistical independence, unlike in general distributions.
Question 3: What does the Ljung-Box Q-statistic test for in financial time series?
- Normality of residuals
- Presence of autocorrelation in residuals (Correct answer)
- Heteroskedasticity in error terms
- Cointegration between two series
Correct answer: Presence of autocorrelation in residuals
The Ljung-Box test checks whether any of a group of autocorrelations of a time series are different from zero.
Question 4: A portfolio has returns that follow a distribution with excess kurtosis of 3. How does this compare to a normal distribution?
- It has lighter tails than normal
- It has the same tail weight as normal
- It has heavier tails than normal (Correct answer)
- It is perfectly symmetric with no skew
Correct answer: It has heavier tails than normal
Excess kurtosis of 3 means total kurtosis of 6 (since normal has kurtosis 4 in some conventions, or excess kurtosis of 3 means leptokurtic with fatter tails than normal).
Question 5: In OLS regression, what assumption is violated when the error variance changes systematically with an independent variable?
- Multicollinearity
- Autocorrelation
- Heteroskedasticity (Correct answer)
- Endogeneity
Correct answer: Heteroskedasticity
Heteroskedasticity occurs when the variance of the error term is not constant across observations, violating the OLS assumption of homoskedasticity.
Question 6: Which numerical method uses random sampling to estimate mathematical quantities and is widely used in option pricing?
- Newton-Raphson method
- Finite difference method
- Monte Carlo simulation (Correct answer)
- Gaussian quadrature
Correct answer: Monte Carlo simulation
Monte Carlo simulation uses repeated random sampling to compute results, making it particularly effective for pricing path-dependent options.
Question 7: If the characteristic function of a random variable X is φ_X(t) = e^(iμt - σ²t²/2), what distribution does X follow?
- Poisson distribution
- Exponential distribution
- Normal distribution with mean μ and variance σ² (Correct answer)
- Cauchy distribution
Correct answer: Normal distribution with mean μ and variance σ²
The characteristic function e^(iμt - σ²t²/2) is exactly the characteristic function of the normal distribution N(μ, σ²).
QFC Quantitative Finance Statistical & Mathematical Analysis 2
Which test is used to assess whether a time series is stationary by testing for a unit root?