For those who want to learn more, I found the following links extremely useful:
The Python library that is being used is pykalman.
In the Quantopian notebook, the meat of the code is here:
start = '2012-01-01' end = '2015-01-01' y = get_pricing('AMZN', fields='price', start_date=start, end_date=end) x = get_pricing('SPY', fields='price', start_date=start, end_date=end) delta = 1e-3 trans_cov = delta / (1 - delta) * np.eye(2) # How much random walk wiggles obs_mat = np.expand_dims(np.vstack([[x], [np.ones(len(x))]]).T, axis=1) kf = KalmanFilter(n_dim_obs=1, n_dim_state=2, # y is 1-dimensional, (alpha, beta) is 2-dimensional initial_state_mean=[0,0], initial_state_covariance=np.ones((2, 2)), transition_matrices=np.eye(2), observation_matrices=obs_mat, observation_covariance=2, transition_covariance=trans_cov) # Use the observations y to get running estimates and errors for the state parameters state_means, state_covs = kf.filter(y.values)
Question 1: How to pick delta? Why delta / (1 - delta) * np.eye(2)?
Where does a delta of 1e-3 come from? And why not just do:
trans_cov = delta * np.eye(2) # How much random walk wiggles
Perhaps this is something that must be optimized using some cross-validation, although I'm not sure what metric to use. If anyone has any insight, would greatly appreciate it.
Question 2: How is observation_covariance = 2 decided?
I understand we're talk abouting prices here, and $2 move for a stock 'feels' like a good estimate for the variance of the price of Amazon, but is there a better way to select this rather than gut?
Question 3: Why is this approach better than just doing some rolling beta?
At the end of the day, to do some rolling beta you must decide what the lookback window is?
I understand the appeal of Kalman because you don't decide the lookback window, but you do need to decide transition_covariance? And changing this can have the same effect as increasing/decreasing your lookback window.
Question 4: Best practices
Are there best practices that I should be aware of? The only thing I can find was this on Quora.
Any help would greatly be appreciated. Feel free to leave me a comment if you have any questions for me.