ML versus IV estimates of spatial SUR models: evidence from the case of Airbnb in Madrid urban area

Abstract

In recent decades we have seen an increased interest in the use of seemingly unrelated regressions models (SUR) in a spatial context, with compelling case studies in different fields. This upsurge has favoured the development of new and more efficient inference techniques. At present, the user has a basic toolkit to deal with this kind of model, that is, however, in need of improvement. This paper focuses on the question of estimating, quickly and accurately, spatial SUR models. The most popular procedure is maximum likelihood (ML) which guarantees precision at the cost of a high computational burden. This is especially true in cases of large sample size and strong spatial structure. We explore simpler estimation algorithms such as instrumental variables (IV), which expedites calculation at the cost of a certain loss in quality of the estimates. We focus on the importance of sample size and the trade-off between accuracy and speed. To that end, we perform a comprehensive simulation experiment in which we compare ML and IV algorithms, looking for their strengths and weaknesses. The paper includes two applications to the case of Airbnb in the urban area of Madrid. First, we estimate a spatial SUR hedonic model of accommodation prices, using micro-data for three different cross sections à la Anselin, that is, considering temporal correlation plus spatial structure. Then, the data are aggregated by neighbourhoods. We specify a spatial SUR model with two equations (apartments and rooms) also using three cross sections. In both cases, the models are estimated by ML and IV using the spsur R package (Angulo et al. in spSUR: spatial seemingly unrelated regression models. R Package version 1.0.0.3, 2019), with the aim of illustrating its capabilities.

Appendix: Additional results on the Monte Carlo

Figures 3, 4, 5, 6, 7, 8 add details to Sect. 3. Here, we include the estimation bias for the parameters of the experiment, using classical boxplots. For reasons of space, we only distinguish between type of equation, SLM or SDM. The data in the graphs refer to the difference between the true and the estimated value of the corresponding parameter for each iteration; so, at the very least, the boxplot should be centred on zero. Moreover, the IV algorithm has been simulated for a sample size of $n=5000$, to check for its consistency. The results, available upon request, confirm the low consistency rate of these estimates.

This is what happens with the boxplots of estimates of the slope coefficients, i.e. $\beta _{2g}$ parameters. Figure 3 reveals several important things: (1) the boxes are centred around a value of zero, which confirms that the estimates are unbiased, (2) the estimates are also consistent, which is clear from the tendency to compress the boxplots around zero, (3) there are almost no differences between the results of the SLM and of the SDM equations; the shape of the two boxplots is basically the same, (4) both algorithms produce strange results which are well outside the upper and lower whiskers; however, this propensity decreases with sample size, (5) 400 observations can be taken as a threshold to significantly reduce the risk of outliers, and (6) the IV estimates appear to be more affected by the incidence of outliers which are of greater size than in the ML approach.

Figure 4 describes the estimation of the intercepts, i.e. $\beta _{1g}$ parameters; it is obvious that the shape of the boxes changes a lot. The results point to the existence of a slight bias for small sample sizes ($n=25,49$), which disappears with large n. Moreover, the distribution of the ML estimation errors is asymmetric and skewed towards the range of positive values. The bias is stronger for the SLM case. Another fact to note is the slow rate of convergence towards the true values. The dispersion of bias concentrates around the value of zero only for large sample sizes, $n=900$, for both algorithms. In sum, there is a high risk of obtaining anomalous estimates for small and medium sample sizes for the intercept terms of the equations.

Similar comments can be made with respect to the estimation of the $\theta _{2g}$ parameters in Fig. 5. The distribution of the estimation errors is centred around a value of zero, but the rate of convergence seems to be slow. There are abundant outliers for small and medium sample sizes, which increases considerably the risk of calculating strange estimates.

Figure 6 synthesizes the estimation of the spatial autocorrelation coefficients, $\lambda _{1}$ and $\lambda _{2}$. The shape of the boxplots confirms the consistency of both algorithms, but also the non-negligible risk of obtaining anomalous estimates even for large sample sizes. The boxes, in the ML case, remain slightly asymmetrical for small sample sizes, $n=25$ and $n=49$, and are not exactly centred on zero. The last result corroborates the tendency of the ML algorithm to underestimate the spatial dependence parameters, as noted previously, for example, by Pace and Barry (1997) or Barry and Pace (1999). The graphs corresponding to IV estimates are blurred because of the presence of outliers; however, we can anticipate that these distributions are centred around zero and, as it is clear in the graphs, the dispersion of the estimates is very large.

Finally, Figs. 7 and 8 summarize the estimation of the terms in $\varvec{\varSigma }$; that is of $\sigma _{1}^2$, $\sigma _{2}^2$ and $\sigma _{12}$. The graphs show a collection of distributions with a similar shape for the ML and IV algorithms, although the dispersion of the second is larger. Overall, these distributions are centred on zero and confirm the consistency of the algorithms. However, there is a large risk of obtaining anomalous estimations for small and medium size samples, $n=25,49,100$, which only decreases significantly for samples of $n=400$ observations. It is obvious that estimation bias is skew to the range of positive values.

Fig. 3

Estimation bias. Slope coefficients

Fig. 4

Estimation bias. Intercepts

Fig. 5

Estimation bias. Theta terms

Fig. 6

Estimation bias. Lambda terms

Fig. 7

Estimation bias. Sigmas

Fig. 8

Estimation bias. Sigmas