Suppose you want to study the behavior of sales of automobiles over a number of years and someone suggests you to try the following models:

yt= B0 + B1 t
yt= à0 + à1 t + à2 t2

Where yt = sales at time t and t = time. The first model postulates that sales is a linear
function of time, whereas the second model states that it is a quadratic function of time.
(a) Discuss the properties of these two models.
(b) How would you decide which model is appropriate between these two models?
(c) In what situation the Quadric Model will be useful.

When analyzing the sales of automobiles over time, the two suggested models, \( y_t = B_0 + B_1 t \) (a linear model) and \( y_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 \) (a quadratic model), present distinct characteristics and implications. Understanding the properties of these models, how to choose between them, and identifying the contexts where the quadratic model is advantageous can guide effective research.

### Properties of the Linear and Quadratic Models

1. Linear Model (\( y_t = B_0 + B_1 t \)):
   - Functional Form: This model assumes a constant rate of change in automobile sales over time. The coefficient \( B_1 \) represents the slope, indicating how much sales increase (or decrease if negative) with each unit increase in time \( t \).
   - Interpretability: The linear relationship implies that sales either consistently increase or decrease without any change in the rate over the observed period. For instance, if \( B_1 \) is positive, this indicates that sales are growing at a steady rate.
   - Simplicity: Linear models are generally easier to ___ ___ _______ ___ _____.
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