What is the standard error of the coefficient?

The standard deviation of an estimate is called the standard error. The standard error of the coefficient measures how precisely the model estimates the coefficient's unknown value. The standard error of the coefficient is always positive.

Use the standard error of the coefficient to measure the precision of the estimate of the coefficient. The smaller the standard error, the more precise the estimate. Dividing the coefficient by its standard error calculates a t-value. If the p-value associated with this t-statistic is less than your alpha level, you conclude that the coefficient is significantly different from zero.

For example, a materials engineer at a furniture manufacturing site wants to assess the strength of the particle board that they use. The engineer collects stiffness data from particle board pieces with various densities at different temperatures and produces the following linear regression output. The standard errors of the coefficients are in the third column.
Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 20.1 12.2 1.65 0.111 Stiffness 0.2385 0.0197 12.13 0.000 1.00 Temp -0.184 0.178 -1.03 0.311 1.00

The standard error of the Stiffness coefficient is smaller than that of Temp. Therefore, your model was able to estimate the coefficient for Stiffness with greater precision. In fact, the standard error of the Temp coefficient is about the same as the value of the coefficient itself, so the t-value of -1.03 is too small to declare statistical significance. The resulting p-value is much greater than common levels of α, so that you cannot conclude this coefficient differs from zero. You remove the Temp variable from your regression model and continue the analysis.

Why would all standard errors for the estimated regression coefficients be the same?

If your design matrix is orthogonal, the standard error for each estimated regression coefficient will be the same, and will be equal to the square root of (MSE/n) where MSE = mean square error and n = number of observations.

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