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

To determine if the quadratic trend \( T_t = \alpha + \beta t^2 \) passes through the moving average filter with weights \( (a_{-1}, a_0, a_1) = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) \), we need to evaluate the behavior of the quadratic trend after applying the filter.

### Step 1: Define the moving average filter
The moving average filter is defined as:
\[
\tilde{T}_t = \frac{1}{3} T_{t-1} + \frac{1}{3} T_t + \frac{1}{3} T_{t+1}
\]

### Step 2: Apply the quadratic trend to the filter
Substitute the quadratic trend \( T_t = \alpha + \beta t^2 \) into the moving average formula:

\[
\tilde{T}_t = \frac{1}{3} \left( \alpha + \beta (t-1)^2 \right) + \frac{1}{3} \left( \alpha + \beta t^2 \right) + \frac{1}{3} \left( \alpha + \beta (t+1)^2 \right)
\]

### Step 3: Expand the terms
Now, expand the terms for \( (t-1)^2 \) and \( (t+1)^2 \):
\[
(t-1)^2 = t^2 - 2t + 1
\]
\[
(t+1)^2 = t^2 + 2t + 1
\]

Now substitute these back into the filter equation:
\[
\tilde{T}_t = \frac{1}{3} \left( \alpha + \beta (t^2 - 2t + 1) \right) + \frac{1}{3} \left( \alpha + \beta t^2 \right) + \frac{1}{3} \left( \alpha + \beta (t^2 + 2t + 1) \right)
\]

### Step 4: Simplify the equation
First, distribute the \(\beta\) across the terms:
\[
\tilde{T}_t = \frac{1}{3} \left( \alpha + \beta t^2 - 2\beta t + \beta \right) + \frac{1}{3} \left( \alpha + \beta t^2 \right) + \frac{1}{3} \left( \alpha + \beta t^2 + 2\beta t + \beta \right)
\]

Now, combine like terms:
\[
\tilde{T}_t = \frac{1}{3} \left( 3\alpha + 3\beta t^2 + \beta + \beta + 0 \right)
\]

\[
\tilde{T}_t = \alpha + \beta t^2 + \frac{2\beta}{3}
\]

### Step 5: Interpretation
The filtered trend \( \tilde{T}_t \) is \( \alpha + \beta t^2 + \frac{2\beta}{3} \), which shows that after applying the moving average filter, the quadratic term \( \beta t^2 \) remains intact, but there is an additional constant \( \frac{2\beta}{3} \).

### Conclusion
Thus, the quadratic trend \( T_t = \alpha + \beta t^2 \) does **not pass perfectly** through the moving average filter. The quadratic term \( \beta t^2 \) is preserved, but there is a shift by a constant \( \frac{2\beta}{3} \), which means the trend is modified slightly by the filter.

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Would you like more details on this conclusion or have any specific questions? Here are some related questions for further exploration:

1. How does a linear trend behave when passed through a moving average filter?
2. What happens when we apply a different set of filter weights, such as \( (0.25, 0.5, 0.25) \)?
3. How do moving average filters affect white noise signals?
4. Can you derive the effect of a cubic trend through the same moving average filter?
5. What is the purpose of using moving average filters in time series analysis?

**Tip:** Moving average filters are commonly used to smooth out short-term fluctuations and highlight longer-term trends in time series data.

Let at ∼ WN(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)?

Explore whether the quadratic trend Tt = α + βt^2 passes through the moving average filter (1/3, 1/3, 1/3). Understand how time series analysis and moving averages affect trends and discover the impact on quadratic terms. Ideal for undergraduate students studying statistics or time series analysis.

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