1

January 2007 Labor Estimates Methodology

USDA NASS Research and Development Division

Stephen Busselberg, Mathematical Statistician

3251 Old Lee Highway, Room 305

Fairfax, Virginia 22030

Phone: 703-877-8000, Ext. 141

Background

Historically, NASS has used probability

sampling to produce labor estimates. NASS

labor estimates for January 2007 consist of

multivariate time series forecasts within 

domains formed by the following crossed

factors: worker type, labor unit, and geographic

region. Worker type is estimated at the level of

field, field and livestock, and all hired workers.

Labor unit includes the wage rate, number of

workers, and number of hours worked.

Geographic regions are composed of groupings

of states that lie within close proximity to each 

other and have similar agricultural production.

Three of the regions contain only one state and

those regions are Hawaii, California, and 

Florida. The United States is also considered a

region.

Regions Labor Unit Worker Type

Northeast I Wage Rates All Hired

Northeast II Number of Workers Field & Livestock

Appalachian I Hours Worked Field

Appalachian II

Southeast

Lake

Corn Belt I

Corn Belt II

Delta

Northern Plains

Southern Plains

Mountain I

Mountain II

Mountain III

Pacific

California

Florida

Hawaii

US

Three worker types, three labor units, and 

nineteen regions equate to 171 required

estimates.

Historical Data

The inputs for the labor time series models

consist of published quarterly data for each 

region, labor unit, and worker type. These data

are available in an uninterrupted time sequence

from January 1997 to December 2006 for hours

worked and number of workers; and from

January 1989 to December 2006 for wage rates.

The published estimates from these years and 

quarters served as the inputs to the time series

modeling, and the ultimate outputs of the

models served as survey indications, which

were used in setting the official estimates for

January 2007. The time series model used in

generating the inputs to the estimation process

produced very consistent results that reflect the

periodic trends of the published estimates.

Methodology

The chosen class of models for the January

2007 labor estimates is Vector Autoregressive

Moving Average (VARMA). The endogenous

dependent variables are the levels of worker

type. Wage rates, number of workers, and

hours worked are assumed uncorrelated.

Schematic plots of the autocovariance structure

and minimum corrected Akaike’s Information

Criterion (AICc) for test values of the

autoregressive and moving average orders

show that the autocovariance structure is

similar across many of the regions.

If the United States region is considered an

averaging out or smoothing over the other

regions, its correlation structure should be

representative of all the other regions combined

and give a good indication of the model

structure to be used for all regions for a given

labor unit. Assuming the autocovariance

structures between the worker types within
2

each labor unit are the same across all regions,

at most three independent time series model

structures were considered appropriate – one

for each level of labor unit. The same model

structure could then be used for each region

within each labor unit. Only the model

parameters differed for computation of the

step-ahead estimates.

Wage Rates

Figure 1

US Wage Rates

(January 1989 - December 2006)

Workers

Field

Field & Livestock

All Hired

W

a

g

e

R

a

t

e

3

4

5

6

7

8

9

10

11

12

Date

JAN1988

JAN1990JAN1992

JAN1994JAN1996

JAN1998

JAN2000JAN2002JAN2004JAN2006JAN2008

The graph in Figure 1 demonstrates that the

levels of worker type would be good candidates

for the endogenous variables in a VARMA

time series model as they are highly correlated

over time. Minimum AICc selection criteria for

the preliminary orders of the autoregressive

(AR) and moving average (MA) polynomials

produced an order of zero for the MA

polynomial for all regions. Further

investigation results conclude any MA

polynomial of positive order does not produce

maximum likelihood coefficients due to lack of

convergence. This condition holds for

estimation of all labor unit levels and therefore

all final models contain only an AR polynomial

or an autoregressive order greater than zero.

Figure 2

Akaike's Information Criterion (Corrected)

-425

-420

-415

-410

-405

-400

-395

-390

1 2 3 4 5 6

Autoregressive Order

Detrended

Integrated

The sums of the AICc statistics over the

regions for preliminary AR orders are shown in

the Figure 2 for two different transformation

options for stationarity. Field workers, field and

livestock workers, and all hired workers appear

to be nonstationary with a linearly increasing

trend. Options for transforming the data to a

wide sense stationary series include integration

through a differencing matrix or a vector of

detrending parameters. The annual cycle of the

wage rates suggests a natural seasonal

differencing of four data points (quarters) and

equates to a first difference lag 4. Figure 2

suggests that a detrended model is superior to

an integrated model for all AR orders, and that

a detrended model of AR order 4 provides the

best fit.

Figure 3

Wage Rate Model AICc by Region

for VAR(4) Detrended Model

U.S.H

A

W

AII

C

A

LIF

O

R

NIA

F

L

O

RID

A

P

A

CIFIC I

A

P

P

A

L

A

C

HIA

N I

D

E

L

TA

M

O

U

N

T

AIN III

N

O

R

T

H

E

A

S

T II

S

O

U

T

H

E

R

N

P

L

AIN

S

N

O

R

T

H

E

A

S

T I

S

O

U

T

H

E

A

S

T I

M

O

U

N

T

AIN I

A

P

P

A

L

A

C

HIA

N II

C

O

R

N

B

E

L

T I

N

O

R

T

H

E

R

N

P

L

AIN

S

LA

K

E

MOUNTAIN IIC

O

R

N

B

E

L

T II

-35

-30

-25

-20

-15

-10

-5

0

The chart in Figure 3 breaks down the summed

AICc for a VAR(4) model by region. The

region with the lowest AICc statistic is the U.S.

It is interesting to note that the three regions

that are comprised of only one state make up

the next three lowest AICc statistics. The final
3

wage rate model takes the following form for

all regions:

!

xt= exp μ + "t + #k lnx ( t$k$μ)

k=1

P

% &

'

( )

*

+

where

!

"

is a 3x3 matrix of autoregressive

coefficients,

!

"

is a 3x1 vector of linear trend

coefficients,

!

μ

is a 3x1 vector of centering

coefficients, and P is equal to the model AR

order of 4. An AR order of 4 uses the last

historical year of wage rate data to estimate a

step-ahead forecast. This model reduces the

autocorrelation in the residuals much more so

than models of other orders, including subsets

such as seasonal models (

!

"1 = "2 = "3 = 0

).

Hours Worked

A plot of US hours worked for all three worker

types shows a process that appears stationary

Figure 4 US Hours Worked

(January 1997 - December 2006)

Workers

Field

Field & Livestock

All Hired

H

o

u

r

s

W

o

r

k

e

d

34

35

36

37

38

39

40

41

42

43

44

45

Date

JAN1997JAN1998JAN1999JAN2000JAN2001

JAN2002JAN2003

JAN2004JAN2005JAN2006

JAN2007

containing a seasonal element similar to the

wage rates with a period of four quarters.

Schematic plots of the autocorrelation function

also show this four quarter seasonal periodicity.

Although the series in Figure 4 appears

stationary, a seasonal differencing of lag 4

(implying a yearly differencing) is recognizably 

appropriate from the autocorrelation function

plots. Preliminary estimates of an 

autoregressive order are at first unclear, as

there is no concrete minimum in the sum of

information criteria over the regions.

Figure 5

Akaike's Information Criterion

0

100

200

300

400

500

600

1 2 3 4 5 6 7 8

Autoregressive Order

AIC

c

-60

-40

-20

0

20

40

60

AIC

Corrected Uncorrected

The summed AICc in Figure 5 suggests an AR

order of zero, and the uncorrected AIC suggests

the higher the order, the better the fit. An

examination of the Schwarz-Bayesian Criterion 

shown in Figure 6 is equally unclear.

Figure 6

Schwarz-Bayesian Criterion

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8

Autoregressive Order

However, adding a seasonal AR order to the

seasonally differenced hours worked series

produces a minimum at 4.
4

Figure 7

Akaike's Information Criterion (Corrected)

20

22

24

26

28

30

32

34

36

38

40

1 2 3 4 5 6 7 8

Seasonal Autoregressive Order

Figure 7 implies a model where the coefficient

matrices of lags 1-3 are zero. Further

examination of the autocorrelations in the

residuals leads to a seasonal AR order of 2 with

a seasonal component of 4, and the resulting

model is VARIMA(2,1,0)4.

Figure 8

Hours Worked Model AICc by Region

for VARIMA(2,1,0)4

-6

-4

-2

0

2

4

6

U.S.H

AWAII

C

A

LIF

O

R

NIA

F

L

O

RID

A

P

A

CIFIC I

D

E

L

T

A

S

O

U

T

H

E

A

S

T I

A

P

P

A

L

A

C

HIA

N

M

O

U

N

T

AIN III

L

A

K

E

NORTHEAST IIN

O

R

T

H

E

A

S

T I

S

O

U

T

H

E

R

N

A

P

P

A

L

A

C

HIA

N

C

O

R

N

B

E

L

T I

N

O

R

T

H

E

R

N

M

O

U

N

T

AIN I

M

O

U

N

T

AIN II

C

O

R

N

B

E

L

T II

The AICc of the individual regions in Figure 8

show a strikingly similar pattern to the wage

rate model (Figure 3) in the rankings of the

AICc’s. The three states and the U.S. that make

up their own region have the lowest scores,

hence best fit.

Number of Workers

The levels of labor type are a summation of

parts with embedded variables for livestock 

workers and other workers. Field workers are

present in all three levels. Livestock workers

could be determined by subtracting field from

field & livestock. “Other workers” could be

calculated by subtracting field & livestock from

all hired workers.

Figure 9

US Number of Workers

(January 1997 - December 2006)

Workers

Field

Field & Livestock

All Hired

N

u

m

b

e

r

o

f

W

o

r

k

e

r

s

300

400

500

600

700

800

900

1000

1100

1200

Date

JAN1996JAN1998

JAN2000

JAN2002JAN2004

JAN2006

JAN2008

A graph of the data for levels of labor type is

displayed in Figure 9. There is a strong

correlation between all three parts which could

be attributed to either the common elements in

each part, or actual correlations between the

field workers, livestock workers, and other

workers, or both. The transformed variables

which show the independent levels for field

workers, livestock workers, and other workers

are displayed in Figure 10. This dissection

explains the field workers as the single major

contributor to the periodicity within the data

and to the correlations between levels shown in

Figure 9.

Figure 10

US Number of Workers

(January 1997 - December 2006)

Workers

Field

Livestock

Other

N

u

m

b

e

r

o

f

W

o

r

k

e

r

s

0

100

200

300

400

500

600

700

800

900

Date

JAN1997JAN1998JAN1999JAN2000JAN2001JAN2002JAN2003JAN2004JAN2005

JAN2006JAN2007
5

Due to the size of the field worker numbers

relative to livestock and other workers, it is not

apparent in the graph above that all three

variables are still highly correlated over time.

Mathematically, the estimates and resulting

information criterion computed from either data

source are the same, whether the original

nested data in Figure 9 is modeled, or the

independent transformed dataset shown in

Figure 10 is used. However, residual

diagnostics prove more conclusive with the

original nested dataset of field, field &

livestock, and all hired workers as the final

model almost completely eliminates significant

correlations. The final model structure for the

number of workers is similar to the structure of

the wage rates model, except that the model

includes an integration vector instead of linear

trend parameters. The final model in Box

Jenkins notation is a VARIMA(4,0,0)x(0,1,0)4.

The AICc statistics for each region present a

different picture for number of workers than for

hours worked or wage rates. The sheer worker

size differences between the regions create

AICc statistics on a non-comparable scale. The

U.S. and California regions are relatively larger

compared with the rest of the regions, and

Hawaii is smaller. This size relationship makes

comparison of the model structure fit not useful

for diagnostics. However, almost complete

elimination of autocorrelation in the residuals

by region is supportive of the model fit.

The region Corn Belt II produces an indication 

that is too low in comparison to historical data

with the chosen model. For this reason, this

region is fitted with a VARIMA(1,0,0)x(0,1,0)4

model which produces a more realistic

indication.
6

APPENDIX: MODEL FITS

Wage Rates

VARIMA(4,0,0)

xt= exp μ + "t + #k lnx ( t$k$μ)

k=1

p

% &

'

(

)

*

+

Hours Worked

VARIMA(2,1,0)4

xt= μ + xt"s + #k xt"sk" x ( t"s(k +1)"μ)

k=1

P

$

Number of Workers

VARIMA(4,0,0)x(0,1,0)4

xt= μ + xt"s + #k xt"k" x ( t"k"s"μ)

k=1

p

$

Explanation of Box - Jenkins (1970) notation

VARMA(p,d,q)x(P,D,Q)s

Parameter Description

p Autoregressive Order

d Differencing Order

q Moving Average Order

P Seasonal Autoregressive Order

D Seasonal Differencing Order

Q Seasonal Moving Average Order

s Seasonal Parameter