Kalman Filtering Example 1: Likelihood Function Evaluation

MATLAB MEX Function Reference

Kalman Filtering Example 1:
Likelihood Function Evaluation

In this example, the log-likelihood function of the SSM is computed using prediction error decomposition. The annual real GNP series, y_t, can be decomposed as

$y_t = \mu_t + \epsilon_t$

where $\mu_t$ is a trend component and $\epsilon_t$ is a white noise error with $\epsilon_t \sim (0, \sigma^2_\epsilon)$ . Refer to Nelson and Plosser (1982) for more details on these data. The trend component is assumed to be generated from the following stochastic equations:

$\mu_t & = & \mu_{t-1} + \beta_{t-1} + \eta_{1t} \ \beta_t & = & \beta_{t-1} + \eta_{2t}$

where $\eta_{1t}$ and $\eta_{2t}$ are independent white noise disturbances with $\eta_{1t} \sim (0, \sigma^2_{\eta_1})$ and $\eta_{2t} \sim (0, \sigma^2_{\eta_2})$ .

It is straightforward to construct the SSM of the real GNP series.

$y_t & = & {H}{z}_t + \epsilon_t \ {z}_t & = & {F}{z}_{t-1} + \eta_t$

where

${H}& = & (1, 0) \ {F}& = & [ 1 & 1 \ 0 & 1 ] \ {z}_t & = & (\mu_t, \beta... ...\eta 1} & 0 & 0 \ 0 & \sigma^2_{\eta 2} & 0 \ 0 & 0 & \sigma^2_\epsilon ]$

When the observation noise $\epsilon_t$ is normally distributed, the average log-likelihood function of the SSM is

$\ell & = & \frac{1}T \sum_{t=1}^T \ell_t \ \ell_t & = & -\frac{N_y}2\log(2\pi)... ...1}2\log(|{C}_t|) - \frac{1}2 \hat{\epsilon}^'_t {C}^{-1}_t \hat{\epsilon}_t$

where C_t is the mean square error matrix of the prediction error $\hat{\epsilon}_t$ , such that ${C}_t = {H}{P}_{t| t-1} {H}^' + {{R}}_t$ .

The following example MATLAB script computes the average log-likelihood function. First, the average log-likelihood function is computed using the default initial values: Z0=0 and VZ0=10⁶I. The second call of function KALCVF produces the average log-likelihood function with the given initial conditions: Z0=0 and VZ0=10^-3I. You can notice a sizable difference between the uncertain initial condition (VZ0=10⁶I) and the almost deterministic initial condition (VZ0=10^-3I) in Output 1.

Finally, the first 16 observations of one-step predictions, filtered values, and real GNP series are produced under the moderate initial condition (VZ0=10I). The data are the annual real GNP for the years 1909 to 1969.


y = [ 116.8 120.1 123.2 130.2 131.4 125.6 124.5 134.3 ...
      135.2 151.8 146.4 139.0 127.8 147.0 165.9 165.5 ...
      179.4 190.0 189.8 190.9 203.6 183.5 169.3 144.2 ...
      141.5 154.3 169.5 193.0 203.2 192.9 209.4 227.2 ...
      263.7 297.8 337.1 361.3 355.2 312.6 309.9 323.7 ...
      324.1 355.3 383.4 395.1 412.8 406.0 438.0 446.1 ...
      452.5 447.3 475.9 487.7 497.2 529.8 551.0 581.1 ...
      617.8 658.1 675.2 706.6 724.7 ];
T = size(y,2);
lead = 0;
Nz = 2;
a = zeros(Nz,1);
F = [1 1; 0 1];
Ny = 1;
b = zeros(Ny,1);
H = [1 0];
var = eye(Ny+Nz)*1e-3;

fprintf('\nLikelihood Evaluation of SSM\n');
fprintf('DATA: Annual Real GNP 1909-1969\n\n');

logl = kalcvf(y, lead, a, F, b, H, var);
fprintf('No initial values are given\n');
fprintf('LOGL = %.2f\n\n', logl);

z0 = a;
vz0 = eye(Nz)*1e-3;
logl = kalcvf(y, lead, a, F, b, H, var, z0, vz0);
fprintf('Initial values are given\n');
fprintf('LOGL = %.2f\n\n', logl);

lead = 1;
z0 = a;
vz0 = eye(Nz)*10;
[logl, pred, vpred, filt, vfilt] = kalcvf(y, lead, a, F, b, H, var, z0, vz0);
fprintf('Filtering and One-Step Prediction\n');
fprintf('  Y         PRED                FILT\n')
format,disp([y(1:16); pred(:,1:16); filt(:,1:16)]')

Output 1: Average Log Likelihood of SSM

Likelihood Evaluation of SSM
DATA: Annual Real GNP 1909-1969

No initial values are given
LOGL = -26313.74

Initial values are given
LOGL = -91883.49

Output 2 shows the observed data, the predicted state vectors, and the filtered state vectors for the first 16 observations.

Output 2: Filtering and One-Step Prediction

  Y         PRED                FILT
  116.8000         0         0  116.7883         0
  120.1000  116.7883         0  120.0997    3.3107
  123.2000  123.4104    3.3107  123.2234    3.1938
  130.2000  126.4172    3.1938  129.5920    4.8826
  131.4000  134.4746    4.8826  131.9381    3.5759
  125.6000  135.5139    3.5759  127.3625   -0.6100
  124.5000  126.7525   -0.6100  124.9012   -1.5607
  134.3000  123.3405   -1.5607  132.3475    3.0651
  135.2000  135.4126    3.0651  135.2379    2.9754
  151.8000  138.2132    2.9754  149.3795    8.7101
  146.4000  158.0896    8.7101  148.4825    3.7761
  139.0000  152.2587    3.7761  141.3621   -1.8201
  127.8000  139.5420   -1.8201  129.8919   -6.7762
  147.0000  123.1157   -6.7762  142.7449    3.3050
  165.9000  146.0499    3.3050  162.3636   11.6833
  165.5000  174.0470   11.6833  167.0227    8.0758

See also

KALCVF performs covariance filtering and prediction

KALCVS performs fixed-interval smoothing

Getting Started with State Space Models

Kalman Filtering Example 2: Estimating an SSM Using the EM Algorithm

References

[1] Nelson, C.P. and Plosser, C.I., Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications, Journal of Monetary Economics, vol 10, pp. 139-162, 1982.

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