Statistical procedures

physical and legal framework within which the market operates is also important for a successful water market. Itmust be physically possible for participants to transfer water corresponding to the traded rights, and also the law must support trades and enable enforcement of the legislation surrounding the trading process. In addition, the law is important in defining the property rights to water.

level was reached where the farmer was indifferent between the two choices. A farmer is risk neutral if the certain amount selected equalled the expected income of the coin toss gamble. For the first question, the gamble was an equal probability of earning Rl 000 000 (xma\) and zero (Xmin) (p=0.5), with an expected income ofR500 000. The second question gamble was an equal probability of earning R800 000 and losing R200 000 (p=0.5), with an expected income of R300 000. These amounts were chosen to resemble turnover amounts that may occur in the local farming operations managed by the study farmers.

Although the Arrow/Pratt absolute risk aversion coefficient has been extensively quoted in literature, it has a major weakness in that it cannot be compared between different studies as the coefficient depends on the scale and range of the data. Nieuwoudt and Hoag (1993) suggested that the coefficient be standardised, a procedure followed by Ferrer (1999) and also adopted in this dissertation. Standardisation was undertaken by converting the distribution (Xmin:S x:S xma·J into a distribution (O:S x*:s l) where Xmin and xma'( are the minimum and maximum values on the x-scale. This provides a unit-less expression of the absolute risk aversion function. The algebraic derivation below shows the sensitivity00.to changes in the scale (whether data are expressed in Rands or Dollars) or range of data.

Let

:. x = Xmin+X (xmax - Xmin)*

where U(x)= _e-^h and U(x*)= _e-A.*x*

.'. 'A*= 'A(Xmax - Xmin) since 'Axmin = constant.

(1)

In this study 'A* is estimated, which is not affected by the range and scale (xmax - Xmin) ofthe data:

4.2.2 Ridge regression

Ridge Regression (RR) allows biased estimation ofthe regression coefficients by modifying the method ofleast squares to remedy a multicollinearity problem. If an estimator has only a small bias and is more precise than an unbiased estimator, it may well be the preferred estimator, since it will have a larger probability of being close to the true parameter (Neter et at.,1996:411; Maddala, 1992). The ridge standardised regression estimators are obtained by introducing into the least squares normal equations a biasing constant K 2: 0 where K usually varies between 0 and 1. Following Neter et al. (1996:412), the ridge trace and the Variance Inflation Factors (VIFs) were used to determine the optimum value of K. This is done by choosing the smallest value ofK where the regression coefficients first become stable in the ridge trace.

4.2.3 Principal component analysis

Principal Component Analysis (PCA) is a multivariate transformation technique with which a set of complex relations can be reduced to a simple canonical form. The purpose ofPCA can also be described as an effort to economize on the number of variables (Jolliffe, 1986).

Principal components are obtained by linear transformations of the observed variables as follows:

(2) where XI, X2 ... Xp are the original variables; the aip are the component loadings such that ai/+ail+ ... +ai/⁼ unity or the eigenvalue for PCi(i.e., normalization). In doing so, PC_I^, the first principal component, makes the greatest contribution to the variance as contained in the p original variables; the second, PC2is chosen to be uncorrelated with the first, and to have as large a variance as possible, etc. The X variates are thus transformed using the

correlation matrix (where variables are measured in widely differing units) to new uncorrelated variates, which account for as much variation as possible in descending order (Nieuwoudt, 1977:78). The number of PCs retained also depends upon whether or not the estimated PCs can be meaningfully interpreted.

4.2.4 Logistic regression

This model uses logit analysis to assess factors that influence a farmer's decision to be a buyer or a seller in the water market. If Buyers are coded 1 and Sellers are coded 0, then the probability that a participant is a buyer(Pi) can be represented by:

P ⁼ E(Y⁼

11

X )⁼ 1 ^-

/ / 1+e-^Zj (3)

where Zⁱ =

j3

⁰

+ j3

nXni and theXnare n variables hypothesised to explain why water is bought or sold.

This equation (3) represents the (cumulative) logistic distribution function. As Ziranges from-00to+00,Pi ranges from

°

to 1. In addition,Piis nonlinearly related toX. HoweverPi is also nonlinearly related to the

f3

's, and the Ordinary Least Squares (OLS) procedure cannot be used to estimate the parameters (Gujarati, 1995:554). The probability of the

/h

participant being a seller is(l-Pi),whileP)(l-PDis known as the odds ratio in favour of the participant being a buyer in the market. The natural log ofPi /(l-PD is:

L, = l{1 _~ ^{P, ]} _~ Z, _~ ¹³⁰

⁺

^fJ,X"

^{+ ... +}

fJ"X",

⁽⁴⁾

Now L, the log of the odds ratio, or logit, is linear in X and, more importantly, in the parameters. However to estimate this model, the values ofthe logit(L_i) must be known. For

(5) estimation purposes, (4) is written as:

L,

~ ^1n[I ~

^{P, ]}

^{= /30}

⁺

^/3,X"

^{+ ... +}

/3"X",

+p,

For data on individual participants, the logits are meaningless, as:

L_i= In[1/0] if the i^1hparticipant is a buyer, and L_i= In[O/l] if the i^thparticipant is a seller.

In this case, the maximum likelihood method is used to estimate the parameters (Gujarati, 1995: 556). A key advantage of the maximum likelihood method is that it does not require the restrictive normality assumption about the distribution of the independent variables in order to make inferences about the estimated model parameters.

The model analyses permanent trades of water entitlements in the water market, since no temporary trades occurred amongst the survey farmers. The main aim of the model is to identify the important factors that contributed to the farmers' decision to either buy or sell water in the market. The farmer type - Buyer or Seller - is hypothesised to be determined by the n attributes (Xs) of the farm business and the farmer. The dependent variable in the analysis (TYPE) was coded using one (1) for farmers who had purchased water entitlements, and zero (0) for farmers who had sold water entitlements within the past five years.