Help understanding regression models with dlm in R

In the "Dynamic Linear Models with R" book, the Regression models section reads: "The static regression linear model corresponds to the case where $W_t = 0$ for any $t$, so that $\theta_t = \theta$ is constant over time."

I am not understanding what this means, because when I fit a dlmModReg with dW = 0 (default values) and then plot the predicted values of the state vectors, my regression coefficients vary over time, and eventually stabilize to a value similar to that of standard least-squares regression.

Plot of regression slope coefficient with dW = 0:

However, if I use dlmMLE to find the MLE of dW before fitting the model, plotting the predicted values of the state vectors results in non-sensible values with multiple large discontinuities.

Plot of regression slope coefficient with dW MLE and no intercept:

Plot of regression slope with dW MLE with intercept:

I've yet to find any literature on how the intercept coefficient gets set, but I'm observing a near perfect linear relationship between the intercept and slope coefficient. The inclusion of an intercept term in the parameter MLE also causes my dV to drop from 16 to 0.003. Can anyone point me to any references which discuss how the intercept term is set on each update?

Plot of intercept and slope with dW MLE:

My questions are:

1. Why are the regression coefficients dynamic/time-varying even with dW = 0 in the first example? If the regression coefficient changes at every time step regardless of whether dW is 0 or not due to the measurement update stage $\beta_t = \hat{\beta_t} + K_t(y_t - \alpha_t - \hat{\beta_t}x_t)$ ($K$ is the Kalman gain), what's the difference between simple linear regression and dynamic linear regression except for some random variation added in the time update equation $\hat{P_t} = P_{t-1} + dW$ ($P$ is estimate error covariance)?
2. Why does the plot of regression coefficients have so many discontinuities when my dW != 0?
3. What is the relationship between the intercept and the slope coefficients? Plotting them reveals a near-perfect linear relationship, but I can't find any literature explaining this. I haven't found the intercept term included in any formulation of the Kalman update equations.

Would anyone be willing to take a look at my data/code?

• The linear relationship only holds when dW = 0, so I think it's just how the intercept/slope evolve over time when performing static linreg at each data point. For now I'm interested in getting this to work with dW != 0 so I'm going to ignore question 3 for now. – tmakino May 30 '13 at 14:18

• If running Kalman filter with $dW = 0$ is equivalent (yet more computationally efficient) to running OLS at each time step on the entire data set, is there an analogy which will help me understand what "dynamic regression" is, namely where $dW != 0$? I've written the Kalman filter equations out by hand and the only difference is in the time update equations, when the a priori estimate error is updated as the sum of the a posteriori estimate error from the previous time step, and dW. – tmakino May 29 '13 at 21:59