accurate statutory valuation john macfarlane university of western sydney
TRANSCRIPT
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
1. Weighted Least Squares
0
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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
0
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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
1. Weighted Least Squares
0
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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
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
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
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.