CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS

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1 HEALTH ECONOMICS, VOL. 6: (1997) HEALTH ECONOMICS LETTERS CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS RICHARD BLUNDELL 1 AND FRANK WINDMEIJER 2 * 1 Department of Economics, University College London and Institute for Fiscal Studies, London, UK 2 Institute for Fiscal Studies, London, UK SUMMARY For small group sizes, the GLS estimator in multilevel models is biased and inconsistent when the random cluster effects are correlated with the regressors. A fixed effects approach, conditioning on the cluster effects, provides consistent estimates for the slope parameters. The two estimators are equivalent when group sizes are large. The same results obtain for two-stage estimation procedures that allow for some of the regressors to be simultaneously determined with the dependent variable. The GLS and fixed effects estimators are applied to data on acute care hospital utilization in the UK, allowing for health authority district effects by John Wiley & Sons, Ltd. Health Econ. 6: (1997) No. of Figures: 0. No. of Tables: 1. No. of References: 7. KEY WORDS multilevel model; correlated group effects; group size; simultaneity INTRODUCTION Consider the simplest two-level model (see, e.g., Goldstein 1 ), the variance components model, defined as y ij = κ + x' ij β + v ij (1) v ij = α j + ε ij where y ij is the dependent variable and x ij is a K-vector of covariates, i = 1,, N j is the level 1 unit, contained in j, the level 2 unit, with j = 1,, M. The total number of observations is N = M j=1 N j. The ε ij are i.i.d. (0, σ 2 ε). For the random effects specification, the α j are i.i.d. (0, σ 2 α) and independent of x ij. For this specification, the generalized least squares (GLS) estimator is the efficient unbiased estimator. However, when x ij and α j are not independent, the GLS estimator is biased and inconsistent for M, when group sizes N j are small. An unbiased and consistent estimator for β in this case can be obtained by conditioning on the α j, i.e. treating them as fixed effects. When group sizes N j are large, the two estimators are equivalent. In some specifications, covariates are included that are considered to be simultaneously determined with the dependent variable. Consider the recursive error components two-equation model y 1ij = y 2ij γ 1 + x' 1ij β 1 + α 1j + ε 1ij (2) y 2ij = x' 1ij π 1 + x' 2ij π 2 + ρα 1j + α 2j + ε 2ij (3) in which the ρα 1j allows for correlation to occur at the level of the district effects and E(ε 1ij ε 2ij ) 0 *Correspondence to: F. Windmeijer, Institute for Fiscal Studies, 7 Ridgmount Street, London WC1E 7AE, UK. Contract grant sponsor: ESRC Research Centre, Institute for Fiscal Studies and King s Fund CCC /97/ $17.50 Received 20 December by John Wiley & Sons, Ltd. Accepted 21 March 1997

2 440 R. BLUNDELL AND F. WINDMEIJER allows for simultaneous feedback to occur between the random disturbances ε 1ij and ε 2ij. For ease of exposition, these equations are specified without an intercept. Consistent estimation of the parameters is discussed for the cases that the group specific effects α are or are not correlated with x. Again, for large group sizes, the two estimators are equivalent. ESTIMATION Let the observations be stacked groupwise, then the random effects model (1) can be written as y = X ι ( β ) + v (4) where X ι = {1 x ' ij} with E(v) = 0, and the variance of v is a block-diagonal matrix Ω with the jth block given by Ω j = σ α 2 J Nj + σ ε 2 I Nj where J is a matrix of ones and I is the identity matrix. If E(X ' ιv) = 0, then the OLS estimator of (κ β')' in model (4) is unbiased and consistent but not efficient. The efficient GLS estimator is given by ( ˆ GLS ˆβ GLS ) = (X ι 'Ω 1 X ι ) 1 X ι 'Ω 1 y which is equivalent to pre-multiplying both sides of equation (4) by σ ε Ω 1/2 and estimating the transformed model by OLS. The matrix σ ε Ω 1/2 is block-diagonal with the jth block given by 2 σ ε Ω j 1/2 = Q Nj + θ j 1/2 J Nj (5) where J Nj = J Nj /N j, Q Nj = I Nj J Nj and θ j = σ ε 2 σ ε 2 + N j σ α 2 (6) The unknown variances of α and ε have to estimated consistently in order to obtain a feasible GLS estimator. Health Econ., 6, (1997) An alternative to the random effects is the dummy variable specification. The model is the same in equation (1), but now the α j are fixed constants or, otherwise stated, the model is conditional on α j : E(y ij x ij, α j ) = κ + x' ij β + α j The equivalent of model (4) is y = Xβ + Dα* + ε (7) where α* = (α 1 + κ,, α M + κ)', and D is the matrix of dummy variables for the M groups. In principle, α* and β can be estimated directly by OLS; in practice, however, the number of estimated group effects may become too large to make estimation feasible. The [dummy variable (DV)] OLS estimator of β in model (7), however, is given by ˆβ DV = ( X' X) 1 X'ỹ where X and ỹ are X and y, taken in deviation from the group means. This transformation is equivalent to premultiplying both sides of equation (7) by the symmetric indempotent and blockdiagonal matrix Q, with jth block Q Nj as defined in equation (5), so ỹ = Qy; X = QX ˆβ DV = (X'QX) 1 X'Qy As mentioned above, an important reason to specify the effects α as fixed rather than random is when these effects are correlated with the regressors X. If this is the case and group sizes are relatively small (this is equivalent to a small number of time periods, T, in panel data applications 3 ), the GLS estimator for β will be biased and inconsistent when the number of groups M, whereas the dummy variable (DV) estimator is unbiased and consistent. However, when group sizes are large, the two estimators for β are equivalent. This occurs because when for every j, N j, θ j 0, with θ j as defined in equation (6). From equation (5), it then follows that σ ε Ω j 1/2 Q Nj so the GLS transformed model is equivalent to the fixed effects transformed model, and ˆβ GLS ˆβ DV. Note that κ cannot be estimated 1997 by John Wiley & Sons, Ltd.

3 CLUSTER EFFECTS IN MULTILEVEL MODELS 441 by GLS when the N j, as X' ι Ω 1 X ι becomes singular. When group sizes are small and the group effects are correlated with the regressors, estimates of the variance components in the GLS model based on residuals will be inconsistent for M. In this case the DV estimator can provide information about σ 2 α. There are various ways of utilizing the residuals of the fixed effects model to estimate σ 2 α. Specifying the residuals as e = y ı ˆδ X ˆβ DV where ı is an N-vector of ones, ˆδ = ȳ x' ˆβ DV is an estimator for the constant, where ȳ = (1/N) Σ j Σ i y ij and x = (1/N) Σ j Σ i x ij, then an estimator for σ 2 α is defined by 2,4 ˆσ α 2 = e'pe {M 1 + tr[(x'qx) 1 X'PX] tr[(x'qx) 1 X' J N X]} N j N j 2 /N where P = I N Q and tr is the trace operator. The analysis and conclusions are the same when a random coefficient on a regressor is considered. If the model is specified as y ij = x ' ijβ + z ij (δ + α j ) + ε ij then Q Nj = I Nj z j (z' j z j ) 1 z ' j, θ j = σ ε 2 /[σ ε 2 + (z' j z j )σ α 2 ] and, when the N j and z ' jz j, ˆβ GLS is the same as the OLS estimator for β in the model y = Xβ + Z*δ* + ε where Z* is the extended matrix with the z j, and δ* is the M-vector of parameters (δ + α j ). Simultaneous equations In the error components simultaneous model (2) with E(ε 1ij ε 2ij ) 0, estimation by either GLS or DV will lead to inconsistent results due to the correlation of the explanatory variable y 2ij with both α 1j and ε 1ij. The GLS transformed version of equation (2) is given by Ω 1 1/2 y 1 = Ω 1 1/2 y 2 γ 1 + Ω 1 1/2 X 1 β 1 + Ω 1 1/2 v 1 (8) where Ω 1 is the variance matrix of v 1, a block diagonal matrix with the jth block of σ ε1 Ω 1/2 1 as given in equation (5). Let X = [X 1 X 2 ], the generalized two-stage least squares (G2SLS) estimator is then obtained by first estimating Ω 1/2 1 y 2 by Ω 1/2 1 X ˆπ, with ˆπ = (X'Ω 1 1 X) 1 X'Ω 1 1 y 2. The second stage is to substitute this estimate for Ω 1/2 1 y 2 in equation (8) and estimate this equation by OLS. Denoting Z 1 = [y 2 X 1 ] and δ 1 = (γ 1 β' 1 )', the resulting G2SLS estimator is given by ˆδ 1G2SLS = [Z ' 1Ω 1 1 y(x'ω 1 1 X) 1 X'Ω 1 1 Z 1 ] 1 Z ' 1Ω 1 1 X(X'Ω 1 1 X) 1 X'Ω 1 1 y 1 Again, ˆπ will be unbiased and consistent, and ˆδ 1G2SLS is consistent when group sizes are small only when the group effects are uncorrelated with X. Note that ˆδ 1G2SLS is not unbiased due to the correlation of Z 1 with y 1. The dummy variable two-stage least-squares (DV2SLS) estimator is obtained by premultiplying equation (2) by Q as defined above: Qy 1 = γ 1 Qy 2 + QX 1 β 1 + Qε 1 (9) The first stage estimates Qy 2 by QX ˆπ DV, with ˆπ DV = (X'QX) 1 X'Qy 2 a consistent estimator for π. This estimate is then substituted for Qy 2 in equation (9), which is subsequently estimated by OLS. The resulting DV2SLS estimator for δ 1 is given by ˆδ 1DV2SLS = and [Z ' 1QX (X'QX) 1 X'QZ 1 ] 1 Z ' 1 QX (X'QX) 1 X'Qy 1 plim M ˆδ 1DV2SLS = δ 1 Again, for large group sizes the two estimators will be equivalent using exactly the same arguments as before. If for every group j, N j, then σ ε1 Ω 1 1 Q, and plim {Nj j} ˆδ 1G2SLS = plim {Nj j} ˆδ 1DV2SLS = δ 1. AN EXAMPLE To illustrate the GLS and dummy variable OLS estimators when group sizes are reasonably large, 1997 by John Wiley & Sons, Ltd. Health Econ., 6, (1997)

4 442 R. BLUNDELL AND F. WINDMEIJER Table 1. Regression results OLS Multilevel DVOLS Weighted regression (R 2 =0.4578) Multilevel GLS (R 2 =0.4569) (dependent variable UTILW) B SE B B SE B B SE B CONST OLDALONE S. CARER UNEMP HSIR SMR The GLS results of the multilevel model are taken from Carr-Hill et al. 6 (Table 6.5, p. 93). The number of observations for the two OLS regressions is 4955 and for the GLS regression As the observations are weighted by the population size of the wards, the fixed effects transformed variables are taken in deviations from their weighted sample means. For further details, see Blundell et al. 7 we apply these estimators to the data used by Smith et al. 5 for developing a formula for the regional distribution of the UK National Health Service (NHS) revenues. The final specification in Smith et al. s study 5 was estimated by multilevel GLS. We briefly describe the variables used in the regression; for a more detailed description of the data and modelling procedures, see Carr-Hill et al. 6 The dependent variable is the NHS hospital utilization variable for acute specialties. This is the standardized estimated costs per ward, UTILW, a ward having an average population of These estimated costs are standardized by the expected costs per ward, the expectation being with respect to age and sex. The dependent variable in the models estimated below is the average of two years of data, / The explanatory ( needs ) variables are health and socio-economic factors. The model includes standardized mortality rates, SMR074, and standardized limiting long-term illness rates, HSIR074. Further needs variables included in the model are the proportion of those of pensionable age living alone, OLDALONE, the proportion of dependants in single carer households, S_CARER, and the proportion of the economically active that are unemployed, UNEMP. The total number of observations at ward level available for estimation is The level 2 units are District Health Authorities (DHA). There are 186 DHA districts. The average number of wards in a district is 26.6, the median is 24, the minimum size is 6 and the maximum size is 95. The estimation results are presented in Table 1. The multilevel GLS and dummy variable OLS estimates are practically the same. Compared with the standard OLS results, there are two main differences. The coefficient on the living alone variable, OLDALONE, is twice as large in the standard OLS model as compared with the multilevel specifications, and the coefficient on the single carer household variable, S_CARER, is insignificant for standard OLS, but becomes significant and much larger in the multilevel models. (When a correction for the intra distinct variance is made, the OLS standard errors are about twice as large, e.g. the standard error for S_CARER becomes ) Another difference is the lower coefficient on the mortality rate variable, SMR074, for the latter two models. These differences indicate that the district effects are correlated with the needs variables and that, to avoid bias, the model should be estimated by the dummy variable OLS method. Because the group sizes are fairly large, the multilevel GLS estimation procedure gives results almost identical with those from the multilevel DVOLS model. However, as we have shown, the DVOLS estimator avoids bias even for small group sizes. ACKNOWLEDGEMENTS We are grateful to Harvey Goldstein for helpful comments and Peter Smith and Stephen Martin for providing the data on the health care utilization used in this study. The financial support of the King s Fund and the ESRC research centre at IFS is gratefully acknowledged. Health Econ., 6, (1997) 1997 by John Wiley & Sons, Ltd.

5 CLUSTER EFFECTS IN MULTILEVEL MODELS 443 REFERENCES 1. Goldstein, H. Multilevel Statistical Models, 2nd edn. Kendall s Library of Statistics 3. London: Edward Arnold, Baltagi, B. H. Econometric Analysis of Panel Data. New York: Wiley, Hsiao, C. Analysis of Panel Data. Econometric Society Monographs 11. Cambridge: Cambridge University Press, Amemiya, T. Advanced Econometrics. Oxford: Blackwell, Smith, P., Sheldon, T. A., Carr-Hill, R. A., Martin, S., Peacock, S. and Hardman, G. Allocating resources to health authorities: results and policy implications of small area analysis of use of impatient services. British Medical Journal 1994; 309: Carr-Hill, R. A., Hardman, G., Martin, S., Peacock, S., Sheldon, T. A. and Smith, P. A Formula for Distributing NHS Revenues Based on Small Area Use of Hospital Beds. York: Centre for Health Economics, University of York, Blundell, R., Boyle, S. and Windmeijer, F. Assessing the Formula for Distributing NHS Revenues. King s Fund Report, by John Wiley & Sons, Ltd. Health Econ., 6, (1997)