# Testing equality of coefficients in the same model

#### moocowcn

Hello, can anyone help me with the following problem?

I have a model called Svensson model, which is used to fit yield curves.

The model reads

$$\displaystyle r\left(\tau\right)=\left[ \begin{matrix} \beta_0\\\beta_1\\\beta_2\\\beta_3 \end{matrix} \right]'\left[ \begin{matrix} 1\\ \lambda_1\left(1-e^{-\tau/\lambda_1}\right)/\tau\\ \lambda_1\left(1-e^{-\tau/\lambda_1}\right)/\tau-e^{-\tau/\lambda_1}\\ \lambda_2\left(1-e^{-\tau/\lambda_2}\right)/\tau-e^{-\tau/\lambda_2}\\ \end{matrix} \right].$$

In this model, $$\displaystyle r\left(\tau\right)$$ is the bond yield at maturity $$\displaystyle \tau$$, $$\displaystyle \beta_0, \beta_1, \beta_2, \beta_3,\lambda_1\text{ and }\lambda_2$$ are the six parameters that need to be estimated.

The components represent the level, slope, first and second hump/trough of a yield curve. As the third and fourth components share the same form, they may be highly correlated which will cause estimation problem. So we want to avoid the fourth component if possible. To do this, we need to test whether $$\displaystyle \lambda_1\text{ and }\lambda_2$$ are statistically different.

Does anyone know the way to test the equality of these two coefficients?

Thanks.

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