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 )?
Tˆ t = P1 r=−1 arTt+r = α + βt2 + 2 3 β 6= Tt The quadratic trend Tt does not pass through the moving average filter.

Can you please explain the answer?

In this problem, we're dealing with a quadratic trend \( T_t = \alpha + \beta t^2 \), which is being passed through a moving average filter \( (a_{-1}, a_0, a_1) = \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) \). The goal is to check if the quadratic trend passes through this moving average filter unchanged.

### Moving Average Filter Application
The moving average filter is defined as:

\[
\hat{T}_t = \sum_{r=-1}^{1} a_r T_{t+r}
\]

For this case, the weights \( a_r \) are \( \frac{1}{3} \), and the filter applies a weighted sum of \( T_t \) over the time points \( t-1 \), \( t \), and \( t+1 \).

\[
\hat{T}_t = \frac{1}{3} T_{t-1} + \frac{

Let at ∼ W N(0, σ2 ) 1. Does the quadratic trend Tt = α + βt^2 pass through the moving average filter (a−1, a0, a1) = ( 1/3, 1/3, 1/3 )?
Tˆ t = ∑_{r=−1}^{1} a_rT_{t+r} = α + βt^2 + 2/3 β ≠ Tt. The quadratic trend Tt does not pass through the moving average filter.

Understand why a quadratic trend (Tt = α + βt^2) does not pass unchanged through a moving average filter with weights (1/3, 1/3, 1/3). This problem demonstrates the effect of filtering on quadratic trends, using key concepts from time series analysis.

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