Accurate Statutory Valuation JOHN MacFARLANE University 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 %)

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 Approaches 1.Weighted (penalised) least squares; 2.Asymmetric loss function  Asymmetric Approaches

1.Weighted Least Squares Minimise: where λ i = 1 ifx i < θ = λ ifx i ≥ θ λ = 1normal least squares, unbiased λ > 1over-estimates λ < 1under-estimates λ ≥ 0 Non-linear as λ is a function of θ.

Example 1 Comparable land values (n=3): 1. $280,000; 2. $300,000; 3. $320,000.

1.Weighted Least Squares

Summary of Results Example 1 λ = 0.1 λ = 0.5 λ = 1 λ = 2 :

Example 2 Comparable land values (n=3): 1. $280,000; 2. $280,000; 3. $340,000.

Summary of Results Example 2 λ = 0.1 λ = 0.5 λ = 1 λ = 2 :

Example 3 Comparable land values (n=4): 1. $280,000; 2. $300,000; 3. $320,000; 4. $380,000

1.Weighted Least Squares

Summary of Results Example 3 λ = 0.1λ = 0.5 λ = 1 λ = 2 :

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

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:

Example 1 Comparable land values (n=3): 1. $280,000; 2. $300,000; 3. $320,000.

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%.

Example 3 Comparable land values (n=4): 1. $280,000; 2. $300,000; 3. $320,000; 4. $380,000

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%.

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?