# How to get statistics for structural parameters after estimating a reduced form model

Let say I have some structural model $$y_t = \alpha + (1+\psi)x_t$$. I estimate the model $$y_t = \alpha + \beta x_t + e_t$$ using OLS. I can back out the structural parameter $$\psi$$ by hand.

How about standard errors and t-statistics for $$\psi$$? Can some econometric software do that? Or do I need to do that by hand? By hand I mean use the model $$(y_t-x_t) = \alpha_1 + \psi_1 x_t + u_t$$ to get standard errors for $$\psi$$?

Update: I checked in stata, SE for $$\psi_1$$ and $$\beta$$ are the same. So I guess that is it. Thus, standard errors for reduced-form and structural parameters are the same?

It depends$$\ldots$$ In your setting, the structural parameter $$\psi$$ is simply $$1$$ minus the value of the reduced form parameter $$\beta$$: $$\psi = 1 - \beta.$$ In such case, the mean of the estimator $$\psi_n$$ will be simply the mean of $$1 - \beta_n$$ and they will have the same variance (as the variance does not change if we add a fixed number or change sign).
If we have a general (non-linear) transformation: $$\psi = g(\beta),$$ then the most common way to proceed (and to get asymptotic consistent standard errors) is by using the delta-method, which is based on a first order Taylor expansion (wiki). In particular, if $$g$$ is $$C^1$$ and if: $$\sqrt{n}(\beta_n - \beta) \to^D {\cal N}(0, \sigma^2),$$ we will have that: $$\sqrt{n}(g(\beta_n) - g(\beta)) = \sqrt{n}(\psi_n - \psi) \to^D {\cal N}(0, \sigma^2 \cdot [g'(\beta)]^2).$$ So the asymptotic mean of $$\sqrt{n}(\psi_n - \psi)$$ will be equal to zero (which means that $$\psi_n$$ is consistent and the asymptotic variance is the asymptotic variance of $$\beta_n$$ multiplied by the derivative $$g'(\beta)$$ squared.
Notice that if $$g$$ is a simple translation then $$g' = \pm 1$$ so the square is $$1$$ and the variance will be equal to $$\sigma^2$$.
To estimate $$\sigma^2 \cdot [g'(\beta)]^2$$ you can use the consistent plugin estimator: $$\hat \sigma^2_n \cdot [g'(\beta_n)]^2.$$ where $$\hat \sigma^2_n$$ is a consistent estimator of $$\sigma^2$$.