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The question asks to find the autocorrelations and partial correlations for the MA(2) process given by:

\[
x_t = v_t - \theta_1 v_{t-1} - \theta_2 v_{t-2}
\]

Where \( v_t \) is a white noise process, and \( \theta_1 \) and \( \theta_2 \) are constants.

### Autocorrelation for MA(2) Process:
For an MA(2) process, the autocovariance function \(\gamma(k)\) for lag \(k\) is given by:

1. **Lag 0:**
   \[
   \gamma(0) = E(x_t^2) = \sigma_v^2 (1 + \theta_1^2 + \theta_2^2)
   \]
   where \( \sigma_v^2 \) is the variance of the white noise process \( v_t \).

2. **Lag 1:**
   \[
   \gamma(1) = E(x_t x_{t-1}) = -\theta_1 \sigma_v^2 (1 + \theta_2)
   \]

3. **Lag 2:**
   \[
   \gamma(2) = E(x_t x_{t-2}) = -\theta_2 \sigma_v^2
   \]

4. **Lag \( k \geq 3 \):**
   For lags greater than 2, the autocovariance is 0:
   \[
   \gamma(k) = 0 \quad \text{for} \, k \geq 3
   \]

### Autocorrelation Function:
The autocorrelation function (ACF) is obtained by normalizing the autocovariance function:
\[
\rho(k) = \frac{\gamma(k)}{\gamma(0)}
\]
Thus, for an MA(2) process:

1. **\(\rho(0) = 1\)**
2. **\(\rho(1) = \frac{-\theta_1 (1 + \theta_2)}{1 + \theta_1^2 + \theta_2^2}\)**
3. **\(\rho(2) = \frac{-\theta_2}{1 + \theta_1^2 + \theta_2^2}\)**
4. **\(\rho(k) = 0 \, \text{for} \, k \geq 3**

### Partial Autocorrelation Function:
The partial autocorrelation function (PACF) for an MA(q) process cuts off after lag q. Therefore, for the MA(2) process:

- **PACF at lag 1** is non-zero.
- **PACF at lag 2** is non-zero.
- **PACF at lag \(k \geq 3\)** is zero.

Would you like further details or clarification on any part of this explanation? 

Here are 5 follow-up questions related to this topic:
1. How is the MA(1) process different from the MA(2) process in terms of autocorrelation?
2. How do we interpret the partial autocorrelation function (PACF)?
3. What is the difference between autoregressive (AR) and moving average (MA) processes?
4. How is the white noise assumption critical to calculating autocorrelations in the MA process?
5. How do we estimate \(\theta_1\) and \(\theta_2\) in an MA(2) model?

**Tip:** The partial autocorrelation function (PACF) is useful in model identification. For an MA process, it cuts off after the order of the process, helping you detect the correct order (q) when fitting a model.

Find the autocorrelations and partial correlations for the MA(2) process: x_t = v_t - θ1 v_{t-1} - θ2 v_{t-2}.

Explore how to find the autocorrelations and partial correlations for the MA(2) process x_t = v_t - θ1 v_{t-1} - θ2 v_{t-2}, including key formulas and concepts in time series analysis.

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