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I have a quadratic linear regression of the form:

y = B0 + B1x^2

I need to do a hypothesis test that the marginal effect of an additional unit of x at a specific level of x is greater than 100.

I know that the marginal effect of x at any point is given by 2*B1*x. However, the regression output provides the standard errors for B1, not 2*B1*x.

To calculate the new standard error, do I simply multiply the standard errors by (2*x)^2?

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Basic properties of variance include $\text{Var}(kY) = k^2 \text{Var}(Y)$.

So $\text{Var}(2x_0Y) = (2x_0)^2 \text{Var}(Y)$.

Consequently, multiplying by $2x_0$ (for some given constant $x_0$ multiplies both mean and standard error by $2x_0$.

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