Suppose that Zt = (at + at−1 + at−3)/3
(a) Show that Zt is weakly stationary.
(b) Find ρk, k = 0, 1, 2, 3, ...
(c) Find Var 
1
5
P5
t=1 Zt

.

### (a) Show that \( Z_t = \frac{1}{3}(a_t + a_{t-1} + a_{t-3}) \) is weakly stationary

To show that \( Z_t \) is weakly stationary, we must prove that its mean is constant over time and its autocovariance depends only on the lag \( k \), not on \( t \). This involves the following steps:

1. **Mean**:  
   Let the process \( a_t \) be a weakly stationary process, meaning:
   \[
   E(a_t) = \mu \quad \text{and} \quad \text{Cov}(a_t, a_{t-k}) = \gamma_a(k)
   \]
   where \( \mu \) is the mean and \( \gamma_a(k) \) is the autocovariance function of \( a_t \). Now, let's compute the mean of \( Z_t \):
   \[
   E(Z_t) = E\left(\frac{1}{3}(a_t + a_{t-1} + a_{t-3})\right)
   \]
   Using the linearity of expectation:
   \[
   E(Z_t) = \frac{1}{3} \left(E(a_t) + E(a_{t-1}) + E(a_{t-3})\right) = \frac{1}{3} (\mu + \mu + \mu) = \mu
   \]
   Hence, the mean of \( Z_t \) is constant and does not depend on \( t \).

2. **Autocovariance**:  
   Next, let's compute the autocovariance \( \gamma_Z(k) = \text{Cov}(Z_t, Z_{t-k}) \). For \( Z_t = \frac{1}{3}(a_t + a_{t-1} + a_{t-3}) \), we have:
   \[
   \gamma_Z(k) = \text{Cov}\left(\frac{1}{3}(a_t + a_{t-1} + a_{t-3}), \frac{1}{3}(a_{t-k} + a_{t-k-1} + a_{t-k-3})\right)
   \]
   Using the bilinearity of covariance:
   \[
   \gamma_Z(k) = \frac{1}{9} \sum_{i=0,1,3} \sum_{j=0,1,3} \text{Cov}(a_{t-i}, a_{t-k-j}) = \frac{1}{9} \sum_{i=0,1,3} \sum_{j=0,1,3} \gamma_a(k + i - j)
   \]
   Since \( \gamma_a(k) \) only depends on the difference between the time indices, \( \gamma_Z(k) \) will only depend on \( k \), not \( t \), confirming that \( Z_t \) is weakly stationary.

### (b) Find \( \rho_k \) (the autocorrelation function)

The autocorrelation function \( \rho_k \) is given by:
\[
\rho_k = \frac{\gamma_Z(k)}{\gamma_Z(0)}
\]
From the expression derived earlier for \( \gamma_Z(k) \):
\[
\gamma_Z(k) = \frac{1}{9} \sum_{i=0,1,3} \sum_{j=0,1,3} \gamma_a(k + i - j)
\]
Now, we need to find \( \gamma_Z(0) \):
\[
\gamma_Z(0) = \frac{1}{9} \sum_{i=0,1,3} \sum_{j=0,1,3} \gamma_a(i - j)
\]
We then divide \( \gamma_Z(k) \) by \( \gamma_Z(0) \) to get \( \rho_k \).

If the \( a_t \) process is white noise (i.e., \( \gamma_a(k) = 0 \) for \( k \neq 0 \)), the autocovariance simplifies, and \( \rho_k \) will take specific values, but without more information on \( a_t \), we leave the general form as above.

### (c) Find \( \text{Var}\left(\frac{1}{5} \sum_{t=1}^{5} Z_t\right) \)

Let \( \bar{Z}_5 = \frac{1}{5} \sum_{t=1}^{5} Z_t \). We want to find \( \text{Var}(\bar{Z}_5) \). Using the properties of variance:
\[
\text{Var}\left(\bar{Z}_5\right) = \frac{1}{25} \text{Var}\left( \sum_{t=1}^{5} Z_t \right)
\]
Since the variance of a sum is:
\[
\text{Var}\left( \sum_{t=1}^{5} Z_t \right) = \sum_{t=1}^{5} \sum_{s=1}^{5} \text{Cov}(Z_t, Z_s) = \sum_{t=1}^{5} \sum_{s=1}^{5} \gamma_Z(t-s)
\]
Thus:
\[
\text{Var}\left(\bar{Z}_5\right) = \frac{1}{25} \sum_{t=1}^{5} \sum_{s=1}^{5} \gamma_Z(t-s)
\]
This sum depends on the autocovariance \( \gamma_Z(k) \), which we derived in part (b).

### Let me know if you'd like more details or examples!

### Questions:
1. Can you clarify whether \( a_t \) is a white noise process?
2. Would you like to see explicit calculations for specific lags in \( \rho_k \)?
3. Should I assume independence for further simplifications in part (b)?
4. Do you need help computing specific examples of autocovariance sums?
5. Is there any specific property of the process you want to explore further?

**Tip**: Weak stationarity simplifies many calculations in time series, making the assumption useful for estimating long-run behavior.

Suppose that Zt = (at + at−1 + at−3)/3
(a) Show that Zt is weakly stationary.
(b) Find ρk, k = 0, 1, 2, 3, ...
(c) Find Var (1/5 Σt=1 to 5 Zt).

Explore the weak stationarity and autocorrelation function of the time series Zt = (at + at−1 + at−3)/3. Understand how to show weak stationarity, find the autocorrelation ρk, and compute the variance of the sum over time. This problem covers key concepts in time series analysis and is suitable for undergraduate students.

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