MODELLING AUSTRALIAN BANK BILL RATES :
A KALMAN FILTER APPROACH
By
Ramaprasad Bhar
School of Finance and Economics
University of Technology, Sydney
P.O. Box 222, Lindfield, NSW 2070
Tel. 02 330 5422 Fax. 02 330 5515
ABSTRACT
This paper examines the applicability of the Kalman Filter technique to forecast future spot
interest rates, based upon the expectation hypothesis of the term structure of interest rates, in the
Australian bank bill market. In this approach, regression estimates are based on the last period's
estimate together with data from the current period. In contrast to constant parameter models,
this allows effective use of information underlying the process driving the evolution of the
parameters. For the period tested, forecasting accuracy of such a time-varying parameter model
shows marked improvement over a constant parameter model.
Acknowledgements:
I am particularly indebted to one anonymous referee for very insightful comments on the earlier
versions of the paper, and would like to thank the editor, Rob Brown, for helpful suggestions.
Australian Bank Bill Rate: Kalman Filter Approach
1.
INTRODUCTION:
A number of recent research studies have found that Treasury Bill futures provide a better
indication of future spot rates than the forward rates implied by the current term structure of
interest rates. Notable among these studies are Cole, Impson and Reichenstein [1991], Hafer,
Hein and MacDonald [1992], Kamara [1990] and MacDonald and Hein [1989]. Studies like Fama
[1984], Hamburger and Platt [1975] and Kane [1983] suggest that the poor performance of the
implied forward rate as a predictor of the future spot rate may be due to several possibilities such
as the existence of a risk premium, policy changes or, possibly, inadequate model specification.
Use of futures market data to forecast future spot rates is restricted to the futures contract
delivery periods. Another approach, in this respect, is to regress future spot rates (ex-post basis)
on some period ahead forward rates implicit in the term structure of interest rates. This Ordinary
Least Square (OLS) method assumes constancy of parameters over the sample period and this
assumption is a potential source of model mis-specification, as shown by Chiang and Kahl [1991].
From the forecasting point of view, this may lead to the least cost solution but may result in suboptimal forecast error.
In this study standard statistical techniques are applied to establish time variation of the
regression parameters in the Australian bank bill market. To account for this time variation of the
parameters in the model, the Kalman Filter technique is employed as an adaptive method of
estimation and forecasting. The result shows marked improvement in forecasting ability over the
time invariant parameter model.
The structure of the paper is as follows. Section 2 describes the Australian bank bill market and
in Section 3 alternative models are developed. Data used in the model estimation and forecasting
2
Australian Bank Bill Rate: Kalman Filter Approach
are described in Section 4 and the analysis of result follows in Section 5. The paper is then
concluded in Section 6.
2.
THE AUSTRALIAN BANK BILL MARKET:
The physical bank accepted bills market is a significant short-term financial market in Australia.
These bills are issued on a discount basis for a period of up to 180 days but the 90-day market is
the most active. These bills have a face value of $500,000 and are usually traded in lots of $5
million. Rates are quoted on yield p.a. basis with 365 days in a year. The relationship between
the face value (F) and the market value (P) is given by,
P = (365 * F)/(365 + yield * days to maturity)
(1)
The data in Table 1 show the average of weekly assets ( bills and certificate of deposits ) of the
authorised dealers between the years 1981 and 1992. The relative popularity of bills over CDs is
evident until the late 1980s. The introduction of interest rate futures contracts based upon bank
bills also reflects the importance of this instrument in the Australian financial market. Carew
[1991, page 117], however, points out that the changes in the statutory reserve deposit (SRD)
requirement introduced in 1988, may shift the relative importance towards CDs. There is some
indication of this in Table 1 during the years 1990 - 1992. Carew [1991, page 118] also argues that
"... the substantial volume of bills lines written for several years and still outstanding means that
bills of exchange will be around for some time.".
3.
MODEL BASED ON THE EXPECTATION HYPOTHESIS:
The expectation hypothesis argues that a forward interest rate corresponding to a certain period
is equal to the expected future spot interest rate for that period. Muth [1961] identifies several
3
Australian Bank Bill Rate: Kalman Filter Approach
key properties for this expectation to be "rational expectation". One of them is the property of
unbiasedness.
To model this expectation, following Friedman [1980], the bank bill rate is
regressed on k-period ahead forecast:
r t = α + β t-k f t + ε 1,t
(2)
where r t is the actual bill rate at time t and t-k f t is the forecast of r t made at t-k. α and β are
parameters of the model and ε 1 is the usual regression error term. Equation (2) is referred to as
the time invariant expectation model in this paper. The unbiasedness property is examined by
the null hypothesis: H0: (α,β) = (0,1). Also, the residuals in the regression equation should be free
from serial correlation.
This joint test of the parameters is equivalent to testing the pure
expectations hypothesis.
An implicit assumption in this model is that the behaviour of the
parameters is time invariant.
So, failure to reject the null hypothesis may imply that the
estimated parameters are sensitive to the sample period used.
For the model to be of practical use, it should be capable of capturing the dynamics of market
behaviour over time. One way to achieve this would be to employ an adaptive filtering technique
where the estimates of the parameters at time t would be related to the estimate of the previous
period in some simple way.
This is an heuristic approach and lacks a sound theoretical
foundation since it does not take into account the underlying stochastic process describing the
change (see for more detail Kahl and Ledolter [1983]).
In order to apply a more rigorous
technique the parameters are explicitly allowed to follow a random process over time and the
resulting method is referred to as a Kalman Filter.
A Kalman Filter commonly refers to estimation of state space models where there are two parts, i)
the transition equation and ii) the measurement equation. The transition equation describes the
evolution of the state variables (i.e. the parameters) and the measurement equation describes how
4
Australian Bank Bill Rate: Kalman Filter Approach
the observations are actually generated from the state variables. Regression estimates for each
time period in this case are based upon the previous period's estimates and data up to and
including the current time period. The model is defined with the following equations:
r t = α t + β t t-k f t + ξ t
where ξ t ∼ N (0, σ2 )
(3)
α t = α t-1 + η1 t and β t = β t-1 + η2 t , where
(4)
η1 t ∼ N ( 0, σ12 ) and η2 t ∼ N ( 0, σ22 ) ,
(5)
and the initial values of α and β are assumed to follow, α0 ~ N( 0,σα2 ) and β0 ~ N( 0,σβ2 ). The
variables r t and t-k f t are as defined in relation to equation (2) and ξ and η are classical well
behaved disturbances associated with the measurement equation and transition equations
respectively.
Prior values of the coefficients and the variances are usually obtained from a
regression of the initial observations of the sample.
Appropriate starting values or initial
conditions are crucial in Kalman Filter implementation. Harvey [1990, pp. 121-122] points out that
when the transition equation is non-stationary (as it is in this study) the initial distribution of the
state variables should be specified in terms of diffuse prior. Harvey [1990, pp. 121-122] also
suggests that by setting σα2 and σβ2 to a large but finite number, a good approximation can be
obtained. In the implementation of the Kalman Filter in this paper, both σα2 and σβ2 are set to
10000. The time invariant model corresponds to the special case where σ12 = σ22 = 0. The
Kalman Filter1 approach should, thus, provide a much improved estimate of the relationship
between the future spot rate and the forward rate.
In modelling interest rates, a random walk process is also suggested by various researchers. For
example, Berger and Craine [1989] show that "the random walk model might be a good
1The method of estimation in Kalman Filter is maximum likelihood conditional on the data observed up to that point.
In that sense it can be viewed as a Bayesian method.
5
Australian Bank Bill Rate: Kalman Filter Approach
approximation for forecasting future long-term interest rates" although it is not suited for testing
market efficiency. In this paper an AR(1)2 model based on equation (6) is estimated to provide
another basis for comparison with the Kalman Filter approach:
r t = α + β r t-1 + ε 2,t
(6)
where α and β are constant parameters similar to the time invariant model; r t-1 is the actual bill
rate at time t-1; ε 2 is the usual regression error term.
The forward rates used in the time invariant and the Kalman Filter approaches are computed
from the Australian 90-day and 180-day bank bill rates. The method used is similar to that
developed by Stigum [1981] for use with discount securities rather than coupon equivalent yields.
This is referred to as the "bill parity" method and is given by:
dr* = [ 1 - ( 1 - dr1 t1/365 ) / (1 - dr2 t2/365) ] * 365/(t1 - t2)
(7)
where, dr1 is the discount rate on the longer term to maturity instrument which has t1 days to
maturity, dr2 is the discount rate for the shorter term to maturity instrument which has t2 days to
maturity (t1 > t2); dr* is the forward rate for the period t2 and t1. The "bill parity" method
provides an accurate estimate for discount securities such as bank bills, since it is constructed to
exactly match the holding period returns of two strategies. The first is buying the longer-term bill
and the second is buying the shorter-term bill and rolling the investment over into the next shortterm bill.
2Random walk is a special case of an AR(1) process. This was correctly pointed by an anonymous referee and the
reference is Kendall and Ord [1990, page 57].
6
Australian Bank Bill Rate: Kalman Filter Approach
In order to compare the three approaches i.e. the time invariant parameter, AR(1) and the Kalman
Filter, each of the three models is examined for general forecast accuracy with one step ahead
forecasts beyond the sample period. The accuracy of forecast is measured in terms of root mean
square error (RMSE), mean absolute error (MABE) and Theil's U-statistic. To establish whether
there are statistically significant differences in relative predictive abilities, the Ashley, Granger
and Schmalensee [1980] (AGS) method is then employed. Further explanation of the AGS test is
included in Section 5 where the results are analysed.
4.
DATA USED IN THE STUDY:
Daily closing rates (end of day) for Australian bank accepted bills for two maturities, 90 days and
180 days were obtained from the Reserve Bank of Australia. The data cover the period January
1990 to March 1993. Figure 1 reveals the general nature of bank bill rates during this period,
containing month-end figures only. The falling interest rate outlook over this entire period is also
evident from the fact that the 180-day rates are typically below the 90-day rates.
The "bill parity" relationship, as explained earlier, is applied to compute the 90-day rate in 90
days’ time i.e. t-kft, in terms of the notation of this paper, where k = 90. This implied forward
rate is the regressor in both the time invariant and the time varying parameter models. To allow
computation of the forward rate, the sample period used to estimate the model parameters begins
in April 19913. In all, 520 observations between 2nd April 1990 and 24th September 1992 are
utilised for parameter estimation and 100 observations thereafter are used for forecasting.
predictive accuracy of the models are tested over the forecast period.
The
Table 2 reports the
descriptive statistics of the estimation and the forecast periods for both the 90-day and 180-day
bill rates.
3This is because there are no observations for the regressor, the 90-day forward rate, for the first 90 days of the data.
7
Australian Bank Bill Rate: Kalman Filter Approach
5.
ANALYSIS OF THE RESULTS: 4
Table 3 provides the statistics of the time invariant parameter model. The adjusted R-square of
0.96 is sufficiently high indicating reasonable explanatory power of the relationship expressed by
equation (2). The estimated parameters are also highly significant as can be seen from the tstatistics.
Granger and Newbold [1974] point out that as most macro economic data are
integrated, the standard significance tests in regressions involving levels of such data may be
misleading. They suggest that a critical t-value of 11.25, rather than the standard normal value of
1.96, be used for significance tests. The parameter β of the time invariant model still appears
significant even when the higher critical t-value is used. Furthermore, the least square estimators
are unbiased and the Gauss-Markov theorem holds whether or not the regressor is stochastic (see
Greene [1993], page 184).
The parameter estimates of the time invariant model, however, appear to be inefficient for two
reasons.
First, as
Table 3 confirms, the OLS residuals are serially correlated. Second, the
assumption of absence of heteroscedasticity is violated. White's direct test for the presence of
heteroscedasticity (White [1980]) and the Breusch-Pagan statistic (Breusch-Pagan [1979]) both
support the presence of heteroscedasticity.
The F-statistic in Table 3, for the joint test of the null hypothesis that α = 0 and β = 1, indicates
that the hypothesis can not be accepted. This is consistent with Chiang and Kahl [1991]. Further,
testing constancy of the parameter estimates with a Chow test (see Greene [1993] pp. 211-212)
4All statistical computations were made using the package TSP/386 on a i486 personal computer running at 33MHZ.
5One of the referees correctly points out that the high R-square may be spurious given the values of DW statistic and
Q(10) in Table 3. The Granger and Newbold (1974) approximation may also be optimistic, particularly, due to the strong
large sample property obtained by Phillips (1986). The parameter β in Table 3 remains significant even after applying
the correction suggested by Phillips (1986).
8
Australian Bank Bill Rate: Kalman Filter Approach
establishes that these parameters are not time invariant within the sample period6. As part of the
residual diagnostic in Table 3, the Augmented Dickey-Fuller test suggests that the residual series
is I(1) i.e. it has a unit root. This indicates that the structural relationship represented by the time
invariant parameters in equation (2) are non-stationary.
Table 4 provides statistics for the AR(1) model. This model has a much smaller sum of squared
residuals than the time invariant parameter model and consequently a higher adjusted R-square
value. The diagnostic statistics suggest that the residuals are largely serially uncorrelated and
there is no evidence of heteroscedasticity. The t-statistic of the estimated parameter α is not
statistically significant, although the joint test hypothesis that α = 0 and β = 1, is significant,
implying rejection of the hypothesis.
The results in Table 4 are similar to those obtained by Chiang and Kahl [1991] but offer much
stronger support for the AR(1) model. Chiang and Kahl [1991] use quarterly observations rather
than daily observed rates used here7.
The results for the Kalman Filter approach in Table 5 show time variation of the parameters.
Quarterly values of mean α t and mean β t and the corresponding standard deviations covering
the entire sample period are given in this table.
The final estimates of the Kalman Filter
procedure for α t and β t show that these are significantly different from 0 and 1 respectively.
Both these parameter series exhibit a first order serial correlation coefficient exceeding 0.99.
These observations suggest that the parameters α t and β t are both time varying and the slope
coefficient is generally not equal to 1.
6The actual hypothesis tested is that the parameters are the same across the mid-point of the sample.
7It may be that daily data exhibit stronger AR(1) behaviour than quarterly data.
9
Australian Bank Bill Rate: Kalman Filter Approach
This result is consistent with the existence of a risk premium as suggested by Lauterbach [1989].
However, the average of α t (0.09134) is not equal to the intercept parameter of the time invariant
model ( -0.0022) as suggested by Garbade and Wachtel [1978]. The average of
β t (0.07028) is
also not equal to the slope parameter of the time invariant model (0.9229). Furthermore, it may
be argued that a piecewise OLS approach as suggested by Fama [1984] is applicable here but the
time variation of the parameters observed in this study points to a continuous variation rather
than step changes.
The performance of the three models is compared using two approaches. The first, based on the
estimation period, compares the sum of squared residuals and residual variance. The second,
using a forecast period, compares the models using the root mean squared error (RMSE), mean
absolute error (MABE) and Theil's U-statistic. In the estimation period comparison, the Kalman
Filter approach achieves both the minimum sum of squared residuals and the minimum error
variance, which represents an order of magnitude improvement over the other two models. This
is followed by the random walk model.
The second comparison reported in Table 6 analyses the one step ahead forecasting accuracy of
the three models. In terms of the three measures, the time invariant parameter model performs
poorly. The performance of the AR(1) and the Kalman Filter approach is similar. To identify any
significant difference in the relative predictive abilities documented in Table 6, the Ashley,
Granger and Schmalensee [1980] (AGS) procedure is employed. The AGS test is a pairwise
examination of statistical difference between two respective mean squared errors (MSEs).
Essentially, the test involves estimating the regression equations shown in Table 7, and the
hypothesis tested is also defined in that table. Since it is a pairwise test and there are three
techniques involved, Table 7 analyses all three possible combinations. Of interest, however, is the
comparison between the random walk and the Kalman Filter approaches since Table 6
establishes the poor performance of the time invariant parameter model . The last panel in Table
10
Australian Bank Bill Rate: Kalman Filter Approach
7 compares these two techniques and the statistics of a3 and b3 establish the dominance of the
random walk model, although the coefficient values are small.
6.
CONCLUSIONS:
This paper examines the relationship between forward rates and future spot rates in the
Australian bank bill market with the help of time varying parameters. Of the two models which
relate the implied forward rates to future spot rates, there is evidence to reject the time invariant
parameter model though the Kalman Filter technique appears to capture the essence of time
variation in the parameters.
In the Kalman Filter approach both the intercept parameter ( α t ) and the slope parameter ( β t )
are random walk (plus noise) in nature and the measurement equation is a linear transformation
of these parameters. Further, the estimation period analysis suggests rejection of the expectation
hypothesis. This result also differs from the AR(1) model, where the model basically relates
"tomorrow's" rate to "today's" rate.
In terms of forecasting ability, both the AR(1) and the Kalman Filter approaches perform better
than the time invariant parameter model. There is also some evidence that the AR(1) model is
better able to forecast than the Kalman Filter approach. In this respect, the result differs from the
findings of Chiang and Kahl [1991] in their study of US Treasury Bill rates. Perhaps one reason
for this difference is the use of daily data whereas Chiang and Kahl use
area, however, requires further investigation.
11
quarterly data. This
Australian Bank Bill Rate: Kalman Filter Approach
REFERENCES
Ashley, R., Granger, C. W. J, Schmalensee, R., [1980], "Advertising and Aggregate Consumption:
An Analysis of Causality", Econometrica, 48, 1149-67.
Berger, A. N., and Craine, R., [1989], "Why Random Walk Models of the Term Structure Are Hard
To Reject", Journal of Business and Economic Statistics, 7, 161-167.
Breusch, T., and Pagan, A., [1979], "A Simple Test for Heteroscedasticity and Random Coefficient
Variation", Econometrica, 47, 1287-1294.
Carew, Edna, [1991], The Financial Markets in Australia, Allen & Unwin, North Sydney.
Chiang, T. C. and Kahl, D. R., [1991], "Forecasting The Treasury Bill Rate: A Time Varying
Coefficient Approach", Journal of Financial Research, 4, 327-336.
Cole, S. C., Impson, M., Reichenstein, W., [1991], "Do Treasury Bill Futures Rates Satisfy Rational
Expectation Properties?", Journal of Futures Markets, 11, 591-601.
Fama, E. F., [1984], "The Information in the Term Structure", Journal of Financial Economics, 13, 509528.
Friedman, B. M., [1980], "Survey Evidence on the 'Rationality' of Interest Rate Expectations",
Journal of Monetary Economics, 7, 453-465.
Garbade, K., and Wachtel, P., [1978], "Time Variation in the Relationship Between Inflation and
Interest Rates", Journal of Monetary Economics, 4, 755-765.
Granger, C and Newbold, P., [1974], "Spurious Regressions in Econometrics", Journal of
Econometrics, 2, 111-120.
Greene, W. H., [1993], Econometric Analysis, Macmillan Publishing Company, New York.
Hafer, R. W., Hein. S. E., MacDonald, S. S., [1992], "Market and Survey Forecasts of the ThreeMonth Treasury Bill rate", Journal of Business, 65, 123-138.
12
Australian Bank Bill Rate: Kalman Filter Approach
Hamburger, M. J. and Platt, E. N., [1975], "The Expectation Hypothesis and the Efficiency of the
Treasury Bill Market", Review of Economics and Statistics, 57, 190-199.
Harvey, A. C., [1990], Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge
University Press, Cambridge.
Kahl, D. R. and Ledolter, J., [1983], "A Recursive Kalman Filter Forecasting Approach",
Management Science, 29, 1325-1333.
Kamara, A., [1990], "Forecasting Accuracy and Development of a Financial Market: The Treasury
Bill Futures Market", Journal of Futures Markets, 10, 397-405.
Kane, E. J., [1983], "Nested Tests of Alternative Term Structure Theories", Review of Economics and
Statistics, 65, 115-123.
Kendall, M. and Ord, J. K., [1990], Time Series, Oxford University Press, New York.
Lauterbach, B., [1989], "Consumption Volatility, Production Volatility, Spot-Rate Volatility, and
the Returns on Treasury Bills and Bonds", Journal of Financial Economics, 24, 155-179.
MacDonald, S. S., and Hein, S. E., [1989], "Futures Rates and Forward Rates as Predictors of NearTerm Treasury Bill Rates", Journal of Futures Markets, 9, 249-262.
Muth, J. F., [1961], "Rational Expectations and the Theory of Price Movements", Econometrica, July,
315-335.
Phillips, P., [1986], "Understanding Spurious Regressions", Journal of Econometrics, 33, 311-340.
Stigum, M., [1981], Money Market Calculations: Yields, Break-Evens, and Arbitrage, Homewood, Ill.,
Dow-Jones-Irwin.
White, H., [1980], "A Heteroscedasticity-Consistent Covariance Matrix Estimator and a Direct
Test for Heteroscedasticity", Econometrica, 48, 817-838.
13
Australian Bank Bill Rate: Kalman Filter Approach
Table 1
Authorised Dealers - Selected Assets a
Year
Bills b
CDs c
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
258
265
361
449
306
286
367
502
308
383
405
161
25
85
177
231
213
207
115
214
274
422
597
226
a Average of weekly figures in $ millions.
Source: Lewis, M. K., and Wallace, R. H., [1993], The Australian Financial
System Longman Chesire, South Melbourne, page 372.
b Commercial bills accepted or endorsed by banks.
c Includes some other bank securities.
14
Australian Bank Bill Rate: Kalman Filter Approach
Table 2
Dataset Descriptive Statistics
90 Day Bill Rate
Statistics
Mean
Std. Dev.
Skewness
Kurtosis(Excess)
Maximum
Minimum
180 Day Bill Rate
Estimation
Forecast
Estimation
Forecast
0.10285
0.03008
0.07637
-1.21163
0.15300
0.05450
0.05811
0.00172
-1.20196
3.59484
0.06000
0.05240
0.10172
0.03007
0.08388
-1.18760
0.15270
0.05320
0.05827
0.00218
-1.94675
2.44954
0.06130
0.05180
Estimation period : 2nd April 1990 - 24th September 1992 (520 observations)
Forecast period : 28th Sept 1992 - 22nd March 1993 (100 observations)
15
Australian Bank Bill Rate: Kalman Filter Approach
Table 3
Statistics of Time Invariant Expectation Modela
Regression:
Sample size
Adjusted R-squared
Sum of squared residual
Residual Diagnostic
DW-statistic
Ljung-Box Q(10)
Augmented Dickey-Fuller test
Breusch-Pagan heteroscedasticity test
White's heteroscedasticity test
Error variance
Co-efficients
α
β
H0: α = 0, β = 1; F-test
Chow-test (sample split at mid point)
520
0.96
0.01523
0.0498
3775.14 (0.000)
-2.26 (0.514)
15.32 (0.000)
25.06 (0.000)
0.2941E-04
-0.0022 (0.022)
0.9229 (0.000)
1123.03 (0.000)
29.06 (0.000)
a
r t = α + β t-90 f t + ε1,t
r t = 90 Day Rate at t, t-90 f t = 90-day forward rate at t implied by 90day and 180-day rate at t-90, ε1 = Regression residual
Numbers in parentheses are p-values.
16
Australian Bank Bill Rate: Kalman Filter Approach
Table 4
Statistics of AR(1) Walk Model a
Regression:
Sample size
Adjusted R-squared
Sum of squared residual
Residual Diagnostic
DW-statistic
Ljung-Box Q(10)
Augmented Dickey-Fuller test
Breusch-Pagan heteroscedasticity test
White's heteroscedasticity test
Error variance
Co-efficients
α
β
H0: α = 0, β = 1; F-test
Chow-test (sample split at mid point)
520
0.99
0.00029
1.79
12.78 (0.236)
-20.48 (0.000)
0.93 (0.334)
1.55 (0.462)
0.5612E-06
-0.41E-04 (0.697)
0.9986 (0.000)
15.74 (0.000)
1.06 (0.347)
a
r t = α + β r t-1 + ε2, t
r t = 90-day rate at t, r t-1 = 90-day rate at t-1 , ε2 = Regression
residual
Numbers in parentheses are p-values.
17
Australian Bank Bill Rate: Kalman Filter Approach
Table 5
Statistics of Kalman Filter Approach a
Sum of squared residual
Error variance
Mean α t
Mean β t
Final estimate of α t
0.0001782
0.0000003
0.09134
0.05377
0.04829
(0.000) b
0.07028
(0.001) b
Final estimate of β t
Time variation of α and β
Period
Mean α
Std. Dev. α
Mean β
Std. Dev. β
Apr. - Jun. '90
Jul. - Sep. '90
Oct. - Dec. '90
Jan. - Mar. '91
Apr. - Jun. '91
Jul. - Sep. '91
Oct. - Dec. '91
Jan. - Mar. '92
Apr. - Jun. '92
Jul. - Sep. '92
0.15604
0.14994
0.12094
0.10801
0.09934
0.09161
0.07611
0.06750
0.06244
0.05259
0.00621
0.00365
0.01046
0.00364
0.00435
0.00298
0.00542
0.00149
0.00333
0.00203
-0.03250
-0.05494
0.03588
0.07174
0.07889
0.08032
0.07905
0.09070
0.06618
0.07744
0.03691
0.01679
0.04417
0.01543
0.00484
0.00231
0.00539
0.00499
0.00757
0.00270
a
r t = α t + β t t-90 f t + ε3, t
r t = 90-day rate at t, t-90 f t = 90-day forward fate at t implied by 90-day and
180-day rate at t-90, ε3 = Regression residual
α t and β t time varying co-efficients; b p-values for t-distribution testing
parameter = 0.
18
Australian Bank Bill Rate: Kalman Filter Approach
Table 6
Forecasting Accuracy a
RMSEb
MABEc
U-Statd
Time Invariant Expectation
0.00712
0.00627
0.06462
AR(1) Model
0.00038
0.00024
0.00330
Kalman Filter Model
0.00055
0.00026
0.00475
a
Forecast period is 100 days after 24th September 1992; One step ahead
forecast by the random walk model and Kalman Filter. Forecast by time
invariant model relies on parameters obtained from the estimation period.
b Root Mean Squared Error
c Mean Absolute Error
d Theil's U statistic
19
Australian Bank Bill Rate: Kalman Filter Approach
Table 7
AGS Test Comparing Mean Squared Error (MSE)a
Comparing e1 and e2
e1,t - e2,t = a1 + b1 [ (e1,t + e2,t) - (me1 + me2,)] + u1,t
H0: MSE 1 > MSE 2, Requires a1 and b1 non-negative and
at least one of them strictly positive
a1 -0.00609 (-84.73)
b1 0.93493 (47.25)
Adjusted R-square 0.96
DW statistic 1.54
Comparing e1 and e3
e1,t - e3,t = a2 + b2 [ (e1,t + e3,t) - (me1 + me3 )] + u2,t
H0: MSE 1 > MSE 3, Requires a2 and b2 non-negative and
at least one of them strictly positive
a2 -0.00629 (-63.88)
b2 0.87434 (33.18)
Adjusted R-square 0.92
DW statistic 1.13
Comparing e2 and e3
e2,t - e3,t = a3 + b3 [ (e2,t + e3,t) - (me2 + me3 )] + u3,t
H0: MSE 2 > MSE 3, Requires a3 and b3 non-negative and
at least one of them strictly positive
a3 -0.00019 (-7.30)
b3 -0.19829 (-6.47)
Adjusted R-square 0.29
DW statistic 1.39
a
Ashley, Granger, Schmalensee (1980) test for statistical difference between
two respective mean squared errors. This requires that mean errors are
positive for both series. If it is positive only for one series then two times the
mean of the negative forecast error series is added to the dependent
variable. This adjustment will allow the test to measure the mean difference
in absolute forecast errors.
Error defined as (Model Forecast - Market Observed)
e1 error for Time Invariant Expectation Model; me1 mean for this error
e2 error for AR(1) Model; me2 mean for this error
e3 error for Kalman Filter Model; me3 mean for this error
u1, u2, u3 regression error; t-stats are in parentheses
20
Australian Bank Bill Rate: Kalman Filter Approach
Figure 1
Bank Bill Yield : Estimation & Forecast Periods
a
17
bab90
16
15
14
bab180
13
12
11
10
9
8
7
6
5
Jun '90
Jun '91
Dec '90
Jun '92
Dec '91
Dec '92
a
Yield is expressed as percentage per annum; Month-end figures shown
BAB90 refers to 90-day bank accepted bill
BAB180 refers to 180-day bank accepted bill
21