Chart 3
Basic Forms for Time Series Regression

For any variable X(t) where t is time (i.e., the year in our case) let LX(t) be the natural logarithm of X(t) and let DLX(t) = LX(t) - LX(t-1).

[A] Regressions with Levels of Anthropometric Measures as Dependent Variable

To illustrate these regressions (for levels of height, weight, chest girth, and the BMI for males and females separately at ages 6, 12, and 18) I give two examples, both involving heights for females age 6. Let HF6 = height for females, age 6; HF12 = height for females, age 12; 6NUTI = index for nutrition, #1 averaged over the 6 years beginning in year t ; 6PHMEDI = index for public health and medicine averaged over the 6 years beginning in year t ; and 6CYLFI = index of child/youth labor input for females averaged over the 6 years beginning in year t . Then the basic regression used to estimate the impact of nutrition, health, and medicine and child/youth labor input for age 6 is (£ is the error term)

DLH6 = a₀ + a₁ DL6NUTI(-6) + a₂ DL6PHMEDI(-6) + a₃ DL6CYLFI(-6) + £

and the basic equation used to estimate the impact of the independent variables on height at age 12 is

DLH12 = b₀ + b₁ DLH6(-6) + b₂ DL6NUTI(-6) + b₃ DL6PHMEDI(-6) + b₄ DL6CYLFI(-6) + £

In regressions on DLH18, 1 use DLH12(-6) instead of DLH6(-6).

[B] Regressions with Six-Year Gains in Anthropometric Measures as Dependent Variable

To Illustrate how I calculated the impact of nutrition, public health and medicine, and the child/youth labor input on six-year gains in height, weight, body mass index, and chest girth, I use gains in female height. I demonstrate my procedure with the gain in female height between ages 6 and 12, GHFA, as dependent variable. The basic equation I used is

DLGHFA = c₀ + c₁ DLH6 + c₂ DL6NUT1 + c₃ DL6PHMEDI + c₄ DL6CYLFI + £

TABLE 11
Secular Determinants of Standing Height and Gain in Height, 1901-1979^a

A. Estimates Based on Indexes Covering Entire Period

1907-1979

1907-1940

1945-1979

Height

H(-6)

NUT1

PHM1

CYL1

H(-6)

NUT1

PHM1

CYL1

H(-6)

NUT1

PHM1

CYL1

Males, 6

n.e.

-

-

-03³

n.e.

-

-

-

n.e.

-

^+.013

-.03²

^{Females, 6}

n.e.

-

-

-

n.e.

-

-

-

n.e.

-

+.01³

-.03⁴

^{Males, 12}

-

-

^-

-^.062

-

-

-.35³

-

-

-

+.02¹

-.09¹

Females, 12

-

-

+.02²

-10¹

-

-

-

-

-

-

+.02²

-.10¹

Males, 18

-.13²

-

-

-

-

-

-

-

-.15¹

-

+.01³

-

Females, 18

-

-

-

-

-

-

-

-

-

-

-

-

1901-1979

1901-1940

1945-1979

Gains in Height

H

NUT1

PHM1

CYL1

H

NUT1

PHM1

CYL1

H

NUT1

PHM1

CYL1

Males, 6-12

-4.10¹

-

-

-.29²

-5.26¹

-

-

-1.26¹

-3.44¹

-

+.06¹

-

Females, 6-12

-2.61¹

-

-

-.28¹

-

-

-

-1.50¹

-2.71¹

-

-

-

Males, 12-18

-6.79¹

-

+.04³

-

-5.98¹

-

-

-

-7.43¹

-

-

-

Females, 12-18

-11.4¹

-

-

-

-11.6¹

-

-

-

-10.8¹

-

-

-

NOTES:

^a Based on first differences of log-log regressions. See chart 2 and appendix to chapter 2 for a discussion. Abbreviations roughly follow those given in tables 8, 9, and 10. H = standing height; H(-6) for year t indicates standing height for persons 6 years or younger in the year t - 6; NUT1 for year t is a six-year average for the years t through t + 5 based on the index of nutrition for the entire prewar and postwar period; PHMEI for year t is a six-year average for the years t through t + 5 for the index of public health and medicine; and CYL1 for year t is a six-year average for the years t through t + 5 for the index of child/youth labor input covering the entire prewar and postwar periods. In the regressions on levels, the three indexes are lagged 6 years. In the case of gains, they are not lagged. In the regressions on gains, H indicates the level of height in the year in which the gain begins. Values only reported if they are at least significant at the 15% level (two-tailed test).

TABLE 11 continued

B. Estimates Based on Nutrition and Child/Youth Labor Input Indexes Covering Postwar Period Only^b

1953-1979

1955-1979

Height

H(-6)

CALFA1

PHM1

CYL2

H(-6)

NUT2

PHM1

CYL2

Males, 6

n.e.

-

+.01⁴

-

n.e.

-

+.01³

-

Females, 6

n.e.

-

-

-

n.e.

-

-

-

Males, 12

+.64¹

-

+.01¹

-

+.68¹

-

+.01¹

-.04³

Females, 12

+.21²

+.04²

+.004⁴

-

+.18¹

-

+.01³

-

Males. 18

-.27¹

-

-

-

-

-

+.01³

-

Females, 18

-

-

-

-

-

-

-

-

1953-1979

1955-1979

Gains in Height

H

CALFA1

PHM1

CYL2

H

NUT2

PHM1

CYL2

Males, 6-12

-

-

+.04¹

-

-

-

+.04¹

-.18³

Females, 6-12

-2.31¹

+.16³

+.02⁴

-

-1.91¹

-

-

-

Males, 12-18

-6.94¹

-

+.04³

-

-6.39¹

-.29⁴

+.04²

-

Females, 12-18

-12.96¹

-

-

-

-11.07²

-

-

-

NOTES:

^b See note to panel A for a discussion of the regression format and a set of basic definitions. Additional abbreviations: CALFA1 = index of combined calcium and fat intake; CYL2 = index of child/youth labor input based on postwar data series not available for the prewar period (see tables 8 and 10).

Significance levels (two-tailed tests): 1, 1% level; 2, 5% level; 3, 10% level; 3, 15% level.

TABLE 12
Secular Determinants of Weight and Gain in Weight, 1901-1979^a

A. Estimates Based on Indexes Covering Entire Period

1907-1979

1907-1940

1945-1979

Weight

W(-6)

NUT1

PHM1

CYL1

W(-6)

NUT1

PHM1

CYL1

W(-6)

NUT1

PHM1

CYL1

Males, 6

n.e.

-

-

-.06³

n.e.

-

-

-

-

-

-

-

Females. 6

n.e.

-

-

-

n.e.

-

-

-

-

-

-

-

Males. 12

+.34¹

-

-

-.16¹

-

-.66⁴

-

-1.07³

-

-

+.08¹

-.17¹

Females, 12

-

-

-

-.13³

-

-

-

-

-

+.08¹

-.17¹

Males, 18

-

-

+.02³

-

-

-

-

-

-

-

+.04³

-

Females, 18

-

-

-

-

-

-

-

-

-

-

-

-

1901-1979

1901-1940

1945-1979

Gains in Weight

W

NUT1

PHM1

CYL1

W

NUT1

PHM1

CYL1

W

NUT1

PHM1

CYL1

Males, 6-12

-

-

-

-.38¹

-1.61²

-

-

-2.19¹

+.27³

+.07¹

-

-.89¹

Females. 6-12

-1.59¹

-

-

-.24³

-

-

-

-1.36¹

-

+.04³

-

-.86¹

Males, 12-18

-1.66¹

-

+.06²

-

-.72³

-.72³

-.37³

-.49³

-

+.06²

^-

^-1.431

Females, 12-18

-2.14¹

-

-

-

-1.41¹

-1.41¹

-

-.77¹

-

-

-

-.1.31³

NOTES:

^a Based on first differences of log-log regressions. See chart 2 and appendix to chapter 2 for a discussion. Abbreviations follow those given in tables 8, 9, and 10. W = weight; W(-6) for year t indicates weight for persons 6 years or younger in the year t-6; NUT1 for year t is a six-year average for the years t through t + 5 based on the index of nutrition for the entire prewar and postwar period; PHME1 for year t is a six-year average for the years t through t + 5 for the index of public health and medicine; and CYL1 for year t is a six-year average for the years t through t + 5 for the index of child/youth labor input covering the entire prewar and postwar periods. In the regressions on levels, the three indexes are lagged 6 years. In the case of gains, they are not lagged. In the regressions on gains, W indicates the level of weight in the year in which the gain begins. Values reported only if they are at least significant at the 15% level (two-tailed test).

TABLE 12 continued

B. Estimates Based on Nutrition and Child/Youth Labor Input Indexes Covering Postwar Period Only^b

1953-1979

1955-1979

Weight

W(-6)

CALFA1

PHM1

CYL2

W(-6)

NUT2

PHM1

CYL2

Males, 6

n.e.

-

-

-

n.e.

-

-

-

Females, 6

n.e.

-

-

-

n.e.

-

-

-

Males, 12

-

-

+.04²

-

-

-

+.04²

-

Females, 12

-

+.15²

-

-

-

-

+.03³

-

Males, 18

-

-

-

-

-

-

-

-

Females, 18

+.87²

-

-

-

-

-

-

-

1953-1979

1955-1979

Gains in Weight

W

CALFA1

PHM1

CYL2

W

NUT2

PHM1

CYL2

Males, 6-12

-72²

+.19²

+.06¹

-

-.73³

-

+.07¹

-

Females, 6-12

-89¹

+.32¹

+.03³

-

-

-

+.05¹

-

Males, 12-18

- 1.22¹

-

+.06²

-

-.94²

-

+.07¹

-

Females, 12-18

-

-

-

-

-

-

-

-

NOTES:

^b See note to panel A for a discussion of the regression format and a set of basic definitions. Additional abbreviations: CALFA1 = index of combined calcium and fat intake; CYL2 = index of child/youth labor input based on postwar data series not available for the prewar period (see tables 8 and 10).

Significance levels (two-tailed test): 1, 1% level; 2, 5% level; 3, 10% level; 4, 15% level.

TABLE 13
Secular Determinants of Body Mass Index, 1901 - 1979a

(Estimates Based on Indexes Covering Entire Period)

1907-1979

1907-1940

1945-1979

BMI

B(-6)

NUT1

PH

CYL1

B(-6)

NUT1

PH

CYL1

B(-6)

NUT1

PH

CYL1

Males, 6

n.e.

-

-

-

n.e.

-

-

-

n.e.

-

-

-

Females, 6

n.e.

-

-

-

n.e.

-

-

-82³

n.e.

-

-

Males, 12

+.31²

-

-

-

+.39³

-

-

-

-.46⁴

-

-

-

Females, 12

-.42¹

-

-

-

-.50²

-

-

-

-

-

-

+.06²

Males, 18

-

-

-

-

-

-

-

-

-.21³

-

-

-

Females, 18

+.17⁴

-.16⁴

-

-

-

-

-

-

+.35³

-

-

-

1901-1979

1901-1940

1945-1979

Gains in BMI

BMI

NUT1

PH

CYL1

BMI

NUT1

PH

CYL1

BMI

NUT1

PH

CYL1

Males, 6-12

-7.13¹

-

-

-

-8.69¹

-

-

-4.39²

-6.35¹

-

-

-

Females, 6-12

-11.0¹

1.37⁴

-

-

12.0¹

-

-

-

-5.94¹

-

-

+.53³

Males., 12-18

-5.03¹

-

-

-

-4.60¹

-

-

-

-5.72¹

-

-

-

Females, 12-18

-3.09¹

-

-

-3.20¹

-2.49¹

-

-1.39¹

-

-

-

-

-

NOTES:

a Based on first differences of log-log regressions. See chart 2 and appendix to chapter 2 for a discussion. Abbrevations roughly follow those given in tables 8, 9, and 10. B(-6) indicates BMI for persons six years younger in the year t - 6; NUT1 for year t is a six-year average for the years t through t + 5 based on the index of nutrition for the entire prewar and postwar period; PH for year t is a six-year average for the years t - 5 through t for the index of public health and medicine; and CYL1 for year t is a six-year average for the years t through t + 5 for the index of child/youth labor input covering the entire prewar and postwar periods. For the regressions on levels, the three indexes are lagged 6 years. In the case of gains they are not lagged. In the regressions on gains H indicates the level of height in the year in which the gain begins. Values reported only if they are at least significant at the 15% level (two-tailed test). See tables 11 and 12 for significance test indicators.

TABLE 14
Secular determinants of Chest Girth, 1901-1979^a

(Estimates Based on Indexes Covering Entire Period)

1907-1979

1907-1940

1945-1979

Chest Girth

c(-6)

NUT1

PH

CYL1

C(-6)

NUT1

PH

CYL1

C(-6)

NUT1

CYL1

Males, 6

n.e.

-

-

-

n.e.

-

-

-

n.e.

-

-

-

Females, 6

n.e.

-.12²

-

-

n.e.

-

-

-

n.e.

-

-

-

Males, 12

-.22⁴

-

.01⁴

-

-

-

.11⁴

-

-

-

.01¹

-

Females, 12

-

-

-

-

-

-

-

-

-

-

-

+.13¹

Males, 18

-

-

-

-

-

-

-

-

-

-.12³

-

-

Females, 18

-

-

-

-

-

-

-

-

+.94¹

-

-

-

1901-1979

1901-1940

1945-1979

Gains in Chest Girth

CG

NUT1

PH

CYL1

CG

NUT1

PH

CYL1

CG

NUT1

PH

CYL1

Males, 6-12

-3.32¹

-

-

-

-6.49¹

-

-

-.95²

-3.19¹

-

.05¹

-

Females, 6-12

-.3.82¹

-

-

-

-4.34²

-1.55³

-

-1.38¹

-3.30¹

-

-.04³

Males, 12-18

-4.37¹

-

-

-

-4.36¹

-

-

-.69³

-3.24²

-.51³

-

-

Females, 12-18

-5.25¹

-

-

-

-4.16¹

-1.24⁴

-

-.74⁴

-

-

-

-

NOTES:

^a Based on first differences of log-log regressions. See chart 2 and appendix to chapter 2 for a discussion. Abbreviations follow those given in tables 8, 9, and 10. C(-6) indicates chest girth for persons six years younger in the year t - 6; NUT1 for year t is a six-year average for the years t through t + 5 based on the index of nutrition for the entire prewar and postwar period; PH for year t is a six-year average for the years t through t + 5 for the index of public health and medicine; and CYL1 for year t is a six-year average for the years t through t + 5 for the index of child/youth labor input covering the entire prewar and postwar periods. In the regressions on levels, the three indexes are lagged 6 years. In the case of gains, they are not lagged. In the regressions on gains, H indicates the level of height in the year in which the gain begins. Values reported only if they are at least significant at the 15% level (two-tailed test). See tables 11 and 12 for significance test indicators.