USING THE KALMAN FILTER TO ESTIMATE SUB- POPULAT ION S · USING THE KALMAN FILTER TO ESTIMATE SUB- POPULAT ION S Howard E. Doran No. 44 - March 1990 ISSN ISBN 0157-0188 0 85834 873

USING THE KALMAN FILTER TO ESTIMATE

SUB- POPULAT ION S

Howard E. Doran

No. 44 - March 1990

ISSN

ISBN

0157-0188

0 85834 873 X

Department of EconometricsUniversity of New EnglandARMIDALE NSW 2351

I. Introduction

Population estimates are important for many aspects of government,

planning, demographic study and historical analysis. The purpose of this

paper is to show how the Kalman filter provides an appropriate, but so far

unexploited, methodology to disaggregate populations into sub-groups of

interest.

The example which is used is that of estimating the populations of the

eight Australian states and territories, given the total population of

Australia at a particular annual date (eg. 30th June) is known, and that the

state populations are observed intermittently after a population census. The

Kalman filter appears to have several advantages over methods that are

currently in use, as follows:

(i

(ii

It is model based, and therefore does not rely on intuition or

special knowledge.

Subject to the model providing an adequate description of reality,

the estimates are statistically optimal in the minimum mean square

eFFOF sense.

(iii

(iv)

The filter provides a measure of the accuracy of the estimates.

Such a measure is notably absent from most existing procedures.

Once the unknown parameters of the system have been estimated,

population estimates can be produced instantly and automatically as

new observations become available.

(v)

(vi)

The estimates always satisfy the condition that they sum to the

(observed) country total, without the need for "fudging".

Estimates are produced from less information than is often used.

It is the author’s opinion that adoption of this technique would provide

valuable standardization (at least in the Australian context) and lessen the

burden of data collection.

2. A Model for the Estimation of Sub-Populations

Let Xit be the population of state i (i = 1,2 ..... k) at time t. For

example, in the Australian context, Xlt represents the population of New South

Wales (NSW) at the 30th June in year t. As there are eight states (or

territories), k = 8.

We will assume the following model:

Xit = X. + bi + ci + ~[It) i = 1,2 kl, t-I , t-I t ......

where

is the natural increase (births minus deaths),

is the unobserved systematic growth over and above natural increase.

The main component of bi,t_1 is the share to statp i of national net

migration,

(~)~it is a random variable, with zero mean which takes account of

non-systematic effects, such as temporary moves between states. This

variable also includes a correction factor to allow for the fact that

natural increase is measured on 31st December, whereas Xit refers to

population numbers on 30th June.

Furthermore, we allow b. to evolve according to a random walk processzt

with drift. Thus,

bit = ~i + b.+ (2)

i,t-I ~it1,2 .....k (2.2)

2

where ~i is a parameter.

It is convenient to define the following vectors:

st = (Xlt, X2t ..... Xkt, blt, b2t ..... bkt)’,

ct = (Clt, c2t ..... Ckt, O, 0 .....

~ = (0, 0 ..... O, ~1’ ~2 .....~k

1), ~(1) C1) (2) (2) (2)~t = (~ t ~2t ..... ~kt ’ ~lt ’ ~2t ..... ~kt 3’

Thus, (2.1) and (2.2) may be written in matrix form as

where

st = A~t-i + ct + ~ + ~t’

Ik IkA =

0 Ik

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

Equation (2.7) is known as the ’transition equation’ of the system.

variables Xit and b. it

variable ct is.

The second element of the model is the ’observation equation’ which

describes how observations relate to the unobserved vector variable st.

need to distinguish between census and non-census years.

The

(and hence at) are generally not observed, but the

We

(a) Census years

In a census year each state population is observed.

denote the vector of observations.

given by

[XI X2t Xkt]’Yt = t ......

We will use Yt to

Thus, in census years Yt is a k-vector,

(2.8)

From the definition (2.3) of at, we thus have

Yt = Zt~t’

where

Zt being a kx2k matrix.

(2.9)

(b) Non-census years

In non-census years, the state populations are not observed. However,

what is known is the total population of the country. Thus, in these years

k

Yt = E Xiti=l

It follows that Yt is a scalar, again given by

Yt = Zt~t ’

where now

and Jk is a k-vector of ones.

(2.1o)

In these cases Zt is a Ix2k matrix.

In summary, the system consists of the transition equation (2.7) together

with the observation equation

(2.10)Yt = Zt~t’

where Zt has the form (2.9) in census years and (2.10) in non-census years.

4

The Kalman Filte~

The Kalman filter, derived by Kalman (1960), has been described by

numerous authors [see, for example, Harvey (1981), Chow (1975), Aoki (1976),

Hannah (1970)]. It is discussed here for convenience.

Consider the linear dynamic system

and

at = At~t-1 + ct + g + ~t ’ (3.1)

Yt = Zt~t + ~t ’ (3.2)

where (3.1) is known as the ’transition equation’, describing how the state

vector st (m×l) evolves, and (3.2) relates the observed variable Yt (nt×1)l to

st. In the above, ct and Yt are observed at all t, # is a parameter vector

while the state vector at is never observed. The random variables ~t and ~t

have zero mean, are mutually uncorrelated and have covariance matrices Qt and

Ht, respectively. The matrices At, Zt, Qt’ Ht and the vector ~ are (for the

moment) assumed known for all t.

We define the dimensions of the above vectors and matrices as follows:

at’ ct’ ~’ ~t are m×l;

Yt’ Dt are ntxl;

At’ Qt are mxm;

Zt is nt×m; and

Ht is ntxnt.

In most applications the row dimension of Yt is fixed.

This has no implications on the theory.

Here it varies.

5

The Kalman filter provides an optimal (in the minimum mean square error

sense) linear estimator at of the unobserved state vector at, together with^

its error covariance Pt’ provided that initial values a0 and PO are known.

2time t, observations Cl,C2 ..... ct and yl,y2 .... Yt are available.

At

Three different prediction errors play a role in the Kalman filter.

(i

(ii

(iii

^

at - at, the error in the optimal predictor. This error has covariance

matrix Pt"

^

8t = at - at[t-l’ where at[t_lis the optimal predictor of at given

observations up to t-l. We denote the covariance matrix of 8t as

vt = Yt - Ytlt-l’

observation Yt"

where Ytlt_1

From (3.2),

is the one-step-ahead prediction of the

^

= (3.3)Ytlt-1 Zt atlt-I¯

The covariance matrix of ~t is denoted by Ft.

Assuming that at-i and Pt-i are known then, schematically, the filter

works as follows:

Pt-i --+ Pt{t-i --) Ft ---+ Pt

^ ^

~t-I --÷ ~tlt-i ~ vt ~

It is worth noting that the recursion which delivers Pt from Pt-i only uses

2Inclusion of the variable ct in the model is not essential, but is

required in our application.

the known matrices At, Qt’ 7"t and Ht. It does not require the observations Yt

and ct. On the other hand, the recursion which produces at from (~t-i does

utilize the observations on Yt and ct, together with the covariance matrices

Ptlt_1 and Ft.The actual equations of the Kalman filter are now given.

Covariance recursion

Ptlt-I

Ft

Pt

= At Pt-i At + Qt ’

= Zt Ptlt-1 Zt + Ht ,

:Ptlt_1 - Ptlt_1Z[ Ftl Zt Ptlt_1

(3.4)

(3.5)

(3.6)

State prediction recursion

^ ^

at[t-I = At ~t-i + ct + ~ ’

^

et = Yt - Zt ~t~t-I ’

^

st = ~tlt-I + Gt ~t ’

(3.7)

(3.8)

(3.9)

where Gt = Pt]t-i Z~ F~1

and is known as the Kalman gain.

(3.1o)

Of particular importance in this application is the case when the

observation equation (3.2) has no noise. That is,

Yt = Zt st ’

implying that Ht m O.

This may be interpreted as the state vector st satisfies the known, time

varying, linear constraint.

Zt st = Yt(3.11)

7

Now, premultiplying (3.9) by Zt we have

Zt at = Zt ~t[t-I + Zt Gt (Yt - Zt~tlt-I

From (3.10)

Zt Gt = (Zt Ptlt_1 Z[)FtI = Ft F~1

= Int

(by (3.5) when Ht m O)

Thus,^

Zt at = Yt ’ (3.12)

establishing that the estimator at satisfies the same restriction as at at

all t. In our application this means that in non-census years the estimated

state populations always total to the observed national population, and in

census years they equal to observed state populations.

Kalman Smoothing^

When the filter has been applied over all observations we have at, Pt

^

(t = 1,2 ..... T). Though at is optimal given yl,y2 ..... Yt’ only sT and PT have

used all the available information. The purpose of smoothing is to revise

st and Pt in the light of all the T observations and works by moving

backwards through time from t=T to t=l. We will denote by st the smoothed

estimator of at, having error covariance matrix ~t" These are given

(see Harvey (1981, 115)) by

~t = st + P[(~t+l - At+l~t - Ct+l - ~i) ,(3.13)

~t = Pt + P[(~t+l - Pt+iIt)P["’(3.14)

where

8

and

~ = Ptp-I

Pt At+l t+IIt ’

^

~T = ~T ’ ~T = PT "

t = T-1,T-2 ..... 1 (3.15)

If we premultiply (3.6) by Zt, it is clear by (3.5) that when Ht = O,^

ZtPt = O. Thus, from (3.15), ZtP[ = 0 and hence, by (3.13), Zt~t = Zt~t.From

(3.12) it follows that the smoothed estimates ~t also satisfy the constraints

on at.

Starting Values

As the Kalman filter is a recursive procedure, there is always a starting

value problem. In this application, we can minimize this problem by

commencing in a census year.

Let us suppose that at t=O (census year) Ptlt_1 has the form:

Pll P12

P21 P22

where each of the Pij are kxk blocks. Then, because by (2.8), Zt = [Ik, O] in

census years, it is easily shown by application of (3.4), (3.5) and (3.6) that

Pt =

0 0

-10 P22 - P21 Pll P12

(3.16)

Thus, if the recursion is started at a census year, the first k elements of at

(see (2.3]) are known, as are all the elements of Pt except the lower

right-hand corner k×k block. We follow Harvey’s suggestion [Harvey (1981,

i13, 205-206)] and set

~0 = (XIo’ X20 .....Xko, O, 0 ..... 0)’ (3.17)

and

PO =

0

0

0

~ Ik

where ~ is a very large number and the Xio are actual census observations.

This is equivalent to setting a diffuse prior probability distribution on the

unknown part of SO, namely blo, b20 ..... bko.

It is now necessary to show that the effect of the diffuse prior PO

eventually is ’worked out’ of the system.

At t = 1, by (3.4),

ik Ik ]o = + 0(1)Ik Ik

where 0(I) stands for a matrix of bounded coefficients.

Ft = Jk’ 0 Pll0 Jk’ 0

Thus, from (3.5),

"’ Jk + 0(I) = mk + 0(I)= ~ Jk

Hence by (3.6),

Pl = ~ + 0(i) ,Mk Mk

where Mk is symmetric-idempotent, given by

and has the property that 3k Mk = Mk Jk O.

(3.19)

It can now easily be shown by induction that at any time t-i prior to

the next census,

~[ (t-l)2Mk (t-l)MkPt-i = (t_l)Mk Mk

+ 0(I) (3.20)

i0

Suppose now that the next census takes place at time t.

dropping the subscript k,

Pt~t-I 0 I (t-l)M M

I

I

Then, by (3.20),

]o / + o(1)

]I

t2M tM ]: < + 0(I)

tM M

Using earlier notation,

o

Pt = o o ]-1P22 - P21 Pll P12

where now

Pll = ~t2(M + O(s)) ;

P12 = P21 = ~tM + 0(1) ,

P22 = ~M + 0(1).

-1

By considering the product (M + O(e))(M - O(e)) it can be seen that the

matrices M(M + O(s))-IM, M(M + O(e))-I and (M + O(s))-IM each differ from M by

terms of the order of ~. It then follows, after simple algebra, that

-1P22 - P21 Pll P12 = 0(1)

Thus, though the effect of < persists up to the next census, it is then

’purged’ from the system by the census observations.

4. Model Estimation

In the foregoing, we have assumed that the matrices Qt and Ht are known.

A much more usual situation, and relevant here, is where Qt and Ht are

constant over time, but unknown. In such cases the observations Yt and ct

(t = 1,2 ..... T) must be used to estimate these matrices, together with the

11

vector

It can be shown [see Harvey (1981, 16)] that the log-likelihood function

can be written in the form

T T1 x loglFtl 1 , F~I vtlog L(y, c; 8) = constant - ~ - ~ Z ut

,t=l t=l

(4.1)

where @ is a parameter vector containing ~ and the unknown elements of Q and

H.

By starting with an initial choice of 8, a search routine can be used to

find the value of ~ which maximises log L. This value is then used to compute

the Kalman predictors. The Kalman recursion is commenced at the first census

year (1933), but the likelihood function is only evaluated from the second

census (1947). According to the earlier discussion, this overcomes the

starting value problem. A flow chart of this procedure is given in Figure 1

below.

untilmax

achieved

0 0 0

>)

[Year t= 1933 l_

I, Kalman filter

It~1933, ¯ ; ¯ ,1984

to 24

inL = Z inLt [

Figure 1

This procedure was used to estimate two models, according to assumptions

made about the generating process for bt, using observations over the period

1933-84 (Australian Bureau of Statistics (1986, Tables A6 and B2)). First,

the random walk with drift, as in (2.2) above was assumed and, second, the

conventional random walk (without drift) was estimated. This latter case is

equivalent to restricting the drift model by the constraint ~i = 0

(i = 1,2,...,k). The maximum values of the log-likelihood function were

217.71 (drift) and 208.35 (no drift), respectively. Thus, the likelihood

ratio statistic (LR) was given by

LR = -2(208.35 - 217.71) = 18.72.

This value is significant at the 5 per cent level (X~(0.5) 15.51),

indicating a clear preference for the drift model. All subsequent results are

based on this model.

The estimates of the parameters of the drift model are shown in Table i.

Two features of this table are of significance:

(2)(i) Apart from Victoria and South Australia, the values of var(~i )

are effectively zero. This will be seen to result in zero standard^

errors of bit.

(ii) South Australia and Tasmania have negative drift constants. This

seems to imply that net migration to these states is falling. Such

a conclusion is consistent with the discussion by the ABS on state

population growth rates (Australian Bureau of Statistics (1986,

6-7)).

13

TABLE 1: Maximum Likelihood Estimates of the Parameters of the Drift Model

~ [ 1010State i var(~ I)) x 1010 var(~ 2)) × gi x 102

NSW 1 13.59 I.i0 x 10-6 6.37

VIC 2 7.20 1.15 1.45

QLD 3 1.97 2.61 x 10-8 5.35

SA 4 2.33 x 10-8 2.03 x 10-I -2.46 x I0-I

-9WA 5 I. 17 6.84 x I0 3.64

TAS 6 0.95 x 10-1 9.68 x 10-10 -4.90 × 10-2

NT 7 1.05 x 10-2 5.94 x 10-13 6.35 x 10-1

-2 -12ACT 8 2.42 × I0 9.51 × I0 1.32

5. Discussion of Results

The results of ABS and Kalman filter estimation of the state populations

for the years 1947-84 are given in Appendix i and Appendix 2. For

convenience, we will discuss only the years 1959-61.

(a) Unsmoothed estimates

The results, prior to smoothing, for the years 1959-61 are shown in

Table 2. The following points should be noted:

(i) For the non-census years 1959 and 1960 the state predictions, denoted

by X, add up to the known Australian population (AUST) apart from a^

small rounding error. This verifies the relationship Zt~t = Yt

discussed earlier.

14

Table 2: Population Estimates for 1959-61. Kalman Estimates (X) Unsmoothed

year- 1959

NSW

ABS 3759833X 3737969

se(X) 67767b 16551

se(b) 0c 46952

census - 0

VIC QLD SA WA TAS NT ACT AUST

2785904 1468236 920897 712069 339375 24088 46069 10056479

2748014 1491036 923799 741390 336951 26528 50787

67359 30343 22941 33996 6712 7088 14146

21895 13916 13439 9450 -127 1651 3422

13738 0 6502 0 0 0 0

39477 22842 13161 11228 6182 642 1370

error

-2110

% error

-0.02

year u 1960

NSW

ABS 3832452X 3800116

se(X) 75790b 17187

se(b) 0c 51343

census m 0

VIC QLD SA WA TAS NT ACT AUST

2857388 1495926 945319 722079 343909 25572 52367 10275020

2807803 1527632 950245 761861 342999 28815 55546

76606 33290 28582 37314 7354 7765 1550121832 14451 13374 9814 -132 1715 3554

14032 0 6894 0 0 0 0

41385 23880 14583 11348 6192 749 1537

error

-3513

% error

-0.03

year ~ 1961

NSW

ABS 3918500X 3918500

~e(X) 0b 17824

~e(b) 0c 48593

error 49853% error 1.27

census - 1

VIC QLD SA WA TAS NT ACT AUST error

2930365 1527513 971486 746749 350339 44480 58827 10548267

2930365 1527513 971486 746749 350339 44480 58827

0 0 0 0 0 0 0

28390 14986 12265 10177 -137 1778 3685

11550 0 3810 "0 0 0 0

40042 22489 13148 11337 6023 1172 1601

59344 -38451 -6717 -36275 1279 13200 -18112.03 -2.52 -0.69 -4.86 0.37 29.68 -3.08

% error

The numbers in the column ’error’ are the one-step-ahead prediction

errors vt defined in (3.3) With this model,

Yt’t-ll = 7t(A~t-i + ct + ~) = ~ [~i,t-I + bi +i--i

and this is compared with the actual observed Australian total

population at time t. Thus, for 1960, Ytlt-I = I0,278,533 yielding

vt = I0,275,020 - i0,278,533 = -3,513.

The ’% error’ expresses this error as a percentage of the Australian

15

(ii)

(iii]

(iv)

population. These percentage errors may be used to assess the quality

of the estimates. We note from Appendix I that of the 38 years for

which estimates were computed, only in 1950 and 1951 were the

percentage errors greater than one per cent.

In the census year 1961, the Kalman estimates are identical to ABS,

which in this case are the observed census values. Again, this^

demonstrates that Zt~t = Yt" As the estimates in census years coincide

with the observed values, the standard errors are identically zero. In

1961, a census year, the one-step-ahead prediction errors are given for

each state. These errors come from comparing the one-step-ahead

^

prediction X.1,t_l + bi,t-I + cit with the observed census value for

each state. The error is also expressed as a percentage of the state

census population.

The 1961 error of almost 30 per cent for the Northern Territory (NT)

strongly suggests that non-census year estimates for this state may be

poor. In fact, an examination of Appendix 1 shows that estimation of

the populations of the two smallest states ACT (Australian Capital

Territory) and NT is not reliable. The reason for this is that the

non-census year observation, namely, the total population of Australia,

is dominated by the largest states and is insensitive to changes in ACT

and NT populations. In fact, a 1 per cent change in the population of

NSW has a greater impact on the national total than a i00 per cent

change in the ACT or NT populations.

The standard errors of the estimates b. for all states except Victoria

(Vic) and South Australia (SA) are always zero. This is because the

(2)variances of ~it were effectively zero for all states except Vic and

SA. This is equivalent to stating that for all except Vic and SA, bit

16

(v)

can be adequately modelled by the deterministic process

bit = ~i + b. . l,t-1

Assessment of the relative merits of the ABS and Kalman filter

estimates can be done by comparing the estimates in the years

immediately prior to a census year with the census year observations.

It appears that there is little to choose between the estimates except

for the small states NT and ACT. Generally speaking, ABS estimates

appear to be superior here. However, in two respects the Kalman filter

has important advantages. First, standard errors of estimates are

produced enabling confidence intervals for the predictions to be

formulated. Second, the one-step-ahead prediction errors for time t

provide an objective measure of the worth of the estimates in t-l.

This is particularly valuable when t corresponds to a census year.

(b) Smoothed estimates

The smoothing algorithm has been described in equations (3.13), (3.14)

and (3.15). The smoothed estimates for the years 1959-61 are shown in

Table 3.

We note the following:

From a historical perspective, the smoothed estimators are undoubtedly

superior to both the unsmoothed Kalman estimators and the ABS

estimators. This is because the smoothed estimators use all the

observations yl,y2 ..... YT’ whereas the previous estimators use only

information available in the current period. An example of the effect

of smoothing can be seen by examining the estimates for NT in 1959-61.

In 1959, both Kalman and ABS predicted the NT population to be about

17

25,000. In 1960 this rose to about 26,000-28,000. However, in 1961,

the census result showed a population of about 44,000. Clearly, the

Table 3: Population Estimates for 1959-61. Kalman Estimates (X) are Smoothed

year u 1959 census ~ 0

NSW VIC QLD SA WA TAS NT ACT AUST

ABS 3759833 2785904 1468236 920897 712069 339375 24088 46069 10056479

X 3757560 2782933 1461638 918605 713016 337775 35857 49090

se(X) 32120 28797 16092 3742 17988 3586 3787 7549

b 16551 23816 13916 12464 9450 -127 1651 3422

se(b) 0 7802 0 2079 0 0 0 0

year = 1960 census = 0

NSW VIC QLD SA WA TAS NT ACT

ABS 3832452 2857388 1495926 945319 722079 343909 25572 52367

X 3823182 2850066 1492301 944231 727744 343986 40006 53499

se(X) 24760 22108 12462 2228 13928 2778 2933 5847b 17187 23577 14451 12671 9814 -132 1715 3554

se(b), 0 7823 0 2228 0 0 0 0

AUST

10275020

year ~ 1961 CenSUS = 1


ABS 3918500 2930365 1527513 971486 746749 350339 44480 58827

X 3918500 2930365 1527513 971486 746749 350339 44480 58827

se(X) 0 0 0 0 0 0 0 0b 17824 22950 14986 12884 10177 -137 1778 3685

se(b) 0 7814 0 2018 0 0 0 0

AUST

10548267

population would not have jumped some 50 per cent in one year. Rather,

the pre-census estimates would appear to have been far too low.

When we consider the smoothed results, which are optimal given all the

observations, we see that the estimates for 1959 and 1960 are about

36,000 and 40,000, respectively. These figures, obtained by working

backwards from the 1961 census results, certainly seem much more

reasonable.

18

(ii) A major effect of smoothing is that the standard errors of estimates

are reduced. This is particularly marked in the years preceding a

census. Thus, for example, in NSW the standard error for 1960 is

reduced from 75,800 to 24,800. Comparing the results in Appendix 1

with those of Appendix 2 shows that this effect is quite general.

Smoothing does not affect the standard errors in immediate post-census

years very much.

6. An Extension: Sub-State Estimation

A slight modification of the above methodology could be used to estimate

populations at the sub-state level. Suppose, for example, that population

estimates for the electoral divisions of NSW are required. In principle, the

only difference here is that the NSW total is not known exactly. However, an

estimate is now available, together with its standard error.

Suppose there are ~ electora! divisions in NSW, and denote by dit

thpopulation of the i division at time t. Then

the

E dit = Xlti=I

in non-census years, and dit is observed in census years.

is not observed in non-census years. However,

In this case, Xlt

Xlt = Xlt - nt

where Dt’ the prediction error, has a variance which has been estimated. For

example, from Table 3, in 1960, var(Dt) = (24,760)2 = 6.13 × 108. If we use^

the notation Yt = Xlt’ we have

Yt = Edit + ~]t ’i=l(6.1)

19

OF,

where now

(6.2)

at =(dlt, d2t ..... d~t, blt, b2t ..... b~t)’

and

(6.3)

The observation equation (6.2) is identical to (3.2) and the variance of Dt’

denoted by Ht, can be treated as known for all t.

In census years, dit is observed and the observation equation becomes

where

Yt = Ztat(6.4)

Zt = [ I~, O]

^

The Kalman filter can now be used to find at in an optimal way.

however an important difference in this case. Because Ht is not zero in

non-census years,

There is

in these years and so

Edit ~ Xlti=l

7. Summary

The Kalman filter provides a suitable methodology for estimating

sub-populations under certain conditions. Once the unknown parameters of the

system have been estimated, population estimation is quick, convenient and

optimal. Standard errors of predictions are computed routinely.

20

References

Aoki, M. (1976), Op£gmm~ ~~ ~ ~7~t

~, North-Holland, Amsterdam.

Australian Bureau of Statistics (1986),

Canberra.

Chow, G.C. (1975), ~ o~ ~o~ o~ ~~ @co/~ ~en%~, John

Wiley and Sons, New York.

Hannan, E.g. (1970), ~ugggp£e ~7~n~ £~, John Wiley and Sons, New York.

Harvey, A.C. (1981), ~%~n~ ~e]~ ~o~, Philip Allan, Oxford.

Kalman, R.E. (1960), "A New Approach to Linear Filtering and Prediction

Problems", ~~ ~ ~ o~ ~tt~c ~£r~, 82, 35-45.

21

APPENDIX i - ABS and Unsmoothed Kalman Population Estimates, 1934-84


NSW VIC QLD .qA WA TAS NT ACT

ABS 2984837X 2984837

se(X) 0b 8912

se(b) 0C 36830

error 3164% error 0.i]

2054700 ]i06414 646072 502479 257077 ]0867 ]69042054700 1106414 646072 502479 257077 10867 16904

0 0 0 0 0 0 0359 7493 -]677 5088 -68 889 1842

11966 0 4677 0 0 0 024273 ]7395 9121 8245 4450 206 60990035 -57094 7705 -36397 -]0365 -284 -6362

4.38 -5.]6 ].]~ -7.24 -4,03 -2.6] -37.64

AUST

7579358

year - 1948 census = 0

NSW VIC QLD

ABS 3015762X 3033468

se(X) 27434b 9548

s~(b) 0c 39447

SA WA TAS NT ACT

2092319 1131199 661152 515073 261205 12253 197902080935 1131652 653555 516259 261476 11982 19430

25098 13480 4662 35074 3001 3169 6318759 8028 -1664 5452 -73 953 1974

12468 0 5236 0 0 0 024881 17586 9668 8720 4720 253 718

AUST

7708761

error

5436

% error

0.07

year ~ 1949 census - 0

NSW VIC QLD

ABS 3092620X 3116694

Be(X) 39988b 10185

se(b) 0c 40626

SA WA TAS NT ACT

2142985 1159123 679304 532190 267061 13390 213852128999 1161399 662570 535696 266316 13403 22984

37557 19100 9589 21369 4244 4482 89384352 8563 -1119 5815 -78 I016 2106

12910 0 5734 0 0 0 027488 18628 10565 9169 4775 314 831

AUST

7908066

error

68331

% error

0.86

year - 1950 census - 0

NSW VlC QLD

ABS 3193371X 3227291

se(X) 50356b 10821

se(b) 0c 40136

SA WA TAS NT ACT

2208077 ]196184 709546 557095 275901 ]4691 238232206819 1195807 675014 559877 271349 15109 27426

48397 23435 14856 26234 5199 5489 1095111311 9099 87 6179 -83 1080 223713300 0 6187 0 0 0 027106 18546 10278 9505 4789 289 853

AUST

8178696

error

127391

% error

1.56


NSW VIC QLD

ABS 3278031X 3318027

se(X) 59641b 11458

se(b) 0c 42157

SA WA TAS NT ACT

2276574 1227700 732429 580342 286192 ]5608 248912280923 1228254 688363 581680 276279 16728 31518

58517 27108 20435 30361 6003 6339 1264916551 9634 1050 6542 -88 1143 236913646 0 6602 0 0 0 030415 19781 10833 10203 5336 358 962

AUST

8421775

error

90842


NSW VIC QLD

ABS 3339454X 3390358

se(X) 68237b 12095

se(b) 0c 43182

SA WA TAS NT ACT

2344490 1259477 755052 599857 296298 15463 263592347361 1259929 702200 601305 281633 18348 35321

68230 30359 26305 34018 6712 7088 1414719376 10169 1575 6906 -93 1207 250113954 0 6988 0 0 0 030910 19775 ]i]93 10789 5184 343 663

AUST

8636458

error

45974

% error

0.53

year = 1953 census - 0

NSW VIC QLD

ABS 3383791X 3446813

se(X) 76328b 12731

se(b) 0c 40680

SA WA TAS NT ACT

2395253 1291409 775780 620546 304079 15852 286442399060 1290017 715132 619183 286731 19907 38516

77684 33308 32445 37340 7354 7765 1550319709 10704 1591 7269 -98 1270 263214230 0 7348 0 0 0 032105 19831 11047 10563 5073 406 738

AUST

8815362

error

3124

% error

0.04

year = 1954 census = i

NSW VlC QLD

ABS 3423528X 3423528

ae (X) 0

SA WA TAS NT ACT

2452340 1318258 797093 639770 308751 16468 303142452340 1318258 797093 639770 308751 ]6468 30314

0 0 0 0 0 0 0

AUST

8986530

error %,error

APPENDIX 1 (cont.)

b 13368 19727so(b) 0 11586

c 41853 33808error -76697 1465

% error -2.24 0.06

11240 13419 7633 -102 1334 27640 3856 0 0 0 0

21(144 10957 11243 5599 395 734-2205 69322 2754 ]7044 -5]16 -11573-0.17 8.70 0.43 5.52 -31,07 -38.18

year ~ 1955

NSW

ABS 3490748X 3488470

so(X) 27361b 14004

so(b) 0c 41649

V]C QLD SA WA TAS NT ACT

2517228 1350016 819566 657114 314093 18209 327492511216 1351717 821560 660143 314303 3826(! 34058

24990 13478 3847 1507(* 3001 3169 631720681 11775 13483 7996 -107 ]397 289612134 0 4525 0 0 0 034506 20222 ]1370 I]343 5590 448 909

AUST

9199729

erzor

18183 0,20

year - 1956

NSW VIC OLD SA WA TAS

ABS 3554256X 3558105

so(X) 39807b 14641

so(b) 0c 46138

NT ACT

259346"7 ]38159o 848556 674528 318469 ]955q 351342575423 1385402 846721 681634 319865 20194 3821637279 ]9095 8032 21361 4244 4482 893822164 12310 13638 8360 -112 1461 302712614 0 5102 0 0 0 036332 22083 11959 ]1626 5764 530 965

AUST

9425563

error

27666

year = ]957

NSW

ABS 3624968X 3620634

so(X) 50067b 15277

so(b) 0c 47694

census ~ 0

VIC QI,D SA WA TAS NT ACT

2656256 1413084 873165 687604 326129 2106(I 378642635238 1420008 8"72386 701890 325527 22197 4225447928 23426 12635 26221 5198 5489 ]095022499 12845 13642 8723 -117 1524 315913036 0 5614 0 0 0 037643 22416 12303 11176 5859 590 1092

AUST

9640137

error

3683

year = 1958

NSW

ABS 3691953X 3677315

so(X) 59254b 15914

se(b) 0c 45616

VIC OLD SA WA TAS NT ACT2718481 1439198 896802 699564 333065 22096 411662689866 145451] 89"7958 720824 331234 24274 4634857849 27096 17616 30343 6003 6339 1264821871 13381 13486 9087 -122 1588 329013408 0 6077 0 0 0 037166 23249 12428 11613 5844 671 ]169

AUST

9842333

error

-14134

year ~ 1959 census

NSW VIC

- 0

QLD SA WA TAS NT ACT

ABS 3759833 2785904 1468236 920897 712069 339375 24088 46069X 3737969 2748014 1491036 923799 741390 336951 26528 50787so(X) 67767 67359 30343 22941 33996 6712 7088 ]4146b 16551 21895 13916 13439 9450 -127 1651 3422so(b) 0 13738 0 6502 0 0 0 0c 46952 39477 22842 13161 11228 6182 642 1370

yesr - 1960 census = 0


ABS 3832452 2857388 ]495926 945319 722079 343909 25572 52367X 3800116 2807803 1527632 950245 761861 342999 28815 55546so(X) 75790 76606 33290 28582 37314 7354 7765 15501b 17187 21832 14451 13374 9814 -132 1715 3554so(b) 0 14032 0 6894 0 0 0 0c 51343 41385 23880 14583 I]348 6192 749 1537

AUST

10056479

AUST

10275020

error

-2110

error

-3~13

% error

-0,02

% error

-0.03

year - 196]

NSW VIC OLD SA WA TAS NT ACT

ABS 3918500 2930365 1527513 971486 746749 350339 44480 58827X 3918500 2930365 1527513 971486 746749 350339 44480 58827so(X) 0 0 0 0 0 0 0 0b 17824 28390 14986 12265 10177 -137 1778 3685so(b) 0 11550 0 3810 0 0 0 0c 48593 40042 22489 13148 11337 6023 1172 1601error 49853 59344 -38451 -6717 -36275 1279 13200 -1811% error 1.27 2.03 -2.52 -0.69 -4.86 0.37 29.68 -3.08

AUST

10548267

error % error

year = 1962 census - 0

APPENDIX 1 (cont.)

NSW VIC

ABS 3986947 2983057X 3963861 2987242

se(X) 27354 24981b 18460 26792

se(b) 0 12102c 46862 38728


]55098] 987495 765961 355667 46003 661721562449 996714 765027 356108 47301 63586

13478 3801 15070 3001 3169 631715522 12054 10541 -142 1842 3817

0 4487 0 0 0 022663 13184 11378 5711 1095 1677

AUST

i0742~91

error

-39349

% error

-0.37


NSW VIC QLD

ABS 4049997X 4018002

se(X) 39791b 19097

se{b) 0c 41021

SA WA TAS NT ACT

3040840 1577866 1010740 788343 360726 48460 733993045562 1599285 1021714 785227 361615 50169 68800

37257 19094 7945 21361 4244 4482 893725873 16057 11888 10904 -147 1906 394912586 0 5068 0 0 0 037441 20460 12001 10323 5077 1209 1591

AUST

109503’78

error

-22102

year ~ 1964 census = 0

NSW VIC QLD

ABS 4107914X 4077009

se(X) 50040b 19734

se(b) 0c 39147

SA WA TAS NT ACT

3]05518 1610697 1038019 808442 364310 51462 803323108041 1635669 1045562 806285 366540 53278 74313

47892 23426 12513 26220 5198 5489 1095025897 16592 11845 11268 -152 1969 408013012 0 5584 0 0 0 035518 19432 12145 9965 4491 1142 1802

AUST

11166702

error

-2330

% error

-0.02


NSW VIC QLD

ABS 4175437X 4138598

se(X) 59219b 20370

se(b) 0c 37211

SA WA TAS NT ACT

3164368 1644533 1067570 825524 367904 59857 884643171825 1672021 1069713 827936 370895 56408 80265

57800 27095 17463 30342 6003 6339 1264826373 17127 11877 11631 -156 2033 421213387 0 6050 0 0 0 035334 18002 11016 10291 4241 1231 1876

AUST

11387664,

error

6084

% error

0.05


NSW VIC QLD

ABS 4237900X 4237900

se(X) 0b 21007

se(b) 0c 39227


SA WA TAS NT ACT

3220216 1674323 1094983 848099 371435 56503 960313220216 1674323 1094983 848099 371435 56503 96031

0 0 0 0 0 0 025017 17662 12427 11995 -161 2096 434411475 0 3392 0 0 0 037111 19955 11314 ]1243 4318 1393 2024

-13317 -32828 2376 -1760 -3546 -3170 9677-0.41 -1.96 0.22 -0.21 -0.95 -5.61 10.08

AUST

11599497

% error


NSW VIC QLD

ABS 4295238X 4286669

se(X) 27331b 21643

se(b) 0c 39892

SA WA TAS NT ACT

3274339 1699981 1109779 879178 375243 61835 1034773276064 1710559 1118644 869575 375528 59922 102112

24957 13477 3386 15069 3001 3169 631724224 18198 12321 12358 -166 2160 447512036 0 4140 0 0 0 040260 19111 11290 12072 5032 1540 2154

AUST

11799078

error

-21392

% error

-0.18


NSW VIC QLD

ABS 4359324X 4339573

se(X) 39733b 22280

se (b) 0c 45370

SA WA TAS NT ACT

3324179 1728995 1121810 915041 379648 67536 1120943335008 1746827 1142102 892680 380346 63569 108526

37194 19092 7161 21358 4244 4482 893723556 18733 12203 12722 -171 2223 460712527 0 4768 0 0 0 042058 20789 12639 13403 5135 1788 2490

AUST

12008634

error

-17011

% error

-0.14


NSW VIC QLD

ABS 4441187X 4414172

se(X) 49948b 22916

se(b) 0c 44846

SA WA TAS NT ACT

3385042 1763086 1139332 954845 384892 72961 1216613405829 1787189 i167172 919875 385349 67626 115799

47782 23423 11410 26216 5198 5489 1095024446 19268 12281 13085 -176 2287 473912958 0 5316 0 0 0 042683 20474 12478 14074 5010 2015 2880

AUST

12263013

error

14549

% error

0.12

APPENDIX 1 (cont.)

year = 1970

NSW

CenSus = 0

VIC QLD SA WA TAS NT ACT

ABS 4522329 3444935 1792742 1157986 991353 387719 78810 131467X 4482391 3473356 1826988 1191955 947]06 390186 71932 123430se(X) 59094 57637 27091 16080 30336 6003 6339 12648b 23553 24646 19803 12265 13449 -181 2350 4870se(b) 0 13338 0 5807 0 0 0 0c 56774 44899 23630 ]3309 16432 5025 2194 3441

AUST

12507349

error

1024

year - 1971 census


ABS 4725502 3601351 1851484 1200113 1053833 398072 85734 151168X 4725502 3601351 1851484 1200113 1053833 398072 85734 151168se(X) 0 0 0 0 0 0 0 0b 24190 34288 20339 8469 13812 -186 2414 5002se(b) 0 11460 0 3369 0 0 0 Oc 53809 41775 22573 ]2019 14779 4617 2264 3365error 162783 58449 -]8~38 -]7417 76845 3~,41 9257 19426% error 3.44 1.62 -].O? -].4~ 7.29 0.76 lO.R(~ 12.85

.AUST

]3067265

% error

year - 1972 census ~ 0


ABS 4795105 3661253 1898477 ]214627 1082016 400307 92080 ]59791X 4788959 3669451 1892642 1220501 1080190 402423 90323 159171se(X) 27328 24954 13477 3362 15069 3001 3169 6317b 24826 33246 20874 8343 14176 -191 2477 5133se(b) 0 12023 0 4121 0 0 0 0c 46486 36171 21291 10473 12699 3990 2336 3394

AUST

13303664

error

-27130

% error

-0.20

1973 census = 0


ABS 4841897 3707652 1951950 1228474 1101040 403086 97126 173305X 4837509 3724266 1932062 ]238916 1103565 406094 94995 167128se(X) 39727 37185 19092 7118 21358 4244 4482 8937b 25463 31253 21409 8074 14539 -196 2541 5265se(b) 0 12516 0 4751 0 0 0 0c 42375 35078 19758 9905 12505 3939 2276 3500

,AUST

13504538

error

-44854

% error

-0.33

1974 census - 0


ABS 4894052 3755725 2008339 1241537 1127597 406150 102923 186240X 4895951 3783564 1972096 1256593 1129166 409786 99755 175658se(X) 49937 47767 23422 11350 26215 5198 5489 10950b 26099 30392 21944 7913 14903 -201 2604 5397se(b) 0 12948 0 5302 0 0 0 0c 40538 32217 20060 9957 12410 3656 1637 3508

year 1975 census - 0


ABS 4932015 3787440 2051361 1265263 1154947 410087 92868 199006X 4934533 3821782 2010716 1273031 1152165 413084 103822 183859se(X) 59079 57616 27090 16004 30335 6003 6339 12648b 26736 27146 22480 7361 15266 -205 2668 5528se(b) 0 13329 0 5794 0 0 0 0c 36534 29646 18144 8901 12971 3320 2116 3508

1976 census - 1


ABS 4959587 3810425 2092374 1274069 i]78341 412313 98227 207739X 4959587 3810425 2092374 1274069 1178341 412313 98227 207739se(X) 0 0 0 0 0 0 0 0b 27373 18922 23015 3375 15630 -210 2731 5660se(b) 0 11457 0 3367 0 0 0 0c 37830 29846 18631 9405 12814 3546 1836 3675error -38217 -68150 41033 -15225 -2062 -3887 -10380 14843% error -0.77 -1.79 1.96 -1.19 -0.17 -0.94 -]0.57 7.15

AUST

13722570

, AUST

13892995

AUST

1’4033083

;

error

-19657

error

-62616

% error

-0.14

% error

-0.45

% error

year - 1977

NSW

ABS 5001887X 4995351

se(X) 27328

census - 0


3837363 2129838 1286118 1204365 415031 103937 2136873843074 2130469 1286645 1202259 415484 102612 21633624953 13477 3361 15069 3001 3169 6317

.AUST

14192233

error

-54929

% error

-0.39

APPENDIX 1 (cont.)

b 28009 16667so(b) 0 12020

c 37565 29480

23550 3144 15993 -215 2795 57920 4120 0 0 0 0

17962 8780 12879 3519 2183 4 3380


NSW VIC QLD

ABS 5053789X 5038358

so(X) 39726b 28646

se (b) 0c 38559

SA WA TAS NT ACT

3863758 2172046 1296204 1227850 417641 ]09979 2179803874747 2169259 1298172 122"7662 418662 107450 224941

37183 19092 7115 21358 4244 4482 893714693 24085 2878 16357 -220 2859 592312514 0 4750 0 0 0 028509 18856 8752 12498 3588 2268 3523

AUST

14359255

error

-44466

% error

-0.31


NSW VIC QLD

ABS 51]i129X 5079128

so(X) 49935b 29282

se(b) 0c 39376

SA WA TAS NT ACT

3886405 2214770 1301108 1246610 420"755 ]]4148 2207963898165 22090]i 1308950 1252453 421882 ]12413 233724

47764 23422 11346 26215 5198 5489 1095012011 24621 2468 16720 -225 2922 605512946 0 5301 0 0 0 028568 18604 8860 12504 3353 2]05 3454

AUST

14515729

orror

-55304

% error

-0.38

year - 1980


ABS 5171526X 5133871

s~(X) 59077b 29919

so(b) 0c 41856

3914302 2265934 1308396 1269067 423589 ]18244 2242903926648 2250558 ]319568 1279538 424932 I]7354 242885

57612 270911 ]5999 3033~ 6003 6339 ]264810474 25156 2182 17084 -230 2986 618713328 0 5793 0 0 0 030195 21897 9549 13904 3865 2276 .3289


NSW VIC QLD

ABS 5234888X 5234888

so(X) 0b 30555

so(b) 0c 41460


SA WA TAS NT IACT

3946916 2345207 1318768 1300055 427223 122615 22~75803946916 2345207 1318768 1300055 427223 122615 227580

o o o o o o o8452 25691 -958 17447 -235 3049 6318

11456 0 3367 0 0 0 029257 22472 8753 14065 3604 2333 3172

-20402 47595 -12532 -10472 -1345 -2 -24782-0.52 2.03 -0.95 -0.81 -0.31 0.00 -10.89

AUST

14695357

AUST

14923260

error

-31055

% error


NSW VIC QLD

ABS 5307947X 5328182

so(X) 27328b 31192

so{b) 0c 42783

SA WA TAS NT ACT

3994121 2419569 1328737 1336910 429751 129428 23~93~3996277 2395940 1326713 1334842 430713 128132 23~607

24953 13477 3361 15069 3001 3169 631710331 26226 -834 17811 -240 3113 ~645012020 0 4120 0 0 0 ; 030596 25024 9972 14818 3747 2405 3311



ABS 5360366X 5388722

so(X) 39726b 31829

so(b) 0c 41060

4037597 2471622 1341521 1364454 432614 133875 2365894028586 2445570 1335614 1365406 434145 133567 247032

37183 19092 7115 21358 4244 4482 89379214 26762 -1002 18174 -245 3176 6581

12513 0 4750 0 0 0 ; 030007 22994 9844 13118 3559 2641 3284

AUST

15178408

AUST

15378646

error

39713

error

-26473

% error

0.26

% error

-0.17

year ~ 1984 census - 0


ABS 5412039X 5440697

so(X) 49935b 32465

so(b) 0c 22943

4078457 2507048 1353916 1383664 437370 138825 2445684052157 2492803 1343781 1393483 437342 139255 256373

47764 23422 11346 26215 5198 5489 109507122 27297 -1332 18538 -250 3240 6713

12946 0 5301 0 0 0 019440 11648 5933 8228 -324 17 ********

AUST

15555895

error

-43751

% error

-0.28

APPENDIX 2 - ABS and Smoothed Kalman Population Estimates, 1934-84



ABS 2984837 2054700 ]]06414 646072 502479 257077 10867 16904X 2984837 2054700 II06414 646072 502479 257077 ]0867 ]6904

se(X) 0 0 0 0 0 0 0 0b 8912 15360 7493 7231 5088 -68 88° 1842

se(b) 0 7973 0 2505 0 0 0 0

AUST

7579358

year - 1948 census - o

NSW VIC OLD SA WA TAS NT ACT

ABS 3015762 2092319 1131199 661152 515073 261205 ]2259 ]9790X 3016333 2091007 I130579 662426 515702 263876 ]1213 17621

se(X) 24769 22114 ]2462 2505 13929 2778 2933 5847b 9548 17636 81128 9273 5452 -73 953 ]974

se(b) 0 79]0 0 2144 0 0 (I 0

AUST

7708761

year - 1949 census -


ABS 3092620 2142985 1159129 679304 532190 267061 13390 21385X 3084271 2145664 1159475 681369 53486’7 271127 ]1876 19414

se(X) 32136 28808 16093 4099 ]7988 3586 3787 7549b 10185 19605 8563 10919 5815 -78 1016 2]06

se(b) 0 7857 0 1836 0 0 0 0

AUST

7908066



ABS 3193371 2208077 ]196184 709546 557095 275901 ]4691 23823X 3187544 2220519 1193994 702854 559999 278615 12874 22293

se(X) 35289 31699 ]7631 4879 19709 3929 4149 8270b 10821 20872 9009 12052 6179 -83 1080 2237

se(b) 0 7826 0 1731 0 0 0 0

AUST

8178696



ABS 3278031 2276574 1227";00 732429 580342 286192 15608 24891X 3273645 2288189 1226876 725186 583168 286015 ]3802 24891

se(X) 35287 31697 17631 4822 19709 3929 4149 8270b 11458 21641 9634 12733 6542 -88 1143 2369

se(b) 0 7823 0 1873 0 0 0 0

AUST

8421775



ABS 3339454 2344490 1259477 755052 599857 296298 15463 26359X 3339836 2349419 1258804 748753 603928 293831 14721 27162

se(X) 32130 28805 16093 3958 17988 3586 3787 7549b 12095 22179 10169 13041 6906 -93 1207 2501

se(b) 0 7840 0 2144 0 0 0 0

AUST

8636458



ABS 3383791 2395253 1291409 775780 620546 304079 15852 28644X 3384993 2401104 1288523 772989 622146 301363 15546 28695

se(X) 24764 22112 12462 2364 13928 2778 2933 5847b 12731 22752 10704 13056 7269 -98 1270 2632

se(b) 0 7860 0 2364 0 0 0 0

AUST

8815362



ABS 3423528 2452340 1318258 797093 639770 308751 ]6468 30314X 3423528 2452340 1318258 797093 639770 308751 16468 30314

se(X) 0 0 0 0 0 0 0 0b 13368 23417 11240 12796 7633 -102 1334 2764

se(b) 0 7855 0 2332 0 0 0 0

AUST

8986530


NSW VIC QLD SA WA TAS NT ACT AUST

APPENDIX 2 (cont.)

ABS 3490748 2517228 1350016 819566 657114 314091 ]8209 32749X 34~1880 2518538 1345776 82(}848 65439(! 314465 20122 33706se(X) 24769 22109 12462 2332 ]3928 2778 2933 5847b ]4004 23855 11775 ]259] 7996 -107 ]397 28~6se{b) 0 7822 0 2(i2o n 0 0 0

a199729

year = 1956

NSW

ABS 3554256X 3566444

se(X) 32124b ]464]

0

census = o


2593467 1381590 848556 674528 318469 19555 351342588564 ]373704 844811 67036] 320197 23928 3755028799 ]6093 3846 ]7988 3586 3787 754923998 123]i~ 12420 8360 -]12 1461 3071779(I (I 18]~ O 0 n O

AUST

9425563

year ~ 1957

NSW VZC QLD SA WA

ABS 3624968X 3633578

no(X) 35275b 15277

se(b) 0

TAS NT ACT2656256 ]413084 873165 687604 326129 21060 ~78642~54708 1402512 8~0192 6~5048 326028 278n2 ~1266

31687 ]’76~] 4~,(’4 1~’I09 3929 414~ 827023994 12~45 12329 ~723 -]17 }~f’4 3]~

7775 0 1712 0 0 0 0

AUST

year = 1958

NSW VIC QLD SA WA TASABS 3691953X 3693124

se(X) 35274b 15914

se(b) 0

ACT

271848] 1439198 896802 699564 933065 22096 411662717601 1431009 893825 698145 331894 31737 4499431686 ]763] 4563 19708 3929 4149 827023958 13381 12351 q087 -122 ]588 32~07781 0 1839 0 0 0 0

AUST

0842333

= 1959

NSW

ABS 3759833X 3757560

se(X) 32120b 16551

se(b) 0

census = o

vic QLD SA WA TAS NT ACT

2785904 1468236 920897 712069 339375 24088 460692782933 1461638 918605 713016 337775 35857 4909028797 16092 3742 17988 3586 3787 754923816 13916 12464 9450 -127 ]651 34227802 0 2079 0 0 0 0

AUST

10056479

year = 1960

NSW

ABS 3832452X 3823182

se(X) 24760b 17187

se(b) 0


2857388 1495926 945319 722079 343909 25572 523672850066 1492301 944231 727744 343986 40006 5349922108 12462 2228 ]3928 2778 2933 584723577 14451 12671 9814 -132 1715 35547823 0 2228 0 0 0 0

AUST

10275020

year : 1961

NSW

ABS 3918500X 3918500

se(X) 0b 17824

se (b) 0

census = 1


2930365 1527513 971486 746749 350339 44480 588272930365 1527513 971486 746749 350339 44480 588270 0 0 0 0 0 022950 14986 12884 10177 -137 1778 36857814 0 2018 0 0 0 0

AUST

10548267

year = 1962

NSW

ABS 3~86947X 3972045

se(X) 23878b 18460

se(b) 0

census = 0

VIC OLD SA WA TAS ACT

2983057 ]550981 987495 765961 355667 46003 661722984627 1555864 99"?520 764650 355398 46666 6551721281 12038 2018 13454 2684 2834 564922544 15522 ]2894 ]0541 -142 1842 38177789 0 ]632 0 0 0 0

AUST

10742291

year = 1963

NSW

ABS 4049997X 4033406

se(X) 29322b 19097

se(b) 0

census = 0

VIC QI,D SA WA TAS NT ACT

3040840 1577866 1010740 788343 360726 48460 733993041322 1585999 1023600 784326 360190 48893 7263926192 14746 2961 16481 3287 3471 691922254 16057 12669 10904 -147 1906 39497777 0 1448 0 0 0 0

AUST

10950378

APPENDIX 2 (cont.)

year = 1964

NSW

CenSus = 0

VIC OLD SA WA TAS NT ACT

ABS 4107914 3105518 ]610697 1038019 808442 3649]0 51462 80332X 4100365 3101474 1615770 ]o48271 804971 364404 51366 80077s~{X) 29321 26192 ]4746 2932 ]648] 3287 34’1] ~QI9b 19734 21952 ]6592 12170 i]268 -]52 ]969 4080se(b) 0 7786 0 1649 0 0 0 0

AUST

]]]66702

year = 1965 census

NSW VIC

= 0


ABS 4175437 3164368 1644533 ]067570 825524 367904 53857 88464X 4171259 3161813 1645672 1072587 826418 368055 53868 87989se(X) 23876 21281 12038 1962 13454 2684 2834 5649b 20370 21578 17127 11379 i]631 -156 2033 4212se{b) 0 7807 0 1962 0 0 0 0

AUST

]1387664

year = 1966 censu~

NSW VIC

ABS 423790(I 3220216 1674323 ]094983 848099 371435X 4237900 3220216 1674329 ]I}94983 848099 371435se(X) 0 0 0 0 0 0b 21007 21166 17662 10307 11995 -]61se(b) 0 7812 0 1948 0 0

5650356503

02096

0

9603196031

04344

0

AUST

11599497


NSW VIC QLD SA WA TASABS 4295238X 4285021

se(X) 238’76b 21643

se(b) 0

NT ACT3274339 1699981 ]]09779 879178 375243 61835 ]u34773272479 1702643 ]116606 879684 375945 61560 ]~513721281 12038 1948 13454 2684 2834 564920907 18198 9415 12358 -166 2160 44757803 0 1627 0 0 0 0

AUST

11799078

year = 1968

NSW

ABS 4359324X 4336051

se(X) 29320b 22280

se(b) 0

census = 0

VIC QLD SA WA TAS NT ACT3324179 1728995 ]121810 915041 379648 67536 1120943328847 1730968 1137312 912862 381177 g6842 ]1457026192 14746 2887 16481 3287 3471 691920768 18733 8693 12722 -171 2223 46077808 0 1442 0 0 0 0

AUST

12008634

year = 1969

NSW

ABS 4441187X 4410993

se(X) 29321b 22916

se(b) 0

year ~ 1970

NSW

ABS 4522329X 4479456

se(X) 23876b 23553

se {b) 0

census = 0


3385042 1763086 ]139332 954845 384892 72961 1216613395169 1763655 1158646 950472 386608 72548 12491926193 14746 2886 16481 3287 3471 691920542 19268 8080 13085 -176 2287 47397837 0 1627 0 0 0 0

census = o

NTVIC QLD SA WA TAS ACT

3444935 1792742 ]]57986 991353 387719 78810 1314673458817 1795566 11";9206 988102 391872 78503 13562321282 12038 1946 13454 2684 2834 564920304 19803 7597 ]3449 -]81 2350 48707882 0 1946 0 0 0 0

AUST

ij263013

AUST

12507349

year ~ 1971

NSW

ABS 4725502X 4725502

se (X) 0b 24190

se (b) 0

VIC QI,D ~A WA TAS NT

3601351 1851484 1200113 1053833 398072 85734 1511683601351 1851484 1200113 1053833 398072 85"734 1511680 0 0 0 0 0 018112 20339 6664 13812 -186 2414 50027918 0 1950 0 0 0 0

,’AUST

]3067265

year = 1972

NSW

ABS 4795105X 4790738

census = 0

VIC QLD SA WA TAS

3661253 1898477 1214627 1082016 4003073658531 1901986 1218798 1081226 401698

NT

9208088306

ACT

]59791162377

AUST

13303664

APPENDIX 2 (cont.)

se(X) 23878 21285b 24826 15988

se{b) 0 7952

12038 1950 ]3454 2684 2834 564920874 5656 14176 -191 2477 5133

0 1635 0 0 0 ~

year ~ 1973 census - ¢)

NSW V]C QI, D

ABS 4841897X 4840422

se(X) 29325b 25463

s~ (b) 0

SA WA TAS NT ACT

3707652 ]951950 ]228474 I]01040 403086 97126 ]733053703850 IQ5067] 1234928 1105539 404642 o0~57 173522

26200 14246 2901 ]6481 3287 341] 691914038 21409 4606 14539 -]O6 2541 5265

8017 0 ]444 0 0 0 0

AUST

]35045~8

year ~ 1974 census ~ 0

NSW VIC OLD

ABS 4894052X 4901224

se(X) 29326b 26099

SA WA TAS NT ACT

3755725 2008339 ]241537 1127597 406150 ]02923 ]862403752927 21100120 ]249441 1132266 407613 93703 185273

26202 ] 4~ 46 2~12 ] 648] 3287 347] 691912088 2] ~44 3462 ] 4903 -2(11 26114 5397

8] 2~ I~ ] 628 O 0 0 ~I

AUST

13722570


NSW VIC QLD

ABS 4932015X 4937823

se(X) 23880b 26736

0

SA WA TAS NT ACT

3787440 2051361 ]265263 ]154947 410087 92868 ]990063786473 2047630 1262862 ]155723 410166 95730 196585

21287 12038 1972 ]3454 2684 2834 564910410 22490 2305 ]5266 -205 2668 5528

8277 h ]972 0 0 0 0

AUST

13892995

year = 1976

NSW

ABS 4959587X 4959587

se(X) 0b 27373

se(b) 0

V IC QLD SA WA TAS NT 6CT

3810425 2092374 1274069 1178341 412313 98227 2077393810425 20~2374 ]274~69 I]78341 412313 98227 20~7739

0 0 0 0 0 cl 09140 23015 11"16 ] 5630 - 2 lI.l 2’1~18452 0 2(158 0 0 0

AUST

14033083


NSW VIC QLD

ABS 5001887X 5002717

se(X) 23891b 28009

se(b) 0

SA WA TAS NT ACT

3837363 2129838 1286118 1204365 415031 103937 2136873835028 2140171 1284652 1200398 415224 102621 211418

21307 12038 2058 13455 2684 2834 56498233 23550 292 15993 -215 2795 57928682 ~ 1763 0 0 ~ 0



ABS 5053789X 5051715

se(X) 29353b 28646

se(b) 0

3863758 2172046 1296204 1227850 417641 109979 2179803860920 2188497 1293725 1223729 418134 107460 215071

26246 14746 3187 16482 3287 3471 69197624 24085 -366 16357 -220 2859 59239011 0 1482 0 0 0 0

AUST

14192233

AUST

14359255

year ~ 1979

NSW

ABS 5111129X 5095700

se(X) 29361b 29282

se(b) 0


3886405 2214770 ]301108 1246610 420755 ]14148 2207963882136 2237451 1302112 1246020 421070 112406 218831

26254 14747 3346 ]6483 3287 347] 69197393 2462] -775 16720 -225 2922 ’60559463 0 1661 0 0 0 0

AUST

14515729

year = 1980 c~nsus ~ 0

NSW VIC QLD

ABS 5171526X 5153401

se(X) 23903b 29919

se(b) 0

SA WA ]’AS NT ACT

3914302 2265934 1308396 1269067 423589 I]8244 2242903908895 2288168 ]310198 1270566 423835 117329 222963

21319 12039 2373 13455 2684 2834 56497395 25156 -979 17084 -230 2986 6187

10047 0 2373 0 0 0 0

AUST

14695357

year = 1981 census ~ i

APPENDIX 2 (cont.)

NSW VIC OLD SA WA TAS NT A<! T

ABS 5234888 3946916 2345207 ]318768 1300055 427223 ]22615 22"7580X 5234888 3946916 2345207 ]318768 13~O055 42’7223 ]22615 227580

se (X) 0 0 0 0 0 0 0 0b 30555 7386 256u l - 1049 ] 7447 - 235 311~9 63 ] 8

so (b) 0 10758 o 3350 0 (I II 0

AUST

]4923260


NSW VIC OLD

ABS 5307947X 5329892

se(X) 27261b 31192

se(b) 0

SA WA TAS NT ACT

3994121 2419569 ]328737 ]336910 42975] ]29428 2319383~94274 23~6146 ]326473 ]335105 430722 128)43 237650

24852 ]3475 3350 ]5066 3001 3169 63177106 26226 -1201 1781] -240 3113 6450

11468 0 4100 0 0 0 0

AUST

]z]7~408

year - 1983 census ~ O

NSW VIC QLD

ABS 5360366X 5390994

se(X) 39622b 31829

so(b) 0

SA WA TAS NT ACT

403"7597 2471672 ]341521 ]364454 432~]4 133875 2365804025977 2445844 ]335244 1365~55 434158 ]33582 247089

37038 191189 7100 2135~ 4244 4482 89376978 267~2 -1308 18]’14 -245 3176 6581

12192 0 4735 0 0 0 0

AUST

]5378646

year = 1984 census ~ 0

NSW V]C QLD

ABS 5412039X 5440697

se(X) 49935b 32465

se(b) 0

SA WA TAS NT ACT

4078457 2507048 ]353916 1383664 437370 ]38825 2445684052157 2492803 1343781 1393483 437342 139255 256373

47764 23422 11346 26215 5198 5489 109507122 27297 -1332 18538 -250 3240 6713

12946 0 5301 0 0 0 0

AUST

15555895

WORKING PAPERS IN ECONOMETRICS AND APPLIED STATISTICS

The Prior Likelihood and Best Linear Unbiased Prediction in StochasticCoefficient Linear Models. Lung-Fei Lee and William E. Griffiths,NOo 1 - March 1979.

Stability Conditions in the Use of Fixed Requirement Approach to ManpowerPlanning Models. Howard E. Doran and Rozany R. Deen, No. 2 - March1979.

A Note on A Bayesian Estimator in an Autocorrelated Error Model.William Griffiths and Dan Dao, No. 3 - April 1979.

On R2-Statistics for the General Linear Model with Nonscalar CovarianceMatrix. G.E. Battese and W.E. Griffiths, No. 4 - April 1979.

Construction of Cost-Of-l,ivtng Index Numbers - A Unified Approach.D.S. Prasada Rao, No. 5 - April 1979.

Omission of the Weighted First Observation in an Autocorrelated RegressionModel: A Discussion of Loss of Efficiency. Howard E. Doran, No. 6 -June 1979.

Estimation of Household Expenditure Functions: An Application of a Classof Heteroscedastic Regression Models. George E. Battese andBruce P. Bonyhady, No. 7 - September 1979.

The Demand for Sawn Timber: An Application of the Diewert Cost Function.Howard E. Doran and David F. Williams, No. 8 - September 1979.

A New System of Log-Change Index Numbers for Multilateral Comparisons.D.S. Prasada Rao, No. 9 - October 1980.

A Comparison of Purchasing Power Parity Between the Pound Sterling andthe Australian Dollar - 1979. W.F. Shepherd and D.S. Prasada Rao,No. i0 - October 1980.

Using Time-Series and Cross-Section Data to Estimate a Production Functionwith Positive and Negative Marginal Risks. W.E. Griffiths andJ.R. Anderson, No. ii - December 1980.

A Lack-Of-Fit Test in the Presence of Heteroscedasticity. Howard E. Doranand Jan Kmenta, No. 12 - April 1981.

On the Relative Efficiency of Estimators Which Include the InitialObservations in the Estimation of Seemingly Unrelated Regressionswith First Order Autoregressive Disturbances. H.E. Doran andW.E. Griffiths, No. 13 - June 1981.

An Analysis of the Linkages Between the Consumer Price Index and theAverage Minimum Weekly Wage Rate. Pauline Beesley, No. 14 - July 1981.

An Error Components Model for Prediction of County Crop Areas Using Surveyand Satellite Data. George E. Battese and Wayne A. Fuller, No. 15 -February 1982.

Networking or Transhipment? Optimisation Alternatives for Plant Loca{Decisions. H.I. Toft and P.A. Cassidy, No. 16 - February 1985.

Oia~.Ino:~tio Toots for the Partial Adjustment and Adai,tiuoModels. H.E. Doran, No. 17 -February ].985.

A Further Consideration of Causal Relationships Between Wages and Prices.

A Monte Carlo Evaluation of the Power of Some Tests For Heteroscedasticity.

W.E. Griffiths and K. Surekha, No. 19 - August 1985.

A Walrasian Exchange Equilibrium Interpretation of the Geary-KhamisInternational Prices. ~.~. Rr~a~a ~ao, ~o. 20 - OCtober 1985.

Using O~rbin’s h-Test to Validate the Partial-Adjustment Model.H.E. Doran, No. 21 - November 1985.

An Investigation into the 5~nall Sample Properties of Covariance Matrixand Pre-Test Estimators for the Probit Model. ~±11~a~ ~. Gr±~E~th~,R. Carter Hill and Peter J. Pope, No. 22 - November 1985.

A Bayesian Framework for Optimal Input Allocation with an UncertainStochastic Production Function. ~±11±am E. Gr±~±~hs, ~o. 23 -February 1986.

A Frontier Production Function for Panel Data: With Application to theAustralian Dairy Industry. ~.a. Coe11± an~ ~.E. Battese, ~o. 24 -February 1986.

Identification and Estimation of Elasticities of Substitution for Firm-Level Production Functions Using Aggregative Data. George ~. Ba~teseand Sohail J. Malik, No. 25 - April 1986.

Estimation of Elasticities of Substitution for CES Production FunctionsUsing Aggregative Data on Selected Manufacturing Industries in Pakistan.George E. Battese and Sohail J. Malik, No.26 - April 1986.

Estimation of Elasticities of Substitution for CES and VES ProductionFunctions Using Firm-Level Data for Food-Processing Industries inPakistan. George E. Battese and Sohail a. Malik, No.27 - May 1986.

On the Prediction of Technical Efficiencies, Given the Specifications of aGeneralized Frontier Production Function and Panel Data on Sample Firms.George E. Battese, No.28 - June 1986.

A General Equilibrium Approach to the Construction of Multilateral IndexNumbers. D.S. Prasada Rao and a. Salazar-Carrillo, No.29 - August1986.

Further Results on Interval Estimation in an AR(1) EPror Model.W.E. Griffiths and P.A. Beesley, No.30 - August 1987.

Bayesian Econometrics and How to Get Rid of Those Wrong Signs.Griffiths, No.31 - November 1987.

H.E. Doran,

William E.

Confidence Intervals for the Expected Average Marginal Products ofCobb-Douglas Factors With Applications of Estimating Shadow Pricesand Testing for Risk Aversion. Chris M. Alaouze, No. 32 -September, 1988.

Estimation of Frontier Production Functtons a~zd the Effic~encies ofIndian Fazvns Using Panel Data from ICRI~A’I"s Village Level Sbudies.G.E. Battese, T.J. Coelli and T.C. Colby, No. 33 - January, 1989.

Estimation of Frontier Production Functions: A Guide to the. ComputerProgram, FRONTIER. Tim J. Coelli, No. 34- February, 1989.

An Introduction to Australian Economy-Wide Modelling. Colin P. Hargreaves,NO. 35- February, 1989.

Testing and Estimating Location Vectors U~e~, Heteroskedasticity., ..William Griffiths and George Judge, No. 36 - February, 1989.

The Management of Irrigation Water During DrOught. Chris M. Alaouze,NO. 37- April, 1989.

An Additive Property of the Inverse of the Sunuivor Funct.Lon and theInverse of the Distribution Function of a Strictly Positive RandomVariable with Applications to Water Allocation Problems.Chris M. Alaouze, No. 38 - July, 1989.

A Mixed Integer Linear Progra,~ning Evaluation of Salinity and WaterloggingControl Options in the Murray-Darling Basin of Australia.Chris M. Alaouze and Campbell R. Fitzpatrick, No. 39 - August, 1989.

Estimation of Risk Effects with Seemingly Unrelated Regressions andPanel Data. Guang II. Wan, William E. Griffiths and Jock R. Anderson,

~ No. 40 - September, 1989. ~

The Optimality of Capacity Sharing in Stochastic Dynamic ProgrammingProblems of Sha~,ed Reservoir Operation. Chris M. Alaouze, No. 41 -

November, 1989.

Confidence Intervals for Impulse Responses from VAR Models: A Comparisonof Asymptotic Theory and Simulation Approaches. William Griffiths andHelmut Ldtkepohl, No. 42 - March 1990.

A Geometrical~Expository Note on Hausman’s Specification Test. Howard E. Doran,No. 43 - March 1990.

Using The Kalman Filter to Estimate Sub-Populations. Howard E. Doran,No. 44 - March 1990.

Documents

USING THE KALMAN FILTER TO ESTIMATE SUB- POPULAT ION S · USING THE KALMAN FILTER TO ESTIMATE SUB- POPULAT ION S Howard E. Doran No. 44 - March 1990 ISSN ISBN 0157-0188 0 85834 873