Answered

In a quadratic regression $Y=\beta_0+\beta_1 X+\beta_2 X^2+u$, the turning point (maximum or minimum) occurs at $X=$ ____ .


A$-\beta_1 /\left(2 \beta_2\right)$
B$-\beta_1 /\left(2 \beta_2\right)$
C$-\beta_0 / \beta_1$
D$-\beta_2 /\left(2 \beta_1\right)$
E$\beta_1 /\left(2 \beta_2\right)$

Verified Answer :

  • The quadratic regression model is given by \(Y = \beta_0 + \beta_1 X + \beta_2 X^2 + u\).
  • To find the turning point, find the first derivative of \(Y\) with respect to \(X\), which is \(\frac{dY}{dX} = \beta_1 + 2\beta_2 X\).
  • Set the derivative to zero and solve for \(X\): \(\beta_1 + 2\beta_2 X = 0 \implies X = -\frac{\beta_1}{2\beta_2}\).
  • The correct option is \(\boxed{a}\).

Explanation

  1. Analysis of the Quadratic Model

The given equation is a quadratic model for regression: \(Y = \beta_0 + \beta_1 X + \beta_2 X^2 + u\), where \(\beta_0, \beta_1,\) and \(\beta_2\) are constant coefficients and \(u\) represents the error term. A turning point of a quadratic function (its vertex, whether a minimum or maximum) occurs where the function's rate of change is zero. Mathematically, this corresponds to the point where the derivative of \(Y\) with respect to \(X\) is equal to zero.

  1. Differentiating the Function

To find the turning point, we differentiate the function \(Y\) with respect to \(X\):
\[\frac{dY}{dX} = \frac{d}{dX}(\beta_0 + \beta_1 X + \beta_2 X^2 + u)\]
The derivative of a constant (\(\beta_0\) and \(u\)) is zero, the derivative of \(\beta_1 X\) with respect to \(X\) is \(\beta_1\), and the derivative of \(\beta_2 X^2\) with respect to \(X\) is \(2\beta_2 X\). Thus:
\[\frac{dY}{dX} = \beta_1 + 2\beta_2 X\]

  1. Solving for the Turning Point

To find the turning point, we set the first derivative equal to zero and solve for \(X\):
\[\beta_1 + 2\beta_2 X = 0\]
Subtract \(\beta_1\) from both sides:
\[2\beta_2 X = -\beta_1\]
Divide by \(2\beta_2\) (assuming \(\beta_2 eq 0\)):
\[X = -\frac{\beta_1}{2\beta_2}\]
This result matches the expression provided in options (a) and (b). Thus, the turning point occurs at \(X = -\frac{\beta_1}{2\beta_2}\).

Examples

In economics, this concept is used to analyze the "cost curve" of a firm. If a firm's total cost function is modeled by \(C(Q) = \beta_0 + \beta_1 Q + \beta_2 Q^2\), where \(Q\) is the quantity produced, the firm often seeks the production level that minimizes average cost or identifies the point of diminishing marginal returns. By finding the turning point of the cost function, managers can optimize production to ensure cost efficiency and maximize profitability.

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