Final review questions
Question 1
1.1
Show that a causal filter $\{a_j\}_{j=0}^{\infty}$ passes an arbitrary quadratic polynomial $m_t = c_0 + c_1 t + c_2 t^2$ if the following three conditions hold:
$$ \sum_{j=0}^{\infty} a_j = 1, \qquad \sum_{j=0}^{\infty} j a_j = 0, \qquad \sum_{j=0}^{\infty} j^2 a_j = 0. $$1.2
Find a causal filter $\{a_0, a_1, \ldots, a_6\}$ with $a_j \neq 0$ for $j=0,\ldots,6$ (and $a_j=0$ otherwise) that passes arbitrary quadratic polynomials and eliminates seasonal components with period $5$.
Question 2
2.1
Let $\{X_t\}$ be an MA$(2)$ process with parameters $\theta_1$, $\theta_2$, and $\sigma^2$,
$$ X_t = Z_t + \theta_1 Z_{t-1} + \theta_2 Z_{t-2}, $$where $\{Z_t\}$ is zero-mean $\mathrm{WN}(\sigma^2)$. Find $\gamma_X(h)$ and $\rho_X(h)$ for $h = 0, \pm 1, \pm 2, \ldots$.
2.2
Find a symmetric filter $\{a_{-3},\ldots,a_3\}$ with $a_j \neq 0$ for $|j|\le 3$ and $a_{-j}=a_j$ that passes arbitrary cubic polynomials and eliminates seasonal components with period $2$ or period $3$.
Hint. The coefficients $a_0,a_1,a_2,a_3$ should satisfy
- $a_0 + 2a_1 + 2a_2 + 2a_3 = 1$
- $a_1 + 4a_2 + 9a_3 = 0$
Question 3
Let $\{X_t\}$ be an ARMA$(1,1)$ process with $\phi=\tfrac12$, $\theta=\tfrac12$, and $\sigma^2=1$:
$$ X_t - \tfrac12 X_{t-1} = Z_t + \tfrac12 Z_{t-1}, \qquad \{Z_t\} \sim \mathrm{WN}(0,1). $$The autocovariance function is
$$ \begin{array}{l|l} \gamma_X(0) = \sigma^2 \left( 1 + \frac{(\theta + \phi)^2}{1 - \phi^2} \right) & \text{with } \phi = \tfrac12,\; \theta = \tfrac12,\; \sigma^2 = 1 \\[4pt] \gamma_X(1) = \sigma^2 \left( \theta + \frac{(\theta + \phi)\phi}{1 - \phi^2} \right) & \gamma_X(0) = \tfrac{7}{3} \\[4pt] \gamma_X(h) = \phi^{h-1} \gamma_X(1),\quad h \ge 2 & \gamma_X(1) = \tfrac{5}{3} \\[4pt] & \gamma_X(h) = \left(\tfrac12\right)^{h-1} \left(\tfrac{5}{3}\right) \end{array} $$3.1
Give $P(X_2 \mid X_1)$ (best linear predictor of $X_2$ from $X_1$, also written $P_1 X_2$). Compute the mean squared error.
3.2
Find $P(X_3 \mid X_2, X_1)$ and its MSE.
3.3
Compute $P(X_n \mid X_2, X_1)$ and the minimum MSE. Find the limit of this MSE as $n \to \infty$.
3.4
Let $X_t = Z_t + Z_{t-1} + Z_{t-2}$ with $\{Z_t\} \sim \mathrm{WN}(0,\sigma^2)$. Compute $P(X_3 \mid X_2, X_1)$ and $P(X_4 \mid X_2, X_1)$.
3.5
Let $d_1, d_2$ be distinct positive integers and let $\{s_t^{(d_1)}\}$, $\{s_t^{(d_2)}\}$ be seasonal components with periods $d_1$ and $d_2$. Consider $X_t = s_t^{(d_1)} + s_t^{(d_2)}$. True or false (justify): $\nabla_d X_t$ with $d = d_1 d_2$ eliminates both seasonal components.
3.6
Let $X_t = A(-1)^t$ where $A$ is a zero-mean random variable with $\mathrm{Var}(A)=\sigma^2$. Find the best linear predictor of $X_2$ based on $X_1$ and the MMSE.
Question 4
4.1
Suppose $\{X_t\}$ has two seasonal components, of periods $12$ and $28$. What is the smallest $d$ such that $\{\nabla_d X_t\}$ has no seasonal components? (Answer given: $84$—explain why.)
More generally, if $\{X_t\}$ has $k$ seasonal components with periods $d_1,\ldots,d_k$, the smallest such $d$ is $\mathrm{lcm}(d_1,\ldots,d_k)$. Justify using $\nabla_d X_t = X_t - X_{t-d}$ and periodicity when $X_t$ has seasonality of period $d$.
4.2
Find a causal filter with at most three nonzero coefficients such that the filtered output (input $X_t$) has neither trend nor seasonal components.
4.3
Consider the moving-average smoother of order $1$ applied to $\{Z_t\}$:
$$ X_t = \tfrac13 (Z_{t-1} + Z_t + Z_{t+1}). $$Find the ACVF $\gamma_X(h)$ for all lags $h$.
4.4
Let $\{s_t\}$ and $\{r_t\}$ be seasonal components with periods $a$ and $b$, where $a,b$ are distinct positive integers with no common prime factors. Let
$$ X_t = s_t r_t + Y_t, \qquad Y_t = \phi Y_{t-1} + Z_t $$be AR$(1)$ with $|\phi|<1$ and $\{Z_t\} \sim \mathrm{WN}(0,\sigma^2)$. Let $d = ab$. Is $\nabla_d X_t$ stationary? If yes, find its ACF; if not, explain why.