Econometric analysis of CE surveys. The Economic Model The individual (respondent) can choose between N alternatives, each alternative can be described.

Econometric analysis of CE surveys

The Economic Model The individual (respondent) can choose between N alternatives, each alternative can be described with a vector of attributes, a i. 1. We assume that the chosen alternative is consumed in a given quantity. For example a certain number of days. 2. For now we assume that only one alternative can be chosen. However, this could include a possibility of opt-out 3. We assume weak complementarity. This means that the attributes of the non-selected alternatives does not affect utility (note this assumption may not always be satisfied) The indirect utility of alternative i for individual k Alternative i is chosen iff

Introducing the error term We want to introduce an error term in order to capture unobservable behavior. In terms of probabilities we have Now we need to: 1. Decide (test) the functional form of the utility function 2. Make an assumption about the distribution of the error term.

Functional form of the utility function Common assumption I: the error term enters the utility function as an additive term. Common assumption II: the utility function is a linear function of the attributes There is a trade-off between the benefits of assuming a less restrictive formulation and the complications that arise from doing so. This is especially relevant for the way income enters the utility function. A simpler functional form (e.g. linear in income) makes estimation of the parameters and calculation of welfare effects easier, but the estimates are based on restrictive assumptions.

The Multinomial Logit Model Historically the most common model. Main reason is simplicity to estimate. However, the last 10 years or so a rapid development of other models, computer capacity and algorithms has made this model somewhat less important. Suppose we have a choice set with J alternatives. The probability that individual k chooses alternative i can then be expressed as We assume that the error terms have an extreme value type I distribution (iid), the variance of this distribution is

The Multinomial Logit Model cont. It can be shown that the choice probability can then be expressed as Which is a very simple and nice expression! But as we will see this will come with some ”costs”. Note also, as with the CVM model: the parameters are normalised with a scale paramters. This complicates the interpretation of models, and in particular a comparion among models.

Some properties Of any discrete choice model 1. The true parameters are confounded with the scale parameter 2. Only the utility difference matters. Consequently, there must be a difference between the alternatives in order to estimate the parameter 3. This means that we can only include M-1 alternative specific constants and that socio-economic characteristics must be interacted with the constants or the attributes. Of the MNL model 4. The alternatives are independent (because of the IID assumption). Results in the IIA property 5. Limited modeling of taste variation. Unobserved heterogeneity is captured via the error term in a simple fashion. However, socio-economic variables can account for observed heterogeneity.

The IIA property The ratio of choice probabilities between two alternatives in a choice set is unaffected by what other alternatives that are available in the choice and the levels of the attributes of the other alternatives May or may not be satisfied, in many cases not. With many alternatives this is a useful property. Can be tested with the Hausman-McFadden test (1984). Essentially: If IIA is satisfied then the ratio of choice probabilities should not be affected by whether another alternative is on the choice set or not. Hence, one way of testing IIA is to remove one alternative and re-estimate the model and compare the choice probabilities.

Estimation and Interpreting the results We will use t he Wetland study discussed in the chapter on choice experiments. A mail survey to Swedish housholds about possible development of wetland areas (both for biodiversity and recreation reasons). AttributeDescriptionVariableLevels Total cost (Cost)The total cost for the individualCost200, 400, 700, 850 Surrounding vegetationForest or meadow-landMeadowForest, Meadow BiodiversityThe wetland can contain different numbers of both rare and more common species. Medbio Highbio Low, Medium, High, FishThe design of the wetland area can improve the conditions for fish species FishNo, Yes Fenced waterlineThe water is surrounded with a 1m fence in order to prevent drowning accidents. FenceNo, Yes CrayfishIntroduction of Swedish crayfish and allow fishing.CrayfishNo, Yes Walking facilitiesConstruction of the wetland area for outdoor lifeWalkNo, Yes

The choice situation Each respondent made at most 4 choice situations. There was always an opt-out situation

The data set-up. The data needs to be arranged in a specific way for all multinomial discrete choice models in Limdep. Each row in the data set represents one of the alternatives of a choice set. So if there are three alternatives (as in this case) each choice set will have three rows in the data set.

Commands in Limdep We use the NLOGIT command in limdep, together with the Model command, where we specify the utility function for each of the alternatives. nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio* medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio* medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio* medbo+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk$ Note 1: Choice is the choice indicator. Note 2: In this particular case we assume that the two ”new” alternaives have a common alternative specific constant. Note 3: The number of ASC cannot be higher than the number of alternatives minus one.

Output The coefficients can be interpreted in terms of sign and significance. Note that the experiment was generic so all the coefficents are generic. Cost: negative, so an increase in cost reduced the likelihood that the alternative is chooosen. Fish: positive, so improved conditions for fish population increases the likelihood that that alternative is choosen. Multinomial Logit CoefficientStandard error Intercept0.11950.3384 Cost-0.00120.0000 Meadow-0.05180.3967 Highbio0.78350.0000 Medbio0.59060.0000 Fish0.40510.0000 Fence-0.19460.0016 Crayfish-0.13010.0339 Walk0.75320.0000

Interpretation continued Intercept (alternative specific constant): positive. This means, in this case, that all else equal is it more likely that respondents choose one of the new alternatives instead of the current state (simple ponds). Can we compare the coefficients for different attributes within this model? Yes and No. A meaningfull comparison of coefficients require that they are measured on the same scale. In this particular case all variables are dummy variables so it is actually possible. But be carefull here! Can we compare the coefficients of this model with the coefficients from another model? Yes and No. You can compare sign and significance. But you cannot compare the size of the coefficients. This because all the coefficients are scaled with an unknown scale paramter. And without further information we cannot say anything about the scale paramter (we will get back to this....).

Adding socio-economic characteristics... We cannot add socio-economic characteristics directly into the model (since they are constant across alternatives). Two options: 1. Interact them with the alterantive specific constants 2. Interact them with the attributes.

Data setup with socio-economic characteristics The socio-economic characteristics are repeated for the individual

In Limdep 1. Interact them with the alterantive specific constants nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_male*male+b_cost*cost+b_meadow*meadow+b_highbi* highbio+b_medbio*medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_male*male+b_cost*cost+b_meadow*meadow+b_highbi* highbio+b_medbio*medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio* medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk$ 2. Interact them with the attributes. create;malecray=male*crayfish$ nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk+b_mcray*malecray/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+bwalkt*walk+b_mcray*malecray/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk+b_mcray*malecray$

The results So men are less likely to choose the new alternative Men dislike crayfish more (but insignificant)

Testing for IIA You can test for IIA, but in my experience this does not make much sense for generic choice experiments, and often we get problems with attributes with to little variation when we drop an alternative. With our data we actually have this problem: sample;all$ nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa1+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish* fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa2+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish* fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio*medbio+b_fish*fish+ b_fence*fence+b_cray*crayfish+b_walkt*walk; ias=base$

Welfare measures

Willingness to pay Main purpouse of the analysis is to obtain some welfare measures (although in some cases we are also interested in predicting the number of visitors, for example). Two common types of measures 1. Marginal WTP 2. ”Total” WTP, or WTP for a certain scenario. The first is perhaps the most common in CE, the second is the traditional welfare measure used in travel cost RUM and, of course, in CV studies. “The standard” in the literature is to focus on marginal WTP.

1. Marginal WTP The most common type of welfare measures derived from choice experiments are marginal rates of substitution (or marginal WTP). Let us assume a linear utility function The MRS between an attribute and money is Thus it is simply the ratio of the coefficients. Note that, since this is a ratio, the scale parameters are canceled. So we can compare across models. Note: This welfare measure is not comparable to welfare estimates from CV. This is the marginal WTP for one attribute.

Sources of variation in marginal WTP We have two sources of variation i. Variation across individuals. The utility function can include covariates. ii. Uncertainty from the randomness of parameters. The parameters are obtained from maximum likelihood. There is no preference uncertainty since the error term does not enter the MRS expression. Variation across individuals can be obtained by including socio-economic characteristics that are interacted with the attributes. Then we can calculate either individual level MRS:s or mean MRS for different socio-economic groups. Note: interaction with the ASC does not affect marginal WTP! Uncertainty from randomness of parameters can be handled in a number of ways: Delta method, Krinsky-Robb, Bootstrapping

i. Variation across individuals Variation across individuals can be obtained by including socio-economic characteristics that are interacted with the attributes. Then we can calculate either individual level MRS:s or mean MRS for different socio-economic groups. Note: interaction with the ASC does not affect marginal WTP!

ii. Randomness of parameters We want to find the standard deviation of an expression that is a non-linear function of a number of parameters. Delta method A first-order Taylor series of the WTP expression WALD command in Limdep Krinsky-Robb Draw a number of times from the asymptotic normal distribution of the parameter estimates and calculate the welfare measure for each of these draws. Bootstrapping Draw a number of times (with replacement) from the observations, re- estimate the model and calculate the welfare measure for each of these draws.

Randomness of parameters in Limdep We can use a command called Wald in Limdep to get the standard errors of a non-linear function using either the Delta-method or Krinsky-Robb (In the previous version of Limdep KR was not available). Delta method Probit;lhs=yes;rhs=one,nbid$ wald;labels=b1,b2;start=b;var=varb; fn1=b1/b2$ Krinsky-Robb Probit;lhs=yes;rhs=one,nbid$ wald;labels=b1,b2;start=b;var=varb; fn1=b1/b2; K&R;Pts=1000$

2. Total Willingness to pay The main difference is that we do not know which alternative the individual will choose, and therefore we do not know, with certainty, the utility before and after the change. Therefore we must formulate this as an expected utility problem instead. Previously we had If the utility function is linear in income WTP (CV) is, under certainty, given by the following expression Simplify to Solve for WTP

Continued… But in a choice experiment we do not know with certainty the utility before and after (we do not know which alternative they would have chosen). Note, that we are now talking about alternative-specific designs. So instead we can write In the case of a MNL model this can actually readily be computed. Given a choice set S, the expected value of the maximum utility is (the log-sum) (Small and Rosen 1981): So the WTP formula is

WTP without the log-sum formula The previous derivation relied on an alternative-specific view on choice experiment. Not surprising since the models originates from transport and travel cost literature. With a generic choice experiment, we can instead talk about utility before and after the change (just as we did with the CV data) Mean WTP is thus We can easily extend this to involve non-linear income effects.

An example Let us go back to our Wetland experiment. Given the functional form, the marginal WTP is simply going to be the ratio of the attribute coefficient and the cost coefficient. Since the experiment was generic we have that mean WTP is However, we have an ASC, that we should take into consideration. The ASC is the “effect” on utility that is not captured by the attributes. We just need to remember where the ASC is. In this case it was for the two improved choices. So if we go from the status-quo to a new state we should include this ASC in the welfare measure. So for a particular example (We go from status quo to High biodiversity and improved Fish), we have

In Limdep nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio *medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio *medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio *medbio+b_fish*fish+b_fence*fence+b_cray*crayfish+b_walkt*walk$ ?= Marginal WTP wald;labels=9_b;start=b;var=varb; fn1=-b3/b2; fn2=-b4/b2; fn3=-b5/b2; fn4=-b6/b2; fn5=-b7/b2; fn6=-b8/b2; fn7=-b9/b2$ ?= WTP wald;labels=9_b;start=b;var=varb; fn1=-(b1+b4+b6)/b2$

Results WTP The total WTP is 1,134 SEK. Thus, this is the mean WTP to go from status quo to a wetland with High Biodiversity and improved conditions for the fish population. This “value” includes the improvements in biodiversity and the fish population + the ASC. Interpretations of the ASC 1. A welfare effect not covered by the attributes. For example, people want an improvement in general. Or that people think/know that there are other positive effects of the improvements 2. People want/do not want changes in general. Inertia etc Question for us: In what cases should we include this effect as a welfare measure or not? My answer: Depends on whether this effect is an artifact of the survey or not.

Results marginal WTP Interpretation: People are on average willing to pay 351 SEK to improve the Fish stock. Can we compare MWTP? Yes, but only if they are measured in the same dimension (as in this case). So most important: High biodiversity and Walking facilities. AttributeMarginal WTPP-value Meadow-44.93710.399 Highbio679.2240.000 Medbio512.0080.000 Fish351.1320.000 Fence-168.710.003 Crayfish-112.7740.042 Walk652.8950.000

Some extensions

A. Comparing models: A warning! It is impossible to directly compare coefficient estimates across different discrete choice models. The reason for this is that the coefficients are scaled with an unknown scale parameter. Neither is it straightforward to pool data sets. The reason is that the two data sets might have different scale parameters (i.e. different variances). If we were naively to pool the two data sets, ignoring the fact that they are not strictly independent when derived from the same sample of individuals, then we are implicitly assuming that the variances are the same across all alternatives. We cannot use the results of Cameron. Remember that we have not modelled this in WTP space, but in utility space. (But actually this is where the latest development in CE econometrics is, to model this in WTP space). But for now we have another approach that we can use.

Estimating the relative scale parameter It is however possible to estimate the ratio of scale parameters if we pool two or several data sets. Suppose we have two data-sets with the following underlying indirect utility functions The choice probabilities are Pooling the data is the same as imposing the restriction that the parameters are the same. Since the parameters are confounded with the scale parameter, this is not a straightforward test.

Continued… What we can do is to pool the data under a restriction that the parameters are equal but allow for different scale factors. This would allow us to estimate the ratio of the scale parameters, or equivalently by setting one of the scale parameter to one, estimating the relative scale factor. This can be conducted with a grid-search procedure proposed by Swait and Louviere (1993). Ignoring the source-specific constants the data matrix for the pooled model would be So one of the data sets is multiplied by the relative scale factor in order to account for the possible difference in scale. The log-likelihood function is concave in the scale-ratio parameter so there is a unique maximum. This means that we can conduct a one-dimensional grid search over possible values to estimate the relative scale parameter.

Doing the grid search in practice Step 1. Determine a start value for the relative scale parameter; say 0.5. Step 2. Multiply one of the data sets with that scale parameter, i.e. multiply the right-hand side variables with 0.5 Step 3. Estimate the pooled model with the re-scaled data. Step 4. Store the log-likelihood function. Step 5. Go back to step 1 but try another parameter values, say 0.55. Cover the relevant range, say 0.5-1.5 to begin with.

Results of a grid search The scale parameter that maximizes the log-likelihood is then imposed on the data, and the data is pooled. From this pooled data we can do a likelihood ratio test to test the hypothesis of equal parameters.

B. Extending the econometrics The Multinomial Logit model is restrictive and there are a number of extensions that we could consider. These extensions “solve” different problems of the MNL model. One set of models relaxes the IIA assumption to some extent: - Nested Logit - Heteroskedastic Extreme Value One set of models relaxes IIA and the homogeneity assumptions -Mixed Logit models -Latent class models (finite mixture models) One set of models handles the panel data nature - Binary Probit/Logit (for binary CE) - Mixed Logit models We will only look at the mixed logit and latent class models.

C. Mixed Logit Models The advantages of mixed models are (i) the alternatives are not independent, i.e. the model does not exhibit IIA (ii) there is an explicit account for unobserved heterogeneity. (iii) it is possible to extend to panel data The utility function for individual k Each individual’s coefficient vector is the sum of the population mean and individual deviation. The individual deviation is assumed to have a particular distribution, usually with mean zero and a standard deviation. If the error terms are IID type I extreme value, the model is called a mixed logit model (or random parameter logit) The individual deviation term is a random term with mean zero. It can take on a number of distributional forms such as normal, lognormal or triangular.

The probabilities Let us assume that the coefficients (the preferences) vary in the population with a distribution with density First we can express the choice probabilities for a given set of preferences (beta vector). This is called the conditional probability The we have the unconditional probability. This is the integral of the standard logit probabilities over all possible values of beta The choice probability in mixed logit is a weighted average of the logit formula at different values of beta, with the weights given by the density function

Estimation I Estimation is complex since the integrals cannot be evaluated analytically. Therefore we have to rely on some type of simulation method, called simulated maximum likelihood estimations. These estimation methods have been developed the last 10 years or so. And the computer capacity has of course improved dramatically.

Estimation II In brief the simulator can be described as follows. From a given distribution,θ, individual specific values of β are drawn. From each draw we approximate the choice probability using the standard logit. The average of Z such draws is the approximate choice probability for individual k, denoted L k. The simulated log likelihood is then, where subscript n index sampled individuals

Determining random parameters Determine which parameters that should have a random distribution There is a specification test, but rather “simple”. For example it does not allow for test where we consider the panel data nature of the data. Determine the distribution of each parameter Difficult. First off all we should see if we can use economic theory, for example implementing restrictions on the distribution. We can run a jack-knife type of procedure where we re-estimate the model taking away one respondent at a time.

Random parameters in RPL models Note that different parameters can have different distribution and some parameters can be fixed. Two common distributions are (i) Normal distribution (ii) Log-normal distribution Others are (iii) Triangular (iv) Uniform (v) Truncated distributions The log-normal distribution is different from the others since it restricts all respondents to have the same sign of the coefficients. However, we know that the log-normal distribution can have a huge impact on for example mean WTP. So be careful when using this distribution. Common practice: Keep the cost-coefficient fixed. Then we know the distribution of MRS and we avoid exploding MRS:s. We will get back to this.

Doing this in Limdep We need to specify which parameters that are random: This is done with the “fcn” statement. We need to specify how many draws to make in the simulation: This is done with the pts command. == Normal distribution nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk; rpl;pts=25;fcn=b_highbi(n),b_medbio(n)$ ?== Log-normal distribution nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk;rpl;pts=25;fcn=b_highbi(l),b_medbio(l)$

Results Normal The preferences has a mean and a standard deviation. Note this standard deviation is a standard deviation due to preference heterogeneity, do not confuse this with the standard deviation due to maximum likelihood estimation. Marginal WTP is here calculated as the ratio (so we do not take the randomness into consideration)

A warning The log-normal distribution forces everybody to have a positive preference. So if you expect a negative preference, estimate the model with the negative values of that attribute In order to compare the results of say a normal and log-normal distributions we have to convert the results for the log-normal distribution.The software estiamtes Mean: Median Standard deviation

Log-normal results

D. Extensions of the mixed logit We will look at: - Panel data - Estimating the distribution of WTP and individual level parameters - Heterogeneity in means -Type of simulation method

D1. Panel data Most often we observe an individual making several choices. Natural to assume that the preferences are stable over the experiment. Consequently, the utility coefficients are allowed to vary among individuals but they are constant among the choice situations for each individual If we only have ASCs that are randomly distributed, then we would have a random effects model. In limdep Specify panel with the “;pds=xx”, where x is the number of choice sets an individual made. create; ID1=Ndx(ID,0)$ matrix;grpsize=gsiz(ID1)$ create;numbs=grpsize(ID1)/3$?3 alternatives Pds=numbs;

D2. Distribution of WTP Our assumption about the distribution of the parameters implies a distribution of WTP. Given a fixed cost coefficient, the distribution of WTP is the same as for the attribute. Note: So we can talk about a population distribution of WTP, where the distribution is the distribution of preferences among in the population. Again: this is NOT the distribution due to parameter uncertainty. We still need to address that uncertainty too, with Krinsky-Robb, Delta method or bootstraping.

Swedish outage cost study AttributeCoefficientP-value Fixed parameters Cost-0.00820.000 Random parameters 4 hour weekday-0.07950.001 8 hour weekday-0.18830.000 24 hour weekday-0.85070.000 4 hour weekend-0.26160.000 8 hour weekend-0.32050.000 24 hour weekend-1.11280.000 Standard deviation of random parameters 4 hour weekday0.02240.411 8 hour weekday0.00620.873 24 hour weekday0.69390.000 4 hour weekend0.01650.552 8 hour weekend0.43150.000 24 hour weekend0.80950.000

WTP estimates Mean marginal WTPStdv marginal WTP 4 hour weekday 9.64 (2.85) 2.71 (3.30) 8 hour weekday 22.83 (4.04) 0.76 (4.74) 24 hour weekday 103.16 (4.56) 84.15 (4.72) 4 hour weekend 31.73 (3.06) 2.00 (3.36) 8 hour weekend 38.86 (4.06) 52.33 (4.42) 24 hour weekend 134.96 (5.20) 98.16 (6.37)

A plot of 24 hour outage WTP A certain proportion of the population with negative WTP according to the model. In this particular case 0.085: calculate;list;phi(0,134.96,98.16)$ ) Sample;1-5000$ calc;j=1$ create;obsnumb=trn(-5000,0)$ proc sample;all$ reject;obsnumb#j$ calc;rnnb=rnn(0,1)$ create;wtp=134.96+rnnb*98.16$ calc;j=j+1$ endproc exec;n=5000$ sample;1-5000$ kernel;rhs=wtp$

Individual level WTP (or type specific WTP) Is it possible to say were in the distribution a specific individual belongs? In a sense: yes. The information we have about the preferences is the individuals choices. Thus their choices reveals something about the specific taste that an individual has. Distinguish between - Distribution of preferences/tastes in the population - Distribution of preferences/tastes in the subpopulation who make particular choices. We will not go through the details, but it is possible to derive expectations about the random parameters for difference subpopulations who make particular choices.

In Limdep You add “;parameters” to the Nlogit statement. This will create a matrix called beta_i, which contains the individual level estimates. Copy to excel and deal with it there or make the calculations in limdep. nlogit;lhs=val;rhs=z;choices=alt1,alt2;rpl;pds=numbs;pts=500; fcn=ca1(n),cano(n),cab(n),ca3(n),ca4(n),ba1(n),bano(n),bab(n),ba3(n),ba4(n);parameters$ sample;all$ create;d1=0$ …. create;d11=0$ namelist;oo=d1,d2,d3,d4,d5,d6,d7,d8,d9,d10,d11$ create;oo=beta_i$ sample;1-395$ create;owtp1=-d1/d11$ histogram;rhs=owtp1;limits=-10,0,1,2,3,4,5,6,7$

Individual levels for outage costs Share of responses with “negative” sign decreases drastically.  Perhaps our concern about inconsistencies from a normal distribution are overrated. % reverse sign MeanStd dev MinMaxIndivPop 4 hour weekday9.630.119.1910.820.000 8 hour weekday22.830.0222.7222.910.000 24 hour weekday103.042.34-40.01218.90.0080.110 4 hour weekend31.720.0931.3732.270.000 8 hour weekend38.9518.10-34.32103.10.0240.229 24 hour weekend134.955.32-37.11262.40.0070.085

A histogram for 24 hour

D3. Heterogeneity in means The mixed logit allows for unobserved heterogeneity, and an estimation of the distribution for this. But we can combine this with observed heterogeneity by allowing the random parameters to interact with socio- economic characteristics. nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(new2)=alfa+b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*f ence+b_cray*crayfish+b_walkt*walk/ U(base)=b_cost*cost+b_meadow*meadow+b_highbio*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk; rpl=male; pts=25;fcn=b_highbi(n),b_medbio(n)$

D4. A small digression: random number generators With simulated maximum likelihood we need to draw random numbers. In many other cases we also wish to generate random numbers. For example when conducting Monte Carlo simulations. Many econometric programs contain pre-programmed random number generators for various distributions. These in turn usually rely on random draws from a uniform distribution: using the so-called inversion method.

The inversion method Suppose that a random variable x has the pdf f(x) and the cdf F(x). Further let u be a uniformly distributed variable. The inversion method use the following “fact”: We can transform any value of u into a value of x by This means that by finding the inverse of the cdf of the original distribution and calculating the value of the inverse at a random draw of a uniformly distributed variable we can obtain a random draw of the variable of interest. Example An extreme value distribution with scale parameter set to 1 has the cdf Now we want to solve for x

Drawing numbers but not completely random Ordinary random number generators are actually not that smart, we call them pseudo-random generators. They tend to leave “holes”. A development of other quasi-random generators, like Halton sequences (Do NOT ask me how they work ). In order to get Halton draws we add “;Halton” to the statement. How many draws? pts=xx. For testing: Around 25. For final models: 500-1000 Results can change going from 50 to 500 draws, in particular for complicated models.

Difference between random and pseudo-random The Pesudo-random leaves much less holes, than a random number generator.

Some warnings RPL models does not “solve” all our problems. Just because we now can estimate these models it is not obvious that we should estimate them. It is not obvious that this approach to heterogeneity is the best approach. The choice of distribution and which parameters that should be random is still not a simple choice. The understanding of the simple Multinomial Logit is still crucial. We can expect a lot more research on the issue of the scale parameter in addition to the development of mixed models. The “hottest” topic right now is modeling in WTP space.

E. Latent Class Model (finite mixture model).In latent class models is discrete, with taking on a finite set of values. It is assumed that respondents belong to different preference classes that are defined by a small number of segments Suppose S segments exist in the population each with different preference structures and that individual k belongs to segment s (s = 1,...,S). The conditional indirect utility function can now be expressed as. For simplicity, let us write the deterministic part of utility as v ik =βZ i, where again Z i is a vector of attributes. The preference parameters (β) vary by segment, so that we can write the indirect utility function as 68

E. Latent Class Model (finite mixture model) The probability of choosing alternative i depends on the segment that one belongs to and can be expressed as: Let us now assume that there is a process describing the probability of being included in a particular segment, and thus us a function of demographic (and other) information. Following Boxall and Adamowicz (1999), Swait (1994), and Gupta and Chintagunta (1993), that process can be specified as a separate logit model to identify segment membership as: 69

continued Let P iks be the joint probability that individual k belongs to segment s and chooses alternative i. This is also the product of the probabilities defined in the two previous equations: P iks = P ik|s P ks. The probability that individual k chooses i becomes the key component in the finite mixture or latent class approach: Note that the ratio of probabilities of selecting any two alternatives would contain arguments that include the systematic utilities of other alternatives in the choice set. This is the result of the probabilistic nature of membership in the elements of S. The implication of this result is that IIA need not be assumed (Shonkwiler and Shaw 1997). 70

Number of classes One issue with latent class models is the choice of number of classes, S. The determination of the number of classes is not part of the maximization problem and it is not possible to use conventional specification tests such as a likelihood ratio tests). What sometimes is used is some sort of information criteria (Scarpa and Thiene, 2005) as well as stability of the parameters in the segments as tools to assess the best number of classes to represent the data. It is possible to estimat LCM using Nlogit 71

Doing this in Limdep - We need to specify if the class probabilities varies with some covariates and which : This is done with the “LCM=xxx”. Without covariates only include “LCM”. - We need to specify how many classes: This is done with the pts command. nlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk/ U(new2)=b_cost*cost+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk/ U(base)=alfa; lcm=male;pts=2$ LCM: + It’s semiparametric. We don’t have to make any distributional assumptions of taste heterogeneity. - if the taste heterogeneity is, for example normal distributed, the LCM will be miss-specified.. 72

LCM results I

LCM results II

F. WTP space Typical traditional model: a random utility model with random parameter. Train and Weeks (2005) calls a this a model in preference space. They suggest another way to specify the random parameter logit model by directly specifying the distribution of WTP instead. They show how this can be done with a reparameterization of the original model (just as Cameron with CVM!) These two approaches are equivalent, and there are only a few empirical studies comparing them; see e.g. Scarpa et al., 2008; Train and Weeks, 2005.

Standard formulation, preference space The standard formulation Where we normalize the scale parameter to one. Standard to assume that the cost coefficient is fixed, since this facilitates estimation and since the distribution of the marginal willingness to pay is the distribution of the corresponding attribute. This implies that the scale parameter is the same for all respondents. An assumption that can be questioned for good reasons. Advisable to let the price coefficient be randomly distributed. There are two additional problems with that: (i) the distribution of the marginal WTP of the attribute could be intractable; for example, a normal distribution of the attribute coefficient and a log-normal distribution of the cost coefficient, (ii) Second, uncorrelated utility coefficients translate into marginal WTPs that are correlated.

WTP space instead Parameterized in terms of WTP instead of utility. Where w is the marginal WTP for that attribute. So the difference here is that the coefficient vector is a vector of marginal WTPs, since. As discussed by Scarpa et al. (2008), whether this specification of the model results in different WTPs than the model in preference space is an empirical question (and would depend on the possible variation of the scale parameter).

Doing this in Limdep -Limdep has a special fix for this using the GMXlogit command. -We write (*n) in the random part for the cost coefficent. create;costs=cost/1000$ gmxlogit;lhs=choice;choices=new1,new2,base; Model: U(new1)=b_cost*costs+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk/ U(new2)=b_cost*costs+b_meadow*meadow+b_highbi*highbio+b_medbio*medbio+b_fish*fish+b_fence*fence +b_cray*crayfish+b_walkt*walk/ U(base)=alfa; fcn=b_cost(*n),b_meadow(n),b_fish(n);pds=numb$ 78

Results

Econometric analysis of CE surveys. The Economic Model The individual (respondent) can choose between N alternatives, each alternative can be described.

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Econometric analysis of CE surveys. The Economic Model The individual (respondent) can choose between N alternatives, each alternative can be described.

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Presentation on theme: "Econometric analysis of CE surveys. The Economic Model The individual (respondent) can choose between N alternatives, each alternative can be described."— Presentation transcript:

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