I have 100 geographical regions in a country. For each region the total number of houses and the number of vacant houses have been collected yearly over 20 years. I have also some other economic indicators at the country level (GDP, interest rate etc.). Now, given the forecasts for these indicators for next year, I want to forecast next year vacancy rate.

I have first used an auto-regressive mixed-effect model in R (package lme4) where the vacancy rate (computed as the ratio of vacant houses over the total number of houses) in a region depends on the last year's vacancy rate, the mean vacancy rate of neighboring regions, GDP and interest rate.

The problem with this model is that the vacancy rate can go outside the range [0,1], which obviously does not make sense. I need to restrict the range of the vacancy rate: a simple fix is restricting ex post.

Does anybody have experience with such models? I think that I can use some mixed multinomial logit model probably.

I would appreciate if you could provide guidance along with some R code.

  • $\begingroup$ If you want an answer to include R code, it would probably help if you include your current R code. Just the lmer() call should be enough. $\endgroup$
    – onestop
    Commented Jan 25, 2011 at 12:45

2 Answers 2


One of the tricks in modelling percentages is to use the logit transformation. Then instead of modelling percentage $p_i$ as linear function you model the logit transform of this percentage:

\begin{align} y_i=\log\frac{p_i}{1-p_i} \end{align}

In R you will need to create new transformed variable and use it as a dependent variable in lmer.

You might look into modelling directly the number of empty houses instead of percentages, then you will not have a problem with non-sensical values. I suggest using log transformation for that. This of course means that you might get more non-vacant houses than there are houses, but this can be used as an indicator of model inadequacies. If on the other hand you have for some regions full booking in historical data meaning that demand was larger than supply, you might want to look into censored regression models.


It would seem to make sense to use a generalized linear mixed model with family=binomial and a logit or probit link. This would restrict your fitted values to the range (0,1). I don't know whether you can combine that with an autoregressive error structure in lmer4 though.

  • $\begingroup$ the OP observes percentages, so the dependent variable is not binomial. Can you use it in glm anyways? $\endgroup$
    – mpiktas
    Commented Jan 25, 2011 at 12:55
  • $\begingroup$ The OP said the vacancy rate is computed as the ratio of vacant houses over the total number of houses, which seems to imply he has these data. Vacant houses would then be a binomial outcome, with total houses as the denominator. $\endgroup$
    – onestop
    Commented Jan 25, 2011 at 13:05
  • $\begingroup$ @mpiktas I have the house numbers. Actually, what I need is to simulate/forecast the probability for a house in a region to be vacant... $\endgroup$
    – teucer
    Commented Jan 25, 2011 at 14:08
  • $\begingroup$ is binomial "model" not problematic, given that the economic indicators are at country level so the same for all the 100 regions? $\endgroup$
    – teucer
    Commented Jan 25, 2011 at 22:24

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