Working Papers in Econometrics and Applied Statistics

Transcription

1 T h e U n i v e r s i t y o f NEW ENGLAND Working Papers in Econometrics and Applied Statistics Finite Sample Inference in the SUR Model Duangkamon Chotikapanich and William E. Griffiths No April 999 Working Papers in Econometrics and Applied Statistics Department of Econometrics University of New England Armidale, NSW 235, Australia. ISSN ISBN X

2 FINITE SAMPLE INFERENCE IN THE SUR MODEL Duangkamon Chotikapanich School of Economics and Finance Curtin University of Technology GPO Box U 987, WA ChotikapanichD@cbs.curtin.edu.au Phone: Fax: William E Griffiths School of Economics Studies University of New England Armidale NSW wgriffit@metz.une.edu.au Phone: Fax: Keywords and phrases: Bayesian inference; Gibbs sampler; g-prior; measuring parameter uncertainty. JEL Classification: C, C33 5 February 999

3 3 ABSTRACT The effect of adding equations on coefficient uncertainty in a system of seemingly unrelated regression equations is analyzed. In contrast to Fiebig and Kim (999), who use a sampling-theory approach, our criterion for measuring parameter uncertainty is the marginal posterior standard deviations of the coefficients. Using an improper noninformative prior, and two proper, mildly informative priors, we evaluate posterior standard deviations for systems of different sizes and compare them with SUR and maximum likelihood standard errors. We find that, despite the additional uncertainty created by additional unknown covariance parameters, adding equations generally improved estimation precision whenever proper priors are used.

4 4 FINITE SAMPLE INFERENCE IN THE SUR MODEL INTRODUCTION A frequently used model in econometrics is the general linear model with unknown error covariance matrix. Typically, it has the form y = X β + e e ~ N (0, Ω) () where y, X, β and e are ( T ),( T K ),( K ) and ( T ) ; and Ω is an unknown disturbance covariance matrix whose elements are dependent on a fixed number of parameters that does not change as T changes. Estimation of β is usually achieved via an "estimated" generalized least-squares (EGLS) or a maximum likelihood (ML) estimator, both of which can be written as ˆ β = ( X Ωˆ X ) X Ωˆ y (2) The difference between EGLS and ML, and between alternative EGLS estimators, lies in the choice of an estimate for Ω. Except in special cases, the finite sample properties of βˆ are not available. Sampling theory inference for βˆ is normally based on the asymptotic approximation ( ˆ β, ( X Ω X ) ) βˆ ~ N (3) The Bayesian inference analog of this result arises when a noninformative uniform prior on β is used, and the analysis is carried out conditional on Ωˆ. Specifically, the

5 5 conditional posterior probability density function (pdf), f ( β y, Ωˆ ), is distributed as normal with mean βˆ and covariance matrix ( X ' Ωˆ X ). Therefore, when drawing inferences on β, we use the result: ( ˆ βˆ, ( X ' Ω X ) ) β ~ N (4) Despite the different interpretations of the origins of (3) and (4), inference from the two approaches is essentially the same if the same Ωˆ is employed. Details of the Bayesian approach can be found in Zellner (97) or Poirier (995). Basing inferences on the covariance matrix ˆ C = ( X ' Ω X ) is less than desirable in both the sampling theory and Bayesian approaches. In sampling theory inference C may not be an accurate reflection of how βˆ varies in repeated samples. In the Bayesian approach, where inference is conditional on an observed sample, the fact that C does not accurately measure repeated sample variation is not the prime concern. However, C is still not a satisfactory measure of uncertainty about β because it does not recognize the related uncertainty about Ω. Because the interpretation of finite sample accuracy is not the same for the sampling theory and Bayesian approaches, solutions for achieving greater finite sample accuracy are also different. A popular sampling theory solution is to use bootstrapping to find more accurate standard errors or t statistics. See Atkinson and Wilson (992) and Fiebig and Kim (999) for details in the context of seemingly unrelated regression (SUR) models, upon which we focus later in this paper. The Bayesian solution to recognizing the additional uncertainty created by not knowing Ω is to focus on the marginal posterior pdf f ( β y), rather than the conditional posterior pdf f ( β y, Ω). The relationship between the two pdfs is given by f ( β y) = f ( β y, Ω) f ( Ω y) d Ω (5)

6 6 Uncertainty in Ω is recognized by using f ( Ω y) as the weighting function in the weighted average of the conditional posterior pdfs f ( β y, Ω). Before the recent explosion in Markov Chain Monte Carlo techniques, many integrals of the form defined in (5) were troublesome; users of Bayesian inference often settled for the second best approximation in (4), not utilizing the marginal posterior pdf. For reviews of Markov Chain Monte Carlo applications in econometrics, and access to the literature, see Koop (994), Albert and Chib (996), Chib and Greenberg (996) and Geweke (998). One special case of the model defined in (), and the one with which we are concerned in this paper, is the SUR model (Zellner, 962). In this model efficiency gains in sampling-theory estimation of β can be achieved by joint estimation of a system of equations with correlated error structure. Suppose the coefficients of a single equation are of interest. For a known covariance structure Ω, adding equations (with different explanatory variables) to the system will improve the precision of estimation of the coefficients of interest. When Ω is unknown and has to be estimated, the effect of adding equations is less clear cut. One would expect gains in efficiency when the number of equations is small relative to the number of observations. However, for a fixed number of observations, one would expect adding equations to eventually lead to a decline in finite-sample efficiency through increasingly less precise estimation of Ω. This question has been investigated in depth by Fiebig and Kim (999) for several alternative sampling-theory point and interval estimators for β. In this paper we investigate the same question from the standpoint of Bayesian inference. In this case, the spread of the marginal posterior pdf f ( β y) is a measure of the precision of estimation or the post-sample precision of our knowledge about β. Accordingly, we examine how the posterior standard deviations of elements in β change as more equations are added to a system with a fixed number of observations. Such an examination provides guidance on whether adding equations leads to an increase or decrease in information about coefficients of interest. Since performance in repeated samples is not of prime importance in Bayesian inference, our analysis is based only on one sample. However, sensitivity of

7 7 results to other samples would be of interest in any more extension study. The model and inference procedures are described in Section 2. Section 3 contains details of data generation and the specific characteristics of the model we utilize. Results and concluding remarks appear in the Sections 4 and 5, respectively. 2. THE SUR MODEL AND ESTIMATOR Consider the general SUR model comprising M equations each containing T observations: y = X β + e i =, 2,, M (6) i i i i where y i and e i are of dimension ( T ), X i is ( T Ki ) and β i is ( K i ). Combining all equations into one big model yields y y y 2 M = X X 2 X M β β β 2 M + e e e 2 M (7) The model (7) may be expressed in compact form as y = X β + e (8) where the dimensions of y, X, β and e are ( MT ),( MT K ),( K ) and ( MT ), respectively, with K = M K i i=. It is assumed that the ( MT MT ) covariance matrix of the disturbance vector e is given by

8 8 E( ee' ) = Ω = Σ (9) I T where Σ is the ( M M ) contemporaneous covariance matrix and I T is an identity matrix of dimension ( T T ). It is also assumed that e is normally distributed. To proceed with Bayesian estimation, we begin with the likelihood function for the unknown β and Σ given by T 2 L( β, Σ y) Σ exp tr S Σ 2 (0) where S is an ( M M ) matrix with (i,j) element equal to y X β )'( y X β ). Using the conventional noninformative prior pdf (see Zellner, 97) ( i i i j j j f ( Σ) Σ ( M + ) 2 () we obtain the joint posterior density function for the unknown parameters as ( T + M + ) 2 f ( β, Σ y) Σ exp tr S Σ 2 (2) The conditional posterior pdf for β given Σ is β Σ β βˆ f (, y) exp ( ) X ( Σ I ) ( β βˆ T X ) (3) 2 where ˆ β = [ X ( Σ IT ) X ] X ( Σ IT ) y. The conditional posterior pdf for Σ given β is an inverted-wishart density which takes the same form as equation (2)

9 9 ( T + M + ) 2 f ( Σ β, y) Σ exp tr S Σ 2 (4) Finally, the marginal posterior pdf for β, obtained by integrating Σ out of the joint pdf in (2), is f ( β y) S T 2 (5) We are interested in the posterior standard deviations of elements in β, obtained from equation (5), and how these standard deviations change as the dimension of S increases (more equations are added). Unfortunately, closed form expressions for these standard deviations cannot be derived. They can be estimated, however, using Markov Chain Monte Carlo (MCMC) techniques. These techniques provide a way of drawing observations from the marginal pdf f ( β y), so that the sample standard deviations of the observations give the necessary information. The MCMC technique that we use is Gibbs sampling (see, for example, Percy 992), where draws of β and Σ are taken iteratively from the conditional posterior pdfs in (3) and (4). Each draw of β comes from a multivariate normal distribution conditioned on the previous draw of Σ ; and each draw of Σ comes from an inverted Wishart distribution, conditioned on the previous draw of β. Following a suitable burn-in period, after which the Markov Chain has converged, the observations on β are drawings from the marginal posterior pdf f ( β y). Another way to proceed is to apply a Metropolis-Hastings algorithm (another MCMC technique) directly to equation (5). See Griffiths and Chotikapanich (997) for an application along these lines. From a Bayesian perspective, traditional sampling theory SUR estimation uses the conditional posterior pdf in (3), conditioned on Σ = Σˆ. The resulting covariance matrix for β is ( ˆ [ X Σ IT ) X ], an expression which underestimates the uncertainty in β by ignoring the related uncertainty in Σ. In fact, a relationship

10 0 between the variances of the conditional and marginal posterior pdfs for β provides an alternative way of estimating the posterior standard deviations. Let β (), β (2),, β (N ) and () Σ, Σ (2),, Σ (N ) be the post burn-in observations from the Gibbs sampler. Then, the marginal posterior covariance matrix for β can be estimated as the sample covariance matrix N Vˆ ( β y) = ( β( i) β)( β( i) β) N i= (6) or as the average of the conditional posterior covariance matrices plus the sample covariance matrix of the conditional means N ˆ V ( ) [ ( ) ] ( ˆ 2 β y = X Σ( i) IT X + β( i) β)(ˆ β( i) β) N N i= N i= (7) In equation (7) ˆ ( ) ( ) ( ) β i = [ X ( Σ i IT ) X ] X ( Σ i IT ) y is the conditional posterior mean and β can be taken as the sample average of the β ( ) or the β (i). In our numerical work differences between V ( β y) and V ( β y), and between the sample ˆ i averages of β ( ) and β (i), were negligible. ˆ ˆ2 ˆ i Since it is our objective to investigate how Vˆ ( β y) changes as M, and hence K, increase for a fixed T, it is important to know the conditions under which the marginal posterior pdf f ( β y) is a proper density. Properness is not guaranteed when the improper prior ( M + ) 2 f ( β, Σ) Σ is used. Griffiths and Chotikapanich (999) have shown that a necessary condition for f ( β y) to be proper is T > K, K being the total number of coefficients in the system. Assuming that each equation has at least two coefficients, it is clear that this condition will be violated long before the number of equations approaches the number of observations. Thus, if we are to follow the lead of Fiebig and Kim (999), and examine scenarios where M is close to T, a proper

11 prior pdf is required. We employed the improper prior in () for values of M such that T > K, and two versions of the following proper prior pdf for all values of M f ( β, Σ) = f ( β) f ( Σ) (8) with g f ( β) exp{ ( β b) X X ( β b)} 2 (9) f ( Σ) Σ ( ν+ M + ) 2 exp{ 2 tr AΣ - } (20) Equation (9) is in the form of the g-prior introduced by Zellner (986). We have chosen the least squares estimator b = ( X X ) X y as the prior mean and ( g X X ) as the prior covariance matrix. Using the data to specify the prior mean and covariance matrix may lead to some objections. However, our objective is to examine how uncertainty about elements in β changes as more equations are added. Setting the prior mean equal to the least squares estimator avoids a situation where uncertainty increases because of conflict between prior and sample information. Furthermore, least squares estimates are independent of the size of the system, and the effect of the prior information can be minimized by setting g sufficiently small. Equation (20) is an inverted Wishart prior pdf; setting values for the prior parameters ν and A, as well as g, is discussed in the next section. Analysis of the SUR model with a more general independent normal-inverted Wishart prior can be found in Percy (996). For Gibbs sampling the conditional posterior pdfs for β and Σ are required. The one for β is where ˆ f ( β Σ, y) exp{ ( β β ) ( ) ( ˆ X Σ IT X β β )} (2) 2

12 2 Σ = Σ + g I M (22) and the conditional posterior mean for β is ˆ β = [ X ( Σ IT ) X ] X ( Σ IT ) y (23) The effect of the g-prior, centred at the least-squares estimate, is to add g to the diagonal elements of Σ. It is interesting that the Ullah and Racine (992) estimator, that performed well in the experiments of Fiebig and Kim (999), was obtained by adding a constant to the diagonal elements of Σ. The conditional posterior pdf for Σ is the inverted Wishart pdf ( ν + T + M + ) 2 f ( Σ β, y) Σ exp{ tr[( A + S) Σ ]} (24) 2 Equations (2 and (24) provide the basis for the draws of β and Σ from the Gibbs sampler. Again, the marginal posterior covariance matrix for β can be estimated from the sample covariance matrix of the draws, or from the sum of (a) the sample mean of the conditional posterior covariance matrices ( Σ I ) X [ X T ] and (b) the sample covariance matrix of the βˆ. 3 SETUP AND DATA FOR NUMERICAL WORK Our setup, model and data originate from Fiebig and Kim (999) who adopted the Batlagi and Griffin (983) model for gasoline demand. The sample of data comprises M = 8 countries each of which has T = 9 annual data points. The model assumes a specification where the parameters of the demand equation vary across countries. The basic model specification for each country used by Fiebig and Kim is:

13 3 Gas Y PMG Car ln = α + β ln + β2 ln + β3 ln + e (25) Car Pop PGDP Pop where Gas/Car is motor gasoline consumption per car, Y/Pop is per capita real income, P / P is real motor gasoline price and Car/Pop is the stock of cars per MG GDP capita. Because there was a high degree of correlation between Y/Pop and Car/Pop, and because having too many explanatory variables in each equation limits the number of equations that can be included when examining the noninformative prior case, we dropped Car/Pop from the model. To ensure we were estimating a correctly specified model we generated the data on Gas/Car based on the remaining two variables. The model becomes: Gas Y PMG ln = α + β ln + β2 ln + e Car Pop P (26) GDP The data were generated with α = 0, β = and β2 = 0. 5 for all equations. The disturbances in all equations were given a common variance of 0.05 and a common contemporaneous correlation of 0.6. These settings mean that the covariance matrix of the least squares estimator for the ith equation is 0.05( X i X i ), or a precision matrix of 20 X i X i. For the g-prior for β we set g = 2 so that the information from the prior is small (one-tenth) relative to that from the data. For the inverted Wishart prior on Σ we used two settings: () ν = M A =

14 4 (2) ν = M A = I M In the first setting ν is set equal to the smallest value for which the prior mean E(Σ) exists, and E ( Σ) = A is equal to the true value of Σ which generated the data. Making ν dependent on M ensures the marginal priors for different sized systems are consistent with the joint prior for the largest system where M = 8. The second setting is a less informative prior. In this case ν = M is the smallest value for which f (Σ) is proper. The prior mean does not exist, but the setting for A is such that the prior medians for each of the σ ii are equal to the true value of Making A diagonal means that gains in precision from error correlation will only occur if the data suggest they should. 4. RESULTS The chosen coefficients of interest are the slope coefficients (the income and price elasticities) in the first two equations. For these coefficients, we examine how SUR and ML standard errors, and the posterior standard deviations from the different priors, change as more equations are added to the system. As mentioned earlier, the SUR and ML standard errors can be viewed as conditional posterior standard deviations, each one conditioned on a different Σ. These results appear in Tables to 4, and are presented graphically (with the exception of ML which, when it existed, was close to SUR) in Figures to 4. The Bayesian results are obtained from a Gibbs sample of size 50,000 which followed a burn-in of 0,000 draws. The ML results and the Bayesian results from an improper prior do not go beyond M = 6 because larger values of M lead to violation of the condition T > K which is necessary for the likelihood function to be bounded. The results can be summarized as follows.. The standard errors obtained from SUR decline as the number of equations is increased. This gives the false impression that estimation is becoming

15 5 more accurate as more equations are added. From the tables, the ML standard errors are very similar to those for SUR and they behave in the same way up to 6 equations. For this reason, the plots for the ML standard errors are omitted. 2. The posterior standard deviations obtained when an improper prior is used are always greater than their SUR and ML counterparts. This reflects the fact that they recognize the uncertainty in the estimation of Σ. (The posterior standard deviation for M = is also greater than that from SUR and ML. This reflects the fact that posterior inferences are based on the t distribution, whose standard deviation is reported, while the sampling theory inferences are based on the normal distribution conditional on elements of Σˆ ). 3. From the perspective of Bayesian inference with an improper prior, the optimal number of equations can be viewed as that for which the posterior standard deviation is smallest. It is under these circumstances that the greatest amount of information about the parameter has been conveyed by the data. Using this criterion, the optimal number of equations is different for different parameters. For the income elasticity in equation the optimal number of equations is 3, with joint estimation becoming worse than single equation estimation from 5 equations onwards. For the price elasticity in the same equation the posterior standard deviation is smallest for M = 6. For the two coefficients in equation 2 the optimal M is 4. However, joint estimation is still preferable to single equation estimation for M = As expected, the posterior standard deviations from the improper prior are all greater than their corresponding counterparts from both proper priors. Also, as expected, the posterior standard deviations from proper prior 2 are greater than the corresponding ones from the proper prior. 5. The values for M = 0 in the proper prior and 2 columns are the prior standard deviations. A comparison of these with the posterior standard

16 6 deviations indicates how much precision has been added through the introduction of data. 6. The original motivation for setting up proper priors that convey mild information was to investigate further what might be the optimal number of equations without encountering problems with singularities and improper posteriors. What we discovered, however, was that there is a general tendency for the standard deviations to decline even as the number of equations becomes relatively large. The decline is not monotonic; for example, the addition of equation 9 increased the posterior standard deviations of the coefficients in equation, and the addition of equation 2 increased the posterior standard deviations of the coefficients in equation 2. However, overall, including extra equations did provide additional information on the coefficients all the way up to M = 8. It seems that providing quite mild prior information helps utilize the information provided by the error correlations in a very effective way. 5. CONCLUDING REMARKS Bayesian estimation provides a vehicle for finite sample inference in models where inference is traditionally performed using asymptotic approximations which do not adequately reflect all parameter uncertainty within a model. The SUR model is an interesting example of such a model because adding equations provides additional information through a correlated error structure and, at the same time, creates greater uncertainty through the need to estimate more unknown covariance parameters. Our numerical work indicates how information depends on the number of equations in the system. It also shows how misleading conditional inferences can be. The fact that adding equations generally adds information when mild prior information is utilized, even when M is large relative to T, is an interesting result. From a sampling theory standpoint, it may be productive to investigate the repeated-sample performance of

17 7 the conditional and unconditional posterior means that result from our informative priors.

18 8 References Albert, J.H. and S. Chib (996), "Computation in Bayesian Econometrics: An Introduction to Markov Chain Monte Carlo" in R.C. Hill (ed.) Advances in Econometrics Volume A: Computational Methods and Applications, JAI Press, Greenwich. Atkinson, S.E. and P.W. Wilson (992), "The Bias of Bootstrapped versus Conventional Standard Errors in the General Linear and SUR Models". Econometric Theory, 8, Baltagi, B.H. and J.M. Griffin (983), Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures, European Economic Review, 22, Chib, S. and E. Greenberg (996), "Markov Chain Monte Carlo Simulation Methods in Econometrics", Econometric Theory, 2, Fiebig, D.G. and J.H.Kim (999), "Estimation and Inference in SUR Models when the Number of Equations is Large", Econometric Reviews, forthcoming. Geweke, J. (998), "Using Simulation Methods for Bayesian Econometric Models: Inference, Development and Communication", Econometric Reviews, forthcoming. Griffiths, W.E. and D. Chotikapanich (997), "Bayesian Methodology for Imposing Inequality Constraints on a Linear Expenditure System with Demographic Factors", Australian Economic Papers, 36,

19 9 Griffiths, W.E. and D. Chotikapanich (999), "A Sample Size Requirement for SUR Estimation" Working Papers in Econometrics and Applied Statistics, Department of Econometrics, The University of New England. Koop, G. (994), "Recent Progress in Applied Bayesian Econometrics", Journal of Economic Surveys, 8,, -34. Percy, D.F. (992), "Prediction for Seemingly Unrelated Regressions", Journal of the Royal Statistical Society, B 54, Percy, D.F. (996), "Zellner's influence on Multivariate Linear Models", in D.A. Berry, K.M.Chaloner and J.K. Geweke (eds.), Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner, John Wiley and Sons, New York. Poirier, D.J. (995), Intermediate Statistics and Econometrics: A Comparative Approach, MIT Press, Cambridge. Shazam user reference manual version 8.0 (997), McGraw-Hill, New York, NY. Ullah, A. and J. Racine (992), "Smooth Improved Estimators of Econometric Parameters", in W.E. Griffiths, H. Lutkepohl and M.E. Bock (eds.), Readings in Econometric Theory and Practice: A Volume in Honor of George Judge, North- Holland, Amsterdam. Zellner, A. (962), An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias, Journal of American Statistical Association, 57, Zellner, A. (97), Introduction to Bayesian Inference in Econometrics, Wiley, New York, NY.

20 20 Zellner, A. (986), On Assessing Prior Distributions and Bayesian Regression Analysis with g-prior Distributions, in P.K. Goel and A. Zellner (eds.), Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, North Holland, Amsterdam.

21 2 Table : Standard Errors for Income Elasticity (Equation ) No. of Eqns SUR ML Improper Proper Proper

22 22 Table 2: Standard Error For Price Elasticity (Equation ) No. of Eqns SUR ML Improper Proper Proper

23 23 Table 3: Standard Errors for Income Elasticity (Equation 2) No. of Eqns SUR ML Improper Proper Proper

24 24 Table 4: Standard Errors for Price Elasticity (Equation 2) No. of Eqns SUR ML Improper Proper Proper

25 25 Figure : Standard Error and Posterior Standard Deviation for Income Elasticity (Equation ) 0.85 Improper Proper2 Proper 0.05 SUR Figure 2: Standard Errors and Standard Deviations for Price Elasticity (Equation ) Improper Proper 2 Proper 0.2 SUR

26 Figure 3: Standard Errors and Posterior Standard Deviations for Income Elasticity (Equation 2) 0.35 Improper Proper Proper 0.5 SUR Figure 4: Standard Errors and Posterior Standard Deviations for Price Elasticity (Equation 2) 0.75 Improper Proper 2 Proper SUR