Math Problem Statement

Let a nonlinear regression model of the type Y=β0+β1X+β2X^2+ε1 approximate the true relation between X and Y, then derive an expression for β0, β1, and β2. If instead of a nonlinear model, we assume a linear regression model of the type Y=α0+α1X+ε2, then compare the coefficients of the linear and nonlinear model.

Solution

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1. Nonlinear Regression Model:

The nonlinear regression model provided is: Y=β0+β1X+β2X2+ϵ1Y = \beta_0 + \beta_1 X + \beta_2 X^2 + \epsilon_1 where β0\beta_0, β1\beta_1, and β2\beta_2 are the regression coefficients, XX is the independent variable, and ϵ1\epsilon_1 is the error term.

2. Linear Regression Model:

The linear regression model given is: Y=α0+α1X+ϵ2Y = \alpha_0 + \alpha_1 X + \epsilon_2 where α0\alpha_0, α1\alpha_1 are the coefficients, and ϵ2\epsilon_2 is the error term.

Objective:

  • Derive expressions for the coefficients β0\beta_0, β1\beta_1, and β2\beta_2 in the nonlinear model.
  • Compare the coefficients α0\alpha_0, α1\alpha_1 from the linear model with those from the nonlinear model.

Steps for Solution:

1. Nonlinear Model Coefficient Derivation:

For the nonlinear model, the coefficients β0\beta_0, β1\beta_1, and β2\beta_2 can be obtained through ordinary least squares (OLS) estimation. Here, the relationship is quadratic in XX. To estimate the coefficients, you would typically solve a system of equations based on minimizing the sum of squared residuals: Residuals: R=(Y(β0+β1X+β2X2))2\text{Residuals: } R = \sum (Y - (\beta_0 + \beta_1 X + \beta_2 X^2))^2 Taking the partial derivatives of RR with respect to β0\beta_0, β1\beta_1, and β2\beta_2 will give you the normal equations that help in solving for the coefficients.

2. Linear Model Coefficient Estimation:

For the linear model, using OLS, the coefficients α0\alpha_0 and α1\alpha_1 can also be estimated by minimizing the residual sum of squares: Rlinear=(Y(α0+α1X))2R_{\text{linear}} = \sum (Y - (\alpha_0 + \alpha_1 X))^2 Again, you would solve the normal equations formed by taking the partial derivatives of this expression with respect to α0\alpha_0 and α1\alpha_1.

3. Comparison of Coefficients:

Once the coefficients β0\beta_0, β1\beta_1, β2\beta_2, α0\alpha_0, and α1\alpha_1 are derived, you can compare:

  • α0\alpha_0 with β0\beta_0, which will likely be close if the quadratic term in the nonlinear model has a small effect.
  • α1\alpha_1 with β1\beta_1, considering the linear approximation assumes β20\beta_2 \approx 0.
  • Additionally, β2\beta_2 captures the curvature of the nonlinear model, which the linear model does not account for.

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Related Questions:

  1. How do we apply ordinary least squares (OLS) in the nonlinear case?
  2. What assumptions are necessary for these regression models to work?
  3. How does multicollinearity affect the estimates of β0\beta_0, β1\beta_1, and β2\beta_2?
  4. Can we generalize this model to higher-degree polynomials?
  5. What happens if XX follows a non-normal distribution?

Tip:

For practical applications, the nonlinear model might be a better fit if the relationship between XX and YY is truly quadratic, but the linear model can still offer useful approximations for small datasets or where simplicity is key.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Linear and Nonlinear Models
Ordinary Least Squares (OLS)
Quadratic Equations

Formulas

Nonlinear Model: Y = β0 + β1X + β2X^2 + ε1
Linear Model: Y = α0 + α1X + ε2
Sum of Squared Residuals: R = Σ(Y - (β0 + β1X + β2X^2))^2

Theorems

Ordinary Least Squares (OLS)
Normal Equations for Linear Regression

Suitable Grade Level

Advanced Undergraduate or Graduate Level