18 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.
Q1 MediumWeak (covariance) stationarity requires:
The full joint distribution to be shift-invariant
Constant mean and variance, and autocovariance depending only on the lag
Independent observations
A zero mean
Q2 MediumAn AR(\(p\)) process is stationary if and only if:
All \(|\phi_i|\lt 1\)
The roots of \(\phi(z)=0\) lie strictly outside the unit circle
The roots of \(\theta(z)=0\) lie inside the unit circle
The mean is zero
Q3 EasyThe \(d\) in ARIMA(\(p,d,q\)) specifies:
The number of AR lags
The number of times the series is differenced to reach stationarity
The number of MA lags
The forecast horizon
Q4 MediumVolatility clustering shows up statistically as:
Autocorrelated returns
Autocorrelated squared (or absolute) returns despite uncorrelated returns
A linear trend in prices
Constant conditional variance
Q5 MediumIn the Kalman filter, the predict step changes the state covariance by:
Shrinking it by the measurement
Inflating it by the process noise \(Q\)
Setting it to zero
Leaving it unchanged
Q6 MediumTwo \(I(1)\) series are cointegrated if:
Both have unit roots
Some linear combination of them is stationary (\(I(0)\)
They are positively correlated
Their VAR is stationary in levels
Q7 MediumFor a stationary AR(1) with parameter \(\phi\), the autocorrelation at lag \(h\) is:
\(\phi/h\)
\(\phi^{|h|}\)
\(1/\phi^h\)
0 for \(h\ge1\)
Q8 MediumThe Yule-Walker equations relate:
MA coefficients to the PACF
AR coefficients to the autocorrelations via \(\rho(k)=\sum_i\phi_i\rho(k-i)\)
The mean to the variance
Residuals to the forecast
Q9 MediumFor a stationary AR(1), as the forecast horizon \(h\to\infty\), the forecast-error variance:
Grows without bound
Approaches the unconditional variance \(\sigma^2/(1-\phi^2)\)
Shrinks to zero
Stays equal to \(\sigma^2\)
Q10 MediumFor a covariance-stationary GARCH(1,1), the unconditional variance is:
\(\omega\)
\(\omega/(1-\alpha-\beta)\)
\(\alpha+\beta\)
\(\omega/(\alpha+\beta)\)
Q11 MediumThe Kalman gain \(K_t\) is large when:
Measurement noise \(R\) is large
The predicted state uncertainty is large relative to measurement noise
The state is perfectly known
The process noise \(Q\) is zero
Q12 MediumA regression of one independent random walk on another typically shows:
Low \(R^2\) and an insignificant slope
High \(R^2\) and a significant-looking slope despite no real relationship
Stationary residuals
A negative \(R^2\)
Q13 MediumAn AR(2) process is best identified by:
An ACF that cuts off after lag 2
A PACF that cuts off after lag 2
Both ACF and PACF cutting off after lag 2
Neither function showing structure
Q14 MediumA stationary AR(1) can equivalently be written as:
A finite MA(\(1\)
An infinite, convergent MA(\(\infty\)) of past shocks
A random walk
A non-invertible MA
Q15 MediumOver-differencing a series (taking \(d\) larger than needed) typically:
Improves stationarity for free
Introduces a non-invertible MA unit root and inflates variance
Removes all autocorrelation
Has no effect
Q16 MediumA GARCH(1,1) fit gives \(\alpha+\beta=0.99\). This means:
Volatility shocks decay almost instantly
Volatility is highly persistent, with a long half-life
The mean return is 0.99
The model is misspecified
Q17 MediumFor a linear-Gaussian model, the Kalman filter produces:
A biased estimate
The exact Bayesian posterior and the minimum-MSE estimate
Only a point estimate with no uncertainty
The smoothed estimate using future data
Q18 MediumYou estimate a pair's hedge ratio on the full history, then backtest trading the resulting spread over that same history. This is:
A valid out-of-sample test
Look-ahead bias: the ratio used future data relative to early trades
Regime change
Survivorship bias