I have data on monthly gross flows in the labor market. For three labor market states, I can calculate 9 transition rates and create a Markov transition matrix where each row gives a value of one.
| $e_{t+1}$ | $u_{t+1}$ | $n_{t+1}$ | $ \sum $ | |
|---|---|---|---|---|
| e_t | $e_t e_{t+1}$ | $e_t u_{t+1}$ | $e_t n_{t+1}$ | 1 |
| u_t | $u_t e_{t+1}$ | $u_t u_{t+1}$ | $u_t n_{t+1}$ | 1 |
| n_t | $n_t e_{t+1}$ | $n_t u_{t+1}$ | $n_t n_{t+1}$ | 1 |
However, the transition rates exhibit seasonality. I want to adjust the data for seasonality, but I want the rows to still sum to one after adjustment. In many economic papers, the ratio to moving average method is used. This method results in row totals that are very close to, but not exactly equal to, one. Also, this method loses too much of the volatility in the data that cannot be attributed to seasonality (e.g., in the early stages of Covid).
I use STL for the time series. After the adjustment, I calculated the difference between 1 and the sum of each row and added a third of the difference to each cell so that the row returns to 1.
| $e_{t+1}$ | $u_{t+1}$ | $n_{t+1}$ | $ \sum $ | |
|---|---|---|---|---|
| e_t | $(e_t e_{t+1})_{sa}+\left( \frac{1-(e_t e_{t+1})_{sa}-(e_t u_{t+1})_{sa}-(e_t n_{t+1})_{sa}}{3}\right)$ | $(e_t u_{t+1})_{sa}+\left(\frac{1-(e_t e_{t+1})_{sa}-(e_t u_{t+1})_{sa}-(e_t n_{t+1})_{sa}}{3}\right)$ | $(e_t n_{t+1})_{sa}+\left(\frac{1-(e_t e_{t+1})_{sa}-(e_t u_{t+1})_{sa}-(e_t n_{t+1})_{sa}}{3}\right)$ | 1 |
| u_t | $(u_t e_{t+1})_{sa}+\left(\frac{1-(u_t e_{t+1})_{sa}-(u_t u_{t+1})_{sa}-(u_t n_{t+1})_{sa}}{3}\right)$ | $(u_t u_{t+1})_{sa}+\left(\frac{1-(u_t e_{t+1})_{sa}-(u_t u_{t+1})_{sa}-(u_t n_{t+1})_{sa}}{3}\right)$ | $(u_t n_{t+1})_{sa}+\left(\frac{1-(u_t e_{t+1})_{sa}-(u_t u_{t+1})_{sa}-(_t n_{t+1})_{sa}}{3}\right)$ | 1 |
| n_t | $(n_t e_{t+1})_{sa}+\left(\frac{1-(n_t e_{t+1})_{sa}-(n_t u_{t+1})_{sa}-(n_t n_{t+1})_{sa}}{3}\right)$ | $(n_t u_{t+1})_{sa}+\left(\frac{1-(n_t e_{t+1})_{sa}-(n_t u_{t+1})_{sa}-(n_t n_{t+1})_{sa}}{3}\right)$ | $(n_t n_{t+1})_{sa}+\left(\frac{1-(n_t e_{t+1})_{sa}-(n_t u_{t+1})_{sa}-(n_t n_{t+1})_{sa}}{3}\right)$ | 1 |
So my question is threefold: Is this a valid approach? If not, is there a better way to account for the seasonality of the transition rates of a Markov transition matrix? Would it be a good alternative to seasonally adjust the gross flow data before calculating the transition rates?
