Post on 22-Feb-2016
description
transcript
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:
ˆ2axn
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?