I am trying to fit an unobserved components model for revenue and transactions for a firm where I also use some exogenous variables that capture economic conditions. The UCM decomposes a time series into trend, seasonality, an idiosyncratic component, and up to 3 cycle(s) using a Kalman filter state space approach with the parameter estimated by MLE.
The basic idea is \begin{equation} y_t = \tau_t + \gamma_t + \psi_t+\beta x_{t-k}+\varepsilon_t \end{equation} where
- $y$ is the dependent variable (revenue or transactions)
- $\tau$ is the trend component
- $\gamma$ is the seasonal component
- $\psi$ is the cycle
- $\beta$ is a vector of regression coefficients on explanatory variables $x$
For my explanatory variables, I have been using lags of the moving averages of monthly Chicago Fed National Economic Activity Indices or the daily Philly Fed Aruoba-Diebold-Scotti Business Conditions Index. Both of these indices measure the common component(s) of many different economic time series using a dynamic factor framework similar to PCA. Their average values are zero. When the indices are positive, the economic activity growth is above trend or average. When the negative, growth is below trend.
The predictive performance of this approach is reasonable. The interpretation of the $\beta$s, however, is a bit awkward since I have essentially a growth rate on the RHS impacting a level on the LHS. This imposes a restriction that growth has the same effect on the level regardless of the level. This makes little sense to me.
I tried using $\Delta_s \ln y = \ln y_t-\ln y_{t-12}$ as the outcome to make it a growth-on-growth model, but the fit and the forecast quality really suffered
Are there any transformations of the indices or the outcomes that I can try to avoid that restriction?
