Accurate Statutory Valuation

JOHN MacFARLANEUniversity of Western Sydney

Content

Motivation Methodology Examples

Motivation

Much of estimation theory is focussed on (obsessed with?) unbiasedness

There are many situation where unbiased estimation is not relevant: Appointments; Consultation times; Software development time and cost;

Motivation (Property)

Property returns (%) Excess returns and under-performance are not (or should

not be) symmetric Downside risk

Property Tax Assessment MVP – Mean Value Price Ratio (85-100% or 90-100%)

Methodology

Estimation Least Squares; Symmetric Loss Function.

Lead to unbiased parameter (expected value) estimates.

Maximum Likelihood Estimation (MLE) May be biased but are consistent.

Alternative Methodologies

Asymmetric Approaches1. Weighted (penalised) least squares;2. Asymmetric loss function

Asymmetric Approaches

1. Weighted Least Squares Minimise:

where λi = 1 if xi < θ

= λ if xi ≥ θ

λ = 1 normal least squares, unbiased

λ > 1 over-estimates

λ < 1 under-estimates λ ≥ 0

Non-linear as λ is a function of θ.

2( )i ix

Example 1

Comparable land values (n=3):

1. $280,000;2. $300,000;3. $320,000.

$300,000x

1. Weighted Least Squares

0

500

1000

1500

2000

2500

3000

3500

4000

4500

270 280 290 300 310 320 330

Sum

of S

quar

es

Theta

Weighted Sum of Squaresλ = 1 λ = 0.5 λ = 2 λ = 0.1

Summary of ResultsExample 1

λ = 0.1 λ = 0.5 λ = 1 λ = 2 : 285 295 300 305

ˆˆ

ˆi i

i

x

Example 2

Comparable land values (n=3):

1. $280,000;2. $280,000;3. $340,000.

$300,000x

0

1000

2000

3000

4000

5000

6000

7000

8000

270 280 290 300 310 320 330

Sum

of S

quar

es

Theta

Weighted Sum of Squaresλ = 1 λ = 0.5 λ = 2 λ = 0.1

Summary of ResultsExample 2

λ = 0.1 λ = 0.5 λ = 1 λ = 2 : 282.9 292 300 310

ˆˆ

ˆi i

i

x

Example 3 Comparable land values (n=4):

1. $280,000;2. $300,000;3. $320,000;4. $380,000

$320,000x

1. Weighted Least Squares

0

5000

10000

15000

20000

25000

30000

270 280 290 300 310 320 330 340 350 360 370

Sum

of S

quar

es

Theta

Weighted Sum of Squaresλ = 1 λ = 0.5 λ = 2 λ = 0.1

Summary of ResultsExample 3

λ = 0.1 λ = 0.5 λ = 1 λ = 2 : 292.3 310 320 332

ˆ

ˆˆi i

i

x

Reverse Problem

What is the optimal choice of λ for a required level of under-estimation (as inferred by the MVP standard)?

2. Asymmetric Loss Function

Loss Function (LINEX)

Requires a prior distribution for parameters

( ) 1, 0aL e a

If we assume that the data is normally distributed with unknown mean (μ) and KNOWN standard deviation (σ), then it can be shown that the optimal estimate wrt the LINEX loss function is:

ˆ2

ax

n

0

2

4

6

8

10

12

14

16

18

-4 -3 -2 -1 0 1 2 3 4

a = 1

a = 0.4

a = -0.4

a = -1

( ) 1aL e a

0

2

4

6

8

10

12

14

16

18

-4 -3 -2 -1 0 1 2 3 4

a = 1

a=0.25

( ) 1aL e a

Example 1

Comparable land values (n=3):

1. $280,000;2. $300,000;3. $320,000.

$300,000x

If we take the standard deviation to be σ=$20,000 then

That is, for a = 1, we would underestimate the value by about $8,200 or a little under 3%.

ˆ 8165x a

Example 3

Comparable land values (n=4):

1. $280,000;2. $300,000;3. $320,000;4. $380,000

$320,000x

If we take the standard deviation to be σ=$40,000 then

That is, for a = 1, we would underestimate the value by about $14,000 or about 4%.

ˆ 14142x a

Conclusion

We have considered two different approaches to systematically under- or over-estimating values.

They represent different approaches both of which deserve further examination.

Thank you!

Questions?