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As far as I see, the parameters of state space models will be automatically treated as time-invariant and I don't know how to specify a time-varying matrix in a state space model via statsmodels in Python. I have read the document, but still failed to find a way.

statsmodels document: https://www.statsmodels.org/stable/statespace.html

For example, for the simplest local level model

\begin{align*} X_{t+1} &= X_t + V_t \\ Y_t &= X_t + W_t \end{align*}

where $V$ and $W$ are white noises. When estimating this model, the initial state must be specified. However, if I assmue the initial state $X_1$ is an unknown constant (thus a parameter) that should also be estimated, then the model above can be formulated as

\begin{align*} X_{t+1} &= a_t + X_t + V_t \\ Y_t &= X_t + W_t \end{align*}

where for $j \ge 2$, $a_j=0$. And with this formulation, I can initially set $X_1 = 0$ and estimate $a_1$ as if it was the unknown constant $X_1$. Then $a_t$ is a time-varying parameter.

Note that in this simple model, the difficulty mentioned above can be circumvented by formulating as \begin{align*} X_{t+1} &= X_t + V_t \\ Y_t &= b_t + X_t + W_t \end{align*} set $X_1 = 0$, and estimate $b_t$ as an time-invariant parameter. But in general, I can not do so.

class LocalLevel(sm.tsa.statespace.MLEModel):

    def __init__(self, endog):
        super().__init__(endog, k_states=1)

        self['design', 0, 0] = 1.0
        self['transition', 0, 0] = 1.0
        self['selection', 0, 0] = 1.0

        self.initialize_known([0], [[0]])

    # Note the **kwargs argument must be included
    def update(self, params, transformed=True, **kwargs):
        params = super().update(params, transformed, **kwargs)
        self['obs_cov', 0, 0] = params[0]
        self['state_cov', 0, 0] = params[1]
        self['obs_intercept', 0, 0] = params[2]

    @property
    def start_params(self):
        return [1.0, 1.0, 0]

I wonder how can I set 'state_intercept' to be time-varying. It should be k_state * n_obs dimension, but I when I tried to set self['state_intercept'], I found it was treated automatically as k_state * 1 dimension, which is time-invariant.

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The solution is to define the matrix to be time-varying in the constructor:

self['state_intercept'] = np.zeros((self.k_states, self.nobs))

so the full code would be:

class LocalLevel(sm.tsa.statespace.MLEModel):

    def __init__(self, endog):
        super().__init__(endog, k_states=1)

        self['design', 0, 0] = 1.0
        self['transition', 0, 0] = 1.0
        self['selection', 0, 0] = 1.0
        self['state_intercept'] = np.zeros((self.k_states, self.nobs))

        self.initialize_known([0], [[0]])

    # Note the **kwargs argument must be included
    def update(self, params, transformed=True, **kwargs):
        params = super().update(params, transformed, **kwargs)
        self['obs_cov', 0, 0] = params[0]
        self['state_cov', 0, 0] = params[1]
        self['obs_intercept', 0, 0] = params[2]

    @property
    def start_params(self):
        return [1.0, 1.0, 0]
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