April 2011 Farm Labor Estimates Methodology

April 2011 Farm Labor Estimates Methodology
USDA NASS Research and Development Division
Stephen Busselberg, Mathematical Statistician
3251 Old Lee Highway, Room 305
Fairfax, Virginia 22030

1. Background
The Farm Labor Survey provides estimates of quarterly wage rates paid, number of workers
hired, and average number of hours worked per week for field workers, livestock workers,
supervisors, other workers, and all hired farm workers. These estimates are available for the U.S.
and eighteen sub-regions comprised of mutually exclusive groups of states. In 2007, the January
Farm Labor Survey was canceled due to budget shortfall. Official estimates for field workers,
field and livestock workers, and all hired workers were provided by modeling historical data
using time series techniques. In April 2011 the Farm Labor Survey was canceled due to a similar
budget shortfall. This paper details the methodology used for modeling the April 2011 farm labor
estimates. The April 2011 farm labor estimates include wage rates paid, hours per week worked,
and number of workers hired at the U.S. and eighteen regional levels for field workers, field and
livestock workers, and all hired workers.

2. Methodology
The class of models used for the April 2011 farm labor was Vector Autoregressive (VAR). The
dependent variable is a vector of worker types (field, field and livestock, all hired) whose
elements are indexed by j=1,2,3. The region is indexed by i and time is indexed by t. The base
VAR form we will write as
,

, ,

, ,

, ,

,

,

,

, ,

, ,

, ,

,

,

,

, ,

, ,

, ,

,

,

,

For matrix notation the subscript j is omitted, and we write

,

.

follows a Gaussian

,

distribution with mean vector 0 and covariance matrix
of one another.

. The regions are modeled independent

2.1 Wage Rates
The natural log transformation of the wage rate was modeled due to a time variant variance of
the wage rate. A second order Taylor Series approximation was used to estimate the expected
wage rate given the model estimate which equates to the expected natural log of the wage rate;
i.e. if w is the wage rate, we want
but we have ln
since we are modeling ln
.A
first order Taylor Series approximation was used to estimate the variance of the expected wage
rate var
. Estimates of the wage rate for a region i and worker type j at time t were
calculated as follows. Note that if the subscript j is omitted, the notation symbolizes a column
vector of worker types.
2.1.1 The Expected Wage Rate
1
2

1
ln

ln

ln

ln

4

The term

is the model prediction variance of the expected natural log transformation

.

The first order Taylor Series approximation for the prediction variance of the expected wage rate
is
1
2.1.2 Derivation of the Expected Log Wage Rate
is derived as follows:
ln

1
4

We first define x as the natural log of the wage rate and de-trend the natural log of the wage rate
by subtracting out a cubic trend as a function of time. Cubic is the highest order polynomial trend
for any region. We then define u as a seasonal fourth difference to filter out quarterly seasonality
, . Substituting
(See note at the end of this section). Next we model u as VAR with ~
the second line from above into the third yields

Substituting the definition of

ln

into the above and consolidating terms gives

4

ln

ln

ln

ln

4

ln

ln

ln

4

ln

4

4

ln

ln

ln

4

Last we take the expectation of both sides of the above equation
ln
ln

4

ln

ln

4

ln

ln

ln

4

The autoregressive order is determined by region using a minimum Akaike’s adjusted
Information Criterion (AICc).
Note: Sinusoidal seasonal de-trending is also an alternative method for detrending and including
∑
∑
ln
where
a seasonal adjustment; i.e.
sin

cos

. This option was not pursued as it was not favorable for all regions.

2.1.3 Derivation of the Expected Wage Rate
. The

We can approximate the expected wage rate
second order Taylor Series expansion about is
1
2
1
2
where R is the Taylor Series remainder. If we take the expectation of both sides

1
2

1
2
VAR

The above relationship yields
1
2
1

1
2

2.1.4 Derivation of the Variance of the Expected Wage Rate Approximation

For the first order Taylor Series approximation of the variance, we first establish the first order
expansion. It is the first two terms in the second order expansion in section 2.1.3.

We can then subtract the second order approximation for

1
2

from both sides.
1
2

VAR

VAR

Squaring both sides and taking the expectation yields

VAR

1
4

1
4

VAR
VAR

Plugging in the definition of the terms above yields

1

1
4
1
4

2.2 Hours and Number of Workers
Hours and number of workers were modeled using the same seasonal differencing transformation
and are therefore similar in structure. Letting
represent the vector of worker types for hours
per week or number of workers hired, the April 2011 estimate for hours per week or number of
workers is

Φ

This is the vector equivalent of an ARIMA

, 0,0

0,1,0 .

References
Casella, George and Berger, Roger L. Statistical Inference. Pacific Grove, CA: Thomas Learning
Inc., 2002.
Shumway, R.H. and Stoffer, D.S. Time Series Analysis and its Applications. New York:
Springer Science+Business Media, LLC., 2006.