Hedonic Housing Prices in Paris: An Unbalanced Spatial Lag Pseudo-Panel Model with Nested Random Effects

This paper estimates a hedonic housing model based on flats sold in the city of Paris over the period 1990-2003. This is done using maximum likelihood estimation taking into account the nested structure of the data. Paris is historically divided into 20 arrondissements, each divided into four quartiers (quarters), which in turn contain between 15 and 169 blocks (îlot, in French) per quartier. This is an unbalanced pseudo-panel data containing 156,896 transactions. Despite the richness of the data, many neighborhood characteristics are not observed, and we attempt to capture these neighborhood spill-over effects using a spatial lag model. Using Likelihood Ratio tests, we find significant spatial lag effects as well as significant nested random error effects. The empirical results show that the hedonic housing estimates and the corresponding marginal effects are affected by taking into account the nested aspects of the Paris housing data as well as the spatial neighborhood effects. JEL No. C31, C33, R21

1 Introduction This paper estimates a hedonic housing model based on ‡ats sold in the city of Paris over the period 1990-2003.The city is divided into 20 arrondissements which in turn are divided into four quartiers (quarters).The precise address of the property is de…ned within a block (îlot, in french).The number of îlots varies between 15 and 169 by quartier.The average number of îlots per quartier is 66.This is a nested structure which obviously comes from the fact that an îlot belongs to a speci…c quartier which in turn belongs to a speci…c arrondissement.We present our results for all ‡ats sold as well as by ‡at type.Real estate agencies and notaries in France use the following classi…cation: studio (or e¢ ciency), two room ‡ats (F2), three room ‡ats (F3), and four (or more) room ‡ats (F4).Despite its importance, there seems to be only few econometric studies on the real estate market in Paris, see Gravel et al. (1997) who focused on 33 municipalities in one suburb of Paris (département of Val-d'Oise).David et al. (2002) and Laferrère (2003) who constructed housing price indeces based on the value of …xed baskets of apartments or houses in some 300 geographic zones for France.Also Meese and Wallace (2003), Le Blanc and Lagarenne (2004) and Maurer et al. (2004).The last study used monthly data covering more than eighty thousands transactions of the french housing market over the period 1990-1999.More recently, Nappi-Choulet and Maury (2009) focused on two main business districts of Paris and its suburbs, while Fack and Grenet (2010) investigated how housing prices react to the quality of education o¤ered by neighboring public and private schools over the period 1997-2004.In an earlier paper using the same ‡at sales data set (but averaged for each quartier), Baltagi and Bresson (2011) illustrated the use of maximum likelihood estimators for panel seemingly unrelated regressions (SUR) with both spatial lag and spatial error (SARAR) components.This paper di¤ers in many respects from Baltagi and Bresson (2011).Most importantly, the data in the previous paper were averaged at the quartier level to get a panel of 80 quartiers over the period 1990-2003 with a total of 1120 balanced observations.In contrast, this paper is at the îlot level with an unbalanced panel containing 156; 896 transactions over the same period.The data is unbalanced in the sense that there are no sales observed for some years for a particular ‡at type even at the quartier level, see Table 1.Section 2 describes in details the unbalancedness in the data.The Baltagi and Bresson (2011) paper uses Seemingly Unrelated spatial MLE to estimate and test the hedonic housing model for ‡ats with 2, 3 and 4 or more rooms.It focuses on a balanced SURE spatial lag and spatial error (SARAR) model.Since it is at the quartier level, there are no nested e¤ects and clearly di¤erent in its estimation from this paper.In contrast, this paper extends the MLE approach developed by Antweiler (2001) for the unbalanced nested panel data model to the case where there are spatial neighborhood spill-over e¤ects incorporated using a spatial lag dependent variable.We work at the most disaggregated îlot level, and we emphasize the nested structure of the data.One added feature in this paper is that we allow the spatial weight matrix as well as the spatial coe¢ cients to vary over time, and we demonstrate that this is relevant to these types of transactions.
Using Likelihood Ratio (LR) tests, we …nd signi…cant spatial lag e¤ects as well as signi…cant nested random error e¤ects.Baltagi et al. (2001Baltagi et al. ( , 2002) ) derived Lagrange Multiplier and LR tests for the nested unbalanced random e¤ects models with panel data.The LR tests used in this paper are a natural extension to the spatial lag dependent variable model.We recognize that there is a large literature on hedonic housing (see for example, Griliches (1971), Rosen (1974), Halvorsen and Pollakowski (1981), Can (1992), Dubin (1992), Arguea and Hsiao (1993), Dubin et al. (1999), Fingleton (2008), Fingleton and Le Gallo (2008), to mention a few).Our paper emphasizes the importance of nested e¤ects in the Paris housing data as well as the spatial lag e¤ects.The empirical results show that the hedonic housing estimates and the corresponding marginal e¤ects are a¤ected by taking into account the nested aspects of the Paris housing data as well as the spatial neighborhood e¤ects.In fact, this paper shows that the impact of the adjacent neighborhoods becomes relatively small when one takes care of the nested random e¤ects.Section 2 describes this rich and unique Paris housing data and gives some summary statistics.Section 3 sets up the unbalanced spatial lag pseudo-panel data model with nested random e¤ects and derives the ML estimation under normality of the disturbances.Section 4 gives the empirical results, while section 5 concludes.

Data description
We have a unique data set on real estate transactions obtained from the data base "BIEN ", managed by the Notary Chamber of Paris which covers Ile-de-France, i.e. the city of Paris and the Paris region.Although the data for a particular sale is made on a voluntary basis, the rate of coverage is about 83% in the Paris region.Moreover, the database is anonymous, to comply with the French law.The dependent variable is the price in euros for each ‡at sold in an îlot for the period 1990-2003.For each transaction, we have detailed characteristics for this ‡at including surface, number of rooms and bathrooms, ‡oor level, whether it has a balcony, whether it has a garage, a maid's room, and time of construction.Also, its precise localisation (Lambert II grid coordinates).The precise address of the property gives us the îlot where the ‡at is located.Grouping the observations by îlots nested in quartiers which are in turn nested in arrondissements, we get an unbalanced pseudo-panel of 156; 896 real estate transactions.This is a pseudo-panel in the sense that these are not repeated sales on the same ‡at over time, but rather sales of ‡ats in a particular îlot over time.In fact, we do not observe repeated sales for the same ‡at over the period 1990-2003.The data is unbalanced in the sense that there are no sales observed for some years for a particular îlot (see Table 1).More formally, we estimate the following hedonic housing regression: where y taqif is the transaction price (in logs) for ‡at f in îlot i nested in quartier q which in turn is nested in arrondissement a at time t.The indices vary as follows: t = 1; :::; T ; a = 1; :::; N ; q = 1; :::; Q ta ; i = 1; :::; M taq ; and f = 1; :::; F taqi .For our data, T = 14 covering the period 1990-2003; N = 20 arrondissements, and we have sales for all ‡ats at this level of aggregation for all arrondissements for all years considered.This unbalancedness in the observed sales data becomes more serious at the îlot level.From Table 1, one notes that the minimum number of transactions is 6443 observed in 1992 and concerns 2360 îlots.The maximum number of transactions is 17089 observed in 1999 and concerns 3166 îlots.X taqif denotes the vector of K explanatory variables describing the characteristics for this ‡at including: surface (in m 2 ), surface squared, number of rooms (from 2 to 13), number of bathrooms, number of parking lots, number of maid's rooms, ‡oor level, balcony (yes or no), the kind of street (street, boulevard, avenue, place), time of construction (6 categories: < 1850, 1850-1913, 1914-1947, 1948-1969, 1970-1980, 1981-2003), distance (in meters) of the ‡at from the center of the quartier, distance (in meters) of the ‡at from the center of the arrondisse-ment.Unfortunately, some variables of interest like property taxes, crime rates, etc., are not available in this data set.
Put Tables 1 and 2 here Table 1 gives the number of quartiers and îlots in which we observe transactions for all ‡ats (and by ‡at type) for a given year.We dropped studios, so the reported statistics pertain to ‡ats with two rooms, three rooms and four to 13 rooms (hereafter F2, F3, F4m, respectively transactions.This explains the slight di¤erence between the descriptive statistics in the earlier Baltagi and Bresson (2011) paper and the descriptive statistics in Table 2 for this paper.For all transactions, the mean price is about 183; 062 euros 1 , ranging from 50; 000 euros to 3:354 million euros.The mean price per square meter is about 2; 894 euros with a large standard deviation (1; 151 euros).The mean surface of the ‡ats is around 60 m 2 , ranging from 20 m 2 to 400 m 2 : In fact, 29% have a surface less than 40 m 2 ; 33% have a surface between 41 60 m 2 ; 18% have a surface between 61 80 m 2 ; 10% have a surface between 81 100 m 2 ; and 8% have a surface of more than 100 m 2 .The average number of bathrooms is less than one.In fact, 23% of the ‡ats have no bathrooms; 71% have one bathroom; 5% have two bathrooms; and 0:3% have three or more bathrooms.More than 94% of the ‡ats have no maid's room; 4% have one maid's room; and less than 1% have two or more maid's rooms.98% of the ‡ats have no balcony, more than 82% have no garage plot; 16% have one garage plot; 1% have two garage plots; and 0:04% have three or more garage plots.The maximum number of ‡oors is 30 but 95% of the ‡ats are between the ground ‡oor and the 7th ‡oor.Around 5% of the ‡ats were built before 1850; 48% of the ‡ats were built between 1850 1913; less than 15% of the ‡ats were built between 1914 1947; about 14% of the ‡ats were built between 1948 1969; less than 12% of the ‡ats were built between 1970 1980; and only 2:3% were built between 1981 2003.Around 9:7% of the ‡ats are located on an Avenue, 7:7% of these ‡ats are on a Boulevard, 0:8% of these ‡ats are on a Place and the majority 82% of these ‡ats are on a street.The average distance to the center of an arrondissement is around 824 m compared to 429 m to the center of a quartier.Table 2 also shows the descriptive statistics by ‡at type.In fact, the average price for an F2 ‡at is 112; 468 euros, compared to 174; 956 euros for F3 and 345; 241 euros for F4m.The average price per square meter is 2775 euros for F2 compared to 2822 euros for F3 and 3248 euros for F4m.The average square meters are 40 for F2 compared to 61 for F3 and 103 for F4m.The average number of bathrooms is 0:72 for F2 compared to 0:78 for F3 and 1:10 for F4m.The average number of maid's rooms is 0:01 for F2 compared to 0:03 for F3 and 0:21 for F4m.The average number of garage plots is 0:12 for F2 compared to 0:19 for F3 and 0:32 for F4m.1:7% of the F2 ‡ats had a balcony compared to 2:2% for F3 and 3:5% for F4m.Around 5:8% of the F2 ‡ats were built before 1850 compared to 4:7% for F3 and F4m.49:4% of the F2 ‡ats were built between 1850 1913 compared to 48:9% for F3 and 45:5% for F4m.16:7% of the F2 ‡ats were built between 1914 1947 compared to 13:8% for F3 and 12:2% for F4m.Only 2:4% of the F2 ‡ats were built between 1981 2003 compared to 2:0% for F3 and 2:5% for F4m.Around 7:6% of the F2 ‡ats are located on an Avenue compared to 9:7% for F3 and 13:2% for F4m.Around 5:9% of the F2 ‡ats are located on a Boulevard compared to 8:2% for F3 and 10:8% for F4m.Around 0:7% of the F2 and F3 ‡ats are located on a Place compared to 1% for F4m.The majority (85:8%) of the F2 ‡ats are located on a street compared to 81:3% for F3 and 75:1% for F4m.The average distance of an F2 ‡at to the center of an arrondissement is 802m compared to 824m for F3 and 838m for F4m.The average distance of an F2 ‡at to the center of a quartier is 422m compared to 429m for F3 and 434m for F4m. Figure 1 gives the average prices per square meter for all ‡ats (and by ‡at type) by year.The average price of ‡ats seems to follow a J-shape curve.We observe a decrease from 1990 to 1997 and a boom after.This downswing followed by an upswing are more pronounced for the larger ‡ats (F4m).Between 1990 and 1997, housing prices dropped by 24% in constant euros for all districts of Paris for all transactions.This is compared to 19% for F2 ‡ats, 25% for F3 ‡ats, and 34% for F4m ‡ats.In contrast, the period 1998 -2003 exhibited rising prices in Paris: 53% in constant euros for all districts of Paris for all transactions.This is compared to 47% for F2 ‡ats, and 60% for F3 and F4m ‡ats.By 2003, prices were almost 19% higher than they were before the property crash of 1991.Table 3 gives quantiles of the number of transactions per îlot, quartier, and arrondissement, per year.In addition, it reports the quantiles of the number of transactions per year.The average number of transactions per îlot per year is 4 compared to 140 per quartier per year, and 560 per arrondissement per year.The minimum number of transactions per îlot per year is 1, while the maximum number is 103 per year.This is compared to a minimum of 1 per quartier per year and a maximum of 787 per quartier per year.The corresponding minimum number of transactions per arrondissement for a particular year is 34, while the maximum number is 1956 per arrondissement per year.The minimum number of transactions per year is 6; 443 while the maximum number of transactions per year is 17; 098.The distribution of the number of transactions for the three kinds of ‡ats is quite similar.
Put Figure 1 and Table 3 here 3 Unbalanced spatial lag pseudo-panel model with nested random e¤ects The real estate literature emphasizes the importance of neighborhoods in determining the value of a house or a ‡at.While one can try and include as many of the neighborhood characteristics in the regression to capture these e¤ects, most attempts may fall short as many neighborhood characteristics are not observed as in our case.For the importance of spatial e¤ects in housing studies, the reader can refer to Helpman (1998), Fingleton (2008), Fingleton and Le Gallo (2008) and Glaeser (2008) to mention a few.And for spatial econometric methods, the reader can also refer to Anselin (1988), Anselin and Bera (1998), Elhorst (2003), Anselin et al. (2008), andBaltagi (2011) to mention a few.One simple method of capturing the e¤ect of neighbors prices is to use a spatial lag regression equation: where t = 1; :::; T ; a = 1; :::; N ; q = 1; :::; Q ta ; i = 1; :::; M taq ; and f = 1; :::; F taqi .This adds the spatial lag variable y e taqif to the hedonic housing equation considered above.The spatial lag coe¢ cient t may be time varying or constant over time.This spatial lag variable is de…ned as: where w taqip denote the elements of the spatial weights matrices W t which vary with t.These matrices may be large (for instance: (17; 098 17; 098) for year 1999 or (15; 261 15; 261) for year 2000, see Table 1).We employ the contiguity relations in de…ning a neighbor.Unfortunately, the use of a distance weight matrix caused memory problems with such a huge (156; 896 156; 896) W matrix for all the transactions over the all 14 years.The Lambert II grid coordinates allow us to compute contiguity, re ‡ecting the relative position in space of one unit with respect to another unit.Two ‡ats are neighbors (with 1 assigned to a neighbor) if they share a common edge; otherwise, they are not (with 0 assigned to a non-neighbor).Since the mean number of transactions per quartier is around 140 (see Table 4), we consider …rst-order contiguity matrices for the 140 nearest neighbors (i.e.nearest sold ‡ats).We also experimented with the 10 nearest neighbors and the results are available in the online supplement.These contiguity matrices are row-normalized, so that each row sums to 1.In this case, the spatial contiguity matrices are sparse.They are …lled with at least 16; 439 nonzero elements for 1991 and 26; 680 nonzero elements for 1999.Figure 2, which shows the connections between the 140 nearest transactions for 4 speci…c years, con…rms the necessary de…nition of speci…c weight matrices W t per year associated with …xed time e¤ects in order to capture the downswing and upswing of housing prices during the period 1990-2003.
Put Figure 2 here The disturbance term is given by: where ta is the arrondissement e¤ect; taq is the quartier e¤ect naturally nested in the respective arrondissement, and taqi is the îlot e¤ect naturally nested in the respective quartier.These could be …xed or random.
The remainder disturbance term for the particular ‡at is random " taqif IIN (0; 2 " ) : For the random speci…cation, we assume that ta IIN 0; 2 , taq IIN 0; 2 , and taqi IIN 0; 2 (5) independent of each other and among themselves.This unbalanced panel is made up of N = 20 top-level arrondissements, each containing Q ta secondlevel quartiers.The second-level quartiers in turn contain M taq third-level îlots, which contain the innermost F taqi observations on ‡ats.P The number of observations in the higher level groups are The number of top-level groups is N T; the number of second-level groups q=1 M taq : Following Antweiler ( 2001), we use block-diagonal matrices of size (H H) corresponding in structure to the groups or subgroups they represent.They can be constructed explicitly by using "group membership"matrices consisting of ones and zeros that uniquely assign each of the H observations to one of the G (or L or N T ) groups.
Let R be such an (H G) matrix corresponding to the innermost group level.Then the block-diagonal (H H) matrix J can be expressed as the outer product of its membership matrices: J = R R 0 .The inner product R 0 R produces a diagonal matrix L e of size (G G) which contains the number of observations of each group.Similarly, let R be such an (H L) matrix corresponding to the secondlevel groups.Then the block-diagonal (H H) matrix J can be expressed as the outer product of its membership matrices: J = R R 0 .Last, let R be such an (H N T ) matrix corresponding to the top-level groups.Then the block-diagonal (H H) matrix J can be expressed as the outer product of its membership matrices: J = R R 0 .If we pool the observations, the log-likelihood is given by: where W is the spatial weight matrix of size (H H) ; W t is the spatial weight matrix2 of size (F ta F ta ) changing at each time period t.
is the spatial lag matrix of size (T T ) whose elements t change at each time period t: I H is an identity matrix of size (H H).The variance-covariance matrix of the disturbance is de…ned as follows: with Extending the derivations of Antweiler (2001) to the case of the spatial lag model (2), we get: (10) where I t is an identity matrix of size (F ta 8 F ta ) and where p=1 w taqip y taqip X taqif .A gradient of this log-likelihood function (10) can be obtained analytically (see the appendix), but it can also be obtained through numeric approximation.In carrying out this maximization, it is necessary to constrain the optimization such that j t j < 1; the variance 2 " remains positive, and that the variance ratios , and remain non-negative.

Empirical Results
We run ML estimation of (i) the random e¤ects (RE) model ignoring the nested e¤ects, and compare it with (ii) the nested RE (N RE) model.Both (i) and (ii) ignore the spatial lag e¤ect.Model (iii) estimates a spatial lag (S) model ignoring the random e¤ects, while model (iv) estimates a spatial nested (SN ) RE model.First, we run these models on the whole 156; 896 transactions for all ‡ats, and then we run these models by ‡at type (F2, F3, F4m)3 over the period 1990 2003: We also run models (iii) and (iv) again allowing the spatial lag e¤ect to vary by year.These models are denoted by (S t ) and (SN t ), respectively.The following Table summarizes our models4 : More bathrooms or maid's rooms or garage plots increase the value of a ‡at as expected.The larger the distance to the center of a quartier or arrondissement, the lower the price, especially for the RE quartier or arrondissement models.Time dummies also con…rm the observed J-shape curve of the housing prices with a decrease from 1990 to 1997 and a boom after.Table 5 gives the MLE results of the unbalanced nested random e¤ect models for all transactions.The nested random e¤ects model for îlot and quartier levels yields a log-likelihood of ( 8946:68) which is lower than the one for the fully nested îlot, quartier and arrondissement RE model ( 8393:30).The LR test (H 0 : = 0) rejects the null of zero arrondissement e¤ects in the nested error structure, once îlot and quartier are taken into account.Once again, the estimated coe¢ cients have the right sign and are highly signi…cant.The estimated nested random e¤ects are b (= 0:187), b (= 0:131) and b (= 0:768) which represent 8:9%, 6:2% and 36:8% of the total variance.Table 6 gives the MLE results of the unbalanced spatial lag model and the unbalanced spatial nested random e¤ect model for all transactions.We have used the 140 nearest neighbors as it is the average number of transactions per quartier (see Table 3).The spatial lag model without nested e¤ects yields a much smaller log-likelihood value ( 42246:42) as compared to the previ-ous models estimated.The spatial lag coe¢ cient yields an estimate of 0:137 which is signi…cant.The spatial nested speci…cation has a better …t of the housing prices with a higher log-likelihood ( 8378:51) but a much smaller estimated spatial coe¢ cient (0:025) which is still statistically signi…cant.The LR test (H 0 : = = = 0) rejects the null of zero nested random error structure.The nested random e¤ects model seems to dominate the spatial lag e¤ect in terms of maximum log-likelihood values.In other words, once you take care of the nested design in the data set, the marginal gain from the spatial lag e¤ect is minimal for our Paris housing data.As a further diagnostic check, we performed the Pesaran (2013) CD test to check the cross-section dependence of the residuals for the various models estimated. 5As we have an unbalanced pseudo-panel of 156; 896 observations, we cannot directly apply the CD test on the estimated residuals.So, we computed the averages of the estimated residuals at the quartier level.In this case, we get a balanced panel of residuals with 1120 (= 80 14) observations.The CD test on these residuals was computed for the spatial lag model, for the nested random e¤ects model and for the spatial lag with nested random e¤ects model.Results are given at the bottom of Tables 5 and  6.These show that the assumption of no cross-section dependence on the estimated residuals is rejected for the spatial lag model: CD = 4:907 (p-value = 0:000).But when we consider the nested random e¤ects model, the null of no cross-section dependence is not rejected at the 5% level (CD = 1:932, p-value = 0:062) and for the spatial lag with nested random e¤ects model (CD = 1:734, p-value = 0:088).Table 7 gives the MLE results of the unbalanced spatial lag model and the unbalanced spatial nested random e¤ect model allowing the spatial lag coef-…cient to vary over time.For the spatial non nested speci…cation, we note that the estimates of t vary between 0:048 ( for the year 1997) and 0:225 (for the year 2000).We …nd again the J-shape curve of the housing prices with a decrease from 1990 to 1997 and a boom after.Once again, the LR test for this time-varying spatial model rejects the null hypothesis of ignoring the three-level nested error structure (H 0 : = = = 0).Moreover, the LR tests rejects the null hypothesis of a constant spatial lag coe¢ cient (H 0 : t = ) in favor of time varying spatial lag coe¢ cients.Note that some values of t are not signi…cantly di¤erent from zero, with the maximum es-timate of t being 0:07 in the year 2000.Figure 3 shows the time varying di¤erences between the spatial lag coe¢ cients estimated with and without the nested random e¤ects.This …gure clearly shows the dramatic drop in the magnitude of the spatial lag e¤ects when we take care of the nested structure.Whether it is a constant or varying spatial lag model, the unbalanced spatial nested random e¤ects speci…cation in Tables 6 and 7 always show a large improvement in the value of the maximum likelihood than the corresponding model that ignores the nested speci…cation 6 .
Put Tables 4 to 7 here Put Figure 3 here Tables Online_9, Online_10 and Online_11 (provided in the online supplement) give the marginal e¤ects for all transactions for the various estimators considered.Seven explanatory variables in our hedonic housing equation are continuous and the remaining nine are dummy variables.For the non-spatial speci…cations, the marginal e¤ect for a continuous attribute X k is given by the estimated coe¢ cient b k .The marginal e¤ect expressed in euros and computed at the mean value of the b ‡at price is then k :y while that for a dummy variable is computed as exp b k 1 :y.For the spatial speci…cations with one spatial lag coe¢ cient , the marginal e¤ects are decomposed into direct, indirect and total marginal e¤ects.In our case, the matrix of partial derivatives of the dependent variable with respect to the k th explanatory variable is given by: 6 Results for the unbalanced spatial model with and without nested random e¤ects but with constant for the F2 (resp.F3 and F4m) ‡ats transactions are available in the online supplement.These results again show that the spatial nested speci…cation has always a better …t with the improvement in the likelihood value coming largely from taking care of the nested e¤ects rather than the spatial lag e¤ect.LeSage and Pace ( 2009) de…ne the direct e¤ect as the average of the diagonal elements of the matrix on the right-hand side of ( 15), and the indirect e¤ect as the average of either the row sums or the column sums of the non-diagonal elements of that matrix.The total marginal e¤ect is then the sum of the direct and indirect e¤ects.
In order to draw inference regarding the statistical signi…cance of the direct and indirect marginal e¤ects, LeSage and Pace (2009) suggest simulating over 1000 draws of the distribution of these e¤ects using an upper-triangular Cholesky decomposition of the variance-covariance matrix implied by the maximum likelihood estimates.In order to avoid me mory problems with the direct inversion of a huge matrix as I b H W , we follow LeSage W will allow for spatial spillovers only within the blocks.At each time period (t = 1; 2; : : : ; T ) spillovers can impact neighboring îlots (the smallest neighborhood group used in the nested structure).So, each diagonal block (I b 1 Fta W t ) of this matrix contains information for the all transactions at the îlot level in one year and allows us to compute the direct, indirect and total marginal e¤ects.The compu tation of the marginal e¤ects do not require the inversion of the I b H W matrix for every parameter combination drawn from the variance-covariance matrix of the ML estimates7 .The overall direct, indirect and total e¤ects can be approximated by computing the mean values over these 1000 draws.Tables Online_9 and  Online_10 (provided in the online supplement) give the direct, indirect and total marginal e¤ects for all the explanatory variables obtained from (15).From these tables, we can see that the marginal spillover e¤ects due to the neighbors are negligible relative to the direct e¤ects.For instance in Table Online_9, the direct e¤ects of the number of bathrooms (0:117) represent 86:26% of the total e¤ects.This percentage is even bigger (97:44%) for the spatial lag with random e¤ects model (see Table Online_10).Some marginal e¤ects expressed in euros and computed at the mean value of the ‡at price are given in Table Online_11.For example, using the results for the spatial lag model in Table Online_9, the total e¤ects on the price per square meter is 349 euros (= 0:117 2984) for the number of bathrooms.Since the indirect e¤ects are negligible when compared with the total e¤ects, we only give the total marginal e¤ects expressed in euros.Table Online_11 (provided in the online supplement) gives these (total) marginal e¤ects for all transactions based on the various estimators considered: random e¤ects, nested random e¤ects, spatial lag and spatial nested speci…cations, respectively8 .Whatever the attribute, we note large di¤erences between the marginal effects computed from the spatial speci…cation and those computed from the other random e¤ects speci…cations.The empirical results show that the marginal e¤ect for a speci…c housing characteristic is lower on average once the nested e¤ects are taken into account.For the spatial nested speci…cation, the marginal e¤ect of one extra bathroom is estimated to be, on average, 293 euros per square meter (hereafter e.s.m).For one extra maid's room, the marginal e¤ect is around 254 e.s.m.For an extra garage, the marginal e¤ect is 326 e.s.m.The marginal e¤ect of having a balcony is 385 e.s.m.If the property is located on an avenue or a place (as compared to the reference location which is the street), the marginal e¤ect is 118 e.s.m and 184 e.s.m, respectively.If the distance to the center of the quartier is increased by 100 m, the marginal e¤ect is estimated to be 17 e.s.m less on average, while the similar marginal e¤ect for an arrondissement is 13 e.s.m.Since we are using unmatched housing data, we also ran the average transactions at the îlot level.The number of transactions are now reduced from 156; 896 to 39; 057 over 14 periods.The results are very similar when we use averages at the îlot level of the individual transactions, but of course with slightly di¤erent magnitudes9 For example, the spatial e¤ect estimate for is of a higher magnitude when we work on averages at the îlot level (0:233 for the spatial lag model, and 0:054 for the spatial lag with nested random effects model).This should be compared in Table 6 with 0:136 for the spatial lag model for the individual transactions, and 0:025 for the spatial lag with nested random e¤ects model.But, the direct, indirect and total marginal e¤ects are very similar to those found for the individual transactions cases.This reassuring …ndings con…rm the stability of our results when we average at the îlot level.Note that the aggregation tends to slightly over-estimate the impact of the neighborhood without signi…cantly altering the impact of the ‡ats characteristics on the price level per square meter.

Conclusion
This paper uses a unique data set on real estate transactions for the city of Paris over the period 1990-2003.We estimate a hedonic housing model using maximum likelihood estimation taking into account the nested structure of the data.This is done by extending the MLE approach developed by Antweiler (2001) for the unbalanced nested panel data model to the case of a spatially lag dependent variable.The latter is used to capture spatial neighborhood spill-over e¤ects which are important in determining housing prices.Using LR tests, we …nd signi…cant spatial lag e¤ects as well as sig-ni…cant nested random error e¤ects.Our paper emphasizes the importance of nested e¤ects in the Paris housing data as well as the spatial lag e¤ects.In fact, this paper shows that the impact of the adjacent neighborhoods becomes relatively small when one takes care of the nested random e¤ects.In addition, we show that due to the unbalanced pseudo-panel aspect of these transactions, one should allow the spatial weight matrix as well as the spatial lag coe¢ cients to vary over time, and that the likelihood ratio tests con…rm that they …t the Paris housing data better.Future research should be concerned with endogeneity of some of the right hand regressors.For example, endogeneity of ‡oor space and price.See, Fingleton and Gallo (2008) who dealt with the simultaneous determination of house prices and the quantities of the attributes.

APPENDIX: derivation of the gradient of the log-likelihood function
The log-likelihood of the spatial lag regression equation for an unbalanced pseudo-panel data model is given by: " where I t is an identity matrix of size (F ta F ta ) and where p=1 w taqip y taqip X taqif = y taqif t y e taqif X taqif .The analytic gradient r ln l is de…ned by: 0 1 @ ln l 0 ; :::; @ ln l B @ C B 1 @ T B @ ln l ; ::: Antweiler (2001), let us de…ne: where X k;taqif is the k th regressor associated with the observation f in the bottom-level group i belonging to the second-level group q which belongs to the top-level group a.Let us de…ne: Extending the derivation of the gradient to the spatial lag case, we get: , k = 1; :::; K @ 2 > :

AKNOWLEDGEMENTS
We would like to thank Werner Antweiler for his help with SAS IML codes and Annick Vignes (Université Paris II / Sorbonne Universités) for providing us with the data on housing prices in Paris.The data were obtained within the framework of a research contract and are not available for public use.We would like to also thank participants at the Conference on Cross-Sectional Dependence in Panel Data Models (Cambridge University, UK, May 30-31, 2013) and at 19th International Conference on Panel Data (Centre for Econometric Analysis, Cass Business School, London, UK, July 4-5, 2013) for their remarks.Many thanks to the Editor and three anonymous referees for their helpful comments.The usual disclaimer applies.1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 spatial lag spatial lag with nested random effects ::: and store the traces of the matrices I up to and including W 100 : This means that the e¤ects embodied in this decompostion of I b 1 H

Figure 1 -
Figure 1 -Average transactions prices per square meter (in euros) ).The data set includes 156; 896 sales transactions for all ‡ats, of which 73; 945 sales transactions are for F2 ‡ats, 48; 348 sales transactions are for F3 ‡ats, and 34; 603 sales transactions are for F4m ‡ats.Table 2 gives gives some descriptive statistics for the entire 156; 896 transactions and by ‡at type.The raw data set contains 166,362 transactions.After cleaning it from outliers and missing observations, we are left with 156, 896

Table 4
gives the MLE results of the unbalanced random e¤ect (RE) models for all 156; 896 transactions.The …rst column estimates a RE model at the îlot level.The second column estimates a RE model at the quartier level and the last column estimates a RE model at the arrondissement level.The estimated random e¤ects at the îlot level is b (= 0:82) which attributes 45% of the total variance to heterogeneity across the îlots.Similarly, the estimated random e¤ects at the quartier level is b (= 0:79) which attributes 44% of the total variance to heterogeneity across the quartiers.While the estimated random e¤ects at the arrondissement level is b (= 0:53) which attributes 38% of the total variance to heterogeneity across the arrondissements.In order to check the correlation between the random e¤ects and the explanatory variables, we perform Hausman tests.The resulting Hausman test for the îlot RE model is 31:9484 (p-value = 0:2766), while that for the quartier RE model is 28:3545 (p-value = 0:4458); and that for the arrondissement RE model is 3:5463 (p-value = 1:000): These are not signi…cant and we do not reject the null hypothesis of no correlation between the random e¤ects and the explanatory variables.In other words, our data does not reject the RE estimators.The regression results all have the right sign and are signi…cant.

Table 1 -
-Number of quartiers, îlots and transactions per year

Table 3 -
-Number of transactions per îlot, quartier, arrondissement and year

Table 6 -
-Unbalanced spatial and unbalanced spatial nested models (all transactions, 1 spatial lag, 140 neighbors) (S versus NS) : unbalanced spatial model versus unbalanced spatial nested model (N versus NS) : unbalanced nested model versus unbalanced spatial nested model