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Does anyone know the specific equations for the three parameters in a least-squares quadratic regression? I'm looking for something like $\beta_1=,\beta_2=,\beta_3=$ for each of $y=\beta_1+\beta_2x+\beta_3x^2$. To be clear, the right side of each of these equations should be evaluateable, using the data, to find the parameter. I was able to find the equations for linear regression on line, but google hasn't turned anything up for this.

Thanks in advance

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    as a note - there is a stats-specific SE too http://stats.stackexchange.com/, you can try there as well2016-01-20

2 Answers 2

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For $y =\beta_1 +\beta_2 x +\beta_3 x^2$, let's define $x_1 = x$ and $x_2= x^2.$ Now we can use the equations of multiple linear regression: $S_{11}= \sum_{n=1}^N x_1^2- \frac{(\sum_{n=1}^N x_1)^2}{N}$ $S_{12}= \sum_{n=1}^N x_1x_2- \frac{(\sum_{n=1}^N x_1\sum_{n=1}^N x_2)}{N}$ $S_{22}= \sum_{n=1}^N x_2^2- \frac{(\sum_{n=1}^N x_2)^2}{N}$ $S_{y1}= \sum_{n=1}^N y x_1- \frac{(\sum_{n=1}^N y\sum_{n=1}^N x_1)}{N}$ $S_{y2}= \sum_{n=1}^N y x_2- \frac{(\sum_{n=1}^N y\sum_{n=1}^N x_2)}{N}$ $\overline{x}_1 = \frac{(\sum_{n=1}^N x_1)}{N}$ $\overline{x}_2 = \frac{(\sum_{n=1}^N x_2)}{N}$ $\overline{y} = \frac{(\sum_{n=1}^N y)}{N}$ $\beta_2=\frac{S_{y1}S_{22}-S_{y2}S_{12}}{S_{22}S_{11}-S_{12}^2}$ $\beta_3=\frac{S_{y2}S_{11}-S_{y1}S_{12}}{S_{22}S_{11}-S_{12}^2}$ $\beta_1=\overline{y}-\beta_2\overline{x}_1-\beta_3\overline{x}_2$