Let at ∼ W N(0, σ2
)
1. Does the quadratic trend Tt = α + βt2 pass through the moving average
filter (a−1, a0, a1) = ( 1
3
,
1
3
,
1
3
)?

To determine whether the quadratic trend \( T_t = \alpha + \beta t^2 \) passes through the moving average filter \( (a_{-1}, a_0, a_1) = \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \), we need to apply the moving average filter to the trend and check whether the output reproduces a quadratic trend (or some other recognizable pattern).

### Moving Average Filter
The moving average (MA) filter applies weights \( \frac{1}{3} \) to the points at \( t-1 \), \( t \), and \( t+1 \). This can be written as:

\[
\text{Filtered value at } t = \frac{1}{3} T_{t-1} + \frac{1}{3} T_{t} + \frac{1}{3} T_{t+1}.
\]

### Applying the Filter
The trend is \( T_t = \alpha + \beta t^2 \). Now, let's compute the moving average filter at time \( t \).

1. \( T_{t-1} = \alpha + \beta (t-1)^2 = \alpha + \beta (t^2 - 2t + 1) = \alpha + \beta t^2 - 2\beta t + \beta \).
2. \( T_{t} = \alpha + \beta t^2 \).
3. \( T_{t+1} = \alpha + \beta (t+1)^2 = \alpha + \beta (t^2 + 2t + 1) = \alpha + \beta t^2 + 2\beta t + \beta \).

Now, apply the filter:

\[
\frac{1}{3} T_{t-1} + \frac{1}{3} T_t + \frac{1}{3} T_{t+1} 
= \frac{1}{3} (\alpha + \beta t^2 - 2\beta t + \beta) + \frac{1}{3} (\alpha + \beta t^2) + \frac{1}{3} (\alpha + \beta t^2 + 2\beta t + \beta).
\]

Simplify the expression:

\[
= \frac{1}{3} \left( 3\alpha + 3\beta t^2 + 0\beta t + 3\beta \right) = \alpha + \beta t^2 + \beta.
\]

### Result:
The filtered trend is:

\[
\alpha + \beta t^2 + \beta.
\]

This is a quadratic function of \( t \), but it differs from the original quadratic trend by a constant shift of \( \beta \). Thus, the quadratic trend **does pass** through the moving average filter, but the result is shifted upwards by \( \beta \).

### Conclusion:
The quadratic trend \( T_t = \alpha + \beta t^2 \) passes through the moving average filter \( (a_{-1}, a_0, a_1) = \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \), resulting in a quadratic trend with an additional constant term \( \beta \).

---

Do you want further details or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How does a linear trend behave under a moving average filter?
2. What happens if the filter weights are not symmetric (e.g., \( a_{-1} = 0, a_0 = 1, a_1 = 0 \))?
3. Can we design a filter to exactly reproduce quadratic trends without any shift?
4. How do moving average filters behave with higher-order polynomial trends?
5. What is the general behavior of moving average filters with stochastic (random) processes?

**Tip:** Moving average filters tend to smooth out higher-frequency components, making them useful for noise reduction but altering trends with higher-order terms like quadratic trends.

Does the quadratic trend Tt = α + βt^2 pass through the moving average filter (a−1, a0, a1) = (1/3, 1/3, 1/3)?

Explore whether the quadratic trend Tt = α + βt^2 passes through the moving average filter (1/3, 1/3, 1/3) with detailed step-by-step analysis, key mathematical concepts, and conclusions. This is a useful study for undergraduate students in time series analysis or applied mathematics.

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