# A demand system for a gambling market: on the relative efficiency of restricted estimators for singular systems of equations.

Introduction

Linear allocation or share systems, because of a budget or other constraint, require that the error terms exhibit linear dependency; this has raised the problem of how to deal, in a symmetric way, with the estimation of parameters in a general linear model (GLM) when the covariance matrix of the error is singular. Actually, one is, in fact, dealing with a set of m GLM's whose error terms exhibit linear dependency, and this is the context in which we carry out our analysis.

The solution essentially associated with Barten (1969) entails a rather remarkable procedure that converts the problem to a nonsingular one. However, the literature has generally dealt with a special case--in which all explanatory variables appear in every equation. In that case, the estimator obtained by the Barten procedure automatically satisfies the adding-up condition and, consequently, has properties which do not hold for the general case.

In Schwarz (1978) use was made of the Barten procedure to obtain both the unrestricted and restricted estimators for the general case, where not all equations contain the same identical variables. In a more recent article, Dhrymes and Schwarz (1987), we showed that in the unrestricted case where different equations may contain different variables, the Barten estimator while it exists, it is also sensitive to the auxiliary parameter, k, used to "make" the covariance matrix of the errors nonsingular. We also showed that when the adding-up restrictions are imposed, the Barten estimator does not depend on this auxiliary parameter. Moreover, we showed that when the adding-up restrictions are imposed, the estimator proposed by Theil (1971), making use of the generalized inverse, is identical with the Barten estimator.

The substance of this paper is to show that, in the context of singular systems, the restricted estimator--where the adding-up restrictions are imposed--is efficient relative to the unrestricted estimator. In the final section, both the restricted and unrestricted estimators are applied to a system of demand equations for the New York City bingo market and, indeed, the theoretically predicted result is obtained; namely, the relative efficiency of the former.

Formulation of the Problem

The system we are dealing with can be written in the notation of Zellner (1962) as:

|Mathematical Expression Omitted~

or, more compactly, as

y = X|Beta~ + U

where

|Mathematical Expression Omitted~

The adding-up constraint is:

|Mathematical Expression Omitted~

where

D = (|D.sub.1~|D.sub.2~ . . . |D.sub.m~)

and |D.sub.i~ is a p x |p.sub.i~ matrix obtained by deleting from the |I.sub.p~-matrix those columns which correspond to determining variables not appearing in the ith equation (or, in other words, to coefficients known to be zero).

As a result,

e|prime~|u.sub.t~ = 0 for all t

where

e|prime~ = (1, 1, . . . , 1) |u.sub.t~ = (|u.sub.t1~, |u.sub.t2~, . . . , |u.sub.tm~)|prime~

and, hence,

|Omega~e = |E(|u.sub.t~|u|prime~.sub.t~)~e = E(|u.sub.t~|u|prime~.sub.t~e) = 0

so that the covariance matrix of the system is singular.

In Schwarz (1978) we derived the MLE of |Beta~ in (1), when |Omega~ is known, as

|Beta~ = |(X|prime~||Phi~.sup.*-1~X).sup.-1~X|prime~||Phi~.sup.*-1~y (3)

where

||Phi~.sup.*-1~ = ||Omega~.sup.*-1~|circle cross~I

and

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

and i = (1/|square roo of k~)e, k being any constant; the MLE of |Beta~ obeying the restrictions in (2) was given by:

||Beta~.sub.r~ = |Beta~ + |(X|prime~||Phi~.sup.*-1~X).sup.-1~D|prime~ ||D(X|prime~||Phi~.sup.*-1~X).sup.-1~D|prime~~.sup.-1~(d-D|Beta ~) (4)

To simplify notation, we define

Q = |(X|prime~||Phi~.sup.*-1~X).sup.-1~

which is a symmetric matrix.

Since |Beta~ and ||Beta~.sub.r~ are both unbiased estimators, it can easily be shown that:

Cov(|Beta~) = QX|prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~XQ (5)

and

Cov(||Beta~.sub.r~) = {Q - QD|prime~ ||DQD|prime~~.sup.-1~DQ} X|prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~ X{Q - QD|prime~ ||DQD|prime~~.sup.-1~DQ} (6)

Relative Efficiency of the Restricted Estimator

Turning to the question of whether the restricted estimator, ||Beta~.sub.r~, is efficient relative to the unrestricted estimator, |Beta~, we first require the following general result.

THEOREM 1

Let B be an n x n symmetric positive semi-definitive matrix and A be an n x n idempotent matrix of rank r |is less than or equal to~ n; let T be the matrix of the characteristic vectors of A and define

|Mathematical expression omitted~

such that |B*.sub.11~ is r x r. Let

C = B - A|prime~BA

Then C is positive semi-definite if

|B*.sub.12~ = |B*.sub.21~ = 0

Proof:

We observe that, since A is an idempotent matrix

|Mathematical Expression Omitted~

Hence, it follows that

T|prime~CT = T|prime~BT - T|prime~A|prime~BAT = B*

|Mathematical Expression Omitted~

If |B*.sub.12~ = |B*.sub.21~ = 0, it is clear that T|prime~CT, and hence C, is a positive semi-definite matrix.

We can now prove

THEOREM 2

The restricted estimator, ||Beta~.sub.r~ of (4) is efficient relative to the unrestricted estimator, |Beta~ of (3).

Proof:

We must show that Cov(|Beta~) - Cov(||Beta~.sub.r~) is a positive semi-definite matrix.

Let

B = QX|double prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~XQ (8)

and

A = I - D|prime~|(DQD|prime~).sup.-1~DQ

Then it follows from (5) and (6) that

Cov(|Beta~) = B

and

Cov(||Beta~.sub.r~) = A|prime~BA

Thus, since A is an idempotent matrix, then by Theorem 1, we need only show that

|B*.sub.12~ = |B*.sub.21~ = 0

From (8) it follows that

B* = T|prime~BT = T|prime~QX|prime~||Phi~.sup.* -1~|Phi~||Phi~.sup.*-1~XQT (9)

But it follows from (7) and (8) that

|Mathematical Expression Omitted~

If we postmultiply by QD|prime~, we have

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

where |K.sub.1~ contains the first r rows of T|prime~QD|prime~.

It follows from our discussion of D following equation (2) that the columns of D|prime~ contain only 0's and 1's; hence, each column of D|prime~ merely adds the same corresponding elements of each row of T|prime~Q. Therefore, since each element of the first r rows of T|prime~QD|prime~ is 0, it implies that

|Mathematical Expression Omitted~

where |L.sub.1~ contains the first r rows of T|prime~Q. Hence, it follows from (9) that

|Mathematical Expression Omitted~

which concludes our proof.

An Empirical Application: The New York City Bingo Market

In this section, we apply the estimators in (3) and (4) to a system of demand equations for the bingo market in New York City, to show that the efficiency result derived in the last section holds empirically.

The data base used(2) consists of data filed for 39 bingo occasions conducted on the night of February 3, 1975(3), in which three goods -- boards, books and jackpots -- were sold. All of these occasions had identical prices and prizes for each of these goods, the only variation being in how the total prize on the book was allocated amongst the four games played on the book.

We, therefore, estimate the parameters of the following system of demand equations for the |t.sup.th~ occasion which was specified in Schwarz (1978, Chapter 3):

|y.sub.t1~ = |c.sub.1~ + ||Beta~.sub.11~|Exp.sub.t~ + ||Beta~.sub.21~ |(|Exp.sub.t~).sup.2~ + |u.sub.t1~

|y.sub.t2~ = ||Beta~.sub.12~|Exp.sub.t~ + ||Beta~.sub.22~|(|Exp.sub.t~).sup.2~ + ||Beta~.sub.32~|Var.sub.t~ + |u.sub.t2~

|y.sub.t3~ = |c.sub.3~ + ||Beta~.sub.13~|Exp.sub.t~ + ||Beta~.sub.23~|(|Exp.sub.t~).sup.2~ + ||Beta~.sub.33~|Var.sub.t~ + |u.sub.t3~ (10)

TABULAR DATA OMITTED

where the dependent variables are dollar-expenditure per-player on boards, books and jackpots, respectively; Exp is total dollar-expenditure per-player(4); and Var is the variance of the prizes for the different book games:

|Mathematical Expression Omitted~

where |X.sub.bj~ is the prize for the |j.sup.th~ book game. Since the market shows risk preference, in order for it to be consistent, if one occasion has more variance on the book games, the players should shift somewhat from jackpots to books.

The variable |(Exp).sup.2~ is used to look at the effect of changes in the level of total expenditure on demand behavior, which is especially important in light of the study by Ali (1977). He found that the representative TABULAR DATA OMITTED bettor takes more risk at a race track where the average bet per person per race is lower than at one where it is higher. He took this as "extra evidence supporting the observation that the representative bettor with less capital tends to take more risk" (Ibid., p. 814).

Let us now turn to the estimation of our system. Tables 1 and 2 present, respectively, the unrestricted and restricted estimates together with their estimated standard errors. The latter are derived from the estimated asymptotic covariance matrix of the estimated coefficients and all tests of significance on the elements of the estimated parameter vector are to be based on the normal, not on the t, distribution.

It is clear that the results support our two hypotheses:

(1) As variance increases, and hence the book becomes a riskier good, players shift somewhat from jackpots to books.

(2) As total expenditure per capita increases, the relative share of jackpots decreases while those of the other goods increase(5), or, in other words, the average player is taking less risk. This result is similar to that found by Ali (1977) for race tracks. The reader may or may not also accept his interpretation of more expenditures as having more capital.

But of even more importance for our present purpose: besides the obvious adding-up property of the restricted estimates, the use of the adding-up constraint has led to more precise estimates in the sense of a reduction of the values of their estimated standard error, confirming the theoretical results of the previous section.

Notes

1. This paper is based, in great part, on my doctoral dissertation at Columbia University (Schwarz, 1978), especially Chapters 5 and 6. I am most grateful to my thesis advisor, Professor Phoebus J. Dhrymes, for his guidance and many helpful comments. The suggestion of an anonymous referee regarding the focus of the paper is also gratefully appreciated. All remaining errors are mine alone.

2. See Table 6.2, p. 82, of Schwarz (1978). For a complete description of its derivation and measurement errors, see Ibid., Chapter 6.

3. The same time period is used to make sure that all other variables, e.g. weather, TV shows, etc., were identical for all the occasions.

4. Except for the $1 admission fee which everyone is required to pay.

5. This is obvious if we look at |Delta~|y.sub.i~/|Delta~Exp, i = 1, 2, 3.

References

Ali, M. M. (1977): "Probability and Utility Estimates for Racetrack Bettors," Journal of Political Economy, 85, 803-815.

Barten, A. P. (1969): "Maximum Likelihood Estimation of a Complete System of Demand Equations," European Economic Review, 1, 7-73.

Dhrymes, P. J. and Schwarz, S. (1987): "On the Invariance of Estimators for Singular Systems of Equations," Greek Economic Review, 9, 88-107.

Schwarz, S. (1978): "The Estimation of a Truly Complete System of Demand Equations: An Application to a Gambling Market," unpublished doctoral dissertation, Columbia University.

Theil, H. (1971): Principles of Econometrics, John Wiley and Sons, New York.

Zellner, A. (1962): "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of the American Statistical Association, 57, 348-368.

Linear allocation or share systems, because of a budget or other constraint, require that the error terms exhibit linear dependency; this has raised the problem of how to deal, in a symmetric way, with the estimation of parameters in a general linear model (GLM) when the covariance matrix of the error is singular. Actually, one is, in fact, dealing with a set of m GLM's whose error terms exhibit linear dependency, and this is the context in which we carry out our analysis.

The solution essentially associated with Barten (1969) entails a rather remarkable procedure that converts the problem to a nonsingular one. However, the literature has generally dealt with a special case--in which all explanatory variables appear in every equation. In that case, the estimator obtained by the Barten procedure automatically satisfies the adding-up condition and, consequently, has properties which do not hold for the general case.

In Schwarz (1978) use was made of the Barten procedure to obtain both the unrestricted and restricted estimators for the general case, where not all equations contain the same identical variables. In a more recent article, Dhrymes and Schwarz (1987), we showed that in the unrestricted case where different equations may contain different variables, the Barten estimator while it exists, it is also sensitive to the auxiliary parameter, k, used to "make" the covariance matrix of the errors nonsingular. We also showed that when the adding-up restrictions are imposed, the Barten estimator does not depend on this auxiliary parameter. Moreover, we showed that when the adding-up restrictions are imposed, the estimator proposed by Theil (1971), making use of the generalized inverse, is identical with the Barten estimator.

The substance of this paper is to show that, in the context of singular systems, the restricted estimator--where the adding-up restrictions are imposed--is efficient relative to the unrestricted estimator. In the final section, both the restricted and unrestricted estimators are applied to a system of demand equations for the New York City bingo market and, indeed, the theoretically predicted result is obtained; namely, the relative efficiency of the former.

Formulation of the Problem

The system we are dealing with can be written in the notation of Zellner (1962) as:

|Mathematical Expression Omitted~

or, more compactly, as

y = X|Beta~ + U

where

|Mathematical Expression Omitted~

The adding-up constraint is:

|Mathematical Expression Omitted~

where

D = (|D.sub.1~|D.sub.2~ . . . |D.sub.m~)

and |D.sub.i~ is a p x |p.sub.i~ matrix obtained by deleting from the |I.sub.p~-matrix those columns which correspond to determining variables not appearing in the ith equation (or, in other words, to coefficients known to be zero).

As a result,

e|prime~|u.sub.t~ = 0 for all t

where

e|prime~ = (1, 1, . . . , 1) |u.sub.t~ = (|u.sub.t1~, |u.sub.t2~, . . . , |u.sub.tm~)|prime~

and, hence,

|Omega~e = |E(|u.sub.t~|u|prime~.sub.t~)~e = E(|u.sub.t~|u|prime~.sub.t~e) = 0

so that the covariance matrix of the system is singular.

In Schwarz (1978) we derived the MLE of |Beta~ in (1), when |Omega~ is known, as

|Beta~ = |(X|prime~||Phi~.sup.*-1~X).sup.-1~X|prime~||Phi~.sup.*-1~y (3)

where

||Phi~.sup.*-1~ = ||Omega~.sup.*-1~|circle cross~I

and

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

and i = (1/|square roo of k~)e, k being any constant; the MLE of |Beta~ obeying the restrictions in (2) was given by:

||Beta~.sub.r~ = |Beta~ + |(X|prime~||Phi~.sup.*-1~X).sup.-1~D|prime~ ||D(X|prime~||Phi~.sup.*-1~X).sup.-1~D|prime~~.sup.-1~(d-D|Beta ~) (4)

To simplify notation, we define

Q = |(X|prime~||Phi~.sup.*-1~X).sup.-1~

which is a symmetric matrix.

Since |Beta~ and ||Beta~.sub.r~ are both unbiased estimators, it can easily be shown that:

Cov(|Beta~) = QX|prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~XQ (5)

and

Cov(||Beta~.sub.r~) = {Q - QD|prime~ ||DQD|prime~~.sup.-1~DQ} X|prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~ X{Q - QD|prime~ ||DQD|prime~~.sup.-1~DQ} (6)

Relative Efficiency of the Restricted Estimator

Turning to the question of whether the restricted estimator, ||Beta~.sub.r~, is efficient relative to the unrestricted estimator, |Beta~, we first require the following general result.

THEOREM 1

Let B be an n x n symmetric positive semi-definitive matrix and A be an n x n idempotent matrix of rank r |is less than or equal to~ n; let T be the matrix of the characteristic vectors of A and define

|Mathematical expression omitted~

such that |B*.sub.11~ is r x r. Let

C = B - A|prime~BA

Then C is positive semi-definite if

|B*.sub.12~ = |B*.sub.21~ = 0

Proof:

We observe that, since A is an idempotent matrix

|Mathematical Expression Omitted~

Hence, it follows that

T|prime~CT = T|prime~BT - T|prime~A|prime~BAT = B*

|Mathematical Expression Omitted~

If |B*.sub.12~ = |B*.sub.21~ = 0, it is clear that T|prime~CT, and hence C, is a positive semi-definite matrix.

We can now prove

THEOREM 2

The restricted estimator, ||Beta~.sub.r~ of (4) is efficient relative to the unrestricted estimator, |Beta~ of (3).

Proof:

We must show that Cov(|Beta~) - Cov(||Beta~.sub.r~) is a positive semi-definite matrix.

Let

B = QX|double prime~||Phi~.sup.*-1~|Phi~||Phi~.sup.*-1~XQ (8)

and

A = I - D|prime~|(DQD|prime~).sup.-1~DQ

Then it follows from (5) and (6) that

Cov(|Beta~) = B

and

Cov(||Beta~.sub.r~) = A|prime~BA

Thus, since A is an idempotent matrix, then by Theorem 1, we need only show that

|B*.sub.12~ = |B*.sub.21~ = 0

From (8) it follows that

B* = T|prime~BT = T|prime~QX|prime~||Phi~.sup.* -1~|Phi~||Phi~.sup.*-1~XQT (9)

But it follows from (7) and (8) that

|Mathematical Expression Omitted~

If we postmultiply by QD|prime~, we have

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

where |K.sub.1~ contains the first r rows of T|prime~QD|prime~.

It follows from our discussion of D following equation (2) that the columns of D|prime~ contain only 0's and 1's; hence, each column of D|prime~ merely adds the same corresponding elements of each row of T|prime~Q. Therefore, since each element of the first r rows of T|prime~QD|prime~ is 0, it implies that

|Mathematical Expression Omitted~

where |L.sub.1~ contains the first r rows of T|prime~Q. Hence, it follows from (9) that

|Mathematical Expression Omitted~

which concludes our proof.

An Empirical Application: The New York City Bingo Market

In this section, we apply the estimators in (3) and (4) to a system of demand equations for the bingo market in New York City, to show that the efficiency result derived in the last section holds empirically.

The data base used(2) consists of data filed for 39 bingo occasions conducted on the night of February 3, 1975(3), in which three goods -- boards, books and jackpots -- were sold. All of these occasions had identical prices and prizes for each of these goods, the only variation being in how the total prize on the book was allocated amongst the four games played on the book.

We, therefore, estimate the parameters of the following system of demand equations for the |t.sup.th~ occasion which was specified in Schwarz (1978, Chapter 3):

|y.sub.t1~ = |c.sub.1~ + ||Beta~.sub.11~|Exp.sub.t~ + ||Beta~.sub.21~ |(|Exp.sub.t~).sup.2~ + |u.sub.t1~

|y.sub.t2~ = ||Beta~.sub.12~|Exp.sub.t~ + ||Beta~.sub.22~|(|Exp.sub.t~).sup.2~ + ||Beta~.sub.32~|Var.sub.t~ + |u.sub.t2~

|y.sub.t3~ = |c.sub.3~ + ||Beta~.sub.13~|Exp.sub.t~ + ||Beta~.sub.23~|(|Exp.sub.t~).sup.2~ + ||Beta~.sub.33~|Var.sub.t~ + |u.sub.t3~ (10)

TABULAR DATA OMITTED

where the dependent variables are dollar-expenditure per-player on boards, books and jackpots, respectively; Exp is total dollar-expenditure per-player(4); and Var is the variance of the prizes for the different book games:

|Mathematical Expression Omitted~

where |X.sub.bj~ is the prize for the |j.sup.th~ book game. Since the market shows risk preference, in order for it to be consistent, if one occasion has more variance on the book games, the players should shift somewhat from jackpots to books.

The variable |(Exp).sup.2~ is used to look at the effect of changes in the level of total expenditure on demand behavior, which is especially important in light of the study by Ali (1977). He found that the representative TABULAR DATA OMITTED bettor takes more risk at a race track where the average bet per person per race is lower than at one where it is higher. He took this as "extra evidence supporting the observation that the representative bettor with less capital tends to take more risk" (Ibid., p. 814).

Let us now turn to the estimation of our system. Tables 1 and 2 present, respectively, the unrestricted and restricted estimates together with their estimated standard errors. The latter are derived from the estimated asymptotic covariance matrix of the estimated coefficients and all tests of significance on the elements of the estimated parameter vector are to be based on the normal, not on the t, distribution.

It is clear that the results support our two hypotheses:

(1) As variance increases, and hence the book becomes a riskier good, players shift somewhat from jackpots to books.

(2) As total expenditure per capita increases, the relative share of jackpots decreases while those of the other goods increase(5), or, in other words, the average player is taking less risk. This result is similar to that found by Ali (1977) for race tracks. The reader may or may not also accept his interpretation of more expenditures as having more capital.

But of even more importance for our present purpose: besides the obvious adding-up property of the restricted estimates, the use of the adding-up constraint has led to more precise estimates in the sense of a reduction of the values of their estimated standard error, confirming the theoretical results of the previous section.

Notes

1. This paper is based, in great part, on my doctoral dissertation at Columbia University (Schwarz, 1978), especially Chapters 5 and 6. I am most grateful to my thesis advisor, Professor Phoebus J. Dhrymes, for his guidance and many helpful comments. The suggestion of an anonymous referee regarding the focus of the paper is also gratefully appreciated. All remaining errors are mine alone.

2. See Table 6.2, p. 82, of Schwarz (1978). For a complete description of its derivation and measurement errors, see Ibid., Chapter 6.

3. The same time period is used to make sure that all other variables, e.g. weather, TV shows, etc., were identical for all the occasions.

4. Except for the $1 admission fee which everyone is required to pay.

5. This is obvious if we look at |Delta~|y.sub.i~/|Delta~Exp, i = 1, 2, 3.

References

Ali, M. M. (1977): "Probability and Utility Estimates for Racetrack Bettors," Journal of Political Economy, 85, 803-815.

Barten, A. P. (1969): "Maximum Likelihood Estimation of a Complete System of Demand Equations," European Economic Review, 1, 7-73.

Dhrymes, P. J. and Schwarz, S. (1987): "On the Invariance of Estimators for Singular Systems of Equations," Greek Economic Review, 9, 88-107.

Schwarz, S. (1978): "The Estimation of a Truly Complete System of Demand Equations: An Application to a Gambling Market," unpublished doctoral dissertation, Columbia University.

Theil, H. (1971): Principles of Econometrics, John Wiley and Sons, New York.

Zellner, A. (1962): "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of the American Statistical Association, 57, 348-368.

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Author: | Schwarz, Samuel |
---|---|

Publication: | American Economist |

Date: | Mar 22, 1993 |

Words: | 1981 |

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