A NOTE ON THE SPECIFICATION AND ESTIMATION OF A BIAS IN THE MEASUREMENT OF THE SIZE AND GROWTH OF REAL GDP IN JAMAICA

A NOTE ON THE SPECIFICATION AND ESTIMATION OF A BIAS IN THE MEASUREMENT OF THESIZE AND GROWTH OF REAL GDP IN JAMAICAAuthor(s): Randolph WilliamsSource: Social and Economic Studies, Vol. 22, No. 3 (SEPTEMBER 1973), pp. 384-394Published by: Sir Arthur Lewis Institute of Social and Economic Studies, University of the WestIndiesStable URL: http://www.jstor.org/stable/27856585 .

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384

A NOTE ON THE SPECIFICATION AND ESTIMATION OF A BIAS IN THE MEASUREMENT OF THE SIZE AND GROWTH

OF REAL GDP IN JAMAICA1

By

R. L. Williams

Introduction

In an earlier paper [7] the existence of a bias in the price deflator of capital goods used by the Department of Statistics, Jamaica, was demonstrated. The omission of an index of construction labour costs from the composite index of prices of capital goods introduced a negative bias in the price deflator over the period 1953 to 1966. A new composite index was constructed which included an index of construction labour costs and the difference between the new index and the Department of Statistics

composite index was found to be significant at the 0.99 level. This note investigates the effect of the bias in the price index of capital goods on the measurement of the size and proportional rate of growth of real GDP in Jamaica. Corrected estimates of the size and the rate of growth of real GDP and its principal components between 1950 and 1966 are also presented.

The measurement of national income in Jamaica dates from the very beginning of the techniques of social accounting. Among the early estimates are those of Benham

[5], Deane [2], Mor?is [1] and Thorne [6]. Current and constant price estimates of the main national income aggregates from 1950 are available from official public ations.2 These publications constitute the primary sources of data used in this

study. Although constant price estimates of the main aggregates have been published up to the year 1968 this study terminates at the year 1966. With the cessation of

publication by the Department of Statistics of the statistics of earnings of workers

employed in large establishments [3], extrapolation of the construction labour cost index became necessary and from 1966 the reliability of our findings would be reduced.

The order of presentation of this paper is as follows: Following this intro

duction, Section II contains specifications and estimates of the bias in the measure ment of the size and of the proportionate rates of growth of real gross domestic

product. In Section III the empirical results are discussed and corrected estimates of the proportional rate of growth of real GDP and its main components are

presented. An appendix to this paper contains a description of the time series used and their sources.

Section II. Specification and Estimation of the Biases

Gross Domestic Product in any year can be represented by the following identity.



NOTE ON THE SPECIFICATION AND ESTIMATION OF A BIAS 385

(1) Y=C+G+NE + r + S + SD

where Y stands for gross domestic product

C stands for consumption of households and firms

G stands for consumption expenditure of government

NE stands for exports net of imports (including services but excluding factor

incomes)

/ stands for gross fixed investment

S stands for change in inventories

SD stands for a statistical discrepancy in compiling the accounts.3

All the variables are valued in constant prices with 1960 prices taken as unity, and are measured in units of millions of pounds.

The bias in the price index of gross fixed investment will lead to a bias in the value of / and in the value of Y. The size and direction of the bias in Y in any year,

represented by H, are given by the following equation: -

(2) H = X(A*?A)/A*A

where X stands for gross fixed capital formation in current prices, A stands for the

Department of Statistics price deflator of gross fixed capital formation with 1960

prices equal unity and A* stands for the "true" price deflator. The bias will be

positive, zero or negative as>4* is respectively greater than, equal to or less than A.

Table I contains two estimates of the real Gross Domestic Product as well as an estimate of H from 1950 to 1966. The figures in the second column of Table I are the estimates from the official sources all revalued in 1960 prices. Those in the third column are the estimates of real gross domestic product at 1960 prices after correcting for the bias in the measurement of real gross fixed capital formation.

Assuming that the variables in equations (1) and (2) are all continuous functions of time and defining the proportional rate of change of a variable at time t

as~^: the proportional rate of change of Y at time t is the weighted average of the propor tional rate of change of the components of Y at time t thus:

(3) YI = a C/C + a2 G/G + ̂ NE/NE + f/I + a$ S/S + a6 SD /SD

where a.(i= 1,2,.,6) are the weights and a dot over a variable indicates the first derivative of that variable with respect to time. Defining the weights in

equation (3) as the share of the components in gross domestic product at time t (so



386 social and economic studies

TABLE I Gross Domestic Product at Constant (1960) Prices - 1950-1966

YEAR

(1)

Official Estimates

(2) (?m.)

Corrected Estimates

(3) (?m.)

BIAS

(4) (?m.)

Column (2) as

proportion of (3)

(5)

1950 1951 1952 1953 1S54 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966

95.9320

104.3170 111.5890 127.6220 142.1400 156.6140 173.0310 198.0000 200.7170 202.6850 215.8760 222.0650 225.9040 233.4500 251.7070 271.9470 283.6390

100.3727 108.9649 114.9082 121.3971 138.6063 154.0303 173.7466 198.5296 198.7426 202.9096 215.6230 221.3645 223.2400 230.6543 249.0236 266.9945 277.4191

-4.4407

-4.6479

-3.3192

+6.2249

+3.5337

+1.6837

-0.7156

-0.5296

+1.9743

-0.2246

+0.2530

+0.7005

+2.6640

+2.7957

+2.6834

+4.9525

+6.2199

0.9558

0.9573

0.9711

1.0512

1.0255

1.0108

0.9959

0.9973

1.0099

0.9989

1.0012

1.0032

1.0119

1.0121

1.0108

1.0186

1.0224

Source: See Appendix

that, for example, a1 =

C/Y), equation (3) may be written as follows:

(4) y/y= 1/y(c+g + ne+ l+s + sd)

The error in the measurement of real gross fixed capital formation affects the

value of /, y and y in equation (4). Expressing the "true" value of a variable by an

asterisk over that variable the "true" proportional rate of change of y at time t is given as follows:

(5) y*/y* = i/y*(c+g+?e+?* + s + sd)

The difference between Equation (4) and (5) is an expression of the bias in the measured proportional rate of change in real gross domestic product at time r, thus:

(6) r/r- f*/y* = (i/y-i/r*)(c^

Since (7) / = x/a and both x and a are by assumption continuous functions of time we have:




(8) f/Y = ??L= )AY - X?/YA2 Ydt

Similarly (9) h/Y* = X/A*Y* - XA*/Y*A*2

Inserting (8) and (9) into (6), collecting like terms and using (7) yields:

(10) Y/Y-Y*/Y*= (C+G + ?E + S + SD)(1/Y-1/Y*) + X(1/AY- 1/A* *) + ([*/ *)A*/A* -( / )A/A

From equation (10) we learn that the bias in the measurement of the propor tional rate of growth of real GDP is the sum of two main sources of error: -

(1) an error in the measured proportional rate of growth of all the com

ponents of real GDP due to estimating incorrectly the share of the components in GDP.

(2) the difference between the proportional rate of growth of the "true" price index of capital goods, A*, weighted by the "true" share of gross fixed capital for

mation in real GDP, and the proportional rate of growth of the Department of Statistics price index of capital goods A, weighted by the share of gross fixed capital formation in real GDP estimated by the Department of Statistics.

The direction of the total bias is not known a priori from equation (10) since all the terms on the right hand side may take either positive or negative sign.

Using the first difference of a variable to approximate the rate of change of that variable with respect to time and expressing the change in year t as a percentage of the value of the variable in year t? 1 the bias specified in equation (10) was estimated for the years 1951 to 1966. The estimate of the first term on the right hand side of

equation (10) representing the contribution to the total bias of the error in the share of private consumption, government consumption, net exports, change in inventory and th? statistical discrepancy in GDP was found to be negligible, either individually or in the aggregate, except in 1953. Only the estimate of the total bias and the last three terms of equation (10) are presented in Table II.

? Discussion

The results presented in Table I are consistent with the expectations that the bias in the Department of Statistics price deflator of capital formation imparts a bias in the opposite direction in the estimates of the size of real GDP in Jamaica over the

period 1950 to 1966. The direction of the bias is given by the sign of (A *?A) in the numerator of equation (2). In the first three years the bias was in the negative



388 social and economic studies

TABLE II Percentage Rates of Growth of Real GDP 1951-1966

Year Y/Y Y*/Y* Column (2) as

proportion of Column (3)

BIAS

Total (a) (a) (a)

(1) (2) 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966

8.7

7.0

14.4

11.4

10.2

10.4

14.4

1.4

1.1

6 3 2.7

0.9

4.4

7.8

8.0

4.3

(3) 8.5

5.5

5.6

14.2

11.8

12.1

14.3

0.1

2.1

6.3

2.7

0.8

3.3

8.0

7.2

3.9

(4) 1.0352

1.2727

2.5714

0.8028

0.8644

0.8595

1.0070

14.0000 0.5238

1.0000

1.0000

1.1250

1.3333

0.9750

1.1111

1.1025

(5) +0.2

+1.5

+8.8

-2.8

-1.6

-1.7

0.1

+1.3

-1.0

0.0

0.0

+0.1

+1.1

-0.2

+0.8

+0.4

(6) -1.15

-0.5

0.2

0.9

0.5

0.7

-0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.2

0.1

0.3

(7) (8) 1.9

2.0

0.4

-1.5

3.0

9.4

2.1

-0.7

1.4

-1.5

2.2

1.1

0.7

0.1

0.7

2.14

0.4

0.2

-7.2

1.2

4.9

12.1

1.9

-1.9

2.5

-1.5

2.1

1.1

-0.5

0.3

0.0

1.9

(a) (Columns (6), (7) and (8) present estimates of the last 3 terms in Equation (10) respectively.

direction, but in recent years the bias has been positive and has been increasing in

magnitude. In 1950, 1951 and 1952 the Department of Statistics price deflator over

estimated the rise in the price of capital goods. The rise in the prices of imported equipment and building material in the early 1950s,and the larger weights attached to

these components in the official price deflator ?(5/8ths and 3/8ths respectively instead of 5/10ths and 3/10ths in the corrected estimate) [7, p. 375]

? led to a

positive bias in the price deflator and a negative bias in the estimate of real GDP.

A decline in the prices of imported equipment and building material occurred in

1953 and in that year the price of local building materials also fell due to the erection of the cement factory at Rockfort near Kingston [4, Table III, p. 9]. The large weights attached to equipment and building materials in the official statistics depressed the

price deflator of capital goods even further. The exclusion of the index of construction labour cost also led to the Department of Statistics price deflator of capital goods

underestimating the true rise in the price of capital goods. Construction labour costs

have been rising faster than the composite index of equipment and building material

except in 1956, 1957 and 1959 [7, Table I]. The negative bias in the price deflator led to a positive bias in the official estimates of real GDP except in 1956, 1957, and 1959.

Between 1950 and 1966 the bias has ranged from minus ?4.6 million to ?6.2

million and has accounted for an error of the order of ? 5 per cent in the official




estimate of real GDP. A test of the hypothesis that the ratio of the official estimate of GDP to the corrected estimate was greater than unity against the null hypothesis that the ratio was unity indicated that the null hypothesis could not be rejected. The value of the t statistic was 0.1377.

The results in Table II indicate the extent and direction of the bias in the measurement of the proportional rate of growth of real GDP between 1951 and 1966. The main source of the difference between the two estimates of the growth rate is the understatement or overstatement of the proportional rate of growth of the price of

capital goods. An understatement of the rate of growth of the price of capital goods tends to introduce a positive bias in the measurement of the rate of growth of real GDP and vice versa. The size of this bias is however influenced by the estimate of the share of gross fixed capital formation in GDP, and is measured by the difference between column (7) and column (8) in Table II. The following example illustrates the

operation of the bias. The Department of Statistics price deflator of capital goods revealed a decline from 0.731 in 1952 to 0.495 in 1953, a proportional rate of change of ?.323. The corrected price deflator showed a rise between 1952 and 1953 from 0.6208 to 0.6304, a proportional rate of change of .019. The share of gross capital formation in GDP in 1953 was 0.224 according to the official estimate and 0.184

according to the corrected estimate. The bias in the proportional rate of growth of GDP from that source is therefore, according to equation (10): (.019)(.184)

?

(?.323)(.224) = .0759 and is shown as the difference between columns (7) and (8) of

Table II.

The arithmetic mean of the total bias over the 16 years 1951 to 1966 is 0.44 per cent and its variance is 5.92. The range is from ?2.8 per cent to 8.8

per cent. The high variability of the error series relative to its mean led to a value of the t statistic of 0.1808. The null hypothesis that the bias was zero could not be rejected.

Table III provides information on the percentage rate of growth of real GDP and its principal components from 1951 to 1966 after correcting for the biases specified in

equation (10). The high rates of growth in the 1950s in the period of establishment of the bauxite mining and alumina refining industries are confirmed. The simple annual

average rate of growth from 1951 to 1957 was 10.28 per cent. From 1958 however the average was 3.82 per cent making an average of 6.65 per cent over the period 1951 to 1966.

The growth in private consumption net of indirect taxes at an annual average rate of 4.21 per cent between 1951 and 1966 has accounted for the largest share of the growth of real GDP. Over the same period government expenditure (excluding capital expenditure) in real terms has grown at a rate of 0.76 per cent. In the 1950s when the real GDP grew at 10.28 per cent the government expenditure (excluding capital) grew only at the rate of 1.02 per cent, while private consumption grew at 6.31

per cent per annum.

The rate of growth of gross fixed investment has been particularly unstable. In the 1950s up to 1957 the annual rate of growth averaged 3.78 per cent. Between 1958



TABLE III Percentage Rates of Growth of Real GDP and its Principal Components 1951-1966

Year

Gross Domestic

Product

Private

Consumption

(Net

tax)

Government Consumption %

Net Exports %

Change in Inventory %

Gross Fixed Investment %

Statistical Discrepancy

1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966

8.56 5.45 5.64

14.17

11.77 12.14

14.26 0.10 2.09 6.26

2.66 0.84 3.32

7.96 7.21 3.90

-0.85 4.73

6.87 9.32 9.57 6.13

8.42

-0.70 1.34 1.62 1.42

4.98 1.06

8.38 2.41 2.62

2.64 -0.38

1.14 0.91 0.39 1.71

0.77 0.43 -0.21

1.00 0.63 0.97

0.66 0.44 0.78 0.29

0.30 0.26 -2.92 -2.47 -1.36

-0.50 -1.43

2.55 1.91 2.68 4.46 2.44

-2.48 -3.90 5.75 1.23

-0.84 0.38

0.27 0.34 1.29

-0.34 1.09

-0.59 0.92

-1.59

0.48 -0.38 0.41

-0.05 -0.58 0.25

7.29 0.45

0.27 6.06

1.87 5.14 5.39 -1.58 -2.15

3.49 -2.42

-1.84

-0.86 3,57 1.27 0.99

0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.28

-0.93

-1.91

-5.33

4.53

-0.48

-2.42 -1.50

I t-1 m 8 H B




and 1966 however the average rate of growth has been almost zero (0.05 per cent). These results have implications for the growth of productivity in Jamaica but a more detailed analysis of the results in Table III would take us far beyond the scope of this note. It is hoped that this analysis will be undertaken in a subsequent paper as more data become available.

FOOTNOTES

*I am grateful to the staff of the Computer Centre, Department of Mathematics, UWI for

computational assistance.

See Appendix to this paper.

3The discrepancy was first published in the 1959 national accounts.

REFERENCES

[1] Central Bureau of Statistics

[2] Deane, Phyllis

[3] Department of Statistics

[4]

[5] National Income of Jamaica

[6] Thorne, A.P.

[7] Wilhams, R.L.

The National Income of Jamaica 1943, 1946, King ston: 1948.

"The Measurement of Colonial National Incomes," National Institute of Economic and Social Research, Occasional Paper XII, Cambridge University Press, 1948.

Employment and Earnings in large establishments 1957-1962 and subsequent issues, Kingston.

Capital Formation in Jamaica 1948-1954, Kingston, 1955.

Memorandum by Frederic Benham, Development and Welfare in the West Indies, Bulletin No. 5, 1944.

"Size Structure and Growth of the Economy of

Jamaica", Social and Economic Studies, U.W.I. Vol.

4, No. 4.

"Gross Fixed Capital Formation in Jamaica 1948-1966: A Constant Price Series", Social and

Economic Studies, December 1971.



392 SOCIAL AND ECONOMIC STUDIES

APPENDIX

Description and Sources of Data

The principal time series used in this study are as follows: -

(1) Gross Domestic Product 1950-1966 (I960 prices)

Estimates of the Gross Domestic Product at factor cost in current and constant

prices are published by the Department of Statistics. The main sources are:

1950-1952 (Current prices) ? National Accounts Income and Expenditure 1950-1957

Kingston 1959, Account 1 p. 66.

1950-1952 ? (1956 prices)

- Jeanette Bethel ? "Some National Income Aggregates for Jamaica at Constant prices" Social and Economic Studies Vol. 10, No.

2, June 1961. Table 276 p. 153.

1953-1960 (Current and 1956 prices) ? National Accounts, Income and Expenditure,

1958, 1959, Tables 5, 5A and National Accounts, Income and Expend iture, 1961.

1959-1966 (Current and 1960 prices). National Income and Product 1968, Table

1, p. 54 and Table 20, p. 59.

From the above sources implicit price deflators of GDP were compiled with 1956 prices and 1960 prices equal unity. The series with 1956 prices as unity was

converted to 1960 base year and against this the 1959-66 price deflator (1960 equal unity) was extrapolated backwards to 1950. The GDP current price series 1950-1966 was then deflated by the 1960 base year price deflator.

(2) Main Components of GDP 1950-1966 (I960 prices)

The time series of the main components of GDP (except capital formation) at 1960 prices, were derived in a manner similar to that described for the GDP series. The time series of private consumption was net of indirect taxes and subsidies. The current

price series of capital formation (X) was derived from the above mentioned sources.

The Department of Statistics implicit price deflator of capital goods with base year 1960 and the corrected price deflator of capital goods with base year 1960 are

published in Williams [7]



TABLE A-1 Principal Components of GDP 1950-I958a (1960Prices)

Components 1950 1951 1952 1953 1954 1955 1956 1957 1958 _ _, _ 2

?. million _

Consumption 101.194 99.740 104.941 113.934 126.904 142.018 153.387 166.248 168.034 ? less Net Indirect Taxes 17.239 16.639 16.677 17.765 19.418 21.258 23.124 21.340 24.521 g

Consumption - W

net indirect taxes 83.955 83.101 88.264 96.169 107.486 120.760 130.263 144.908 143.513 g Government Consumption 10.060 12.716 12.295 13.607 14.717 15.268 17.920 19.271 20.126 Q

Net Exports (includes non-factor services) -9.256 -8.947 -8.660 -12.019 -15.019 -16.911 -17.693 -20.190 -15.125 >

Change in Inventory 1.372 0.527 0.949 1.265 1.687 3.479 2.950 4.861 3.688 Gross Fixed Capital

Formation 14.242 21.568 22.060 22.375 29.735 32.334 40.307 49.680 46.541 _ G.D.p. (factor cost) 100.373 108.965 114.908 121.397 138.606 154.930 173.747 198.530 198.743

~

aConstant (1960) price estimates of the principal components from 1959 are easily accessible in Department of Statistics publications.


w co

S > H

o

>

> co



TABLE A-2 Implicit Price Deflators of GDP and Principal Components 1950-1966 (Base Year i960)

Year

1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966

G.D.P.

(factor cost)

0.731

0.783

0.851

0.836

0.842

0.871

0.916 0.969

0.990 0.979

1.000

1.039 1.064

1.096

1.088

1.093

1.139

Consumption (a) Private Gov't

0.670

0.771

0.850

0.854

0.840

0.852

0.856

0.877

0.926

0.954

1.000

1.012

1.001

1.035

1.048

1.064

1.077

0.666

0.692

0.732

0.779

0.761

0.858

0.904

0.934

0.949

0.991

1.000

1.047

1.076

1.096

1.145

1.175

1.256

Exports (b)

0.739

0.791

0.830

0.865

0.928

0.961

0.934

1.072

1.021

0.992

1.000

1.014

1.072

1.152

1.119

1.053

1.082

Imports (b)

0.688

0.830

0.895

0.809

0.813

0.880

0.953

0.983

0.982

1.002

1.000

1.047

1.141

1.137

1.184

1.207

1.262

Change in

Inventory

0.729

0.949

0.843

0.790

0.889

0.891

0.949

1.049

0.949

0.988

1.000

1.239

1.152

1.189

1.377

1.317

1.104

Gross Fixed Investment

(c) (d) 0.704

0.721

0.731

0.495

0.521

0.631

0.982

1.061

0.978

1.080

1.000

1.091

1.150

1.120

1.140

1.140

1.250

(a) Net of indirect taxes and subsidies

(b) Goods and non-factor services

(c) Department of Statistics implicit price deflators

(d) Corrected price deflators to include the price index of construction labour cost. See [7].

0.4866

0.5638

0.620$ 0.6324

0.5818

0.6640

0.9622

1.0488

1.0184

1.0780

1.000

1.0940

1.1560

1.2020

1.2060

1.2540

1.3990

C/5 o ? >

8

g c/a




Documents

A NOTE ON THE SPECIFICATION AND ESTIMATION OF A BIAS IN THE MEASUREMENT OF THE SIZE AND GROWTH OF REAL GDP IN JAMAICA