A Better Approach to Calculate Approved Yield -

Indexing

Dr. Myles J. WattsProfessor, Montana State University

Agricultural Economics & Economics Department

Economic Consultant, Watts and Associates, Inc.

2

Crop Insurance

• A risk management tool for producers to alleviate financial stress from:– Low yields.– Unexpected price declines.

3

Concerns

• Producers ability to obtain meaningful crop insurance is eroded after a series of poor yielding years.

• After unusually good or bad years, rates are actuarially less sound.

• Technology improvements resulting in increasing yield are ignored by current system.

4

Current Method of Calculating Indemnity Trigger

• Approved yield = simple average of 4-10 years of producer supplied yield history.

• (Approved yield)*(Coverage level) = Indemnity trigger.

• If yield outcome falls below the indemnity trigger, an indemnity is paid.

5

Current Method

• Pros–Easy to administer.–Easy to understand.

6

Current Method• Cons

– Does not account for technological increases• Biased.

– Small sample size• Efficiency of approve yield estimates.

– Series of unusually low yielding years will dramatically lower approved yield, making an indemnity payment less likely.

– Series of high yields increases probability of an indemnity payment.

7

U.S. Corn Historic Yields

0

20

40

60

80

100

120

140

160

1924 1928 1932 1936 1940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Cor n P lanted Y ield T r end

8

U.S. Wheat Historic Yields

0

5

10

15

20

25

30

35

40

45

1918 1922 1926 1930 1934 1938 1942 1946 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 2006

US Wht plyld T r end

9

Wheat Yield, Petroleum County, Montana

0

5

10

15

20

25

30

35

40

1924 1932 1940 1948 1956 1964 1972 1980 1988 1996 2004

Planted Yield Trend 10 year simple average 5 year simple average 2005 yield

10

Objectives

• Develop an approved yield as accurate as possible point estimate of expected yield.– Reduce bias of a simple average.– Increased efficiency over a simple average.– Reduce adverse selection and moral hazard.– Administratively feasible.

11

Proposed Alternative – Indexing• Longer term (e.g. > 50 years) regional data

(NASS) sets used along with producer actual production history.

• Method overview– Statistically estimate trend line from long term

regional data to forecast expected regional yield.– Calculate average of farm level and regional series

for given time period.– Difference between two is added (subtracted) to

(from) expected regional yield to calculate the producers approved yield.

12

Detailed Discussion

• Let

2

2

2

number of farm yield observations number of regional yield observations

annual increase (slope) of yield trend line

farm level variance

variance of trend line residuals or errors

I

r

e

f

TTb

2 2

farm level variance beyond regional variance e

13

Bias

• Simple average of current method is biased because of technology. Bias is

• Indexed yield predicted from linear regression has no bias.

1.2ITb

14

Efficiency

• Variance of simple average = .• Variance of indexed yield has two

components which are orthogonal and additive.– Variance of regional expected yield ( ) in

year =

22Y

IT

2r̂

22 2 2ˆ

2

1

1 4 21 .1r

r rr e eT

r r r

t

T t TT T Tt t

1rT

15

Efficiency cont.

– Variance of difference between farm & regional average yield =

Therefore, variance of Indexed yield is

22 .ff

IT

22 2 2 2

ˆ4 2 .

1fr

Indexed r efr r I

TT T T

16

Efficiency Gain• Large number of observations at regional level

increases the efficiency of indexing.• The efficiency of the simple average and indexing is

the same when the number of observations satisfies

• The Indexed yield will provide a more efficient estimate of the approved yield if the length of the regional data series is greater than approximately four times the length of the farm data series.

1 .

4 2r r

Ir

T TT

T

17

Illustration of EfficiencyLet

• For approved yields to be efficient (stable) and unbiased, the distribution of approved yields must be concentrated around the expected yield (small variance).

2

2

2

900 farm yield total variance

450 regional variance

450 farm level added variancee

f

20 30 40 50 60 70 80

4 225 210 176 159 150 143 139 1356 150 172 138 122 112 106 101 988 113 154 120 103 94 87 83 7910 90 142 108 92 82 76 71 68

Length of Regional Time Series (years)

Indexed Yield Variance

Length of Farm Data

(years)

Simple Average Variance

18

Critical Component Indexing• Estimation of yield trend lines.

• ‘s are parameters to be statistically estimated.

2

4

1

3

Most General Form:

ˆ t otY

t

19

Estimating Trend Lines

• Form is flexible.For example:

IF THEN

Linear

Exponential

Sigmoid

4 0

2 4

2 41, 0

20

Figure 3Illustration of the Effect of

0

50

100

150

200

250

300

350

400

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Time

Expe

cted

Yie

ld

2 1

2 .5

2 0

2 1

2 1.25

2

215 20Y t

21

Figure 4Illustration of the Effect of

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Time

Expe

cted

Yie

ld

2

2

2015100

tYt

2 8

2 4

2 2

2 1

2

22

Figure 5Illustration of the Effect of

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Time

Expe

cted

Yie

ld

4

43

2015 tYt

3 10 3 100 3 1000

3

23

Figure 6Illustration of the Effect of

0

10

20

30

40

50

60

70

80

90

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Time

Expe

cted

Yie

ld

4

42015100

tYt

4

4 3.5

4 4

4 6

24

Model Selection• Model chosen by:

– Standard statistical tests such as F-test.– Visual inspection of graphs.

• Model choice experience– Have estimated 100’s of trend lines for crop yields in

the US and other countries.

Selected Form Proportion of Time Selected

Linear 80%

Sigmoid 15%

No Trend 5%

2 4( 1, 0)

2( 0)

2 4( )

25

Wheat Yield, Petroleum County, Montana

0

5

10

15

20

25

30

35

40

1924 1932 1940 1948 1956 1964 1972 1980 1988 1996 2004

Planted Yield Trend 10 year simple average 5 year simple average 2005 yield

26

Petroleum County, Montana

• Estimated trend line equation is

• Scale t

• Expected regional indexed yield 2005 = 22.6.

9.656

9.656

14.106ˆ 8.487 .988392t

tYt

1918 .10

yeart

27

Adjustment to Indexed Method

• Indexed Farm Yield = Indexed Regional Yield = (Farm Average – Regional Average).

28

Regional (county) and Hypothetical Farm Yields for Petroleum County

Year Example Farm Yield Regional Yield

1995 30.00 20.50

1996 35.00 26.00

1997 40.00 32.80

1998 30.00 36.30

1999 30.00 28.60

2000 15.00 9.10

2001 0.00 7.50

2002 10.00 5.60

2003 26.00 21.40

2004 25.00 21.40

29

Petroleum County Simple Farm Average and Indexed Yield

Number Of

Years*Farm

AverageRegionalAverage

Difference of

AveragesIndexed

Yield

4 15.25 13.98 1.28 23.88

6 17.67 15.60 2.07 24.67

8 22.00 20.34 1.66 24.26

10 24.10 20.92 3.18 25.78

*Most Recent Years