5  Kalman filtering

Estimating unobserved components

5.1 Literature

Alternatives to what follows can be found in Harvey (1989), Hamilton (1994), Kim and Nelson (1999), Durbin and Koopman (2001) or Triantafyllopoulos (2021). There are many books devoted to the Kalman filter as a casual Amazon search mostly from an engineering perspective. However econometricians since Harvey and Pierse (1984) have used it in a way somewhat different from standard engineering applications. We will cover the filter and then look at a simple example if filtering, then develop a maximum likelihood estimation approach.

5.1.1 Reminder of a state-space model

Consider the trend-cycle model \[\begin{equation} y_t = \chi_t + \tau_t + \varepsilon_t \end{equation}\] where the cycle equation is \[\begin{equation} \chi_t = c+\rho_1 \chi_{t-1} + \rho_2 \chi_{t-2}+v_{1t} \end{equation}\] and the trend equation is \[\begin{equation} \tau_t = \tau_{t-1} + v_{2t} \end{equation}\]

In state space this can be written \[\begin{align} y_t &= \begin{bmatrix} 1 & 0 & 1 \end{bmatrix} \begin{bmatrix} \chi_t \\ \chi_{t-1} \\ \tau_t \end{bmatrix} + [1] \varepsilon_t\\ \begin{bmatrix} \chi_t \\ \chi_{t-1} \\ \tau_t \end{bmatrix} &= \begin{bmatrix} c \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} \rho_1 & \rho_2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \chi_{t-1} \\ \chi_{t-2} \\ \tau_{t-1} \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} v_{1t} \\ v_{2t} \end{bmatrix} \end{align}\]

5.2 A useful class of models

We need a framework that nests this type of model (and many more). State-space models are one such framework, amenable to classical (and Bayesian) estimation. Quite a lot of apparatus required before we can apply maximum likelihood.

Key points:

• Many quantities routinely used to build models and analyse policy are unobservable.
• Econometricians face a major problem estimating such models: everything unobserved needs estimating simultaneously (states and parameters).
• Fortunately there is a method we can use: the Kalman filter (Kalman (1960)).
• Has the useful spin-off that we can also use it to calculate the value of the likelihood function.

5.3 What is a filter?

Imagine we had a time-varying parameter model where we estimate \(\beta_t\); this becomes a time series so we have a lot of parameters to estimate: we outline an estimation method here, which we will then characterise as the Kalman Filter.

Simple observation and transition equations \[\begin{align} y_t &= H_t\beta_t + e_t, &var(e_t) = R \\ \beta_t &= \mu + F \beta_{t-1} + v_t, &var(v_t)=Q \end{align}\] where \(\beta_t\) is a vector of stochastic variables, \(y_t\) a vector of measurements and the data forms an information set such that \(\psi_T = \{y_T,y_{T-1},...,y_1\}\).

Note that written this way the shocks are potentially reduced form ones. Typically there will be less shocks then either states (or observables) or it may be that structural shocks enter into one or more relationships. We would then write \(e_t = B\eta_t\) and \(v_t = G\nu_t\) where \(B\) and \(G\) are matrices of appropriate dimension that feed the structural shocks into the right equations. If we assume both \(eta_t\) and \(\nu_t\) only contains independent unit variance shocks, then \(R = B\mathbb{E}(\eta_t\eta_t')B' = BB'\) and similarly \(Q=GG'\).

The reason for using the reduced form shocks is toreduce the amount of notation. We will also (to further reduce notation) mostly set \(\mu = 0\), but it is straightforward to have a vector of constants, which we include in the final formulae.

5.3.1 Three estimation problems

Jazwinski (1970) defines three type of estimation problem

• Smoothing is the problem of estimating \(\beta_k\) for any \(k<T\)
• Filtering is the problem of estimating \(\beta_k\) for \(k=T\)
• Prediction is the problem of estimating \(\beta_k\) for any \(k>T\)

“The object of filtering is to update our knowledge of the system each time a new observation \(y_t\) is brought in.” (Durbin and Koopman (2001))

Notation

If we understand the notation we are a long way to understanding the solution. \[ \begin{align} \beta_{t|t-1} &= \mathbb{E}[\beta_t | \psi_{t-1}] && \hbox{Estimate of } \beta_t \hbox{ conditional on } \psi_{t-1} \hbox{ (predicted)} \\ \beta_{t|t} &= \mathbb{E}[\beta_t | \psi_t] && \hbox{Current } (\psi_t) \hbox{ sample estimate} \hbox{ (filtered)}\\ \beta_{t|T} &= \mathbb{E}[\beta_t | \psi_T] && \hbox{Full } (\psi_T) \hbox{ sample estimate} \hbox{ (smoothed)}\\ P_{t|j} &= \mathbb{E}\left[(\beta_t - \beta_{t|j}) (\beta_t - \beta _{t|j})'\right] && \hbox{Conditional cov. based on } \psi_j,\ j=[t-1,t,T]\\ y_{t|t-1} &= \mathbb{E}[y_t | \psi_{t-1}] = H_t \beta_{t|t-1} && \hbox{Prediction of}\ y_t \hbox{ given } \psi_{t-1}\\ \eta_{t|t-1} &= y_t - y_{t|t-1} = y_t - H_t \beta_{t|t-1} && \hbox{Prediction error given }\psi_{t-1} \\ f_{t|t-1} &= \mathbb{E}[\eta_{t|t-1}\eta_{t|t-1}'] && \hbox{Conditional } (\psi_{t-1}) \hbox{ variance of prediction error} \end{align} \]

In what follows we will calculate some quantities that are consequences of the state space formulation of our problem that will allow us to apply the regression lemma to estimate the unobserved state variables.

5.3.2 Forecasting

Consider the simple first-order VAR model \[\begin{equation} \beta_t = F\beta_{t-1}+v_{t},\quad v_t\sim N(0,Q) \end{equation}\] We can use to make the conditional forecast \[\begin{equation} \beta_{t|t-1} = F\beta_{t-1} \end{equation}\] where \(\beta_{t|t-1}= \mathbb{E}[\beta_t|\psi_{t-1}]\) and \(\psi_{t-1}\) is the information set available at time \(t-1\).

If \(\psi_{t-1}\) includes \(\beta_{t-1}\) we can straightforwardly forecast next period (and the next-but-one period etc) using the model. This is a standard forecasting exercise given any estimated (or even calibrated) economic model.

5.3.3 Uncertainty

How can we assess the associated forecast uncertainty? The forecast covariance \(P_{t|t-1} = var(\beta_t|\psi_{t-1})\) is given by \[\begin{align*} P_{t|t-1} &= \mathbb{E}\left[ (\beta_t - \beta_{t|t-1})(\beta_t-\beta_{t|t-1})'\right] \\ &= \mathbb{E}\left[ (F\beta_{t-1}+v_t-F\beta_{t-1|t-1}) \left(\beta_{t-1}'F'+v_t'-\beta_{t-1|t-1}'F'\right) \right] \\ &= \mathbb{E}\left[ F\left(\beta _{t-1}-\beta _{t-1|t-1}\right) \left( \beta_{t-1}-\beta _{t-1|t-1}\right) ' F' \right] + \mathbb{E}[v_t v_t'] \\ &= F\mathbb{E}\left[\left(\beta _{t-1}-\beta _{t-1|t-1}\right) \left( \beta_{t-1}-\beta _{t-1|t-1} \right)' \right] F' + \mathbb{E}[v_t v_t'] \\ &= FP_{t-1|t-1}F' + Q \end{align*}\] where \(P_{t-1|t-1} = var(\beta_{t-1}|\psi_{t-1})\). The forecast error variance depends on the previous error variance; that value depends on the information set.

If \(\beta_{t-1}\) forms part of the information set \(\psi_{t-1}\) then \[\begin{equation} P_{t-1|t-1} = var \left(\beta_{t-1}|\psi_{t-1}\right) = 0 \end{equation}\] and there is no uncertainty other than from the disturbance terms and \(P_{t|t-1}=P_t=Q\).

If \(\beta_{t-1}\) does not form part of the information set \(\psi_{t-1}\) but \(\beta _{t-2}\) does then \(P_{t-1|t-1}=Q\) and \(P_{t|t-1}=FQF'+Q\). This can be continued backwards; the unconditional (steady-state) covariance of \(\beta_t\) is the limit \(P=FPF'+Q\). We can easily calculate error bands for \(\beta_t\) using the appropriate information set.

5.3.4 Prediction error

We can turn this around, as it must be the prediction errors are given by \[\begin{align} \eta_{t|t-1} &= y_t - \mathbb{E}[y_t|\psi_{t-1}] \\ &= y_t - \mathbb{E}[H_t\beta_t+e_t|\psi_{t-1}] \\ &= y_t-H_t\beta_{t|t-1} \end{align}\] where \(\eta_{t|t-1}\) is uncorrelated with \(\psi_{t-1}\). So the ‘news’ over that contained in \(y_t\) above \(\psi_{t-1}\) is captured by \(\eta_{t|t-1}\). It will be that \(\eta_{t|t-1}\sim N(0,\Sigma_{\eta\eta})\); we need to find an expression for the covariance.

5.3.5 Current-data predictions

Now we find \(\mathbb{E}[\beta_t | \psi_t]\) – the best prediction of the unknown coefficient vector given current information. Using the regression lemma we know that \[\begin{align} \mathbb{E}[\beta_t | \psi_t ] &= \mathbb{E}[\beta_t | \psi_{t-1}, \eta_{t|t-1} ] \\ &= \mathbb{E}[\beta_t | \psi_{t-1}] +\Sigma_{\beta\eta} \Sigma_{\eta\eta}^{-1}\eta_{t|t-1} \\ &= \beta_{t|t-1} + \Sigma_{\beta\eta}\Sigma_{\eta\eta}^{-1} \eta_{t|t-1} \end{align}\] because \(\psi_{t-1}\) and \(\eta_{t|t-1}\) are uncorrelated and \(\eta_{t|t-1}\) is mean zero.

Similarly, we can find \(P_{t|t}\) as the best prediction of the variance of \(\beta_t\) given \(\psi_t\). Using the regression lemma we know that \[\begin{align} P_{t|t} &= \mathbb{E}[(\beta_t-\beta_{t|t})(\beta_t - \beta_{t|t})'|\psi_{t-1}, \eta_{t|t-1}] \\ &= \mathbb{E}[(\beta_t - \beta_{t|t})(\beta_t-\beta_{t|t})'|\psi_{t-1}] - \Sigma_{\beta\eta}\Sigma_{\eta\eta}^{-1}\Sigma_{\eta\beta} \\ &=P_{t|t-1}-\Sigma_{\beta\eta} \Sigma_{\eta\eta}^{-1} \Sigma_{\eta\beta} \end{align}\]

5.3.6 Estimated model covariances

All we need do is plug the relevant expressions into the regression lemma. So, what is \(\Sigma_{\beta\eta}\)? \[\begin{align} \Sigma_{\beta\eta} &= \mathbb{E}[(\beta_t-\beta_{t|t-1}) \eta_{t|t-1}'] \\ &= \mathbb{E}\left[(\beta_t-\beta_{t|t-1}) (y_t-H_t\beta_{t|t-1})'\right] \\ &= \mathbb{E}\left[(\beta_t-\beta_{t|t-1}) (H_t\beta_t+e_t-H_t\beta_{t|t-1})'\right] \\ &= \mathbb{E}[(\beta_t-\beta_{t|t-1}) (\beta_t-\beta_{t|t-1})'H_t'] + \mathbb{E}[(\beta_t - \beta_{t|t-1}) e_t'] \\ &= P_{t|t-1} H_t' \tag{$\Sigma_{\beta\eta}$} \end{align}\] as \(\mathbb{E}[(\beta_t - \beta_{t|t-1})e_t']=0\).

What is \(\Sigma_{\eta\eta}\)? \[\begin{align} \Sigma_{\eta\eta} &= \mathbb{E}[(y_t-H_t \beta_{t|t-1})(y_t-H_t\beta_{t|t-1})'] \\ &= \mathbb{E}[(H_t\beta_t+e_t-H_t\beta_{t|t-1})(H_t\beta_t+e_t-H_t\beta_{t|t-1})'] \\ &= \mathbb{E}[(H_t\beta_t-H_t \beta_{t|t-1})(H_t\beta_t-H_t\beta_{t|t-1})'] + \mathbb{E}[e_t e_t'] \\ &= \mathbb{E}[H_t(\beta_t-\beta_{t|t-1})(\beta_t-\beta_{t|t-1})'H_t'] + R \\ &= H_t P_{t|t-1} H_t' + R \\ &= f_{t|t-1} \tag{$\Sigma_{\eta\eta}$} \label{svv} \end{align}\] where we define \(f_{t|t-1}=\mathbb{E}[\eta_{t|t-1}\eta_{t|t-1}']\). Now we’re ready.

5.4 The Kalman filter

The equations of the filter are

• the conditional expectation depending on \(\psi_{t-1}\);
• an update that uses \(\eta_{t|t-1}\) to obtain the best \(t\)-period prediction now based on \(\psi_t\).

These must be of the form \[\begin{align} \mathbb{E}[\beta_t | \psi_{t-1}] &= \mu + F \mathbb{E} [\beta_t | \psi_{t-1}] \\ \mathbb{E}[P_t|\psi_{t-1}] &= F \mathbb{E}[P_{t-1}|\psi_{t-1}] F' + Q \\ \mathbb{E}[\beta_t | \psi_t] &= \mathbb{E}[\beta_t|\psi_{t-1}] + \Sigma_{\beta\eta} \Sigma_{\eta\eta}^{-1}\eta_{t|t-1} \\ \mathbb{E}[P_t|\psi_t] &= \mathbb{E}[P_t|\psi_{t-1}] -\Sigma_{\beta\eta}\Sigma_{\eta\eta}^{-1}\Sigma_{\eta \beta} \end{align}\] These are specifically \[\begin{align} \beta_{t|t-1} &= \mu +F\beta_{t-1|t-1} &&\hbox{Predicted } \beta \\ P_{t|t-1} &= F P_{t-1|t-1}F' + Q &&\hbox{Predicted } P \\ \eta_{t|t-1} &= y_t-H_t\beta_{t|t-1} &&\hbox{Prediction error} \\ f_{t|t-1} &= H_t P_{t|t-1}H_t' + R &&\hbox{Pred. error variance} \\ \beta_{t|t} &= \beta_{t|t-1}+P_{t|t-1}H_t' f_{t|t-1}^{-1}\eta_{t|t-1} &&\hbox{Updated } \beta \\ P_{t|t} &= P_{t|t-1} - P_{t|t-1}H_t' f_{t|t-1}^{-1}H_tP_{t|t-1} &&\hbox{Updated } P \end{align}\] The filter evaluates these recursively, beginning from \(\beta_0\), \(P_0\). Treatment of these initial condition reflects knowledge/model:

• Stationary models can use the steady-state
• Non-stationary models use something which is often (confusingly) called a diffuse prior (zero mean, large variance)

5.5 The Kalman filter evaluates the likelihood

There is an extremely useful spin-off to using the Kalman filter. For known initial conditions – say \(\beta_0 = b_0\) – the likelihood of a state-space model with \(T\) observations of \(m\) variables is \[\begin{align} \log L(\theta | y) &= \sum_{t=1}^T \log \left(p (\theta | \psi_{t-1})\right) \\ &= - \Phi - \frac{1}{2}\sum_{t=1}^T \left( \log(\det(f_{t|t-1})) + \eta_{t|t-1}' f_{t|t-1}^{-1} \eta_{t|t-1} | \psi_{t-1} \right) \end{align}\] where \(\Phi = \frac{Tm}{2}\log \left( 2\pi \right)\) with \(\theta\) all the non-state parameters to be estimated. We can use the Kalman filter to obtain \(\eta_{t|t-1}\) and \(f_{t|t-1}\) as they are the prediction error and its variance. This is the prediction error decomposition of the log-likelihood.

A maximum likelihood estimate maximizes \(\log L(\theta | y)\) by choice of \(\theta\).

Durbin, J., and S. J. Koopman. 2001. Time Series Analysis by State Space Methods. Oxford: Oxford University Press.

Hamilton, J. D. 1994. Time Series Analysis. Princeton, NJ: Princeton University Press.

Harvey, Andrew C. 1989. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press.

Harvey, Andrew C., and Richard G. Pierse. 1984. “Estimating Missing Observations in Economic Time Series.” Journal of the American Statistical Association 79 (385): 125–31.

Jazwinski, Andrew H. 1970. Stochastic Processes and Filtering Theory. Mineola, N.Y.: Dover Publications Inc.

Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems.” Transactions of the ASME Journal of Basic Engineering 82 (Series D): 35–45.

Kim, Chang-Jin, and Charles R. Nelson. 1999. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications. MIT Press.

Triantafyllopoulos, Kostas. 2021. Bayesian Inference of State Space Models: Kalman Filtering and Beyond. Springer Texts in Statistics. Cham, Switzerland: Springer.