# Estimate Markov process for incomplete market model

Let's say I have an economy in which the agent's budget constraint is:

$$c_t + k_{t+1} = (1+r-\delta)k_t+w_t\varepsilon_t$$

Where $$\varepsilon$$ follows a Markov process with 2 states.

I now want to estimate this Markov process; I do the following:

1. Estimate the log earning residuals.
2. Specify a statistical model for these residuals (e.g. AR(1))
3. Estimate the parameters of the statistical model.
4. Discretize the process with the Tauchen method.

In the last point, I need (i) the estimated persistence, (ii) the estimated variance of the error, (iii) the number of states that I fix to be equal to 2 and (iv) the number of standard deviations above and below the mean of the 2 states.

Let's say that by setting that the good state is 1 std above the mean and the bad is one std below I get:

$$z = (\bar{z},\underline{z}) = (0.27,-0.27)$$

and the Markov matrix is

$$(0.93, 0.07;0.07,0.93)$$

with the associated stationary distribution equal to

$$(0.5, 0.5)$$.

The mean of the vector $$z$$ is obviously zero because the residual have by construction zero mean. However, to plug the Markov process into the model, I have to take the exponential of the vector $$z$$, because I estimated the model using log earnings. However the mean of the vector

$$(\exp{(\bar{z})},\exp{(\underline{z})})$$

It is not 1. Therefore, given the stationary distribution associated with the Markov matrix, the average labor income in the model would not be equal to the average labor income in the data.

What I am doing wrong?