Financial Risk Manager Part 1

Explanation

In a pure seasonal dummy model:

$y_t = \sum_{i=1}^{s} \gamma_i D_{i,t} + \varepsilon_t$

Where:

• $y_t$ is the time series variable
• $s$ is the number of seasons (e.g., 4 for quarterly data, 12 for monthly data)
• $\gamma_i$ are the seasonal factors/coefficients for each season $i$
• $D_{i,t}$ are dummy variables that equal 1 when time $t$ corresponds to season $i$, and 0 otherwise
• $\varepsilon_t$ is the error term

Key Insight: When all seasonal factors $\gamma_i$ are equal (i.e., $\gamma_1 = \gamma_2 = \cdots = \gamma_s$), this means that each season has the same effect on $y_t$. In other words, there is no differential seasonal pattern - the series behaves the same way in all seasons.

Mathematical Interpretation: If $\gamma_1 = \gamma_2 = \cdots = \gamma_s = \gamma$, then:

$y_t = \gamma \sum_{i=1}^{s} D_{i,t} + \varepsilon_t$

But note that for any given time $t$, exactly one $D_{i,t} = 1$ and all others are 0, so $\sum_{i=1}^{s} D_{i,t} = 1$ always. Therefore:

$y_t = \gamma + \varepsilon_t$

This shows that $y_t$ is simply a constant ($\gamma$) plus random noise, with no seasonal variation.

Why the other options are incorrect:

• A: A seasonally adjusted time series would be needed if there WERE seasonality, not when there is no seasonality.
• C: Additional dummy variables would be needed if the model is misspecified or if there are other patterns not captured, not when there's no seasonality.
• D: Since both A and C are incorrect, this option is also incorrect.

Example: If quarterly sales data shows $\gamma_1 = \gamma_2 = \gamma_3 = \gamma_4 = 100$, then sales are consistently 100 units in all quarters, indicating no seasonal pattern.