Kalman ﬁlter Recursions – Main Equations the Prediction and Updating

8/14/2019 Kalman filter Recursions Main Equations the Prediction and Updating

1/20

Kalman filter recursions main equations

The prediction and updating equations leading us to calculate the prediciton errors and the

resulting variance are as follows:

Given at1 the MMSE oft1 at time t 1, with

at1 t1 WS(0, Pt1),

the prediction equations are:

at/t1 = Tat1,

and

at/t1 t

0,

2

Pt/t1

where

Pt/t1 = TPt1T + RQR

Given that at/t1 is MMSE oft at time t 1, the MMSE of zt at time time t 1 clearly is,

zt/t1 = ytat/t1.

The associated prediction error is

zt zt/t1

= t = y

t

t at/t1

+ Nt

the expectation of which is zero. Hence,

var t = E(2

t ) = E

ytt at/t1

t at/t1

y

+ E(N2t )

[since cross product terms have zero expectations]

= 2ytPt|t1yt + 2h = 2ft

The updating equations are given by:

at = at|t1 + Pt|t1yt

zt ytat|t1

/ft,

and

Pt = Pt|t1 Pt|t1ytytPt|t1/ft where ft = y

tPt|t1yt + h.

1


2/20

We have to highlight here the role played by,

1. the prediction error given by

t = zt ytat|t1,

and the variance associated with it, 2ft and

2. the Kalman gain given by

Pt|t1yt.

If prior information is available, that is,

0 WS(a0, P0),

where a0 and P0 are known, the Kalman filter recursions can be used to get the prediction errors

and the variance associated with them.

2


3/20

Series A -- Chemical process concentration readings

15

15.5

16

16.5

17

17.5

18

18.5

1 26 51 76 101 126 151 176

Series D -- Chemical process viscocity readings

0

2

4

6

8

10

12

1 51 101 151 201 251 301

Series E -- Wolfer sun spot numbers

0

20

40

60

80

100

120

140

160

180

1 21 41 61 81

Series F -- Yields from a batch chemical process

0

10

20

30

40

50

60

70

80

90

1 11 21 31 41 51 61


4/20

Series A -- ACF

0

0.1

0.20.3

0.4

0.5

0.6

1 6 11 16

Series A -- PACF

-0.2

0

0.2

0.4

0.6

0.8

1 6 11 16

Series D -- ACF

0

0.20.4

0.6

0.8

1

1 6 11 16

Series D -- PACF

-0.2

0

0.2

0.4

0.6

0.8

1

1 6 11 16

Series E -- ACF

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 6 11 16

Series E -- PACF

-1

-0.5

0

0.5

1

1 6 11 16

Series F -- ACF

-0.6

-0.4

-0.2

0

0.2

0.4

1 6 11 16

Series F -- PACF

-0.6

-0.4

-0.2

0

0.2

0.4

1 6 11 16


5/20

Table1:S

ummaryofModelsFitted

Series

NOBS

FittedMode

ls

ResidualVariance

A

197

z

t

0.9

2

(0.0

4)z

t1

=

1.3

8

(0.7

0)

+

et

+0.5

8

(0.0

8)

et1

0.0

98

D

310

zt

0.8

7

(0.0

3)z

t1

=

1.2

2

(

0.2

5)

+

et

0.0

9

E

100

zt

1.5

6

(0.1

0)z

t1

+

1.0

1

(0.1

6)z

t2

0.2

1

(

0.1

0)z

t3

=

11.4

6

(2.8

7)

+

et

230.4

3

z

t

1.4

1

(0.0

7)z

t1

+

0.7

2

(0.0

7)zt2

=

14.5

3

(2.5

2)

+

et

230.4

3

F

70

zt

+

0.3

4

(0.1

3)z

t1

0.2

0

(0.1

3)zt2

=

58.7

6

(11.0

0)

+

et

117.5

6

Note:Valuesin()under

eachestimatedenotestandar

derrors.

Table2:SummaryofDiag

nosticTestsonResiduals

ofFittedModels

Series

NOBS

FittedModels

Q

DF

A

197

zt

0.9

2zt1

=

1.3

8

+

et

+0

.58et1

25.5

6

18

D

310

zt

0.8

7zt1

=

1.2

2

+

et

8.8

2

19

E

100

zt

1.5

6zt1

+

1.0

1zt2

0.2

1zt3

=

11.4

6+

et

18.0

1

17

100

zt

1.4

1zt1

+

0.7

2zt2

=

14.

53+

et

23.7

2

18

F

70

zt

+

0.3

4zt1

0.2

0zt2

=

58.

76+

et

11.8

8

18

1


6/20

Forecasts from stationary and DS models

100

120

140

160

180

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58

Lead time

Forecasts

Forecast standard error

0

20

40

60

1 6 11 16 21 26 31 36 41 46 51 56

Lead time

Std.error


7/20

CPI -- 1955:1 2002:12

35

235

435

635

835

1035

1235

1 51 101 151 201 251 301 351 401 451 501 551

Logarithm of CPI

3.5

3.7

3.9

4.1

4.3

4.5

4.7

4.9

5.1

5.3

5.5

5.7

5.9

6.1

6.3

6.5

6.7

6.9

1 51 101 151 201 251 301 351 401 451 50

WPI -- 1970:1 2000:12

30

80

130

180

230

280

330

1 51 101 151 201 251 301 351

Logarithm of WPI

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

1 51 101 151 201 251 301


8/20

M1: 1955:1 2002:12

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

1 51 101 151 201 251 301 351 401 451 501 551

Logarithm of M1

7.5

8.5

9.5

10.5

11.5

12.5

13.5

1 51 101 151 201 251 301 351 4

M3: 1955:1 2002:12

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

1 51 101 151 201 251 301 351 401 451 501 551

Logarithm of M3

7.5

8.5

9.5

10.5

11.5

12.5

13.5

14.5

1 51 101 151 201 251 301 351 4


9/20

Export: 1970:1 2000:12

1

30

59

88

117

146

175

1 51 101 151 201 251 301 351

Logarithm of Ex

-0.06

0.44

0.94

1.44

1.94

2.44

2.94

3.44

3.94

4.44

4.94

1 51 101 151 20

Imports: 1970:1 2000:12

1

41

81

121

161

201

1 51 101 151 201 251 301 351

Logarithm of im

-0.05

0.45

0.95

1.45

1.95

2.45

2.95

3.45

3.95

4.45

4.95

5.45

1 51 101 151 20


10/20

Food credit: 1970:4 2003:3

0

10000

20000

30000

40000

50000

60000

70000

1 51 101 151 201 251 301 351

Logarithm of food credit

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

1 51 101 151 201 251

Nonfood credit: 1970:3 2003:3

0

100000

200000

300000

400000

500000

600000

700000

800000

1 51 101 151 201 251 301 351

Logarithm of nonfood cred

8

9

10

11

12

13

14

1 51 101 151 201 251


11/20

Fig1: White Noise

-4

-2

0

2

4

1 18 35 52 69 86 103 120 137 154 171 188

Time

Z_

{t}

Fig 2: MA(1) model. Theta=-0.8

-4

-2

0

2

4

1 18 35 52 69 86 103 120 137 154 171 188

Time

z_

{t}

Fig3: MA(1) model; Theta=0.8

-4

-2

0

2

4

1 18 35 52 69 86 103 120 137 154 171 188

Time

z_

{t}

Fig 4: AR(1) model; Phi = 0.65

-4

-2

0

2

4

6

1 18 35 52 69 86 103 120 137 154 171 188

Time

z_

{t}


12/20


13/20

M1: 1955:1 2002:12

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

1 51 101 151 201 251 301 351 401 451 501 551

Logarithm of M1

7.5

8.5

9.5

10.5

11.5

12.5

13.5

1 51 101 151 201 251 301 351 401 451 501 551


14/20

Empirical applications

We shall apply the unit root test procedures discussed above on some major Indian macro series.Since many Indian studies have confirmed the existence of unit roots in many Indian time series,we shall consider our attempts in this section as a confirmatory exercise, besides being a pedagogicdemonstration of unit root test procedures. Hence we shall employ these test procedures on narrow

money series (M1) only as a demonstration. This is monthly data, ranging from 1955:1 to 2002:12 a sample of 576 observations and this data was collected from the various issues of the RBI bulletin.A plot of the series is attached at the end.

Which of the three models that we have discussed, should one choose to employ? An inspectionof the plots reveals that since all the series are upward trending, Case 3 would be better model.But we have to check if this upward movement is better characterized as a drift or as stationaryfluctuations around a time trend. We shall be using the ADF test procedure as well as demonstratethe application of the PP test. But, as an extended exercise, we also shall estimate the other twocases as well and check for the null if these series can be labelled as pure random walk or randomwalk with drift. We select the correct number of lags using Halls general to specific methodology.

All results are discussed at the 5% level.

The following model was estimated for M1 data by OLS, with the number of lags decided byHalls methodology. (standard errors in parentheses)

m1t = 0.086722(0.030965)

+ 0.000126(0.000045)

t + 0.988506(0.004377)

m1t1 0.018542(0.042389)

m1t1 + 0.024024(0.039460)

m1t2

0.008666(0.039064)

m1t3 0.123152(0.039061)

m1t4 0.099201(0.039087)

m1t5 + 0.085646(0.039208)

m1t6

0.104373(0.039127)

m1t7 0.085999(0.039212)

m1t8 0.127945(0.039152)

m1t9 0.025808(0.039217)

m1t10

+ 0.126156(0.039181)m1t11 + 0.361437(0.039554)m1

t12 + 0.083216(0.042464)m1t13

RSS = 0.158604.

Since the available sample size in this case will be 562, the ADF normalized bias test can be calculatedas:

T( 1)

1 1

2

12

13

=562(0.988506 1)

1 + 0.18542 0.024024 0.361437 0.083216= 6.46.

Refering to Table B.5, Case 4 critical values tabulated in Hamilton, the calculated value of6.46 is

much greater than the asymptotic critical value of21.8. Hence, the unit root of null is accepted.The equivalent t statistic for the same null is calculated as,

(0.988506 1)/(0.004377) = 2.63.

Since the calculated t statistic is greater than the asymptotic critical value of3.41, (refer to TableB.6,Case 4) we again conclude that the null of unit root is accepted. Thus, the test conducted sofar accepts that M1 may be modelled as an ARIMA(13, 1, 0) process. Finally, since the distributionof has been derived maintaining that = 0, we need to conduct an F test to test the joint nullof = 1, = 0. But the calculated F test value of 8.07 is greater than the 5% asymptotic criticalvalue of 6.25 but is less than the 1% value of 8.25. Hence the null of unit root with a possible drift,is rejected at the 5% level, but accpeted at the 1% level.


15/20

Next we shall check if the first two cases fit any better. We shall assume the test regression asabove. The results of the regression exercises are given below:

m1t = 0.000276(0.005121)

+ 1.00093(0.000506)

m1t1, RSS = 0.234373

m1t = 1.00096(0.000083)

m1t1, RSS = 0.234374

Here the F statistic for the joint null = = 0, = 1 against the general alternative that isgiven by the test regression is calculated as

F = (0.234374 0.158604)/15(0.158604/562) = 17.9,

which is much greater than the 5% asymptotic critical value of 4.68 and this means, we can reject thenull of unit root without a drift. Next we compute the F statistic to test the joint null = 0, = 1a follows:

F = (0.234373 0.158604)/14(0.158604/562) = 19.18

which is again much greater than the 5% asymptotic critical value of 6 .25. So once again the null isrejected.

But ADF test is sensitive to the correct lag size. So these rejections have to be viewed in thatlight. Perhaps M1 has a significant time trend, a drift as well as a unit root. So is there a case for aquadratic time trend as against the null ofsecond unit root? Since the plot of the levels data clearlyshows a quadratic trend, I did a preliminary check calculating the autocorrelation values of residualsobtained from models with (1) a linear time trend only and (2) with a quadratic time trend added.But the lag 10 autocorrelation values remained as high as 0.893 and 0.689 for models (1) and (2)respectively even after these regressions. Hence, it looks like, our ADF test with the joint null, at

the 1% significance level, is likely a better model! We shall discuss more about this series later.


16/20

Unit roots versus breaking trends alternative

The null and the alternatives have been parameterized as follows:

Null Hypotheses:

Model (A) zt = + dDT Bt

+ zt1 + et

Model (B) zt = 1 + zt1 +

2 1

DUt + et

Model (C) zt = 1 + zt1 + dD

T Bt

+

2 1

DUt + et where

D

T Bt

= 1 if t = TB + 1, 0 otherwise;

DUt = 1 if t > TB, 0 otherwise

and et could be either an uncorrelated innovation sequence or a stationary process. Instead of

considering the alternative hypothesis that zt is a stationary series around a deterministic linear

trend with time invariant parameters, Perron considers the following alternative models:

Model (A) zt = 1 + t +

2 1

DUt + et

Model (B) zt = + 1t +

2 1

DTt

+ et

Model (C) zt = 1 + 1t + +

2 1

DUt +

2 1

DTt + et

where

DT

t = t TB, and DTt = t if t > TB and 0 otherwise.

Here TB refers to the time of break, that is, the period at which the change in the parameters of the

trend function occurs.

Extension to more general processes: ets correlated

Using the regression under Case 1 and the critical values sited above is valid only for uncorrelated

es. When there is correlation among the error terms, Perron suggests the following model which is

along the lines of ADF testing procedures and uses the literature on outlier specification. He nests

the null and the alternative models in the following fashion:

zt = A + ADUt +

At + dAD

T Bt

+ Azt1 +k

i=1

cizti + et,

zt = B + BDUt +

Bt + BDTt

+ Bzt1 +k

i=1

cizti + et,

zt = C + CDUt +

Ct + CDTt + dCD

T Bt

+ Czt1 +k

i=1

cizti + et.

1


17/20

Figure -- 1 Logarithm of Nominal wages

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

88.2

8.4

8.6

8.8

9

9.2

1 3 5 7 9 11 1 3 15 1 7 19 2 1 23 2 5 2 7 29 3 1 33 3 5 37 3 9 41 4 3 45 4 7 49 5 1 5 3 55 5 7 59 6 1 63 6 5 67 6 9 71

Figure --3 Logarithm of common stock prices

1

1.5

2

2.5

3

3.5

4

4.5

5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

Figure 2 -- Logarithm of quarterly real GNP

6.9

7.1

7.3

7.5

7.7

7.9

8.1

8.3

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151 157


18/20

Figure 4 -- CDF of alpha under the

"crash" model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 5 -- CDF of alpha under the

"breaking trend" model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2


19/20

Table 1

Results of ADF unit root test procedure

Regression: zt = + t + zt1 +k

i=1

izt1 + et

Series k t t t RSS S(e

M11955:1-2002:12 13 0.087 2.80 0.1103 2.84 0.989 -2.63 0.159 0.3101955:1-1980:12 13 0.229 2.95 0.3103 2.84 0.969 -2.92 0.09 0.3101981:1-2002:12 14 0.363 1.72 0.4103 1.60 0.965 -1.66 0.045 0.210

M3 7 0.2120 3.25 0.11103 3.21 0.974 -2.81 0.961 0.1710

CPI1955:1-2002:12 12 0.046 3.46 0.8104 3.32 0.987 -3.22 0.041 0.74101955:1-1974:12 12 0.089 2.11 0.1103 2.26 0.975 -2.09 0.02 0.9101975:1-2002:12 12 0.123 2.48 0.1103 2.36 0.975 -2.41 0.00 0.510

WPI1970:1-2000:12 12 0.151 5.09 0.3103 4.95 0.957 -5.01 0.09 0.8101970:1-1974:8 8 -0.135 -1.1 0.7103 2.20 1.03 1.06 0.004 0.1110

1974:9-2000:12 12 0.153 3.76 0.25103 3.64 0.962 -3.68 0.02 0.6810

Exports1970:1-2000:12 13 0.036 2.70 0.6103 1.57 0.957 -1.52 3.54 0.0111970:1-1985:2 12 0.084 2.69 0.12102 1.31 0.896 -1.40 0.09 0.016

1985:3-2000:12 12 0.147 1.41 0.305103 0.38 0.969 -0.65 0.009 0.5510

Imports 12 0.069 3.96 0.1102 2.39 0.893 -2.43 0.0149 0.014

Food credit1970:4-2003:3 12 0.227 3.92 0.4103 3.54 0.962 -3.89 0.09 0.0101970:4-1987:7 13 0.300 2.50 0.75103 1.90 0.950 -2.35 0.003 0.0121987:8-2003:3 13 0.871 4.79 0.21102 4.18 0.89 -4.66 0.09 0.8510

Non-food credit 12 0.319 3.10 0.5103 3.00 0.962 -3.00 0.001 0.210

Note: k for all models was fixed using Halls general-to-specific method. All data are in logs. S(e)refers to the residual variance.

Table 2

Sample autocorrelations of the detrended series

Series T Variance r1 r2 r3 r4 r5 r6 r7 r8M1 576 0.031 0.99 0.97 0.96 0.95 0.93 0.92 0.91 0.91 CPI 576 0.009 0.99 0.97 0.95 0.93 0.91 0.89 0.87 0.86 WPI 372 0.004 0.98 0.95 0.91 0.86 0.82 0.78 0.74 0.70

Exports 372 0.042 0.74 0.72 0.69 0.70 0.68 0.70 0.70 0.68 Food credit 396 0.164 0.93 0.83 0.76 0.71 0.68 0.64 0.59 0.54

Note: All series were detrended using Model A under the alternative hypothesis.

1


20/20

Table3

ResultsofPe

rronunitroottestproced

ure

Regression,ModelA:zt=+DU

t+t+dD(TB)t+zt1

+

k i=1

izt1+et

Series

T

TB

t

t

t

d

td

t

S(e)

M1

576

312

0.54

0.093

3.05

-0.006

-1.88

0.0001

3.59

-0.003

-0.54

0.988

-2.77

0.0003

CPI

576

240

0.42

0.045

3.51

-0.001

-0.83

0.0001

3.47

0.004

0.79

0.986

-3.39

0.0001

WPI

372

57

0.15

0.131

4.73

-0.006

-2.81

0.0002

4.72

0.001

0.19

0.962

-4.67

0.0001

Exports

372

183

0.49

0.020

1.05

-0.021

-0.76

0.001

1.81

-0.015

-0.35

0.935

-1.78

0.012

oodcredit

396

209

0.53

0.44

5.51

-0.12

-3.82

0.001

4.97

-0.05

-0.43

0.920

-5.45

0.0104

Note:=T

B/T

andkfor

allthemodelswasfixedat12asbefore.Alldataareinlog

s.S(e)isvarianceoftheresid

uals.

Table4

ResultsofZivot-Andrewsunitroottestprocedure

Mo

delA

Mod

elB

Mode

lC

Series

t-statistics

Breakyear

tstatistics

Breakyear

tstatistics

Breakyear

M1

-3.75

1989:2

(410)

-3.35

1980:5

(305)

-3.57

1980:4

(304)

CPI

-4.64

1958:10

(58)

-5.02

1970:11

(191)

-5.12

1968:1

(157)

WPI

-5.68

1972:5

(29)

-5.67

1973:5

(41)

-5.94

1977:2

(86)

Exports

-2.81

1979:5

(113)

-2.20

1985:6

(186)

-3.12

1977:4

(88)

Imports

-3.83

1984:2

(170)

-3.12

1974:2

(50)

-3.89

1972:11

(35)

Foodcredit

-5.72

1987:3

(204)

-4.38

1975:10

(67)

-5.71

1987:3

(204)

on-foodcredit

-5.25

1978:9

(102)

-5.14

1983:8

(161)

-5.52

1980:6

(123)

Kalman ﬁlter Recursions – Main Equations the Prediction and Updating

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Transcript of Kalman ﬁlter Recursions – Main Equations the Prediction and Updating