I am taking a mandatory Time Series class at university. The professor's lecture slides, reading materials, and seminar training are all the more confusing. He also refuses to answer any kind of content-related question outside the classroom (we are taught in smaller groups, circa 15 students). This leaves me out pretty much in the blue.
For those who are experienced in Time Series and EView software (which we exclusively use for the purpose of this class, no R or Python or Excel), I am wondering the following. Professor is currently presenting on the Additive Components of Time Series, with Trend, Seasonality, Cycle, and Non-Systematic, which he writes as:
$$ y_t = T_t + C_t + S_t + a_t $$
In EViews, we worked with a quarterly number of household distribution. He looked at the distribution of scores that looked somewhat S-shaped and so he wrote the following model:
$$ Y_t = c + \beta{_1}t + \beta{_2}t^2 + \beta{_3}t^3 + \gamma{_1}D_{1}t + \gamma{_2}D_{2}t + \gamma{_3}t + \epsilon{t} $$
Expressed in Eviews as:
$$ Households\hspace{0.2cm}c\hspace{0.2cm}t^1 \hspace{0.2cm} t^2 \hspace{0.2cm} t^3 \hspace{0.2cm}d^1 \hspace{0.2cm}d^2 \hspace{0.2cm}d^3 $$
with D being calculated in Eview as:
@seas(1)
@seas(2)
@seas(3)
@seas(4) (purposefully omitted in the model)
And the key table of outputs looks as such:
My key interrogation / problem is that I do not exactly know the kind of model I am working with. It looks like a polynomial regression model, and so I understand the logic of adding a squared and cubic term to accommodate non-linear relationships. Since my professor refuses to answer any student question, I had to rely on asking GPT for clarifications. But GPT is not always accurate. Anyways, here is how I understand the model:
$$ Y_t = c + \beta{_1}t + \beta{_2}t^2 + \beta{_3}t^3 + \gamma{_1}D_{1}t + \gamma{_2}D_{2}t + \gamma{_3}t + \epsilon{t} $$
Is the output of a multiple time regression model with seasonality terms, where:
$\beta{_0}$: Is the time regression intercept, the expected value of household number when controlling for the effects of time and of seasonality.
$\beta{_1}t$: The partial regression coefficient of linear time t, where t is the discrete index value as we fit the distribution one score after the other (t = 1, t = 2, t= 3, ... t = N). GPT tells me that t is not the actual observed values, but the index location of each value (first value in ts gives ts = 1, second t = 2, etc.)
$\beta{_2}t^2$ and $\beta{_3}t^3$ are the same, except that quadratic and cubic terms allow to account for curved increases and decreases. This serves the purpose of improving the fitted regression line when time series plot isn't linear but more S-shaped.
$\gamma{_1}D_{1}t$: The seasonal term for Q1. I understand the coefficient as the adjusted mean difference between Q4 (the reference level) and Q1. It basically says "Q1 is significantly different from Q4, and this influences the scores of Q1 that must be corrected by adding or subtracting a certain amount to the predictions for Q1 values". D stands for dummy variable, meaning that this variable (as written in the formula) becomes = 1 when any value considered happens in Q1 (or Q2 for D2t, Q3 for D3t). When D1t = 1, the seasonal correction is applied by multiplying $\gamma{}$ by 1.
$\epsilon{t}$: The error term that corrects for off-predictions and make Yt the observed ts score.
Anyways, I would seek for help in conceptually clarifying what I am working with. I had a Regression class before that was pretty challenging, but the process and theory were nearly crystal clear. Here, I feel like there are some aspects of a standard regression, but modelling time instead (which is something I never covered before). References (textbooks, tutorials, other topics) are much appreciated.
