Chapter 25. State Space Models and the Kalman Filter

The EViews sspace (state space) object provides a straightforward, easy-to-use interface for specifying, estimating, and working with the results of your single or multiple equation dynamic system. EViews provides a wide range of specification, filtering, smoothing, and other forecasting tools which aid you in working with dynamic systems specified in state space form.

A wide range of time series models, including the classical linear regression model and ARIMA models, can be written and estimated as special cases of a state space specification. State space models have been applied in the econometrics literature to model unobserved variables: (rational) expectations, measurement errors, missing observations, permanent income, unobserved components (cycles and trends), and the non-accelerating rate of unemployment. Extensive surveys of applications of state space models in econometrics can be found in Hamilton (1994a, Chapter 13; 1994b) and Harvey (1989, Chapters 3, 4).

There are two main benefits to representing a dynamic system in state space form. First, the state space allows unobserved variables (known as the state variables) to be incorporated into, and estimated along with, the observable model. Second, state space models can be analyzed using a powerful recursive algorithm known as the Kalman (Bucy) filter. The Kalman filter algorithm has been used, among other things, to compute exact, finite sample forecasts for Gaussian ARMA models, multivariate (vector) ARMA models, MIMIC (multiple indicators and multiple causes), Markov switching models, and time varying (random) coefficient models.

Those of you who have used early versions of the sspace object will note that much was changed with the EViews 4 release. We strongly recommend that you read “Converting from Version 3 Sspace” on page 775 before loading existing workfiles and before beginning to work with the new state space routines.

Background

We present here a very brief discussion of the specification and estimation of a linear state space model. Those desiring greater detail are directed to Harvey (1989), Hamilton (1994a, Chapter 13, 1994b), and especially the excellent treatment of Koopman, Shephard and Doornik (1999).

Specification

A linear state space representation of the dynamics of the n × 1 vector yt is given by the system of equations:

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The one-step ahead prediction error is given by,

and the prediction error variance is defined as:

The Kalman (Bucy) filter is a recursive algorithm for sequentially updating the one-step ahead estimate of the state mean and variance given new information. Details on the recursion are provided in the references above. For our purposes, it is sufficient to note that given initial values for the state mean and covariance, values for the system matrices Ξt , and observations on yt , the Kalman filter may be used to compute one-step ahead estimates of the state and the associated mean square error matrix, {at t − 1, Pt t − 1} , the contemporaneous or filtered state mean and variance, { at, Pt} , and the one-step ahead prediction, prediction error, and prediction error variance, { yt t − 1, t t − 1, Ft t − 1} . Note that we may also obtain the standardized prediction residual, et t − 1 , by dividing t t − 1 by the square-root of the corresponding diagonal element of Ft t − 1 .

Fixed-Interval Smoothing

Suppose that we observe the sequence of data up to time period T . The process of using this information to form expectations at any time period up to T is known as fixed-interval smoothing. Despite the fact that there are a variety of other distinct forms of smoothing (e.g., fixed-point, fixed-lag), we will use the term smoothing to refer to fixed-interval smoothing.

Additional details on the smoothing procedure are provided in the references given above. For now, note that smoothing uses all of the information in the sample to provide

ˆ ≡ ≡ E ( )

smoothed estimates of the states, αt at T T αt , and smoothed estimates of the state

variances, Vt ≡ varT( αt) . The matrix Vt may also be interpreted as the MSE of the

smoothed state estimate ˆ t .

α

As with the one-step ahead states and variances above, we may use the smoothed values to form smoothed estimates of the signal variables,

Lastly, the smoothing procedure allows us to compute smoothed disturbance estimates,

variance matrix:

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n − 1 additional periods using no additional signal information. This procedure is repeated for each observation in the forecast sample, s = t + 1, …, t + n .

Dynamic Forecasting

The concept of dynamic forecasting should be familiar to you from other EViews estimation objects. In dynamic forecasting, we start at the beginning of the forecast sample t , and compute a complete set of n-period ahead forecasts for each period n = 1, …, n in the forecast interval. Thus, if we wish to start at period t and forecast dynamically to

t+ n , we would compute a one-step ahead forecast for t + 1 , a two-step ahead forecast for t + 2 , and so forth, up to an n -step ahead forecast for t + n . It may be useful to

note that as with n-step ahead forecasting, we simply initialize a Kalman filter at time

t+ 1 and run the filter forward additional periods using no additional signal information. For dynamic forecasting, however, only one n-step ahead forecast is required to compute all of the forecast values since the information set is not updated from the beginning of the forecast period.

Smoothed Forecasting

Alternatively, we can compute smoothed forecasts which use all available signal data over the forecast sample (for example, at + n t + n ). These forward looking forecasts may be computed by initializing the states at the start of the forecast period, and performing a Kalman smooth over the entire forecast period using all relevant signal data. This technique is useful in settings where information on the entire path of the signals is used to interpolate values throughout the forecast sample.

We make one final comment about the forecasting methods described above. For traditional n-step ahead and dynamic forecasting, the states are typically initialized using the one-step ahead forecasts of the states and variances at the start of the forecast window. For smoothed forecasts, one would generally initialize the forecasts using the corresponding smoothed values of states and variances. There may, however, be situations where you wish to choose a different set of initial values for the forecast filter or smoother. The EViews forecasting routines (described in “State Space Procedures” beginning on

page 771) provide you with considerable control over these initial settings. Be aware, however, that the interpretation of the forecasts in terms of the available information will change if you choose alternative settings.

Estimation

To implement the Kalman filter and the fixed-interval smoother, we must first replace any unknown elements of the system matrices by their estimates. Under the assumption that the t and vt are Gaussian, the sample log likelihood: