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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?

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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 $\sigma^2$.

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  • $\begingroup$ Thanks. Will try that in MATLAB. Lemme ask though if you've tried this in other econometric software - STATA, EVIEWS, OXMETRICS, R, PYTHON? $\endgroup$ Sep 19, 2021 at 4:37

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