Explanation

When using seasonal dummy variables to model seasonality with four seasons (quarters), we need to understand how dummy variables work:

1. Dummy Variable Concept: A dummy variable takes the value 1 when the observation belongs to the category of interest, and 0 otherwise.

2. Four Seasons Representation: With four seasons, we would typically use three dummy variables (to avoid the dummy variable trap), but the question is asking about the arrangement that would represent the third season specifically.

3. Pattern Analysis:

   • Option A: 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, ... - This represents the first season (1 appears every 4 periods)
   • Option B: 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, ... - This represents the third season (1 appears at positions 3, 7, 11, etc.)
   • Option C: 0, 0, 3, 0, 0, 3, 0, 0, 3, ... - This uses 3 instead of 1, which is incorrect for dummy variables
   • Option D: 0, 1, 0, 0, 1, 0, 0, 1, 0, ... - This represents the second season (1 appears at positions 2, 6, 10, etc.)

4. Why Option B is Correct:

   • The third season occurs at positions 3, 7, 11, etc. in a quarterly time series
   • The pattern 0, 0, 1, 0, 0, 1, 0, 0, 1, ... correctly places 1 at every third position in the repeating 4-period cycle
   • This follows the principle that at any given time, we can be in only one season

5. Key Insight: The pattern shows that the third quarter gets a value of 1, while all other quarters get 0. The sequence repeats every 4 periods, with the 1 appearing in the third position of each cycle.

Note: The explanation provided in the text confirms this: "Dummy variables work in a way such that it's 1 in the quarter/season of interest and 0 otherwise. At any given time, we can be in only one season."