Site aggregation in a Random Utility Model of recreation. (includes appendices)

I. INTRODUCTION

Site aggregation is common in Random Utility Models of recreation. By site aggregation we mean defining a group of recreation sites as a single choice alternative. Common examples are letting counties or regions be choice alternatives. Within these counties or regions there may be dozens or even hundreds of individual sites from which to choose. When the number of potential sites is large, as is frequent in recreation studies, site aggregation is often used to reduce the choice set to a manageable size. Unfortunately, the cost of site aggregation is a loss in estimation accuracy.

We explore the degree of this loss empirically by comparing models that employ site aggregation to a model that does not. Our experiment is with fishing trips to lakes in Wisconsin. We treat lakes as individual sites. There are more than one thousand in our sample. We define aggregate sites in terms of either counties or broader regional aggregates. Our model without aggregation uses a random draw procedure to represent opportunity sets like that used by Parsons and Kealy (1992).

Kaoru and Smith (1990) were the first to investigate the implications of site aggregation in a Random Utility Model. They analyzed marine recreational fishing trips and considered experiments that aggregated 35 individual sites to 23 sites in one case and to 11 sites in another. The estimated models based on the aggregated data portrayed plausible models of behavior. The benefit assessments, however, differed considerably from their counterparts in the disaggregated models.

Our experiment considers a much larger degree of aggregation than Kaoru and Smith's. Our experiments reduce 1,133 sites to 9 in one case and to 61 in another. Our estimation results strongly caution against aggregation of the extreme degree (9 sites) for either portraying behavior or for assessing benefits. But, to a considerable extent, our results for the case with less aggregation (61 sites) confirm Kaoru and Smith's finding--a plausible aggregate model is estimated but the benefits often diverge widely from their counterparts in the disaggregated model.

We begin with Ben-Akiva and Lerman's (1985) theory of aggregation in a Random Utility Model.

II. AGGREGATION IN THE RANDOM UTILITY MODEL

Consider an individual selecting a fishing site, which in our model is a lake. Let L be the set of all possible lakes. Aggregation amounts to partitioning this set into J subsets that do not overlap. Following Ben-Akiva and Lerman's notation, define these subsets as |L.sub.i~ (i= 1, 2, 3, ... , J). Each i is an aggregate alternative, which may be a group of lakes in the same county, in the same watershed, or sharing some common characteristic such as size.

Assume an individual's utility from fishing at lake l is

|U.sub.l~ = |V.sub.l~ + ||epsilon~.sub.l~. |1~

|V.sub.l~ is a deterministic portion of utility which is a function of the characteristics of the lake and ||epsilon~.sub.l~ is a random error. It follows that the utility of visiting aggregate alternative i is just

|Mathematical Expression Omitted~

|U.sub.i~ is the maximum utility among all lakes in group i.

If the ||epsilon~.sub.l~ are independent and identically distributed (iid) Gumbel random variables with mode 0 and scale |mu~, we can decompose equation |2~ as

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the mode of the random variable |Mathematical Expression Omitted~ and ||epsilon~.sub.i~ is distributed Gumbel with mode 0 and scale |mu~ (see Ben-Akiva and Lerman |1985, 106~ for a similar decomposition).

Equation |3~ can be decomposed further to

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~ is the average utility at lakes in aggregate alternative i. |Mathematical Expression Omitted~ and is a measure of the heterogeneity of the lakes in aggregate alternative i. |M.sub.i~ is the number of lakes in the aggregate alternative i--the size of the aggregate alternative. (See Appendix 1 for the derivation of equation |4~.) Equation |4~, the random utility of aggregate alternative i, is the same as Ben-Akiva and Lerman's equation |9.9~. We assume |V.sub.l~ = ||beta~x.sub.l~, where |x.sub.l~ is a vector of site characteristics and |beta~ is a parameter vector to be estimated. Then, |Mathematical Expression Omitted~ where |Mathematical Expression Omitted~ is a vector of the means of the elements in |x.sub.l~ from group |L.sub.i~.

Due to a lack of information or high cost of obtaining such information, site aggregation schemes usually specify the utility of an aggregate alternative ignoring the terms involving 1n |B.sub.i~ and 1n |M.sub.i~ in equation |4~. |Mathematical Expression Omitted~ or some approximate measure of |Mathematical Expression Omitted~ is used alone. The loss in estimation accuracy due to site aggregation is the bias introduced by omitting 1n |B.sub.i~ and 1n |M.sub.i~ in estimation. This suggests a natural test of aggregation bias--compare an estimated model that accounts for these terms with an estimated model that does not. Our experiments do just that.

We consider three variations of equation |4~. First, we estimate the equation assuming we know the heterogeneity (|B.sub.i~) and the number of lakes (|M.sub.i~) in each aggregate alternative. This is our Full Information Model--full …