Standard error for parameters Taylor rule policy reaction function My question is about to find standard error for parameters separately. I estimated monetary policy reaction function by this model: 

$r_t=(1-ρ)α+(1-ρ)βπ_t+(1-ρ)γx_t+ρr_{t-1}+ε_t$

where $r_t$ is interest rate, $π_t$ is inflation rate, $x_t$ is output gap, and $r_{t-1}$ is first lag of interest rate. I want to find standard error for $α$, $β$, $γ$ parameters. I have standard error for $(1-ρ)α$, $(1-ρ)β$, $(1-ρ)γ$ and $ρ$. If someone can show me how to find, it would be very helpful for me.
Thank you for your help.
 A: Define a vector function for the parameters you know
$$F(\alpha,\beta,\gamma,\rho) =\begin{pmatrix}  (1-\rho)\alpha \\(1-\rho)\beta \\(1-\rho)\gamma \\\rho\end{pmatrix} = \begin{pmatrix} \theta_1 \\ \theta_2 \\ \theta_3 \\\theta_4 \end{pmatrix}$$
Solve to find
$$F^{-1}(\theta) =\begin{pmatrix}  \theta_1/(1-\theta_4) \\\theta_2/(1-\theta_4)  \\\theta_3/(1-\theta_4)  \\ \theta_4\end{pmatrix} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \\ \rho \end{pmatrix}$$
You have that the variance covariance matrix of  $\sqrt n(\hat \theta - \theta) \sim \mathcal N(0, \Sigma)$ so
$$\sqrt n(F^{-1}(\hat \theta)  - F^{-1}(\theta_0) ) \sim \mathcal N(0,\Omega)$$
where $$\Omega = \left(\frac{\partial F^{-1}(\theta) }{\partial \theta^\top}\right) \Sigma \left( \frac{\partial F^{-1}(\theta) }{\partial \theta^\top} \right)^\top$$
see delta method.
The jacobian of $F^{-1}$ is in this example easy to derive
$$F^{-1} = \begin{pmatrix} 1/(1-\theta_4) & 0 & 0 & \theta_1/(1-\theta_4)^2\\
0 & 1/(1-\theta_4) & 0 & \theta_2/(1-\theta_4)^2\\
0 & 0 & 1/(1-\theta_4) & \theta_3/(1-\theta_4)^2\\
0&0&0&1
\end{pmatrix}$$
Here is a fairly complete illustration of how to implement the delta method in this example. The approach is as follows
(1) Set the parameter $\alpha,\beta,\gamma,\rho$ used in simulation
(2) Simulate return data
(3) Estimate model using regression equation from question and getting $\hat \theta$ and $\hat \Sigma$
(4) Construct functions for $F^{-1}$ and $J(F^{-1})$ the jacobian of the vector function $F^{-1}$ hence $\frac{\partial F^{-1}(\theta)}{\partial \theta^\top}$
(5) Control that $\frac{\partial F^{-1}(\theta)}{\partial \theta^\top}$ is correct comparing the calculated derivatives with numerica derivatives (I find this step very important because when I often make coding mistakes in coding up $\frac{\partial F^{-1}(\theta)}{\partial \theta^\top}$ and sometimes also mistakes in deriving the derivatives)
(6) Compute $\hat \Omega$ and take the square root of diagonal to get standard errors. There is an implementation of the method in the R package car which I call to compare. 
library(car)
library(numDeriv)

# (1) Set the structural parameters
alpha <- 1
beta <- 1
gamma <- 1
rho <- 0.5

# (2) Simulate some data
T <- 1000

pi_t <- rnorm(T)
x <- rnorm(T)
r_0 <- 0

r <- rep(0,T)
r[1] <- (1-rho)*alpha + (1-rho)*beta*pi_t[1] + (1-rho)*gamma*x[1] + rho*r_0 + rnorm(1)

for (t in 2:T)
    {
        r[t] <- (1-rho)*alpha + (1-rho)*beta*pi_t[t] + (1-rho)*gamma*x[t] + rho*r[t-1] + rnorm(1)
    }
plot(r,type="l")


# (3) Estimate the model
lag_r <- c(r_0,r[1:(T-1)])
model <- lm(r~pi_t+x+lag_r)
summary(model)
theta <- coef(model) # Get parameters
vcov(model) # Get covariance matrix
SIGMA <- vcov(model)


# (4) Make vectorfunction Finv matrixfunction D_Finv (return jacobian of Finv)
F_inv <- function(theta)
    {
        # unpack parameters from vector
        theta_1 <- theta[1]
        theta_2 <- theta[2]
        theta_3 <- theta[3]
        theta_4 <- theta[4]

        alpha <- theta_1/(1-theta_4)
        beta <- theta_2/(1-theta_4)
        gamma <- theta_3/(1-theta_4)
        rho <- theta_4

        out <- c(alpha,beta,gamma,rho)
        names(out) <- c("alpha","beta","gamma","rho")
        return(out)
    }

D_Finv <- function(theta)
    {
        # unpack parameters from vector
        theta_1 <- theta[1]
        theta_2 <- theta[2]
        theta_3 <- theta[3]
        theta_4 <- theta[4]

        D_mat <- matrix(0,4,4)
        D_mat[1,1] <- 1/(1-theta_4)
        D_mat[2,2] <- 1/(1-theta_4)
        D_mat[3,3] <- 1/(1-theta_4)
        D_mat[4,4] <- 1

        D_mat[1,4] <- theta_1/(1-theta_4)^2 
        D_mat[2,4] <- theta_2/(1-theta_4)^2 
        D_mat[3,4] <- theta_3/(1-theta_4)^2 
        return(D_mat)
    }

# (5) Check the manually coded function for jacobian using numerical derivatives
F_inv(theta)
jacobian(F_inv,theta)
D_Finv(theta)


# (6) Compute new covariance matrix
theta <- coef(model)
D_Finv(theta)
OMEGA <- (D_Finv(theta))%*%vcov(model)%*%t(D_Finv(theta))
sqrt(diag(OMEGA))
deltaMethod(model,"pi_t/(1-lag_r)")
deltaMethod(model,"x/(1-lag_r)")

