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Exercise 2: Nested logit model

Kenneth Train and Yves Croissant

2025-07-12

The data setHC frommlogit contains data inR format on the choice of heating and central cooling system for 250 single-family, newly built houses in California.

The alternatives are:

• Gas central heat with coolinggcc,
• Electric central resistence heat with coolingecc,
• Electric room resistence heat with coolingerc,
• Electric heat pump, which provides cooling alsohpc,
• Gas central heat without coolinggc,
• Electric central resistence heat without coolingec,
• Electric room resistence heat without coolinger.

Heat pumps necessarily provide both heating and cooling such that heat pump without cooling is not an alternative.

The variables are:

• depvar gives the name of the chosen alternative,
• ich.alt are the installation cost for the heating portion of the system,
• icca is the installation cost for cooling
• och.alt are the operating cost for the heating portion of the system
• occa is the operating cost for cooling
• income is the annual income of the household

Note that the full installation cost of alternativegcc isich.gcc+icca, and similarly for the operating cost and for the other alternatives with cooling.

1. Run a nested logit model on the data for two nests and one log-sum coefficient that applies to both nests. Note that the model is specified to have the cooling alternatives (gcc},ecc},erc},hpc}) in one nest and the non-cooling alternatives (gc},ec}, `er}) in another nest.

library("mlogit")data("HC",package ="mlogit")HC <-dfidx(HC,varying =c(2:8,10:16),choice ="depvar")cooling.modes <-idx(HC,2)%in%c('gcc','ecc','erc','hpc')room.modes <-idx(HC,2)%in%c('erc','er')# installation / operating costs for cooling are constants,# only relevant for mixed systemsHC$icca[!cooling.modes] <-0HC$occa[!cooling.modes] <-0# create income variables for two sets cooling and roomsHC$inc.cooling <-HC$inc.room <-0HC$inc.cooling[cooling.modes] <-HC$income[cooling.modes]HC$inc.room[room.modes] <-HC$income[room.modes]# create an intercet for cooling modesHC$int.cooling <-as.numeric(cooling.modes)# estimate the model with only one nest elasticitynl <-mlogit(depvar~ich+och+icca+occa+inc.room+inc.cooling+int.cooling|0, HC,nests =list(cooling =c('gcc','ecc','erc','hpc'),other =c('gc','ec','er')),un.nest.el =TRUE)summary(nl)

## ## Call:## mlogit(formula = depvar ~ ich + och + icca + occa + inc.room + ##     inc.cooling + int.cooling | 0, data = HC, nests = list(cooling = c("gcc", ##     "ecc", "erc", "hpc"), other = c("gc", "ec", "er")), un.nest.el = TRUE)## ## Frequencies of alternatives:choice##    ec   ecc    er   erc    gc   gcc   hpc ## 0.004 0.016 0.032 0.004 0.096 0.744 0.104 ## ## bfgs method## 11 iterations, 0h:0m:0s ## g'(-H)^-1g = 7.26E-06 ## successive function values within tolerance limits ## ## Coefficients :##              Estimate Std. Error z-value  Pr(>|z|)    ## ich         -0.554878   0.144205 -3.8478 0.0001192 ***## och         -0.857886   0.255313 -3.3601 0.0007791 ***## icca        -0.225079   0.144423 -1.5585 0.1191212    ## occa        -1.089458   1.219821 -0.8931 0.3717882    ## inc.room    -0.378971   0.099631 -3.8038 0.0001425 ***## inc.cooling  0.249575   0.059213  4.2149 2.499e-05 ***## int.cooling -6.000415   5.562423 -1.0787 0.2807030    ## iv           0.585922   0.179708  3.2604 0.0011125 ** ## ---## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ## Log-Likelihood: -178.12

1. The estimated log-sum coefficient is\(0.59\). What does this estimate tell you about the degree of correlation in unobserved factors over alternatives within each nest?

The correlation is approximately\(1-0.59=0.41\). It’s a moderate correlation.

2. Test the hypothesis that the log-sum coefficient is 1.0 (the value that it takes for a standard logit model.) Can the hypothesis that the true model is standard logit be rejected?

We can use a t-test of the hypothesis that the log-sum coefficient equal to 1. The t-statistic is :

 (coef(nl)['iv']-1)/sqrt(vcov(nl)['iv','iv'])

##        iv ## -2.304171

The critical value of t for 95% confidence is 1.96. So we can reject the hypothesis at 95% confidence.

We can also use a likelihood ratio test because the multinomial logit is a special case of the nested model.

# First estimate the multinomial logit modelml <-update(nl,nests =NULL)lrtest(nl, ml)

## Likelihood ratio test## ## Model 1: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0## Model 2: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0##   #Df  LogLik Df  Chisq Pr(>Chisq)  ## 1   8 -178.12                       ## 2   7 -180.29 -1 4.3234    0.03759 *## ---## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Note that the hypothesis is rejected at 95% confidence, but not at 99% confidence.

2. Re-estimate the model with the room alternatives in one nest and the central alternatives in another nest. (Note that a heat pump is a central system.)

nl2 <-update(nl,nests =list(central =c('ec','ecc','gc','gcc','hpc'),room =c('er','erc')))summary(nl2)

## ## Call:## mlogit(formula = depvar ~ ich + och + icca + occa + inc.room + ##     inc.cooling + int.cooling | 0, data = HC, nests = list(central = c("ec", ##     "ecc", "gc", "gcc", "hpc"), room = c("er", "erc")), un.nest.el = TRUE)## ## Frequencies of alternatives:choice##    ec   ecc    er   erc    gc   gcc   hpc ## 0.004 0.016 0.032 0.004 0.096 0.744 0.104 ## ## bfgs method## 10 iterations, 0h:0m:0s ## g'(-H)^-1g = 5.87E-07 ## gradient close to zero ## ## Coefficients :##              Estimate Std. Error z-value Pr(>|z|)  ## ich          -1.13818    0.54216 -2.0993  0.03579 *## och          -1.82532    0.93228 -1.9579  0.05024 .## icca         -0.33746    0.26934 -1.2529  0.21024  ## occa         -2.06328    1.89726 -1.0875  0.27681  ## inc.room     -0.75722    0.34292 -2.2081  0.02723 *## inc.cooling   0.41689    0.20742  2.0099  0.04444 *## int.cooling -13.82487    7.94031 -1.7411  0.08167 .## iv            1.36201    0.65393  2.0828  0.03727 *## ---## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ## Log-Likelihood: -180.02

1. What does the estimate imply about the substitution patterns across alternatives? Do you think the estimate is plausible?

The log-sum coefficient is over 1. This implies that there is more substitution across nests than within nests. I don’t think this is very reasonable, but people can differ on their concepts of what’s reasonable.

2. Is the log-sum coefficient significantly different from 1?

\begin{answer}[5] The t-statistic is :

 (coef(nl2)['iv']-1)/sqrt(vcov(nl2)['iv','iv'])

##        iv ## 0.5535849

lrtest(nl2, ml)

## Likelihood ratio test## ## Model 1: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0## Model 2: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0##   #Df  LogLik Df  Chisq Pr(>Chisq)## 1   8 -180.02                     ## 2   7 -180.29 -1 0.5268      0.468

We cannot reject the hypothesis at standard confidence levels.

3. How does the value of the log-likelihood function compare for this model relative to the model in exercise 1, where the cooling alternatives are in one nest and the heating alternatives in the other nest.

logLik(nl)

## 'log Lik.' -178.1247 (df=8)

logLik(nl2)

## 'log Lik.' -180.0231 (df=8)

The\(\ln L\) is worse (more negative.) All in all, this seems like a less appropriate nesting structure.

3. Rerun the model that has the cooling alternatives in one nest and the non-cooling alternatives in the other nest (like for exercise 1), with a separate log-sum coefficient for each nest.

nl3 <-update(nl,un.nest.el =FALSE)

1. Which nest is estimated to have the higher correlation in unobserved factors? Can you think of a real-world reason for this nest to have a higher correlation?

The correlation in the cooling nest is around 1-0.60 = 0.4 and that for the non-cooling nest is around 1-0.45 = 0.55. So the correlation is higher in the non-cooling nest. Perhaps more variation in comfort when there is no cooling. This variation in comfort is the same for all the non-cooling alternatives.

2. Are the two log-sum coefficients significantly different from each other? That is, can you reject the hypothesis that the model in exercise 1 is the true model?

We can use a likelihood ratio tests with modelsnl andnl3.

lrtest(nl, nl3)

## Likelihood ratio test## ## Model 1: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0## Model 2: depvar ~ ich + och + icca + occa + inc.room + inc.cooling + int.cooling | ##     0##   #Df  LogLik Df  Chisq Pr(>Chisq)## 1   8 -178.12                     ## 2   9 -178.04  1 0.1758      0.675

The restricted model is the one from exercise 1 that has one log-sum coefficient. The unrestricted model is the one we just estimated. The test statistics is 0.6299. The critical value of chi-squared with 1 degree of freedom is 3.8 at the 95% confidence level. We therefore cannot reject the hypothesis that the two nests have the same log-sum coefficient.

4. Rewrite the code to allow three nests. For simplicity, estimate only one log-sum coefficient which is applied to all three nests. Estimate a model with alternativesgcc,ecc anderc in a nest,hpc in a nest alone, and alternativesgc,ec ander in a nest. Does this model seem better or worse than the model in exercise 1, which puts alternativehpc in the same nest as alternativesgcc,ecc anderc?

nl4 <-update(nl,nests=list(n1 =c('gcc','ecc','erc'),n2 =c('hpc'),n3 =c('gc','ec','er')))summary(nl4)

## ## Call:## mlogit(formula = depvar ~ ich + och + icca + occa + inc.room + ##     inc.cooling + int.cooling | 0, data = HC, nests = list(n1 = c("gcc", ##     "ecc", "erc"), n2 = c("hpc"), n3 = c("gc", "ec", "er")), ##     un.nest.el = TRUE)## ## Frequencies of alternatives:choice##    ec   ecc    er   erc    gc   gcc   hpc ## 0.004 0.016 0.032 0.004 0.096 0.744 0.104 ## ## bfgs method## 8 iterations, 0h:0m:0s ## g'(-H)^-1g = 3.71E-08 ## gradient close to zero ## ## Coefficients :##               Estimate Std. Error z-value  Pr(>|z|)    ## ich          -0.838394   0.100546 -8.3384 < 2.2e-16 ***## och          -1.331598   0.252069 -5.2827 1.273e-07 ***## icca         -0.256131   0.145564 -1.7596   0.07848 .  ## occa         -1.405656   1.207281 -1.1643   0.24430    ## inc.room     -0.571352   0.077950 -7.3297 2.307e-13 ***## inc.cooling   0.311355   0.056357  5.5247 3.301e-08 ***## int.cooling -10.413384   5.612445 -1.8554   0.06354 .  ## iv            0.956544   0.180722  5.2929 1.204e-07 ***## ---## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ## Log-Likelihood: -180.26

The\(\ln L\) for this model is\(-180.26\), which is lower (more negative) than for the model with two nests, which got\(-178.12\).