Kalman Filter. Lawrence J. Christiano

Transcription

1 Kalman Filter Lawrence J. Christiano

2 Background The Kalman filter is a powerful tool, which can be used in a variety of contexts. can be used for filtering and smoothing. To help make it concrete, we will derive the filter here. basic tool for forecasting, and for computing forecast confidence intervals.

3 State Space/Observer Form Canonical representation of data: where x t = Fx t 1 + u t, Eu t u 0 t = Q, Y data t = a + Hx t + w t, Ew t w 0 t = R, w t, u s are iid over time and uncorrelated for all s, t. u t s uncorrelated with past x t s. w t s uncorrelated with x t s at all leads and lags. Eigenvalues of F all less than 1 in absolute value. Let Y t denote demeaned data, Y t Yt data a, Will derive the Kalman filter, whichsolvestheprojection problem: Y t+j t P! " i Y t+j Y t, xt+j t P hx t+j Y t, j > 0 where Y t [Y 1,..., Y t ]. We simplify by setting w t = 0 for all t.

4 Example of Projection Let the log wage rate, w, and log price level, p, be w = z + u p = z + v, where u and v are uncorrelated with each other and with z. All have zero mean. Suppose you observe w, but what you re really interested in is w p. obviously a move in w that reflects z is not interesting to you. You form the projection, P [w p w] aw, where a solves min a E [w p aw] 2

5 Orthogonality Property of Projections: Projection solves a particular optimization problem: First order condition: projection error min a E [w p aw] 2 z } { E[w p aw]w = 0! a = E (w p) w Ew 2 = E (u v)(z + u) E (z + u) 2 = s 2 u s 2 u + s 2 z = s2 u/s 2 z s 2 u/s 2 z + 1 Orthogonality of projections: projection error uncorrelated with information, w, used in computing the projection.

6 The Filter Will compute projections: x t+1 t, Y t+1 t, recursively: * * * )x 1 0, Y 1 0, )x 2 1, Y 2 1,..., )x T+1 T, Y T+1 T Will simultaneously compute measures of uncertainty: i 0 P t+1 t = E hx t+1 x t+1 t ihx t+1 x t+1 t

7 Forecasts of the Data Will focus primarily on forecasting x t because forecasts of Y t easy to read from forecast of x t 2 3 P! uncorrelated with everything in Y " t z} { Y t+j Y t = P 4Hx t+j + w t+j Y t 5 i = HP hx t+j Y t + P! " w t+j Y t = Hxt+j t. Also, = E = E i 0 E hy t+j Y t+j t ihy t+j Y t+j t i 0 hhx t+j + w t+j Hx t+j t ihhx t+j + w t+j Hx t+j t h * * i 0 H )x t+j x t+j t + w t+j ih)x t+j x t+j t H 0 + w t+j = HP t+j t H 0 + R.

8 First Date of the Filter At t = 0 have Y 0 = f, the empty set. So, x 1 0 = P [x 1 Y 0 ] = Ex 1 = 0, the unconditional expectation. Also, ih i 0 P 1 0 = E hx 1 x 1 0 x 1 x 1 0 = V, say, where V denotes the variance of x 1. Compute V by solving the Ricatti equation: V = E [Fx t 1 + u t ][Fx t 1 + u t ] 0 = FVF 0 + Q Most robust way to find V is V = V in: Set V 0 to be any pos. def. matrix, compute V j+1 = FV j F 0 + Q, j = 0, 1, 2,...

9 An Intermediate Date with the Filter Suppose we have x t t 1, P t t 1 in hand. We now receive a new observation, Y t. Want to compute x t+1 t, P t+1 t. We do this in two steps: First, compute x t t, P t t. Second, compute x t+1 t, P t+1 t.

10 First Step for the Filter Basic recursive property of projections: 2 x t t = x t t 1 + P 6 4 forecast error in x t t 1 new information in Y t not in Y t 1 z } { x t x t t 1 z } { Y t Hx t t 1 {z } Y t t 1 This formula is obviously correct in the special case where the information in Y t allows you to compute the forecast error, x t x t t 1, exactly. Has a learning interpretation you update your old guess, x t t 1, about x t using what is new about the information in Y t, i.e., using Y t Hx t t

11 First Step for the Filter Write i x t t = x t t 1 + P hx t x t t 1 Y t Hx t t 1 h i = x t t 1 + a t Y t Hx t t 1, where the matrix, a t, solves *i 2 min E hx a t x t t 1 a t )Y t Hx t t 1 t First order condition associated with optimality: *i h i 0 E hx t x t t 1 a t )Y t Hx t t 1 Y t Hx t t 1 = 0, which again is the orthogonality of projections.

12 First Step for the Filter First order condition implies: i 0 E hx t x t t 1 ihy t Hx t t 1 * 0 = a t E )Y t Hx t t 1 *)Y t Hx t t 1 or, P t t 1 z } ih i { 0H 0 E hx t x t t 1 x t x t t 1 P t t 1 z } *) * { 0H = a t HE )x t x t t 1 x t x 0 t t 1, so that a t = P t t 1 H 0 ) HP t t 1 H 0* 1.

13 First Step for the Filter We conclude x t t = x t t 1 + P t t 1 H 0 ) HP t t 1 H 0* 1 h Y t Hx t t 1 i. With x t t in hand, we move on to P t t : P t t = i 0 E hx t x t t ihx t x t t 2 orthogonal to (Y t Hx t t 1 ) z } ) *{ = E 6x 4 t x t t 1 a t Y t Hx t t *i 0 hx t x t t 1 a t )Y t Hx t t 1 *i h i 0 = E hx t x t t 1 a t )Y t Hx t t 1 x t x t t 1, by orthogonality. 3

14 First Step for the Filter From the previous slide, *i i 0 P t t = E hx t x t t 1 a t )Y t Hx t t 1 hx t x t t 1 = P t t 1 P t t 1 H 0 ) HP t t 1 H 0* 1 HPt t 1 completing the derivation of x t t and P t t. Now we proceed to the second step, to compute x t+1 t and P t+1 t.

15 Second Step for the Filter By linearity of projections: It follows that: =0 z } { x t+1 t = Fx t t + u t+1 t. x t+1 t = forecast, x t+1 t 1, based on t 1 info, Y t 1 z } { Fx t t 1 Kalman gain matrix, K t new information z ) } { + FP t t 1 H 0 HP t t 1 H 0* z } { 1 i hy t Hx t t 1. Next, P t+1 t...

16 Finally, Second Step for the Filter P t+1 t = i 0 E hx t+1 x t+1 t ihx t+1 x t+1 t = h * ih * i 0 E F )x t x t t + u t+1 F )x t x t t + u t+1 = FP t t F 0 + Q = F 1P ) t t 1 P t t 1 H 0 HP t t 1 H 0* 2 1 HPt t 1 F 0 + Q. Done! We now have * * )x 1 0, P 1 0,..., )x T+1 T, P T+1 T and also * * )x 1 1, P 1 1,..., )x T T, P T T

17 Forecasting We have the one-step-ahead forecast and its uncertainty: Then, x T+1 T, P T+1 T =Fx T+1 T =0 z } { z } { x T+2 T = P [x T+2 Y t ] = FP [x T+1 Y T ] + P [u T+2 Y T ] and so on: x T+j T = F j 1 x T+1 T.

18 Forecasting Want measures of forecast uncertainty. For T + 2: P T+2 T = i 0 E hx T+2 x T+2 T ihx T+2 x T+2 T = h * ih * E F )x T+1 x T+1 T + u T+2 F )x T+1 x T+1 T = FP T+1 T F 0 + Q Similarly, for j > 1 i 0 P T+j T = E hx T+j x T+j T ihx T+j x T+j T h * i = E F )x T+j x T+j T + u T+j h * i 0 F )x T+j x T+j T + u T+j = FP T+j 1 T F 0 + Q

19 Forecasting Note, as j!, P T+j T! V x T+j T! 0 These features follow from the fact that the eigenvalues of F are less than unity in absolute value. Message: for observations far in the future, available data not helpful and might as well just guess the uconditional mean, with forecast error variance equal to unconditionarly

20 Smoothing We have reviewed filtering, which is what is used in forecasting (and, calculation of likelihood). Also useful to do smoothing: P [x t Y T ], t = 1, 2,..., T. Smoothing gives the best guess about the value taken on by a variable that is in the model (like the output gap, or the natural rate of interest), but that is not contained among the observed data. Derivations of the Kalman smoother first derive the Kalman filter, as we did, and then derive the smoother as a second step.