Estimation of Unobserved Expected Monthly Inflation Using Kalman Filtering Author(s): Edwin Burmeister, Kent D. Wall, James D. Hamilton Source: Journal of Business & Economic Statistics, Vol. 4, No. 2 (Apr., 1986), pp. 147-160 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/1391314 Accessed: 24/10/2008 09:11 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=astata. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact
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? 1986 AmericanStatisticalAssociation
Estimation
of
Journalof Business & EconomicStatistics,April1986, Vol.4, No. 2
Unobserved
Inflation
Monthly
Expected Kalman Filtering
Using
Edwin Burmeister Department of Economics, University of Virginia, Charlottesville, VA 22901
Kent D. Wall Department of Systems Engineering, University of Virginia, Charlottesville, VA 22901
James D. Hamilton Department of Economics, University of Virginia, Charlottesville, VA 22901 Hamiltondeveloped a techniquefor estimatingfinancialmarketexpectationsof inflationbased on the observedtime-seriespropertiesof interestrates and inflation.The techniqueis based on a state-space representationderivedfroman underlyingvectorautoregressiveprocess of the expected real interestrate and the expected inflationrate on lagged expectationsand lagged values of the observed Treasurybillrate and the actual inflationrate. This articleextends this workin two ways. First,we use monthlydata, since the quarterlydata used by Hamiltonmay obscure many interestingmovements,especiallyfordeterminingthe role of inflationary expectationsin stock pricemovements,and this is one of our primaryinterests.Second, we employ an alternativemethoddeveloped by Burmeisterand Wallfor estimatingthe parametersof the model,and this methodleads to a differentidentification proof.Bothapproachesshare the use of the Kalmanfilterto estimatethe unobservedvariables,inthiscase, expected ratesof inflation. KEYWORDS:Expectedinflation;Unobservedvariables;Kalmanfilter;State space.
1.
The estimatedmonthly expected inflation series exhibits the propertiesone would hope for, as did Hamilton'squarterly series; it is unbiased,rational,and efficient. In particeconomic variables ular, we found no noncontemporaneous in the relevantinformationset thathelpexplainthe difference betweenthe actualinflationseriesandourmonthlyestimated expected inflationseries. The standarderrorof agents forecasting inflation one month ahead is approximately.3% at a monthly rate. MonteCarlosimulationswere conductedto ascertainhow muchof the totaleconometricuncertainty(as definedin Sec. 6) in our estimated inflation series is due to noise in the structuralmodel and how much is due to parameteruncertainty. We found that the total econometric uncertaintyis about .16% per month;about .13% of this is due to structuralnoise and about .09% is due to parameteruncertainty [(.16)2 - (.13)2 + (.09)2]. These numbersareroughlywhat one would expect given Hamilton'squarterlyestimates. One importantapplicationof our estimatedexpected inflation series is presented. We found that both the errorin expected inflation for period t and the change in expected inflation between periods t + 1 and t are statisticallysignificant macroeconomicfactors in the context of the Ross (1976) ArbitragePricing Theory (APT) for explaining security returnsin period t. The presentationproceeds as follows: The model is explained in Section 2, and in Section 3 a state-spacerepresentationis derived. Identificationis establishedin Section 4, with parameterestimates given in Section 5. In Section
INTRODUCTION
Hamilton (1985) developed a technique for estimating financial marketexpectationsof inflationbased on the observed time series propertiesof interestrates and inflation. The technique is based on a state-spacerepresentationderived from an underlyingvector autoregressiveprocess of the expected real interestrateand the expected inflationrate on lagged expectations and lagged values of the observed Treasurybill rate and the actual inflation rate. Hamilton appliedthis techniqueto postwarquarterlydata to generate historical estimates of the rates of inflation anticipatedby bondmarkets.[Theformulationis quitegeneralandincludes previousworkon expectedinflationas specialcases in which most of the parametersare restrictedto zero values; see Hamilton(1985) for details.] This articleextends this work in two ways. First, we use monthly data, since quarterlydata may obscure many interestingmovements, especially for determiningthe role of inflationaryexpectationsin stock price movements,and this is one of our primaryinterests. Second, we employ an alternativemethoddeveloped by Burmeisterand Wall (1982, 1984a,b) for estimating the parametersof the model, and this method leads to a different identificationproof. Both approachessharethe use of the Kalmanfilterto estimatethe unobservedvariables, in this case, expected rates of inflation. [The use of Kalman filtering techniques to estimate unobservedvariablesin economics was introducedby Burmeister and Wall (1982).] 147
Journalof Business & EconomicStatistics,April1986
148
6 we discuss our estimated expected inflation series, and tests of this series are given in Section 7. Finally, an application of our expected inflation series to the Arbitrage Pricing Theory is presented in Section 8, and the article concludes in Section 9 with a brief discussion of further researchsuggested by our work. 2.
THE MODEL
We introducethe following notation: p(t) = the price level in month t as measuredby the implicit deflatorfor nondurableconsumption; 7r(t) = the rate of inflation between months t and t -
7e(t)
l)]/p(t -
I(t) = {n(t -
)} x 100;
p(t -
i(t) = nominal returnon Treasurybills expiring at the end of montht in %, assumedto be known at the beginningof month t; re(t) = expected (ex ante) real interestrate - i(t) -
e(t);
i(t) -
7r(t);
7e(t).
[Onemustbe cautiousto distinguishbetweenthe information set of the economic agents when they form the expectation 7n(t) and the informationset availableto the econometrician who wishes to obtain an estimate of 7e(t). Intuitively, the Kalmanfilter producesthe "best' estimate of n"(t) (given eitherdata over the whole sample or at least data up to and includingtime t) that is consistentwith the model. In other words, given a model of how expectationsand actual realizations interact, it producesthe "best" estimate of unobserved expectationsconsistentwith the other variableshaving their observed realizations.] We note at the outset that there is a conceptualproblem with the timing of rates because of the way in which the price level is measured. Price indices are constructedfor montht by using dataover the entiremonth, and hence they may be taken to measurethe price level approximatelyat the middle of month t. Consequently,the rate of inflation 7r(t)as defined measuresthe percentagerate of change in the price level from approximatelythe middle of month t - 1 to the middle of month t. The nominal interestrate i(t), however, is measuredfrom the beginning to end of montht. Fortunately,however, with monthly data we found that our empirical results are robust with respect to 1-month timing changes in the inflationrate or nominalinterestrate.
[(t
1)
r'(t) = k, + (lr'(t
e(t) = the inflationforecasterrormade by agents - n(t) -
1), i(t - 2), . .. .},
-
-1)
p(t - 2)/p(t - 2)
arenot announceduntilsometimeduringmontht, we assume that agents at the beginningof montht know the price level for month t - 1 and are not surprisedby subsequentannouncements. Following Hamilton(1985), we postulatethatthe following vectorautoregressiveprocesscharacterizesthe evolution of the expected real interestrate and the expected inflation rate:
r(t) = actual (ex post) real interestrate -
1), ...
1) and (t -
informationat time t
month t;
e(t), ne(t -
and it could includeotherinformationas well. Even though
= expected rate of inflation based on the available
where l(t) is the informationset -E[n(t)ll(t)], availableto economic agents at the beginningof
1), 7r(t - 2), ....
i(t), i(t -
1
in % {[p(t) - p(t -
Most important,when i(t) is replaced by i(t - 1) with everythingelse unchanged,thereis essentiallyno alteration in our estimated expected inflation series. The empirical resultsreportedsubsequentlymatchi(t) with 7r(t);hence the expected real rate of interest in month t is defined as the nominalrate of interestduringmonth t minus the expected inflation rate from approximatelythe middle of month t - 1 to the middle of montht. Accordingly,if we take the index t to mean the beginning of month t, we assume that the informationset contains
-
1) +
2re(t -
+ 03r (t-
3) + 04r (t-
+
/27re(t-
2) +
+
1n7(t -
+
474r(t
-
1) +
4) + 3) +
t37r(t^7r(t -
2)
2) +
,ue (t W47e(t ,37E(t-
1) -
4)
3)
(1)
4) + el(t)
and 71r(t + 1) = k + ailr(t) + a2re(t + a4r(t - 3) +
,r V(t) +
- 2) + fle(t + 337Te(t + Y17r(t- 1) +
1) + a3r"(t- 2) (t - 1) 2r'r
- 3) + ',7r(t)
37r(t- 2)
- 3) + e,(t). + '47r(t
(2)
In what follows, we assume that e,(t) and e,(t) are un-
correlatedwith re(t - j), 7"(t - j), and 7r(t - j) for all j > 1; thus the autoregressivedynamicsof real interestrates and expectationsof inflation are assumed to be stable and sufficiently simple to admit a low-ordervector autoregressive representationof the formof (1) and (2). [Foreconomic motivationof a simpleautoregressiveprocessfor realinterest rates,see FamaandGibbons(1982) andLittermanandWeiss (1985).] The key innovationof Hamilton'stechnique,however, is that it does not require EI(t) and E,(t) to be uncor-
related with lagged values of other variables that may be known to agents but not to the econometrician.Thus Equation (2) is not the rule used by agents to forecast inflation, but simply correspondsto the statisticalprojectionof those forecasts re(t + 1) on a strict subset of the variablesby
Inflation Burmeister, Wall,andHamilton: ExpectedMonthly which they are actually determined.The central idea is to rely on the hypotheses(1) and(2) of stable, simpledynamics, anduse ex post dataon i(t + 1), 7z(t + 1) to drawinferences about the unobservedex ante magnitudeIre(t+ 1), as in a standardsignal-noise extraction problem. Agents face a
and i(t) = 'e(t) + (VI - 01)7re(t- 1) + (Y2 -
$2)/ e(t -
forecasting problem in formulating 7re(t + 1), for which i(t + 1) and i(t + 1) are not known. The econometrician, by contrast, faces a signal problem of inferring 7re(t + 1),
+ (V/4 -
14) re(t
and for this latter purpose, use of 7r(t + 1) and i(t + 1) may be helpful. Of course, one uncoversonly an inaccurate estimateof agents' forecasts ne(t+ 1) by this method, since the actual model used by agents in forecastingis presumed to be unavailableto the econometrician.Hamilton (1985) showed that standardfiltering algorithms can be used to produce estimates of the degree of uncertaintyassociated with this problemof inference, which we refer to as "filter uncertainty"in the empiricalresults presentedhere. In additionto assumingthat the dynamicsof real interest rates are stable and relatively simple, we also assume that markets are efficient; that is, all available informationis used by agents to form their expectations. As discussed in detail in Hamilton (1985), this efficiency assumptionimposes orthogonalityconditionson (1) and (2) thatimply that
+
E{[el(t), 82(t), e(t)]'[El(r), ?2(T), e(r)]} 2~ 0 0 =
0
a5
0
0
0
ae
_m
= 0
(fI
-
a1)re(t)
17(t - 1) +
1)
3r(t - 3)
27r(t - 2) +
7r(t) = 7e(t)
(4)
+
e(t),
(5)
which statesthatthe differencebetween actualand expected inflationis the forecasterrore(t). Ourstate-spaceapplication requires the assumptionthat e(t) in Equation(5) be zero mean and white noise, so it is here that we are implicitly imposing the assumption of rational expectations. Subsequenttests on the estimatedseries e(t) will confirmthatthis specificationis consistent with the data. Hamilton(1985) showed that Equations(3), (4), and (5) constitutea state-spacerepresentationof the vector autoregressive model (1) and (2), as follows:
y(t) = Hx(t) + Dz(t) + v(t),
(6)
where x(t) = [7re(t), 7ie(t -
7e(t)
= re(t) into (2) and (1),
+ (/2
-
a2) e(t
-
a3) e(t - 2) + (f4 -
y(t) = [i(t), r(t)]', w(t) = [E2(t),0, 0,
+ a4i(t -
3) + y,n(t) + y27(t
+ y3r(t -
2) + y47r(t -
-
a,
0
I
0 0 0
0 0
0
0
0 0 0
0 0 0
-1
0
0 0O
VI -
1
0 Y\ 0 0 0 0 0 0 0 0 0O 0 0
Ca3 Y4 0O 0O
O
01
0
/4 -
3
2 -
0 02
0
a4
0 0 0 1
0 0 1 0
O
a1 a, G=
aC2 33 -
#2
1
0
0 , 0 k2 0
I'2 Y3 Y4 0 k" 0O 0O 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0O 0O 0 0 0
3 -
0
?3
V4
0
-
04
0
and
1)
3) + k2 +
0,0]',
v(t) = [el(t), e(t)]',
3)
+ ali(t) + a2i(t - 1) + a3i(t - 2)
3), /e(t - 4)]',
7r(t),7(t - 1), 7r(t- 2), 7r(t- 3), 7r(t- 4), 1]',
1) a4)re(t -
1), /re(t - 2), 7re(t -
z(t) = [i(t), i(t - 1), i(t - 2), i(t - 3), i(t - 4),
H= + (f3 -
- 4) + +,i(t -
The second outputequationis simply
THE STATE-SPACE REPRESENTATION
1) =
3)
43)/re(t -
+ -47r(t - 4) + k, + Er(t).
respectively, gives us the state equationand the first output equation, as follows: 7 e(t +
-
+ 02i(t - 2) + 03i(t - 3) + 04i(t - 4)
if t = T
if t # r.
Substitution of i(t) -
2) + (/'3
x(t + 1) = Fx(t) + Gz(t) + w(t),
For some of the Monte Carlo simulationsreportedsubsequently, we also take [el(t), e2(t), e(t)] to be trivariate normal. As observedby Hamilton(1985), FamaandGibbons(1982, 1984) employed a model that is a special case of Equation (1) with 01 = 1 and all other parametersset to zero, and Gessler (1981) estimated0, with all otherparametersset to zero. Similarly, we allowed for a more complex dynamic evolution of expected real interest and inflation rates than that in earlierwork by Mishkin (1981). Indeed, the specificationbased on Equations(1) and (2) (presentedin Sec. 3) entails 29 parametersin a nonlinear estimation problem, a task that pushed our available computingcapacitynear its limit. Thus furthergeneralitywould be difficult to achieve at this time. 3.
149
82(t)
(3)
D =
0 0
01
02
0/3
0
0
0
4
0
0
1l
42
000000
3
44
k,
Journalof Business & EconomicStatistics,April1986
150
Note from the above that the last four state equationsare simply re(t) = ne(t), e(t - 1) = 7e(t 1e(t - 2), and ne(t - 3) = 7e(t - 3).
4.
1),
e(t - 2) =
IDENTIFICATION
There are 29 parametersto be estimatedfrom the state4; ', . . . space specification (6): ca,, . . ., a4; f, . . .. Y4;
41,
. . . , 04;
/i,
. .
,
4;
1I, . .
,
k4;kl, k2; and a,
a,,, and ao. We must verify that the cross-equationrestrictions implicit in (6) from the underlyingrepresentations(1) and (2) are sufficient to identify these 29 parameters.In Hamilton(1985) the unknownparametersin (1) and(2) were identifiedand then estimatedby maximumlikelihoodmethods based on the implicit restrictionsthat they impose on the vector autoregressivemoving average (ARMA) representationfor i(t), n(t). Here, however, we show that identification can be established directly from the state-space representation(6). Thus we follow the approachof Wall (1984) and seek parameterizationsthat are unique in terms of their effect on the first and second moments of the observed dependent variables. This is exactly the approach taken by Hannan (1969, 1971, 1976), Kohn (1979), and othersfor vector ARMA model representations.The details are different, of course, because the characterizationof the equivalenceclasses for state-spacerepresentationsis different from that for vector ARMA representations. More specifically, if we let R denote the variance-covariance(var-cov) matrixfor v(t) and Q denote the var-cov matrixfor w(t), then the first two momentsof y(t) are given by E{v(t)} = /(t)
and E{[y(t) - E{y(t)}][y(s)
-E{y(s)}]'}=
r(t,
s),
where t-1
p(t) = HFE{x(O)} + E F'-'-iGz(i) + Dz(t) i=0
and r(t, s) = HF'- s(s)H'
if t > s
= HP(t)H' + R
if t = s
= HP(t)(F'-s)'H'
if t < s.
Here we employ fl(t) to denote the var-cov matrixfor the state x(t); it evolves accordingto fl(t + 1) = FfI(t)F' + Q, with f(0)
(7)
= E{x(0) - E{x(0)}][x(0) - E{x(0)}]'}. By fo-
cusing on the expressions for u(t) and F(t, s), we can ascertain how observationalequivalence arises and thereby understandthe basic cause of any identificationproblemsin our state-spacemodel. We will first restrictourselves to state-spacerepresentations that are minimal [i.e., completely controllablewith respect to z(t) and w(t) and completely observable]. [This
is completelyanalogousto the left-primeconditionimposed on vector ARMA models for the purpose of establishing identification;see Hannan(1969, 1971), Kohn (1979), or Rosenbrock(1970).] Then we know thatall observationally equivalentstructuresare relatedthroughan n x n nonsingular matrixT, as follows: F2 = T- 'FT; H2 = HIT;
G2 = T-'G,;
D2 = Di;
Q2 = T-'Q,(T-')';
R2 = R,.
Under such a relationship,the model structurespecified by F2, G2, Q2, H2, D2, and R2 is indistinguishablefrom that given by Fl, G,, Ql, H1, D,, and R,, using only first- and second-momentinformation.In fact, the n x n matrixT defines a family of equivalence classes in the space of all state-spacemodels of dimensionn. To obtain an identified model, we must select a structure(parameterization)that selects one and only one member from each equivalence class. Thus it is sufficient for identificationthat enough a prioristructurein F, G, Q, H, D, and R has been specified to precludeany but the trivial transformationT = I. In AppendixA, we show thatthe specification(6) admits only T = I and hence is identified. 5.
ESTIMATIONOF PARAMETERS
The parameterand state variableestimationswere carried out using standardKalman-filteringtechniquesas described in Burmeisterand Wall (1982, 1984a,b) and the references cited therein. To make this articleself-contained,however, AppendixB, which describesthe Kalman-filteringalgorithm we used, is included. Our sample of p(t) and i(t) covered January1964-May 1983;becauseof laggedvariables,the estimationwas carried out over a sample from July 1964 to May 1983, containing 239 monthlyobservations.An initialguess of the 29-element parametervector, 0, was made, and the initial var-cov matrix of the state vector, x(t), was 20I5. Similarly, we set 7e(- r) = r( z) for r = 1, . . .,4; that is, out-of-sample expectedinflationrateswere set equalto actualvalues. With these initial conditions, the value of the negative log-likelihood function after one iterationwas -611.17, and after 36 iterationsit was -621.173. At this point, however, the gradients were between - 1.4 and + 1.5, too large to be acceptable. We then reinitializedthe system with the value of 0 obtainedfrom36 iterations,but withthe var-cov matrix for x(t) again set at 2015. Convergencewas obtained after another 32 iterations, with the final gradients bounded by -.00044 and +.00058 and with a final value of the negative log-likelihood function equal to -621.178. The final 0 estimates and their standarderrors are reportedin Table 1. The Durbin-Watsonstatistics for the two output equations, (5) and (6), are 2.005 and 2.008, respectively. (Of course, these statistics must be interpretedwith care because of the presenceof lagged variables.) On observingthe patternof significantparametersin Table 1 and referringto Equations(1) and (2), it becomes immediately evident that both the expected real interestrate,
Burmeister,Wall,and Hamilton:ExpectedMonthlyInflation Table 1. EstimatedParameterValuesand StandardErrors(SE) Parameter Value al
.23192
a2 a3
- .77227 1.4758
a4 1l
- .99328 .37327
P2
.12408
/3 ,4
1.0941
Y,1
72 Y3 Y4
4)1
02 03
04
SE
Parameter Value
.44520
1
.58946 .55312*
2 3
' .28952* .45798 .55324
' .48001*
4 1
.38766
SE .48763
- .25976 - .93821
.58121 .49609
.90838 - .34229
.28423*
.22746
bifi, +
p, = Ep, +
i=
unanticipated change in
the term structuremeasuredas the returnon government bonds in month t minus returnon Treasurybills in month 1; f3, = UI, = unanticipated inflation measured as 7z(t) - fi(t) (from Table 2); and f4, = DEI, = the difference in expected inflation measuredas i'(t + 1) - 7"(t) (from
t -
Table2). An economicjustificationfor thesefactormeasures is providedin Chen, Roll, and Ross (1983), Roll and Ross (1984), and Burmeisterand Wall (1986). To demonstratethat our inflationaryexpectationswork has a useful application, we computed an ordinaryleast squaresregressionfor the equation RSPI, = Po + lIUPRIS,+ P2UTS, + 33UI, + P4DEI, + 8,
gives 4
bi= o +
over a sample from July 1964 to May 1984, where RSPI, denotesthe returnon the Standard& Poor's (S&P)500 stock indexin montht. [Theuse of ourestimatedexpectedinflation series in variableson the right side of Equation(16) entails a partialequilibriummethodologyandshouldbe viewed with the usual caution. Of course, unobservedexpectationsand observed realizationsare best estimatedjointly in a general
4
+b i=
i fi +bi 1
+
,,
i=l
which is exactly the form of the precedingregressionequation. We obtainedthe following results(for whichthe t statistics are reportedin parentheses): fo = .0079
(3.28) A3 = -1.99
(16)
et
I
(-2.22) R2 = .22
i, = 1.05 f2 = -.412 (5.44) (-2.18) ,/4 = 4.37 (2.37)
F(4, 234) = 18.0
DW = 1.93.
Both /13 and /4 are significant, a strong indication that our estimated expected inflation series has useful economic content (assuming that the APT is correct). Moreover, we conclude that over the July 1964-May 1984 monthly sample period, unanticipated inflation has a negative impact on stock market returns. On the other hand, the difference in expected
Burmeister,Wall,and Hamilton:ExpectedMonthlyInflation Table5. CharacteristicRoots of the Polynomial(11) for the EstimatedParameterValues0 Root
Absolutevalue
1i = 1.038 + .03545i
1.0386
.2 = 1.038 - .03545i 3 = -1.155 /4 = 1.195 + 1.156i /.5 = 1.195 - 1.156i 6 = -.5530 + 1.935i -.5530 - 1.935i 7 - 4.590 ,8
1.0386 1.155 1.6626 1.6626 2.012 2.012 4.590
inflation, which in this model can be interpretedas an unanticipatedexpectationthat the level of inflation will rise, has a positive impact on nominal stock marketreturns.A more complete examinationof the APT, using the expected inflationseries derivedhere, is madein BurmeisterandWall (1986), and a new methodfor jointly estimatingthe b's and the i's is presentedin McElroy,Burmeister,andWall(1985). 9.
CONCLUSION
We have only scratchedthe surface of the researchopportunitiesopened by these new techniques. In particular, in futurework we intendto estimateotherexpectedinflation series by using two alternativemeasuresof the price level: the consumerprice index (CPI) and a constructedmonthly implicit deflator for the gross national product. It will be interestingto learn how these expected inflation measures compare with the one given here and, most important,to see whetherconclusions such as those in Hamilton(1985) about forecast errors and business cycles are robust with respect to these differentmeasuresof the price level. A second importanttask for futurework is to examinethe stabilityof our parameterestimatesover time. Thereis some evidencethatfinancialmarketsunderwenta structuralchange aroundOctober 1979, and we plan to test whetherthe parametervalues agentsimplicitlyuse to forecastinflationalso shifted aroundthis date. Finally, thereare a host of interestingeconomic questions that can be investigatedwith our expected inflation series: What is the relationshipbetween the level of inflation and the varianceof forecast errors?Does the latterrelationship have anythingto do with business cycles? Why have recent real interestrates been so high'?How do inflationaryexpectations influence long-terminterestrates, and does the variance of forecasterrorsplay any role in determiningthe term structureof interestrates?We hope to addresssome of these issues, as well as others, in our futureresearch. Table6. CharacteristicRoots of the Polynomial(14) for the EstimatedParameterValues0 Root
Absolutevalue
i, = 1.048
1.048
,2 = -1.118 A,3= 2.082 - 2.094i ,4 = 2.082 + 2.094i
1.118 2.95 2.95
157
Table7. Tests on the EstimatedInflationForecast Error Definitionof the variablex(t)
R2
F(4, 234)
DW
e(t) Returnon governmentbonds in montht minusreturnon Treasurybillsin month t- 1 Returnon governmentbonds in montht minusreturnon corporatebonds in montht Rate on 4-6-month commercialpaper in montht Returnon Treasurybillsin montht 7n(t) Inflationin montht as measuredby the CPI Returnon the Standard& Poor's500 stock index in montht Returnon corporatebonds in montht Returnon governmentbonds in montht Trade-weightedvalue of the U.S. dollarin montht
-.0086
.49
1.99
.0017
1.10
2.02
.0014
1.87
2.00
-.014
.176
2.00
-.015 -.015
.103 .14
2.00 2.02
.135
1.96
.0023 .014
1.87
1.98
.014
1.85
2.00
.0017
1.10
2.02
-.0035
.887*
1.97
*Thisvalue is F(4, 124) because of data limitations.
ACKNOWLEDGMENTS This articlewas presentedat the Fifth WorldCongressof the EconometricSociety, Cambridge,MA, August 17-23, 1985. We thank the National Science Foundationfor researchsupport(SES-82-18229), IbbotsonAssociatesfor providing some of the data, and JonathanEatonfor comments. APPENDIX A: IDENTIFICATIONOF THE STATE-SPACE MODEL Our state-spacemodel, (6), possesses a structurethat is a special case of that considered in Wall (1984), and its identificationcan be established by the theory developed there. Specifically, we makeuse of Theorem2 of thatpaper, with minor modifications. Proposition 1.
Let {F, G, Q, H, D, R} be as specified
in (6). If /4 - 04 and /4
-
a4
are nonzero and if at least
one element of the first row of G is nonzero, then the statespace model is minimal. Proof. Consider first the controllabilitycondition. The state-spacemodel is completely controllableiff the rankof [G FG F2G .. F4G] is 5. The companionform of F and the special structureof G ensurethatone will always be able to find embeddedin the controllabilitymatrixa 5 x 5 upper triangularsubmatrix.This can be seen by straightforward but tedious calculation. For example, suppose that g,, is nonzero;then we could form an uppertriangularsubmatrix from the first columns from each of G, FG, F2G, F3G, and F4G. This submatrixwould have gi, down the diagonaland hence be of rank 5. If gil = 0, then we could repeat the exercise by using g12. We are guaranteedthat we can form at least one such full-ranktriangularsubmatrixbecause of the assumptionin the proposition.
Journalof Business & EconomicStatistics,April1986
158
Next considerthe observabilitycondition.Here we desire the rankof observabilitymatrix[H' F'H' (F2)'H' * (F4)'H'] to be 5. Once again the companionform of F, the special form of H, and the assumptionsof the propositionyield observability. This follows from the fact that F and the second row of H form an observablepair for a fourth-order state-spacemodel when we dropfrom considerationthe last column of F and the last element of the second row of H. Thusworkingwithjust the secondrow of H, HF, HF2, HF3, andHF4, we can always forma submatrixof rank4. Indeed, it will again be triangularwith 1 in the first position and ,84- a4 down the diagonal. Combiningthis submatrixwith the first row of H and the assumptionin the proposition concerning y/4 - 04 guarantees that we always can find five
linearly independentcolumns in the observabilitymatrix. Proposition2. Let the conditionsof Proposition1 hold. Then the state-spacemodel (6) is identified. Proof. For minimal state-spacemodels, the form of the equivalence relation that leads to lack of identificationis known to be representedby a nonsingularcoordinatetransformationin the statespace (see Wall 1984andthe references cited therein). Specifically, if {F,, G,, Q,, H,, D,, R,} and
where a' denotes the g,, element of G2 and a, denotes the glI element of G,. It is easy to see that comparisonof the first elements on both sides of this equationrequiresc, to be nonzero, whereas all of the subsequentcomparisonsrequire c2 = c3 = C4 = C5 = 0. Thus the vector c has all elements zero except the first. Finally, let us consider the relationshipbetween H, and H2, stipulatedby the matrixequationH2 = HIT. From the second row of this equationwe find that
H2 = HIT,
D2 = D1,
Q2 =
Identificationobtainswheneverthereareenoughrestrictions in the specification that the only matrix T consistent with these and satisfying the above equationsis T = I. In this way we will be assuredof selecting one and only one representativefrom each equivalenceclass. We now show that there are enough restrictionsin (6) to force T = I. Considerfirstthe relationshipbetweenF, andF2. We have T as the solution to the matrixequationTF2 - FIT = 0. It can be shown (see Lukes, in press or Wall 1984) that all solutions to the foregoing equation, with F as specified in (6), must be of the form T = [c, -pl(F)c,
-p2(F,)c, -p3(Fi)c, -p4(F,)c],
where c E N with N the null space of p5(F1)andp,,(F) the characteristicpolynomialof the respectiven x n submatrix of F (where n = 1 correspondsto the upper left corner 1 x 1 submatrixdefined by /i - ac). Now considerthe relationshipbetweenGI and G2. T must satisfy the matrixequationTG2= G,. Fromthe assumptions of the proposition, at least one element of the first row of G is nonzero, so without loss of generality let us assume that it is ac. The first columns on each side of the matrix equationmust satisfy a,
a,
0 0
0 0
0 = [c, -p,(F1)c, -p2(F,)c, -p3(FI)c, -p4(F,)c]
0 ,
0 0
*
*],
APPENDIX B: DESCRIPTION OF THE KALMAN-FILTERINGALGORITHMAND UNOBSERVED VARIABLE ESTIMATION
Ql,
R2 = R,.
*
where the asterisksdenote generallynonzeroentries. Hence cl = 1, and we conclude that the first column of T is the first column of the identity matrix. (We also conclude that the first row of T is the first row of an identity matrix,but this is not needed for the proof.) Insertionof this resultinto each of the remainingfour columns of T reveals that each of these is, in turn, the correspondingcolumn of a 5 x 5 identity matrix. Thus T = I and the model specificationis identified.
alence class, then they are relatedby G2 = T-I'G,
*
= [c
{F2, G2, Q2, H2, D2, R2} are any two members of an equivF2 = T-IFiT,
[1 0 0 0 O]T
0 0 0]
[1
For expositional convenience first consider the problem of estimatingx(t) given the parametersof F, G, F, H, and D, together with the first two momentsof y(t) and z(t). If x(t, r) denotes the minimum mean squarederrorestimate of x(t) given the model and all observed data up through time r, Y* = {y(l), y(2) ....y()}, Zr = {z(l),
z(2),
....
()},
then x(t, t) is producedby the following recursivecomputation: x(t + 1, t) = Fx(t, t) + Gz(t),
(B.1)
P(t + 1, t) = FP(t, t)F' + FQF',
(B.2)
B(t + 1, t) = HP(t + 1, t)H' + R,
(B.3)
?(t + 1, t) = y(t + 1) - Hx(t + 1, t) - Dz(t + 1),
(B.4)
K(t + 1) = P(t + 1, t)H'B-'(t + I, t),
(B.5)
f(t + 1, t + 1) = x(t + 1, t) + K(t + l)?(t + 1, t), P(t + 1, t + 1) = [I - K(t + l)H]P(t + 1, t),
(B.6)
(B.7)
for to - t - T. P(t + 1, t) is the var-cov matrix of the
estimationmatrixerrorin x(t + 1, t); that is, P(t +
, t) =
1, 1) x(t +
+ 1, t)t)]
x [x(t + 1) -
(t + 1, t)]'}.
Burmeister,Wall,and Hamilton:ExpectedMonthlyInflation
B(t + 1, t) is the var-cov matrixof the innovation;that is, B(t + 1, t) = E{jE(t+ 1, t)E(t + 1, t)'}.
The initial values for x(t, t) and P(t, t) are assumed to be known and are given by o(to, to) = x(0) = E{x(to) given all information
at time to}, P(to, to) = P(O) = E{[x(to) -
(O)][x(t) - x()]'}.
Thus P(t, T) is the var-cov matrixof the errorin estimating x(t) given all observationsup throughtimerz t. The vector e(t + 1, t) representsthe innovationsprocess and is analogous to the model residualsused in econometricestimation. Equations(B. 1)-(B.7) constitutethe Kalmanfilter. More efficient estimates of the states can be obtainedby using all of the sample informationavailable, that is, .(t, T). This is referredto as the smoothedestimate. It is derived from the filtered estimate, x(t, t), by means of a reverse "sweep" over the datafromT back to t + 1. Broadlyspeaking, computationis as follows: the recursiveKalmanfilter is employed in reverse time "beginning"at time T using a diffuse "prior"for x(T, t + 1), that is, P(T, T + 1) = x. For any time t in the closed interval[0, T], this reversetime filter producesan estimate, x(t, t + 1), along with its correspondingvar--covmatrix,P(t, t + 1). This representsour best estimate of x(t), using data only over the interval[t + 1, T]. Combiningthis with our forwardtime estimate, x(t, t), using only dataover the interval[0, t], gives us the desired result x(t, T). The method of combinationfollows from a classical result in probabilityand statistics;namely, the optimal combinationof two independentestimatesx(t, t) [with precision matrixP-'(t, t)] and x(t, t + 1) [with precision matrix P-'(t, t + 1)] is
x(t, T) = P(t, T)[P-l(t, t)x(t, t) + P-l(t,
t + l).(t, t + 1)],
with correspondingprecision matrix P-l(t, T) = P-'(t,
t) + P-'(t, t + 1).
Details of the smoothing algorithmsare given in Cooley, Rosenberg, and Wall (1977). Thus once filtered estimates areobtained,they can be revisedby the smoothingalgorithm to produce the most efficient estimates of x(t). [It can be shown that P(t, t) - P(t, T) for to - t - T. This should be
intuitively clear, since by definition x(t, T) uses more informationthan x(t, t). See Jazwinski (1970, chap. 7) or Bryson and Ho (1969, chap. 13).] Using the Kalman filter to generate model residualsenables the formationof a loss function that can be used in parameterestimation. The parametersto be estimatedmay include not only the unknownelements of H, D, andR (the parametersof the behavioralequations)but, moreimportant, those of F, G, F, and Q. The algorithmsfor estimationof the unknowns in this manner are called prediction error
methods and, like the Kalmanfilter, are thoroughlytreated in the control literature(see Ljung and Sorderstrom1983).
159
The algorithmemployed in the presentstudy is outlinedby the following steps: 1. Collect the unknown parametersinto a vector 0 of dimensionN x 1. Denote an initial guess at its true value by 0?, and insertthis into the Kalmanfilter Equations(B. 1)(B.7). Set i = 0. 2. Using the Kalmanfilter Equations(B.1)-(B.7), compute the model innovationssequence {E(t + 1, t); t, < t T - 1}, where ~(t + 1, t) = (t + 1, t, 0') is an implicit function of O'. 3. Form the loss functionJ(0i), where 1 T-l
J(O)
-
2
E ot
[?l(t + 1, t)'A[+ll,^(t+ 1, t) + ln(det i,+ l,)] (B.8)
and A,+II,is some positive definite weighting matrix. 4. Computean improvedestimateof 0, denotedby 0'+, such that J(O'i+) - J(0'). Use 0i+l = 0' - p'M-l'J('i)/dO, wherep' is a (scalar)step size parameterandMi- is a positive definite N x N matrixsuch that in the limit (as i --> c) it tends to the inverse Hessian of J. (See discussion after step 5.) 5. Checkto see whether110'+'- 0'll< 6, and/orIjAJ(Oi+')/ a011- 62. If so, stop; Oi+'is acceptedas the "best" estimate of 0. Otherwise, set 0' to 0i+', i = i + 1, and returnto step 2. If it is assumed that e(t) is normallydistributedfor each t and A,+ i, is set equalto B(t + 1, t), then approximate maximum likelihood estimates are obtained. The approximation involved relates to the prior on the unknowncoefficient vector 0. The differencebetweenour likelihoodfunction and the exact likelihood function is of the order 1/T. Thus asymptotically(T -> cc) there is no difference. (See Pagan 1980 for a discussion of this fact.) The iterative algorithm given earlier requires an initial estimate, 0?; a convergence criterion, 6, and/or ^2;and expressionsfor the componentsof the gradientvector aJ(()/ 0O.The gradientsmay be computednumerically,using simple finite first differences of J(0), or analytically, using a straightforwardapplicationof differentialcalculus to (B.8). The method by which pi and M, are computeddepends on the particularfunctionminimizationalgorithmemployed. A Davidson-Fletcher-Powellvariablemetricalgorithmis used here, since M, ' is then computedautomatically,with aJ(O)/ a0 being the only user-suppliedinformation.On convergence M,' is the inverse Hessian of J(0) (i.e., the information matrix)and yields valuable informationconcerning estimated parameterstandarderrors, correlations(covariances), and identifiability.In particular,once the algorithm converges, a simple scaling of M, ' producesan estimateof the parametervar-cov matrix. This estimate is useful not only in hypothesis tests on elements of 0 but also in examiningidentifiability.Since local identifiabilityandasymptotic nonsingularityof the Hessian matrixare equivalent, a nearly singular M, indicates identificationproblems. In practice, this is most easily tested by convertingthe param-
160
Journal of Business & Economic Statistics, April 1986
eter var-cov matrixto a correlationmatrixand examining the off-diagonal elements. Interparameter correlationsnear unity, say ?.996, lead to a singularconditionsuggestingan overparameterizedspecificationand lack of complete identification. Moreover, a singularHessian for J(O) results in nonconvergenceof the numericaloptimizationalgorithm,so lack of convergence and lack of identificationare highly related. [ReceivedMarch 1985. RevisedJuly 1985.]
REFERENCES Bryson, A. E., and Ho, Y.-C. (1969), AppliedOptimalControl, Waltham, MA: Blaisdell. Burmeister,Edwin, and Wall, Kent D. (1982), "KalmanFiltering Estimationof UnobservedRationalExpectationsWith an Applicationto the GermanHyperinflation,"Journal of Econometrics,20, 255-284. (1984a), "UnobservedRationalExpectationsand the GermanHyperinflationWith EndogenousMoney Supply: A PreliminaryReport," in LectureNotes in Controland InformationSciences (Vol. 63), eds. A. V. Balakrishnanand M. Thoma, New York:Springer-Verlag,pp. 279293. (1984b), "UnobservedRationalExpectationsand the GermanHyperinflationWith EndogenousMoney Supply," workingpaper, University of Virginia, Dept. of Economics;(in press, InternationalEconomic Review). (1986), "TheArbitragePricingTheoryand MacroeconomicFactor Measures,"The Financial Review. Chen, Nai-fu, Roll, Richard, and Ross, Stephen A. (1983), "Economic Forces and the Stock Market:Testing the APT and AlternativeAsset Pricing Theories," CRSP WorkingPaper 119, Universityof Chicago. Cooley, ThomasF., Rosenberg,Barr,and Wall, Kent D. (1977), "A Note on OptimalSmoothingfor Time VaryingCoefficientProblems,"Annals of EconomicSocial Measures, 6, 453-456. Fama,EugeneF., andGibbons,MichaelR. (1982), "Inflation,RealReturns and CapitalInvestment,"Journal of MonetaryEconomics, 9, 295-323. (1984), "A Comparisonof InflationForecasts,"Journal of Monetarv Economics, 13, 327-348. Gessler, Geary (1981), "An EfficientMarketsModel of the ExpectedReal Returnon TreasuryBills," unpublishedmimeographedreport, Federal TradeCommission.
Hamilton,James D. (1985), "UncoveringFinancialMarketExpectations of Inflation,"Journal of Political Economy, 93, 1224-1241. Hannan,E. J. (1969), "TheIdentificationof VectorMixed AutoregressiveMoving Average Systems," Biometrika,56, 223-225. (1971), "TheIdentificationProblemfor MultipleEquationSystems With Moving Average Errors,"Econometrica,39, 751-765. of ARMAX and (1976), "The Identificationand Parameterization State Space Forms," Econometrica,44, 713-723. IbbotsonAssociates (1984), Stocks, Bonds, Bills and Inflation:Quarterly Science (Vol. 2, No. 2), Chicago:Author. Ibbotson,Roger G., and Sinquefield, Rex A. (1982), Stocks, Bonds, Bills and Inflation:The Past and the Future, Charlottesville,VA: The Financial Analysts ResearchFoundation. Jazwinski,A. H. (1970), StochasticProcesses and Filtering Theor!, New York:Academic Press. Kohn, R. (1979), "IdentificationResults for ARMAX Structures,"Econometrica, 47, 1295-1304. Litterman,RobertB., and Weiss, Laurence(1985), "Money, Real Interest Rates, and Output:A Reinterpretationof PostwarU.S. Data," Econometrica, 53, 129-156. Ljung, Lennart,and Sorderstrom,Tovsten (1983), Theoryand Practice of RecursiveIdentification,Cambridge,MA: MIT Press. Lukes, D. L. (in press), "Affine Feedback Controllabilityof Constant Coefficient DifferentialEquations,"SIAMJournal of Optimizationand Control. McElroy, MarjorieB., Burmeister, Edwin, and Wall, Kent D. (1985), "Two Estimatorsfor the APT Model When Factors Are Measured," workingpaper, EconomicsLetters, 3, 271-275. Mishkin, Frederic S. (1981), "The Real Interest Rate: An Empirical Investigation,"Carnegie-RochesterConferenceSeries on Public Policx, 15, 151-200. Pagan, A. (1980), "Some IdentificationandEstimationResultsfor Regression Models WithStochasticallyVaryingCoefficients,"Journalof Econometrics, 13, 341-363. (1984), "TheArbitragePricingTheoryApproachto StrategicPortfolio Planning,"Financial AnalystsJournal (May-June), 14-26. Rosenbrock, Harold H. (1970), State-Space and MultivariableSystems, New York:John Wiley. Ross, Stephen A. (1976), "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory, 13, 341-360. Wall, Kent D. (1984), "IdentificationTheory for Varying Coefficient Regression Models," paper presented at the Sixth Conference of the Society for Economic Dynamics and Control, Nice, France,June.
