Non Linear Model
Let us consider an equation $$Y = 10 + 5{X^2}$$
By putting the values of $$X = 0,1,2,3,4$$ in this equation, we find the values of $$Y$$, as given in the table below. The first and second differences are calculated in the table.
$$X$$

$$Y$$

First differences $$\Delta Y$$

Second differences $${\Delta ^2}Y$$

$$\begin{array}{*{20}{c}} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array}$$

$$\begin{array}{*{20}{c}} {10} \\ {15} \\ {30} \\ {55} \\ {90} \end{array}$$

$$\begin{array}{*{20}{c}} {15 – 10 = 5} \\ {30 – 15 = 15} \\ {55 – 30 = 25} \\ {90 – 55 = 35} \end{array}$$

$$\begin{array}{*{20}{c}} {15 – 5 = 10} \\ {25 – 15 = 10} \\ {35 – 25 = 10} \end{array}$$

The second differences are exactly constant. The general quadratic equation or non linear model is written as
\[Y = a + bX + c{X^2}\] $$\left( {c \ne 0} \right)$$
This is also called the second degree parabola or second degree curve. The graph of the data is shown in the figure below:
This figure is not a straight line; it is a curve, or we say that the model $$Y = 10 + 5{X^2}$$ is nonlinear. Remember that if in certain observed data the second differences are constant or almost constant, we find the second degree curve close to the observed data. We shall face this type of situation in time series.