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 will the Quadric Model be useful

Properties of the Models:

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(a) Linear Model (yt = B0 + B1 t):

• Assumes a constant rate of change in sales over time.
• Provides a straight-line relationship between sales and time.
• B0 represents the intercept, and B1 represents the slope.

(b) Quadratic Model (yt = à0 + à1 t + à2 t^2):

• Allows for a nonlinear relationship between sales and time, introducing curvature.
• à0 is the intercept, à1 is the linear coefficient, and à2 is the quadratic coefficient.
• The quadratic term (t^2) accounts for changing rates of sales growth.

Deciding Between the Models:

(b) To decide which model is appropriate, consider:

• Graphical Analysis: Plot the data and visually assess if a linear or quadratic trend seems more appropriate.
• Statistical Tests: Conduct hypothesis tests comparing the fit of the linear and quadratic models. Commonly, the F-test or likelihood ratio test can be used. If the quadratic model provides a significantly better fit and there is evidence of curvature in the relationship, it might be more appropriate. However, if the linear model is simpler and sufficient to explain the observed trend, it could be preferred.

Usefulness of Quadric Model:

(c) The quadratic model is useful in situations where:

• There are indications of nonlinear growth or decay in the sales data over time.
• The rate of change in sales is not constant but varies with time.
• There is an expectation of an initial acceleration or deceleration in sales before reaching a turning point.

The quadric model allows for a more flexible representation of the relationship between sales and time, capturing potential curvature in the trend. This can be particularly valuable when trying to model complex patterns or turning points in the behavior of sales over a number of years.