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 

൥

ݔଵ,௧

ݔଶ,௧

ݔଷ,௧൩

௜

ൌ ෍቎߶ଵ,ଵ,௞ ߶ଵ,ଶ,௞ ߶ଵ,ଷ,௞

߶ଶ,ଵ,௞ ߶ଶ,ଶ,௞ ߶ଶ,ଷ,௞

߶ଷ,ଵ,௞ ߶ଷ,ଶ,௞ ߶ଷ,ଷ,௞቏

௜

൥

ݔଵ,௧ି௞

ݔଶ,௧ି௞

ݔଷ,௧ି௞൩

௜

௉೔

௞ୀଵ

൅ ൥

ݖଵ,௧

ݖଶ,௧

ݖଷ,௧൩

௜

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

ݔଵ,௧

ݔଶ,௧

ݔଷ,௧൩

௜

ൌ ࢞௜௧. ࢠ௜௧ follows a Gaussian 

distribution with mean vector 0 and covariance matrix ઱௜. The regions are modeled independent 

of one another. 

 
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 ܧሾݓሿ but we have ܧሾlnሺݓሻሿ since we are modeling lnሺݓሻ. A 

first order Taylor Series approximation was used to estimate the variance of the expected wage 

rate varሺܧሾݓሿሻ. Estimates of the wage rate ݓ 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 

ݓෝ௜௝௧ ൌ ܧൣݓ௝௜௧൧ൌ݁࢟ෝ࢚࢐࢏ ൬1 ൅

1

࢚࢐࢏ොܡߪ 2

ଶ ൰

࢟ෝ௜௧ ൌ ܧሾlnሺ࢝௜௧ሻሿ ൌ lnሺ࢝௜௧ିସሻ ൅ ࢻ௜௧ ൅෍઴௜௞ሾlnሺ࢝௜௧ି௞ሻ െ lnሺ࢝௜௧ି௞ିସሻ െ ࢻ௜௧ି௞ሿ

௉೔

௞ୀଵ

ࢻ௜௧ ൌ෍ࢼ௤௜ሾݐ ௤െ ሺݐെ4ሻ ௤ሿ

ଷ

௤ୀଵ

The term ܡߪො೔ೕ೟

ଶ is the model prediction variance of the expected natural log transformation ࢟ෝ௜௝௧. 

The first order Taylor Series approximation for the prediction variance of the expected wage rate 

ݓෝ௜௝௧ is 

ߪ௪ෝ೔ೕ೟

ଶ ൌ ݁ଶ௬ො೔ೕ೟ ൬1 ൅

1

4 ߪ௬ො೔ೕ೟

ଶ ൰ ߪ௬ො೔ೕ೟ଶ

2.1.2 Derivation of the Expected Log Wage Rate

࢟ෝ௜௧ is derived as follows: 

࢞௜௧ ൌ lnሺ࢝௜௧ሻ െ෍ࢼ௤௜ݐ௤

ଷ

௤ୀ଴

࢛௜௧ ൌ ࢞௜௧ െ ࢞௜௧ିସ

࢛௜௧ ൌ ෍઴௜௞࢛௜௧ି௞

௉೔

௞ୀଵ

௧௜ࢠ ൅
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 

(See note at the end of this section). Next we model u as VAR with ࢠ௜௧~ܰሺ૙, ઱௜ሻ. Substituting 

the second line from above into the third yields 

࢞௜௧ െ ࢞௜௧ିସ ൌ ෍઴௜௞ሺ࢞௜௧ି௞ െ ࢞௜௧ି௞ିସሻ

௉೔

௞ୀଵ

௧௜ࢠ ൅

Substituting the definition of ࢞௜௧ into the above and consolidating terms gives 

௤ݐ௤௜ࢼ෍െ ௧ሻ௜࢝lnሺ

ଷ

௤ୀ଴

െ lnሺ࢝௜௧ିସሻ ൅෍ࢼ௤௜ሺݐെ4ሻ௤

ଷ

௤ୀ଴

ൌ ෍઴௜௞ ቎lnሺ࢝௜௧ି௞ሻ െ෍ࢼ௤௜ሺݐെ݇ሻ௤

ଷ

௤ୀ଴

െ lnሺ࢝௜௧ି௞ିସሻ ൅෍ࢼ௤௜ሺݐെ݇െ4ሻ௤

ଷ

௤ୀ଴ ቏

௉೔

௞ୀଵ

௧௜ࢠ ൅

lnሺ࢝௜௧ሻ െ lnሺ࢝௜௧ିସሻ െ෍ࢼ௤௜ሾݐ ௤െ ሺݐെ4ሻ௤ሿ

ଷ

௤ୀଵ

ൌ෍઴௜௞ ቐlnሺ࢝௜௧ି௞ሻ െ lnሺ࢝௜௧ି௞ିସሻ െ෍ࢼ௤௜ሾሺݐെ݇ሻ௤ െ ሺݐെ݇െ4ሻ௤ሿ

ଷ

௤ୀଵ ቑ

௉೔

௞ୀଵ

௧௜ࢠ ൅

lnሺ࢝௜௧ሻ ൌ lnሺ࢝௜௧ିସሻ ൅෍ࢼ௤௜ሾݐ ௤െ ሺݐെ4ሻ௤ሿ

ଷ

௤ୀଵ

൅෍઴௜௞ ቐlnሺ࢝௜௧ି௞ሻ െ lnሺ࢝௜௧ି௞ିସሻ െ෍ࢼ௤௜ሾሺݐെ݇ሻ௤ െ ሺݐെ݇െ4ሻ௤ሿ

ଷ

௤ୀଵ ቑ

௉೔

௞ୀଵ

௧௜ࢠ ൅

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

࢟ෝ௜௧ ൌ ܧሾlnሺ࢝௜௧ሻሿ

ൌ lnሺ࢝௜௧ିସሻ ൅෍ࢼ௤௜ሾݐ ௤െ ሺݐെ4ሻ௤ሿ

ଷ

௤ୀଵ

൅෍઴௜௞ ቐlnሺ࢝௜௧ି௞ሻ െ lnሺ࢝௜௧ି௞ିସሻ െ෍ࢼ௤௜ሾሺݐെ݇ሻ௤ െ ሺݐെ݇െ4ሻ௤ሿ

ଷ

௤ୀଵ ቑ

௉೔

௞ୀଵ
ൌ lnሺ࢝௜௧ିସሻ ൅ ࢻ௜௧ ൅෍઴௜௞ሾlnሺ࢝௜௧ି௞ሻ െ lnሺ࢝௜௧ି௞ିସሻ െ ࢻ௜௧ି௞ሿ

௉೔

௞ୀଵ

ࢻ௜௧ ൌ෍ࢼ௤௜ሾݐ ௤െ ሺݐെ4ሻ ௤ሿ

ଷ

௤ୀଵ

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 

a seasonal adjustment; i.e. ࢟ෝ௜௧ ൌ ࢻ௜௧ ൅ ∑ ઴௜௞ሾlnሺ࢝௜௧ି௞ሻ െ ࢻ௜௧ି௞ሿ ௉೔

௞ୀଵ where ࢻ௜௧ ൌ ∑ ࢼ௤௜ݐ ଷ ௤ ௤ୀ଴ ൅

ࢼ௜ସ sin ଵ

ଶ ݐߨ ൅ ࢼ௜ହ cos

ଵ

ଶ

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

2.1.3 Derivation of the Expected Wage Rate 

 We can approximate the expected wage rate ݓෝൌܧሾݓሿ ൌ ܧൣ݁୪୬ ௪൧ൌܧሾ݁௬ሿ ൌ ܧሾ݂ሺݕሻሿ. The 

second order Taylor Series expansion about ݕ is 

݂ሺݕሻ ൌ ݂ሺݕොሻ ൅ ݂ᇱ

ሺݕොሻሺݕെݕොሻ ൅

1

2 ݂ᇱᇱሺݕොሻሺݕെݕොሻଶ ൅ ܴ

݂ሺݕሻ ൎ ݂ሺݕොሻ ൅ ݂ᇱ

ሺݕොሻሺݕെݕොሻ ൅

1

2 ݂ᇱᇱሺݕොሻሺݕെݕොሻଶ

where R is the Taylor Series remainder. If we take the expectation of both sides 

ܧሾ݂ሺݕሻሿ ൎ ݂ሺݕොሻ ൅ ݂ᇱ

ሺݕොሻܧሾݕെݕොሿ ൅

1

2 ݂ᇱᇱሺݕොሻܧሾሺݕെݕොሻଶሿ

ൎ ݂ሺݕොሻ ൅

1

2 ݂ᇱᇱሺݕොሻVARሾݕሿ

The above relationship yields 

ܧሾݓሿ ൎ ݁௬ො ൅

1

2 ݁௬ො

௬ߪ

ଶ

ൎ ݁௬ො ൬1 ൅

1

௬ොߪ 2

ଶ൰

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. 

݂ሺݕሻ ൎ ݂ሺݕොሻ ൅ ݂ᇱ

ሺݕොሻሺݕെݕොሻ

We can then subtract the second order approximation for ܧሾ݂ሺݕሻሿ from both sides. 

݂ሺݕሻ െ ܧሾ݂ሺݕሻሿ ൎ ݂ሺݕොሻ ൅ ݂ᇱ

ሺݕොሻሺݕെݕොሻ െ ݂ሺݕොሻ െ 1

2 ݂ᇱᇱሺݕොሻVARሾݕሿ

ൎ ݂ᇱ

ሺݕොሻሺݕെݕොሻ െ 1

2 ݂ᇱᇱሺݕොሻVARሾݕሿ

Squaring both sides and taking the expectation yields 

ܧሾሼ݂ሺݕሻ െ ܧሾ݂ሺݕሻሿሽଶሿ ൎ ݂ᇱ

ሺݕොሻଶܧሾሺݕെݕොሻଶሿ ൅

1

4 ݂ᇱᇱሺݕොሻଶVARሾݕሿଶ

ൎ ݂ᇱ

ሺݕොሻଶVARሾݕሿ ൅

1

4 ݂ᇱᇱሺݕොሻଶVARሾݕሿଶ

Plugging in the definition of the terms above yields 

௪ෝߪ

ଶ ൎ ݁ଶ௬ොߪ௬ො

ଶ ൅

1

4 ݁ଶ௬ො

௬ොߪ

ସ

ൎ ݁ଶ௬ො ൬1 ൅

1

௬ොߪ 4

ଶ൰ ߪ௬ොଶ

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 ࢞௜௧ 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 

࢞ෝ௜௧ ൌ ܧሾ࢞௜௧ሿ ൌ ࢞௜௧ିସ ൅෍Φ௜௞ሺ࢞௜௧ି௞ െ ࢞௜௧ି௞ିସሻ

௉೔

௞ୀଵ

This is the vector equivalent of an ARIMAሺܲ௜, 0,0ሻ ൈ ሺ0,1,0ሻସ. 

 
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.