1. Econ 427 lecture 7 slides
Modeling Seasonals
Byron Gangnes

2. Modeling seasonality
Deterministic seasonality refers to perfectly predictable recurring seasonal patterns in a time series
Weather, holidays, agricultural cycles, tradition
Sometimes series are seasonally-adjusted, but many times we want to work with unadjusted series but capture this predictable component
Can use regression models to estimate seasonal components.
Byron Gangnes

3. Estimating seasonal models
We use a set of seasonal dummy variables to allow for predictable recurring patterns in the data
Consider the quarterly case
D1 = (1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0, …)
D2 = (0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0, …)
D3 = (0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0, …)
D4 = (0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1, …)
The “1”s indicate that that period is qtr 1 or qtr 2, etc.
Byron Gangnes

4. General seasonal model
Byron Gangnes

5. Model with trend and cycle
For ex, with a quadratic trend plus seasonal factors:
Can also use these “seasonal” dummies to capture trading day and holiday effects. You have a question on problem set 2 that looks at weekday effects.
Byron Gangnes

6. Forecasting with seasonals
Like the time trend (or trend^2, etc.) the dummy variables have a perfectly predictable pattern, so we know at time T what there values will be in coming periods. Once we have estimates of the parameters, we can easily calculate the optimal forecast:
Byron Gangnes

7. Forecasting with seasonals
Our example model:
At time T+h:
Byron Gangnes

8. Forecasting with seasonals
The expected value of this given the information available at time time T:
All the RHS vars except epsilon are known at time T.
Then we operationalize it by replacing unknown true params with our OLS estimates:
Byron Gangnes

9. Forecasting w/ trend and seasonals
Do example of visns in Eviews.
Byron Gangnes