MONTECARLO

Some Monte Carlo results for the modified logit model

by

Chris M. Alaouze* School of Economics University of New South Wales Sydney NSW 2052 Australia 2033

Abstract This paper presents some Monte Carlo results on the dependence on n and T of the convergence in probability and distribution of key parameters of the modified logit model of Amemiya and Nold and the generalisation of this model to the case where the error term follows an AR(1) process. Results are presented for n in the range 10 to 350 and T in the range 10 to 200.

*

Paper presented at the ESAM98 Conference held at the ANU, July 8-10, 1998.

1

1

Introduction

Amemiya and Nold (1975) proposed a useful generalisation of the Berkson logit model for grouped data (Berkson, 1955; Theil, 1971) by adding an error term to the logit model: Pt = Prob. {Yit = 1 ν t } = 1 / (1 + exp( − (β ′x t + ν t ))

(1)

i = 1, ..., nt; t = 1, ..., T, Yit is a binary random variable which takes the value 1 with probability Pt, and 0 with probability (1 - Pt), xt is a k x 1 vector of explanatory variables which are assumed to be constant, νt is a random error term and β is a k x 1 vector of unknown parameters. Pt is usually estimated from a random sample {Yit, i = 1, ..., nt} taken from the group for which xt applies. Amemiya and Nold assume that νt is an independently and identically distributed (iid) random variable with mean and variance E(νt) = 0, V(νt) = σ2 respectively, so that their modified logit model is applicable for cross sectional studies. Alaouze (1998a, 1998b) has generalised this model to include the case where survey data are collected in a time series, so that νt is temporally correlated.1 Specifically, he assumed that νt is a stationary AR(1) random variable: ν t = ρν t-1 + η t , where ρ < 1, η t is an iid random variable with E( η t ) = 0, V( η t ) = σ 20 and E( η4t ) < ∞. The parameter β can be estimated using feasible GLS (Alaouze, 1998a, 1998b) which in the case where νt is iid reduces to weighted least squares (Amemiya and Nold, 1975). Alaouze (1998a, 1998b) has noted that the asymptotic properties of estimators of the parameters associated with (1) in the case where νt follows a stationary AR(1) process depend on nt and T becoming large. It is straightforward to demonstrate that the asymptotic properties of estimators of the parameters associated with (1) also depend on nt and T becoming large in the case where νt is assumed to be iid. This paper presents some Monte Carlo results which are designed to investigate the nature of the dependence on nt and T of (i) the convergence in probability of estimators of key parameters associated with (1) and (ii) the convergence in distribution of feasible GLS estimators of β. Monte Carlo results are presented for the cases where νt is iid and νt follows a stationary AR(1) process. The Monte Carlo results were obtained using Shazam (version 7.0). The rest of the paper is organised as follows. Section 2 contains an overview of basic statistical results for the modified logit model. Monte Carlo results are presented in Section 3 and the final section contains a summary of the findings of the paper.

2

Basic Statistical Results

Amemiya and Nold noted that once νt is realised, Pt is a parameter rather than a random variable, thus the mean and variance of Yit must be specified conditionally: E(Yit ν t ) = Pt and V(Yit ν t ) = Pt (1 − Pt ) . It is assumed that the random variables {Yit, i = 1, ..., nt, t = 1, ... , T} conditional on ν (the T× 1 vector with elements νt) are iid

2 at each t and Yit is independent of Yjk, t ≠ k. To simplify notation it is assumed that nt = n, t = 1, ..., T. Let p t =

n



i =1

Yit n, , t = 1,..., T then under our assumptions, the conditional random

p t ν t has the following asymptotic normal distribution as n → ∞ : 1 D ( n ) 2 ( p t − Pt ) → N( 0, Pt (1 − Pt )).

variable

Now let p be the T× 1 vector with elements {p t }, P be the T× 1 vector with elements {Pt} and ∆ be the T× T matrix: ∆ = diag. {P1(1-P1), ..., PT(1-PT)}, then under our assumptions the conditional random vector pν has the following multivariate normal asymptotic 1 D distribution as n → ∞ : ( n ) 2 ( p - P) → N(0- , ∆ ) where 0− is a T× 1 vector of zeros. Taking the logit transformation of (1) we obtain log( Pt / (1 − Pt )) = β ′x t + ν t or: log( p t / (1 − p t )) = β ′x t + ε t + ν t

(2)

where ε t = log( p t / (1 − p t )) − log( Pt / (1 − Pt )) . Applying the δ method (Bishop, Fienberg and Holland 1975, pp. 486-502) the asymptotic distribution of εt conditional on 2 ν is: D (n) 1 2 ε t → N(0,1/Pt (1 − Pt )). (3) Since εt is a continuous function of pt, εtν is independent of εkν (t ≠k). Letting ε be the T× 1 vector with elements εt, the conditional random vector εν has the following asymptotic distribution as n → ∞ : D ( n) 1 2 ε →

N(0- , ∆-1 ),

(4)

Amemiya and Nold have noted from an expression which is their counterpart of (3) that since asymptotically the expected value of εt conditional on νt does not depend on νt, εt and νt are asymptotically uncorrelated. It is important to note from (2) and (4) that asymptotically, the expected value of εt conditional on ν does not depend on ν, so that ε is uncorrelated with ν asymptotically. Letting qt = log (pt/(1-pt)) and q be the T× 1 value vector with elements {qt}, (2) yields the following linear model: q = Xβ + ε + ν

(5)

where X is the T× k matrix with rows x ′t , t = 1, ..., T. Assuming that n is large enough for the asymptotic distribution of ( n) 1 2 ε conditional on 3 ν to be applicable , the covariance matrix of q is:

3

(

)

E[E[(q − Xβ)(q - Xβ)′ ν]] = E[E(nεε′ ν)]] / n + 2 n 1 2 E[νE[n 1 2 ε′ ν]] + E[νν′] = E[∆ ] / n + E[νν′]. −1

(6)

Since ∆−1 = diag.{1 / P1 (1 − P1 ), 1 / P2 (1 − P2 ),..., 1 / PT (1 − PT )}, consistent estimation of E[ ∆−1 ] would require the average of a series of Pt, each element of which is obtained from a random sample {Yit} which is obtained from a different realisation of νt with xt fixed. Such a series will generally not be available as only one random sample of Yit is obtained at each t and for only one realisation of νt. As a working approximation, we shall use the following estimator of E[ ∆−1 ] : ∆ −1 = diag. {1 / p (1 − p ), 1 / p 2 (1 − p 2 ),..., 1 / p T (1 − p T )}.

1

1

Let E( νν′) = Ω , then if ν has iid elements, Ω = σ 2 I , and if ν has elements drawn from a stationary AR(1) process, Ω = σ 2 V, where σ 2 = σ 20 / (1 − ρ 2 ) and V is the TxT matrix with elements {ρ

i- j

} , i = 1, ..., T; j = 1, ..., T.

Equation (6) suggests that a feasible GLS estimator of β could be defined if a consistent estimator of Ω could be found. Since ε converges in probability to 0− as n → ∞ ; when n is large, and ν has iid elements, (5) is asymptotically equivalent to the standard linear regression model with iid errors as described for example in Theil (1971, pp.378-381). Theil shows that for this model a consistent estimator of σ 2 (as T → ∞ ) can be obtained from OLS residuals. When n is large and ν is drawn from a stationary AR(1) process, the linear model (5) is asymptotically equivalent to the standard linear regression model with stationary AR(1) errors as described, for example, in Theil (1971, pp.237, 250-257, 405407). In this latter model, a consistent estimator of σ 2 V (as T → ∞ ) can be constructed from OLS residuals, and the resulting feasible GLS estimator is asymptotically normally distributed as T → ∞ (Theil 1971, pp.405-407). Intuitively, this suggests that for the modified logit model as described by (5) and (6), a  could be obtained from the residuals of the OLS regression consistent estimator of Ω Ω

( )

of q on X, and that the feasible GLS estimator of β in (5) based on the estimated 1   would have an asymptotically normal distribution, where covariance matrix  ∆ − 1 + Ω n  these asymptotic properties depend on n → ∞ and T → ∞ . These intuitive results have been rigorously verified for the case where ν is drawn from a stationary AR(1) process by Alaouze (1998a, 1998b), and by similar methods is proven for the case where ν is iid in the appendix of this paper. The following asymptotic results can be proved for the modified logit model with stationary AR(1) errors: (i) The OLS estimator of β in (5), b = ( X ′X) −1 X1q is a consistent estimator of β as n → ∞ and T → ∞ .

4 (ii)

Let ut be element t of the OLS residual vector u obtained from the regression of q on ρ = 

X, u = q - Xb ,

then

the

statistics:

σ 2 =

T ∑ u2 t t =1

T

and

T / (T − 1)  ∑ u 2t  / T are consistent estimators of σ 2 and ρ as t =1 n → ∞ and T → ∞.

(iii)

T ∑ u u t t −1 t =2

The feasible GLS estimator of β:

−1 −1 1   X) −1 X ′ 1 ∆ −1 + Ω   q , (7) β F = ( X ′ ∆ −1 + Ω n  n   is the matrix Ω with σ 2 and ρ replacing ρ and σ 2 , is asymptotically where Ω normally distributed as n → ∞ and T → ∞ . That is: −1  1  −1   D 12  (T) (β F − β) → N(0, p lim ( X ′ ∆ + Ω X / T) −1 ), n  n→∞ , T→∞ as n → ∞ and T → ∞, so that β has approximately the following multivariate

F

normal distribution in finite but large samples: 1   β F ∼ N (β, (X′ ∆ −1 + Ω n 

−1

X) −1 )

Monte Carlo results which are designed to investigate the dependence on n and T of the convergence in probability of ρ and β F , and the convergence in distribution of a standardised version of β are presented in the next section. F

The following asymptotic results can be proved for the modified logit model with iid errors: (iv) The OLS estimator of β in (5), b = ( X ′X) −1 X ′q is a consistent estimator of β as n → ∞ and T → ∞ . (v)

Let ut be element t of the OLS residual vector u obtained from the regression of q on X, u = q - Xb , then the statistics: σ 12 =

1 T 2 T   ∑ u t − ∑ 1 np t (1 − p t ) ,  T  t =1 t =1

(8)

σ 22 =

1  T 2 ∑u  , T  t =1 t 

(9)

and

are consistent estimators of σ 2 as n → ∞ and T → ∞. The estimator σ 12 is the one used by Amemiya and Nold (1975) in forming their WLS estimator of β. In view of the fact that εt and νt are asymptotically uncorrelated and εt is

(

)

approximately distributed as N 0,1 nPt (1 − Pt ) when n is large, the inclusion of the term −

1 ∑ 1 np t (1 − p t ) in Amemiya and Nold’s estimator of σ2 can be seen T t =1 T

5 as correcting for any bias in σ 22 which has as its source the variance of εt for finite n. Since εt is o(1), σ 12 and σ 22 should tend to equality as n → ∞ . (vi)

Since we have two consistent estimators of σ2, we can define two asymptotically equivalent weighted least squares estimators of β.

{(σˆ + 1 np (1 − p )) ,..., (σˆ + 1 np (1 − p )) } and D = diag. {(σ + 1 np (1 − p )) ,..., (σ + 1 np (1 − p )) } , −1 2

2 1

Let D1 = diag.

1

−1 2

2 2

2

1

1

1

−1 2

2 1

T

T

−1 2

2 2

T

T

then we can define the following WLS estimators of β: −1 β = (X ′D D X) X ′D D q , 1

1

1

1

1

(10)

and

−1 β 2 = (X ′D 2 D 2 X) X ′D 2 D 2 q The following results hold for β 1 and β 2 (appendix):



(11)

−1 

( T)1 2 (β 1 − β) → N 0, p lim (X ′D1 D1 X T)  , as n → ∞ and T → ∞ , D





n→∞ ,T→∞

and 

−1 

( T)1 2 (β 2 − β) → N  0, p lim (X ′D 2 D 2 X T)  , as n → ∞ and T → ∞ . D





n→∞ ,T→∞

Monte Carlo results which are designed to investigate the dependence on n and T of the convergence in probability of σ 12 , σ 22 , β 1 and β 2 , and the convergence in distribution of standardised versions of β and β are presented in the next section. 1

3

2

Monte Carlo results

This section reports Monte Carlo results for the estimators ρ and β F which are associated with (5) when νt follows a stationary AR(1) process, and the estimators σ 12 , σ 22 , β 1 and β 2 which are associated with (5) when νt is iid. Monte Carlo results for ρ and β F The purpose of the Monte Carlo results for ρ and β F is to investigate the dependence on n and T of the convergence in probability of ρ and β and the convergence in F

distribution of β F . The Monte Carlo results which are presented in Table 1 are based on 1000 replications of ρ and β for selected values of n and T in the range n=10 to n=350 F

and T=10 to T=200. The Monte Carlo study is based on the following probability model:

{

}

(

(

))

Pt = Pr ob. Yit = 1 ν t = 1 1 + exp − (β + ν t ) , i = 1,..., n; t = 1,..., T ,

(12)

6 ν t = ρν t −1 + η t ,

(13)

where ηt is iid N(0,1), β=0.2 and ρ=0.5. For each n and T, each of the 1000 replications involved: (i)

Generating 50+T observations on νt, with the first observation drawn randomly from a N(0,1.33) distribution and the remaining observations generated using (13) with ηt an iid N(0,1) random variable. The first 50 observations were then discarded to reduce the impact of the first observation on the νt used in the study (Hall and McAleer, 1987, p.9). The remaining T observations are the νt, t=1,…,T used in the study. From (12), the νt, t=1,…,T determine Pt, t=1,…,T.

(ii)

For each t, n realisations of a Bernouilli random variable drawn randomly from a distribution with parameter Pt, {yit} were obtained, and the sample proportion, pt =

n ∑y it i =1

/ n calculated. The {pt} were then used to calculate the logits

(

)

q t = log p t (1 − p t ) . (iii)

, ,...,1) . The residuals Define the 1×T vectors: q ′ = (q 1 , q 2 ,..., q T ) and x ′ = (11 from the OLS regression of q on x were used to compute ρ and σ 2 . Using: q, x, ρ , σ 2 and {pt}, the feasible GLS estimator of β, −1  1    x β F =  x ′ ∆ −1 + Ω    n

−1

−1

1   q and the standardised feasible GLS x ′ ∆ −1 + Ω n 

−1   1 −1 ~   12     , where w  =  x ′ ∆ + Ω x estimator of β: β = β F − 0.2 w    n

(

)

−1

were

calculated.4 The theoretical results outlined in section 2 indicate that β F and ρ converge in ~ probability to β and ρ respectively and that β converges in distribution to a N(0,1) random variable as n → ∞ and T → ∞ . ~ After 1000 replications, we have, for given n and T: ρ k , β Fk and β k , k=1,…,1000. The sample means and standard deviations of { ρ k } and { β Fk } were calculated: β=

ρ=

1000  ∑ β Fk k =1

1000 ∑

k =1

1000

ρ k 1000

and

and

(

1000 σ 1 =  ∑ β Fk − β  k =1

σ 2 =

)

(ρ k − ρ )

1000 ∑  k =1

2

1

2

2 999  , 

1 2 

999  , 

~ and a chi-squared goodness of fit statistic for testing the null hypothesis that { β k } are a random sample from a N(0,1) distribution was calculated as follows. A frequency ~ ~ distribution for { β k } was obtained by counting the number of { β k } which were

7 classified into each of the ten intervals: -1.5 to ≤ -1.0, > -1.0 to ≤ 0.5, > -0.5 to ≤ 0, > 0 to ≤ 0.5, > 0.5 to ≤ 1.0, > 1.0 to ≤ 1.5, > 1.5 to ≤ 2.0, > 2.0. Let " j ~ be the actual number of { β k } which fall into interval j and let Lj be the expected frequency for interval j associated with a random sample of size 1000 from a N(0,1) distribution, then the following statistic: X2 =

10 ∑ j=1

("

j

− Lj

)

2

Lj

~ has an asymptotic (k→∞) χ 29 distribution if { β k } is randomly drawn from a N(0,1) distribution (De Groot, 1975, pp.438-439). The values of X 2 , β , σ 1 , ρ and σ 2 obtained for the selected values of n and T are shown in Table 1. As noted above, the theoretical results indicate that β and ρ converge in k

k

probability to β and ρ respectively as n → ∞ and T → ∞ . Consistency would therefore be indicated if β approached β and σ 1 shrank, and ρ approached ρ and σ 2 shrank as n and T increased. From Table 1, we see that most of the values of β are close to β = 0.2 with some improvement as n and T increase, but the most striking feature is that for n fixed σ 1 falls monotonically with T; when T is fixed, σ 1 does not consistently fall monotonically with n, thus it seems T → ∞ is the critical factor affecting the consistency of β . F

Turning now to ρ and σ 2 , we see that ρ < ρ = 0.5 for all n and T shown in Table 1, suggesting that ρ is downward biased. When n is fixed and T is increased, ρ and σ 2 increase and fall (respectively) with T; when T is fixed and n is increased ρ tends to increase and σ 2 tends to fall with n. When n and T are increased, ρ increases rapidly and σ 2 falls rapidly with n and T. At low values of n and T, ρ seems to have a severe downward bias. For example, when T = 10 and n = 10, ρ = 0.113 and σ 2 = 0.321; when T = 30 and n = 50, ρ = 0.393 and σ 2 = 0.168; when T = 90 and n = 100, ρ = 0.448 and σ 2 = 0.097; when T = 160 and n = 250, ρ = 0.471 and σ 2 = 0.070; and when T = 200 and n = 350, ρ = 0.483 and σ 2 = 0.061. From this we may conclude that ρ underestimates ρ in finite samples and is severely biased when n and T are low.

~ In determining a range of n and T for which the distribution of β is close to its asymptotic distribution, we shall use the 1% critical value of the χ 29 distribution, which ~ is 21.67. If X2 > 21.67 we conclude that the distribution of β is a poor approximation of its asymptotic distribution. From Table 1, we note that X2 tends to be large with many values exceeding 21.67 when T < 100 and n < 100. When T ≥ 100 and n ≥ 100, few X2 statistics exceed 21.67. When T ≥ 160 and n ≥ 250 all X2 statistics are below 21.67. These results suggest that the convergence in distribution of β F does depend on n and T increasing and that the distribution of β closely approximates its asymptotic distribution F

when T ≥ 160 and n ≥ 250.

8

Monte Carlo results for σ 12 , σ 22 , β 1 and β 2 The purpose of the Monte Carlo results for σ 12 , σ 22 , β 1 and β 2 is to investigate the dependence on n and T of the convergence in probability of σ 2 , σ 2 , β and β , and the 1

2

1

2

convergence in distribution of β 1 and β 2 . The Monte Carlo results which are presented in Tables 2 and 3 are based on 1000 replications of σ 12 , σ 22 , β 1 and β 2 for selected values of n and T in the range n=10 to n=350 and T=10 to T=200. The Monte Carlo results are based on the probability model (12), with νt in (12) being iid N(0, 1.3333) and β=0.2. For each n and T, each of the 1000 replications involved: (i)

Generating T independent observations drawn from a N(0, 1.3333) distribution. From (12), the νt, t=1,…,T determine Pt, t=1,…,T.

(ii)

For each t, n realisations of a Bernouilli random variable drawn randomly from a distribution with parameter Pt, {yit} were obtained, and the sample proportion, pt =

n ∑y it i =1

/ n calculated. The {pt} were then used to calculate the logits

(

)

q t = log p t (1 − p t ) . (iii)

Define the 1×T vectors: q ′ = (q 1 , q 2 ,..., q T ) and x ′ = (11 , ,...,1) . The residuals from the OLS regression of q on x were used to compute σ 12 and σ 22 as defined by (8) and (9) respectively. The feasible weighted least squares estimators β and 1

β 2 , as defined by (10) and (11) were computed, as well as the standardised ~ ~ 12 12 1 2 , β1 = β 1 − 0.2 w β2 = β 2 − 0.2 w estimators, and where  1 = (x ′D 1 D 1 x ) w

(

−1

)

 2 = (x ′D 2 D 2 x) . and w

(

)

−1

The theoretical results outlined in section 2 indicate that σ 12 and σ 22 converge in ~ ~ probability to σ2, that β1 and β2 converge in distribution to N(0,1) as n → ∞ and T → ∞. ~ ~ After 1000 replications, we have for given n and T: σ 12k , σ 22 k , β 1k , β 2 k , β1k and β 2k , k=1,…,1000. The sample means and standard deviations of σ 12k , σ 22 k , β 1k and

{β } were calculated:

{ } { } { }

2k

12

σ12

2 1000  = ∑ σ 12k 1000 and sd σ 12 =  ∑ (σ 12k − σ12 ) 999  k =1  k =1 

σ 22

2 1000  = ∑ σ 22 k 1000 and sd σ 22 =  ∑ (σ 22 k − σ 22 ) 999  k =1  k =1 

1000

1000

, 12

,

9

(

1000 β1 = ∑ β 1k 1000 and sd β1 =  ∑ β 1k − β1  k =1 k =1 1000

(

)

1000 1000 β2 = ∑ β 2 k 1000 and sd β 2 =  ∑ β 2 k − β2  k =1 k =1

 999  

2

)

12

 999  

2

, 12

.

{ }

~ Also, chi squared goodness of fit statistics for testing the null hypotheses that β1k and ~ β 2k are random samples from a N(0,1) distributions were calculated. A frequency ~ ~ distribution for { β1k } was obtained by counting the number of { β1k } which were classified into each of the ten intervals: -1.5 to ≤ -1.0, > -1.0 to ≤ 0.5, > -0.5 to ≤ 0, > 0 to ≤ 0.5, > 0.5 to ≤ 1.0, > 1.0 to ≤ 1.5, > 1.5 to ≤ 2.0, > 2.0. Let " 1j ~ be the actual number of { β1k } which fall into interval j and let Lj be the expected frequency for interval j associated with a random sample of size 1000 from a N(0,1) distribution, then the following statistic:

{ }

10

(

X12 = ∑ " 1 j − L j j=1

)

2

Lj

~ has an asymptotic (k→∞) χ 29 distribution if { β1k } is randomly drawn from a N(0,1) distribution. ~ ~ A frequency distribution for { β 2k } was also obtained by counting the number of { β 2k } ~ which were classified into each of the ten intervals used for classifying { β1k } as ~ described above. Let " 2 j be the actual number of { β 2k } which fall into interval j, then the following statistic: 10

(

X 22 = ∑ " 2 j − L j j=1

)

2

Lj

~ has an asymptotic (k→∞) χ 29 distribution if { β 2k } is randomly drawn from a N(0,1) distribution. As noted in section 2, σ 12 is the estimator of σ2 proposed by Amemiya and Nold (1975), and this estimator attempts to correct σ 22 for any bias caused by ε t ≠ 0 in finite samples. ~ The WLS estimators β and β are computed using weights based on σ 2 as described in 1

1

section 2. The Monte Carlo results for

1

σ 12k , σ12

and

sd σ 12 ,

for β 1k , β1 and sdβ1 as well as

X12 are listed in Table 2. ~ The WLS estimators β 2 and β2 are computed using weights based on σ 22 as described in section 2. The Monte Carlo results for σ 2 , σ 2 and sd σ 2 , for β , β and sdβ2 as well 2k

as

X 22

are listed in Table 3.

2

2

2k

2

10 The results presented in Table 2 will be discussed first and then we shall discuss any notable differences between the results in Table 2 and Table 3. Any differences can be attributed to not correcting σ 22 for any bias in estimating σ2 which may be attributed to ε t ≠ 0 in finite samples. Since ε t converges in probability to 0 as n → ∞ , we would expect the results in Table 3 to approach those of Table 2 as n becomes large for fixed T. The values of β1 , sd β1 , σ12 , sd σ 12 and X12 are shown in Table 2. The theoretical results outlined in section 2 indicate that β 1 and σ 12 converge in probability to β and σ 2 as n → ∞ and T → ∞ . Consistency would therefore require that β1 approached β and sd β1 shrank and σ12 approached σ 2 and sd σ 12 shrank as n and T increased. From Table 2 we see that most values of β1 are close to β=0.2 with some improvement as n and T increased. As for sd β1 , we see that for fixed T, sd β1 tends to fall with n, the fall in sd β1 is however weak and not monotonic; for fixed n, sd β1 falls strongly with T, but the fall is not always monotonic; and sd β1 falls strongly as n and T increase but again the fall is not always monotonic. Thus it seems that the most important factor in the consistency of β 1 is T increasing. From Table 2, we see that most values of σ12 are close to σ 2 = 1333 . with some improvement as n and T increase. As for sd σ 12 , for fixed T sd σ 12 tends to fall with n, the fall in sd σ 12 is however weak and not monotonic; for fixed n, sd σ 12 falls strongly with T, but the fall is not always monotonic; and sd σ 12 falls strongly as n and T increase, but again the fall is not always monotonic. Thus it seems that the most important factor in the consistency of σ 12 is T increasing. ~ In determining the range of n and T for which the distribution of β1 is close to its asymptotic distribution, we shall use the 1% critical value of the χ 29 distribution, which ~ is 21.67. If X 12 > 2167 . we conclude that the distribution of β1 is a poor approximation of its asymptotic distribution. From Table 2, we see that no chi-square statistics exceed 21.67 when T ≥ 20 and n ≥ 150 . This suggests that the distribution of β 1 closely approximates its asymptotic distribution when T ≥ 20 and n ≥ 150 . Turning to Table 3, we see that β 2 is close to β=0.2 with some improvement as n and T increase and that sdβ2 falls with n and T but that the most important factor affecting the consistency of β 2 is T increasing. The range of n and T for which X 22 < 2167 . is T ≥ 20 and n ≥ 150 , so that we may conclude that the distribution of β closely approximates 2

its asymptotic distribution when T ≥ 20 and n ≥ 150 . Thus, the asymptotic properties of consistency and convergence in distribution of β 2 are similar to those of β 1 , suggesting that not correcting σ 22 for any bias due to ε t ≠ 0 in finite samples does not affect the consistency and convergence in distribution of the WLS estimator of β computed using weights based on σ 22 .

11

As expected, σ 22 > σ12 which follows directly from the manner in which these statistics . , suggesting that were constructed. For low n and T, σ 22 is much larger than σ 2 = 1333 2 2 σ 2 is an upward biased estimator of σ , with the bias being pronounced for low n and T. From this we may conclude that in finite samples, σ 12 is a better estimator of σ 2 than is σ 22 , but that not correcting σ 22 for any bias that may result because ε t ≠ 0 in finite samples does not affect the asymptotic properties of the WLS estimator of β which is computed using weights based on σ 22 . In comparing the results presented in Table 3 with those presented in Table 2, it may be noted that the results in Table 2 approach those in Table 3 as n increases for fixed T. Thus the results tabulated in Tables 2 and 3 behave as predicted by theory in this respect.

4

Summary

Some Monte Carlo results on the dependence on n and T of the convergence in probability and distribution of estimators of key parameters of the modified logit model were presented in this paper. In the case where νt follows a stationary AR(1) process, it was found that ρ was downward biased in finite samples, severely so for low n and T. The consistency of ρ depends on both n and T increasing. It was also found that for the feasible GLS estimator of β, β consistency depended on both n and T increasing, but F

mainly on T increasing, and that β F closely approximated its asymptotic distribution for n and T in the range T ≥ 160 and n ≥ 250 . In the case where νt is iid, two consistent estimators of σ2, σ 12 and σ 22 are available. The estimator σ 12 is the one proposed by Amemiya and Nold and is computed by correcting σ 22 for any bias caused by ε t ≠ 0 in finite samples. It was found that the asymptotic properties of the feasible weighted least squares estimators of β, β and β (where β is 1

2

1

computed using in forming the weights and β 2 is computed using σ 22 in forming the weights) are not sensitive to which estimator of σ2 is used in forming the weights. σ 12

The consistency of σ 12 and β 1 depends on n and T increasing, but T increasing seems to have the major influence. The asymptotic distribution of β is closely approximated for n 1

and T in the range T ≥ 20 and n ≥ 150 . By construction σ 22 > σ 12 , and as expected σ 22 approaches σ 12 as n and T increase; σ 22 appears to be upward biased, and the bias is severe for low n and T. Thus it seems that σ 12 is a better estimator of σ 2 but that the choice of σ 12 or σ 22 in forming the weights for the feasible WLS estimator of β does not affect the asymptotic properties of the feasible WLS of β in any important way.

12

Appendix: Proofs In this appendix proofs of the consistency of σ 12 and the asymptotic normality of β 1 are provided. Before proceeding to the proofs, we need the three Lemmas outlined below. Lemma 1: Let Zn,T (n = 1, 2, ….; T = 1, 2, …) be a double sequence of random variables P Z uniformly in T as n → ∞ , (ii) Z → P Z as T → ∞, and assume: (i) Z → n,T

T

T

P Z as n → ∞ , T → ∞. then: (iii) Z n , T → Lemma 2: Let Zn,T = Xn,T + Yn,T (n = 1, 2, ...; T = 1, 2, ...) be a double sequence of P 0 uniformly in T as n → ∞ , random variables. The following conditions: (i) X n,T → P Y uniformly in T as n → ∞ , (iii) Y → D Y as T → ∞, imply: (ii) Y → n ,T

(iv) Z n,T

T

T

D Y as n → ∞, T → ∞. →

The proofs of Lemma 1 and Lemma 2 may be found in Alaouze (1998,a). Lemma 3: Let: ( Z (n1,)T , Z (n2,T) ,..., Z (nk,T) ), n = 1, 2, ...; T = 1,2, ... be a double sequence of vector random variables, and let for any real λ i , i = 1,..., k not all zero: (1) (2) (k) D λ 1 Z n,T + λ 2 Z n,T + ,...,+ λ k Z n,T → λ 1 Z (1) + λ 2 Z ( 2 ) + ,...,+ λ k Z ( k )

as n → ∞, T → ∞; where (Z(1), ..., Z(k)) have a joint distribution, F(z1, z2,...,zk). Then the limiting joint distribution function of ( Z (n1,)T , Z (n2,T) ,..., Z (nk,T) ) exists and is equal to F(z1, z2,...,zk). Proof: The proof is similar to the proof of Theorem (xi) in Rao (1973, p. 123). Replace n by n,T in Rao's proof and take the limits (in Rao's proof) as n → ∞ and T → ∞. Consistency of σ 12 We take as a starting point the linear model (5) of section 2: q = Xβ + ε + ν

(A1)

with E( ν) = 0 , E(ν 2 ) = σ 2 , E(ν 4 ) < ∞ and lim X ′X T = Q , where Q is a positive T→∞

definite matrix. Let b be the OLS estimator of β in A1, then the T×1 vector of OLS residuals may be written as u = X( b − β) + ε + ν = Mε + Mν where M = I − X( X ′X) −1 X ′ and I is a T×T

(

)

identity matrix. Let ut be element t of u, then from (8): σ 12 =

T 1 2 u ∑ t − ∑ 1 np t (1 − p t ) T t =1

(A2)

13 T P P P 2 ~ as n → ∞ As n → ∞ , u → Mν and ∑ 1 np t (1 − p t ) → 0 for any T ≥ k , thus σ 12 → σ 1 t =1

for any T ≥ k , where:

T ~ 2 = 1 ∑ ν 2 − ν ′X  X ′X  σ 1 T t =1 t T  T 

−1

X ′ν . T

(A3)

The second term in (A3) converges in probability to zero as T → ∞ and the first term converges in probability to σ2, thus, P ~ 2 → σ 2 as T → ∞ . σ 1

(A4)

P Thus, from Lemma 1, σ 12 → σ 2 as n → ∞ and T → ∞ . To prove the consistency of σ 22 , 1 T replace (A2) with σ 22 = ∑ u 2t and proceed as for the proof of the consistency of σ 12 . T t =1 Asymptotic distribution of β 1   1 −1  From (10), β 1 = (X ′D1 D1 X) X ′D1 D1q , where D1 = diag.  σ 12 +  np p 1 −  1( 1 )   1  σ 12 +    np 1 − p ( )  T T 





1 2

,...,

1 2

  . Now write β 1 as: 

−1 −1 β 1 = β + (X ′D1 D1 X) X ′D1 D1ε + (X ′D1 D1 X) X ′D1 D1 ν .

(A5)

As n → ∞ , the second term on the RHS of (A5) converges in probability to zero, and P ~ )−1 I for any T ≥ k . Thus, as n → ∞ for T ≥ k , D1 →(σ 1 P β 1 − β →( X ′X) −1 X ′ν .

(A6)

Now Theil (1971, pp.378-381) shows that:

(

)

1 D T 2 β − β → N(0, σ 2 Q −1 )

(

as T → ∞ ,

(A7)

)

−1 where β − β = ( X ′X) X ′ν and 0 is k×1 vector with zero elements. Now for any non-

zero, k×1 vector λ 1 2 T λ′

(β − β) → N(0, σ λ ′Q λ) D

2

−1

as T → ∞ .

(A8)

14

From Lemma (2), Lemma 3,

1 2 T

(β

1

(

1 2 T λ′

(β

)

D N(0, σ 2 λ ′Q −1λ ) as n → ∞ and T → ∞ , and from − β → 1

)

D − β → N(0, σ 2 Q −1 ) as n → ∞ and T → ∞ . Thus in finite but large

)

−1 samples, β 1 ~ N β, (X ′D1 D1 X) approximately. The asymptotic normality of β 2 may

be proved in the same manner.

15

Endnotes 1

The papers Alaouze (1998,a) and Alaouze (1998,b) together constitute a revised version of a paper with the same title as Alaouze (1998,a) which was presented to the 1994 Australasian Meeting of the Econometric Society held at the UNE, Armidale, NSW in July. The first Draft of the 1994 paper was written while the author was on study leave in the Department of Econometrics at the University of Sydney. 2

Amemiya and Nold (1975) work with (equation 2, p. 255): log( p t / (1 − p t )) = β ′x t + α t + ν t

where αt = (pt - Pt)/Pt(1 - Pt), instead of (2). However, αt and εt are asymptotically equivalent as n → ∞ , so for all practical purposes, (2) is equivalent to the linear model developed by Amemiya an Nold. This assumption is necessary because the moments of ε do not exist in finite samples. This follows because in finite samples, the moments of ε are determined using the binomial probability distribution and there is a non-zero probability that no positive responses occur or that all responses are positive, in which events log (pt/(1 - pt)) is unbounded or undefined. The practice of working with the asymptotic distribution of n1 2 ε when n is large seems to be standard, see for example Theil (1970) p. 145. 3

4

In 1000 replications it is inevitable that pt = 1 or pt = 0 will occur, in which case q t = log p t (1 − p t ) and diagonal element t of ∆ −1 / n , δ t = 1 np t (1 − p t ) cannot be

(

calculated.

(

)

In

this

event,

)

Z t = log (p t + 1 2 n) (1 − p t + 1 2 n)

[

Cox

(1970,

pp.33-34)

recommends

that and

]

Vt = (1 + 1 n)(1 + 2 n) n(p t + 1 n)(1 − p t + 1 n) be used as substitutes for qt and δt respectively. The rationale for this is that Zt and Vt are computable when pt = 1 or pt = 0

(

)

and that E(Zt) is a close approximation of log (Pt ) (1 − Pt ) and E(Vt) is a close

approximation of the variance of Zt. In the Monte Carlo study, whenever pt = 1 or pt = 0 occurred in a replication, Zt and Vt were calculated instead of qt and δt.

16

References Alaouze, C.M. (1998,a), “A modified logit model for time series with an application to the pricing behaviour of manufacturing firms in Australia”, School of Economics, UNSW, February. Alaouze, C.M. (1998, b), “Proofs of some asymptotic results stated in “A modified logit model for time series with an application to the pricing behaviour of manufacturing firms in Australia”, School of Economics, UNSW, January. Amemiya, T and F.C. Nold (1975), "A Modified Logit Model", Review of Economics and Statistics, (57) pp. 255-257. Berkson, J. (1955), "Maximum Likelihood and Minimum Chi-Square Estimates of the Logistic Function", Journal of the American Statistical Association, (50) pp. 130-162. Bishop, M., S. Fienberg and P. Holland (1975), Discrete Multivariate Analysis, Cambridge: MIT Press. Cox, D.R. (1970), Analysis of binary data, London: Chapman and Hall. De Groot, M.H. (1975), Probability and statistics, Menlo Park: Addison-Wesley. Hall, A.D. and M. McAleer (1987), “A Monte Carlo study of some tests of model adequacy in time series analysis”, Australian National University, Department of Econometrics, Working Paper No.148, March 1987. Rao, C.R. (1973), Linear statistical inference and its applications, New York: Wiley. Theil, H. (1971), Principles of econometrics, New York: Wiley. Theil, H. (1970), “On the estimation of relationships involving qualitative variables”, American Journal of Sociology, (76) pp.103-154.

17 Table 1: Monte Carlo Results for Time Series Logit n

10

20

50

100

150

200

250

300

350

X2

141.35**

272.98**

618.52**

733.61**

807.45**

913.23**

1031.84**

925.21**

1052.61**

β

0.204

0.172

0.232

0.218

0.202

0.199

0.222

0.185

0.169

0.577

0.568

0.585

0.573

0.579

0.596

0.583

0.585

0.604

0.113

0.170

0.190

0.226

0.203

0.230

0.219

0.232

0.217

T

10

σ1 ρ

20

2 σ X2

0.321

0.302

0.294

0.295

0.289

0.298

0.297

0.301

0.311

22.99**

52.00**

128.73**

139.03**

133.24**

292.30**

277.26**

215.35**

317.40**

β

0.184

0.182

0.174

0.187

0.205

0.198

0.207

0.205

0.196

0.428

0.433

0.428

0.419

0.422

0.439

0.429

0.425

0.426

0.231

0.285

0.327

0.352

0.343

0.341

0.349

0.355

0.363

0.226

0.208

0.205

0.212

0.210

0.211

0.218

0.211

0.210

63.26**

52.23**

81.73**

150.03**

144.49**

136.76**

117.26**

σ1 ρ

2 σ X2 β

30

σ1 ρ

2 σ X2 β

60

σ1 ρ

2 σ X2 β

90

σ1 ρ

100

16.75

0.176

0.183

0.187

0.210

0.219

0.196

0.192

0.189

0.186

0.343

0.342

0.352

0.344

0.349

0.359

0.356

0.361

0.358

0.269

0.327

0.375

0.393

0.400

0.399

0.405

0.413

0.417

0.188

0.175

0.171

0.168

0.174

0.172

0.169

0.166

0.165

23.91**

34.75**

30.81**

35.89**

23.30**

18.33

11.91

10.34

19.83

0.203

0.202

0.188

0.199

0.192

0.204

0.204

0.184

0.202

0.261

0.250

0.246

0.255

0.252

0.253

0.247

0.249

0.255

0.316

0.368

0.415

0.435

0.437

0.455

0.451

0.450

0.450

0.131

0.121

0.119

0.116

0.117

0.117

0.112

0.115

0.115

26.65**

9.63

32.71**

38.38**

8.77

31.60**

0.183

0.187

0.191

0.201

0.196

0.196

0.190

0.206

0.215

0.217

0.199

0.201

0.215

0.199

0.211

0.214

0.195

0.203

0.326

0.386

0.430

0.448

0.459

0.459

0.464

0.465

0.465

0.104

0.100

0.094

0.097

0.092

0.098

0.099

0.094

0.093

13.14

15.74

15.07

2 σ X2

29.35**

β

0.192

0.188

0.194

0.198

0.200

0.201

0.196

0.204

0.203

0.188

0.187

0.188

0.189

0.191

0.196

0.197

0.194

0.202

0.331

0.391

0.436

0.454

0.459

0.463

0.467

0.467

0.471

0.102

0.095

0.090

0.092

0.091

0.088

0.086

0.089

0.086

σ1 ρ

2 σ X2 β

110

9.17

σ1 ρ

21.25

15.93

14.20

14.53

14.50

8.35

15.84

11.66

15.03

10.82

14.44

8.16

25.48**

20.95

18.87

16.91

8.34

0.192

0.183

0.201

0.198

0.199

0.191

0.190

0.205

0.209

0.192

0.184

0.180

0.194

0.185

0.191

0.195

0.178

0.188

0.336

0.386

0.437

0.451

0.464

0.464

0.466

0.471

0.478

0.084

0.085

0.085

0.085

0.088

2 σ 0.095 0.090 0.091 0.085 Note: ** indicates significance at the 1% level.

18 Table 1 (Continued) n

10

20

50

100

150

200

250

300

350

X2

5.46

7.44

23.88**

10.69

10.84

11.00

11.58

5.86

8.07

β

0.195

0.195

0.193

0.202

0.201

0.205

0.203

0.200

0.191

0.183

0.176

0.174

0.178

0.182

0.183

0.178

0.181

0.174

0.336

0.387

0.435

0.456

0.465

0.466

0.475

0.472

0.475

0.082

0.082

T

120

σ1 ρ

130

2 σ X2

0.089

0.089

0.084

0.083

0.084

0.078

0.081

37.03**

26.53**

4.51

7.82

7.14

8.20

9.84

β

0.187

0.197

0.197

0.200

0.199

0.198

0.199

0.194

0.197

0.165

0.167

0.168

0.172

0.169

0.165

0.167

0.170

0.176

0.339

0.388

0.436

0.461

0.462

0.464

0.474

0.471

0.468

0.084

0.079

0.076

0.079

0.075

0.080

0.080

2.69

8.84

22.25**

9.72

σ1 ρ

140

0.087

0.083

25.79**

24.12**

β

0.186

0.185

0.195

0.194

0.205

0.192

0.199

0.190

0.205

0.163

0.162

0.155

0.163

0.163

0.165

0.163

0.176

0.163

0.339

0.397

0.443

0.466

0.470

0.470

0.473

0.480

0.474

0.084

0.083

0.078

0.076

0.075

0.074

0.074

0.077

0.077

27.95**

9.68

7.00

9.99

6.89

9.79

0.189

0.188

0.196

0.199

0.189

0.199

0.200

0.204

0.198

0.158

0.149

0.152

0.153

0.159

0.149

0.151

0.153

0.154

0.347

0.396

0.440

0.464

0.470

0.471

0.476

0.475

0.476

0.072

0.070

σ1 2 σ X2 β

σ1 ρ

180

17.98

18.38

16.57

20.20

12.66

18.74

2 σ X2

0.076

0.073

0.070

0.070

0.072

0.070

24.51**

47.03**

6.07

8.59

8.74

24.39**

β

0.190

0.184

0.195

0.203

0.204

0.184

0.202

0.191

0.204

0.147

0.144

0.144

0.145

0.149

0.144

0.152

0.140

0.143

0.344

0.398

0.447

0.465

0.472

0.476

0.480

0.481

0.479

0.075

0.074

0.068

0.065

0.068

0.062

0.069

0.068

0.068

28.26**

9.12

7.79

6.45

7.88

σ1 ρ

200

22.43**

2 σ X2

ρ

160

10.01

12.47

8.51

0.072 12.17

2 σ X2

48.49**

β

0.185

0.186

0.188

0.202

0.198

0.202

0.200

0.194

0.199

0.131

0.143

0.134

0.138

0.134

0.144

0.138

0.139

0.139

0.341

0.398

0.449

0.468

0.472

0.475

0.482

0.480

0.483

0.065

0.061

0.062

0.064

0.061

σ1 ρ

16.52

2 σ 0.074 0.068 0.063 0.065 Note: ** indicates significance at the 1% level.

11.21

10.55

19 Table 2: Monte Carlo Results for WLS, Amemiya-Nold Weights n T

X 12 β1

10

sd β1

σ12 sd σ 1

2

X 12

250

300

350

18.29

13.79

16.68

18.64

46.10**

29.70**

44.09**

44.01**

92.11**

0.182

0.186

0.178

0.193

0.198

0.204

0.221

0.194

0.195

0.381

0.359

0.356

0.354

0.376

0.364

0.365

0.365

0.374

1.183

1.278

1.287

1.245

1.226

1.219

1.182

1.225

1.195

0.709

0.751

0.683

0.645

0.617

0.599

0.571

0.577

0.600

17.88

13.58

25.95**

10.75

12.75

11.21

7.95

20.88

0.194

0.187

0.201

0.189

0.207

0.275

0.267

0.266

0.265

0.248

0.257

0.257

0.251

0.264

1.274

1.409

1.357

1.325

1.299

1.287

1.272

1.279

1.293

0.511

0.539

0.504

0.486

0.445

0.457

0.435

0.429

0.431

23.85**

5.80

3.54

9.17

0.182

0.180

0.192

0.195

0.193

0.188

0.205

0.203

0.198

0.231

0.216

0.214

0.209

0.213

0.216

0.213

0.212

0.211

1.332

1.440

1.389

1.346

1.333

1.322

1.294

1.304

1.309

sd σ 1

0.427

0.450

0.453

0.404

0.387

0.356

0.342

0.362

0.369

X 12

43.71**

53.06**

24.03**

7.61

7.34

5.63

0.177

0.179

0.190

0.196

0.194

0.198

0.200

0.200

0.190

0.160

0.149

0.145

0.146

0.151

0.148

0.157

0.146

0.150

1.350

1.475

1.416

1.360

1.337

1.334

1.339

1.324

1.323

sd σ 1

0.297

0.324

0.296

0.271

0.266

0.261

0.258

0.249

0.250

X 12

66.50**

36.96**

0.177

0.185

0.195

0.189

0.199

0.202

0.192

0.201

0.199

0.129

0.122

0.120

0.123

0.123

0.125

0.130

0.128

0.118

1.364

1.482

1.424

1.363

1.372

1.349

1.343

1.343

1.328

sd σ 1

0.251

0.274

0.262

0.235

0.230

0.212

0.206

0.206

0.200

X 12

56.62**

29.19**

25.33**

0.174

0.186

0.189

0.196

0.208

0.192

0.203

0.203

0.197

0.123

0.119

0.114

0.115

0.112

0.116

0.117

0.120

0.118

1.366

1.481

1.435

1.377

1.361

1.342

1.334

1.334

1.332

sd σ 1

0.230

0.251

0.252

0.213

0.209

0.209

0.190

0.196

0.196

X 12

54.36**

53.62**

22.58**

0.181

0.182

0.194

0.192

0.206

0.199

0.200

0.200

0.196

0.113

0.111

0.109

0.115

0.111

0.111

0.105

0.108

0.108

1.361

1.484

1.417

1.376

1.355

1.350

1.350

1.337

1.337

0.225

0.237

0.219

0.202

0.199

0.195

0.187

0.196

0.193

16.51

β1

sd β1

σ12 2

β1

sd β1

σ12 2

β1

sd β1

σ12 2

β1

sd β1

σ12 2

β1

110

200

0.199

X 12

100

150

0.196

2

90

100

0.179

sd σ 1

60

50

0.190

sd β1

σ12

30

20

11.71

β1

20

10

sd β1

σ12 sd σ 1

2

21.49

Note: ** indicates significance at the 1% level.

9.46

11.95

12.19

16.09

16.21

16.10

15.50

8.66

5.52

3.33

5.79

11.05

16.33

12.79

15.37

10.03

4.88

12.42

11.28

13.22

16.81

15.13

7.73

11.46

12.47

20 Table 2 (Continued) n T

X 12 β1

120

sd β1

σ12 2 sd σ 1 X 12

β1

130

sd β1

σ12 2 sd σ 1 X 12

β1

140

sd β1

σ12 2 sd σ 1 X 12

β1

160

sd β1

σ12 2 sd σ 1 X 12

β1

180

sd β1

σ12 2 sd σ 1 X 12

β1

200

sd β1

10

20

50

100

150

200

250

300

350

68.97**

38.98**

26.83**

6.62

7.06

6.28

9.15

4.88

9.79

0.177

0.186

0.193

0.202

0.195

0.193

0.199

0.204

0.200

0.109

0.108

0.101

0.106

0.103

0.106

0.100

0.103

0.105

1.370

1.487

1.426

1.380

1.365

1.354

1.348

1.344

1.334

0.207

0.227

0.220

0.193

0.190

0.189

0.181

0.184

0.179

65.93**

59.29**

25.03**

7.21

9.57

5.26

3.41

7.12

0.177

0.182

0.189

0.198

0.200

0.198

0.195

0.196

0.200

0.108

0.100

0.100

0.098

0.101

0.102

0.100

0.105

0.101

1.374

1.500

1.428

1.378

1.369

1.342

1.344

1.345

1.338

0.203

0.226

0.215

0.186

0.176

0.171

0.173

0.171

0.162

44.04**

23.75**

33.88**

8.88

4.96

0.186

0.190

0.188

0.198

0.198

0.199

0.197

0.196

0.199

0.104

0.103

0.095

0.097

0.098

0.100

0.096

0.098

0.093

1.380

1.483

1.429

1.380

1.359

1.356

1.343

1.343

1.337

0.197

0.207

0.204

0.185

0.185

0.166

0.170

0.172

0.165

49.73**

39.48**

38.67**

0.180

0.188

0.188

0.195

0.202

0.200

0.200

0.203

0.197

0.099

0.094

0.088

0.090

0.091

0.088

0.091

0.090

0.088

1.372

1.494

1.423

1.386

1.361

1.349

1.357

1.346

1.341

0.180

0.200

0.184

0.170

0.163

0.153

0.156

0.146

0.158

99.15**

48.62**

8.49

9.11

9.61

9.77

0.178

0.185

0.194

0.199

0.197

0.200

0.200

0.197

0.203

0.088

0.086

0.088

0.087

0.084

0.086

0.085

0.085

0.081

1.384

1.483

1.432

1.382

1.368

1.352

1.348

1.340

1.340

0.172

0.181

0.178

0.163

0.150

0.146

0.141

0.149

0.153

78.17**

45.16**

30.12**

21.86**

7.024

5.65

6.16

0.178

0.184

0.195

0.192

0.195

0.200

0.200

0.202

0.196

20.90

10.44

19.75

6.62

17.55

13.47

10.19

15.23

4.20

4.71

17.24

10.62

14.67

15.10

11.28

19.13

0.085

0.086

0.080

0.081

0.080

0.078

0.081

0.084

0.080

σ12

1.385

1.489

1.434

1.381

1.359

1.360

1.353

1.344

1.335

2 sd σ 1

0.161

0.169

0.163

0.145

0.140

0.135

0.134

0.138

0.135

Note: ** indicates significance at the 1% level.

21 Table 3: Monte Carlo Results for WLS n T

10

10

20

50

100

150

200

250

X 22

56.53**

19.26

9.89

13.96

30.01**

24.19**

41.37**

37.14**

78.76**

β2

0.192

0.192

0.179

0.193

0.198

0.205

0.221

0.194

0.195

0.400

0.368

0.358

0.355

0.377

0.364

0.365

0.365

0.374

1.819

1.609

1.417

1.308

1.268

1.249

1.206

1.245

1.212

0.788

0.816

0.715

0.661

0.628

0.606

0.576

0.581

0.604

57.91**

34.43**

8.72

6.64

0.199

0.183

0.197

0.200

0.195

0.187

0.201

0.189

0.207

0.287

0.273

0.268

0.265

0.248

0.257

0.258

0.251

0.264

1.911

1.744

1.489

1.389

1.340

1.318

1.297

1.299

1.311

0.567

0.585

0.528

0.498

0.452

0.462

0.439

0.432

0.434

45.69**

44.05**

6.85

4.37

3.84

8.61

0.190

0.184

0.193

0.195

0.193

0.188

0.205

0.203

0.198

0.241

0.220

0.215

0.209

0.213

0.216

0.213

0.212

0.211

1.971

1.776

1.520

1.409

1.375

1.352

1.319

1.324

1.327

0.472

0.489

0.474

0.413

0.393

0.360

0.345

0.365

0.372

79.15**

76.22**

32.48**

4.97

5.59

9.500

5.74

0.184

0.183

0.191

0.196

0.194

0.198

0.201

0.200

0.190

0.165

0.152

0.146

0.146

0.151

0.148

0.157

0.146

0.150

1.989

1.810

1.547

1.423

1.378

1.365

1.364

1.344

1.341

0.329

0.351

0.310

0.278

0.271

0.264

0.260

0.250

0.252

104.84**

69.29**

29.08**

0.184

0.188

0.196

0.190

0.199

0.202

0.192

0.202

0.199

0.133

0.124

0.121

0.124

0.123

0.125

0.130

0.128

0.118

2.002

1.817

1.555

1.426

1.414

1.380

1.368

1.363

1.346

0.278

0.297

0.275

0.241

0.234

0.214

0.208

0.207

0.202

87.48**

52.95**

34.56**

0.181

0.189

0.190

0.196

0.208

0.193

0.203

0.203

0.197

0.127

0.121

0.115

0.115

0.112

0.116

0.117

0.120

0.118

2.003

1.816

1.567

1.441

1.402

1.373

1.359

1.354

1.349

0.255

0.272

0.264

0.219

0.212

0.211

0.192

0.200

0.197

99.13**

80.60**

34.02**

8.93

0.187

0.186

0.195

0.192

0.206

0.199

0.200

0.200

0.196

sd β2

σ22 2 sd σ2 X 22

β2

20

sd β2

σ22 2 sd σ2 X 22

β2

30

sd β2

σ22 2 sd σ2 X 22

β2

60

sd β2

σ22 2 sd σ2 X 22

β2

90

sd β2

σ22 2 sd σ2 X 22

β2

100

sd β2

σ22 2 sd σ2 X 22

β2

110

sd β2

15.57

25.48**

11.87

10.75

14.72

10.82

10.76

14.37

16.45

15.18

10.03

14.14

7.19

4.95

3.70

16.27

11.53

18.21

300

11.61

9.77

14.21

350

18.50

18.21

15.26

7.47

10.15

15.10

0.117

0.113

0.110

0.116

0.111

0.111

0.105

0.108

0.108

σ22

1.999

1.819

1.548

1.439

1.396

1.380

1.375

1.357

1.354

2 sd σ2

0.250

0.257

0.229

0.207

0.202

0.197

0.189

0.197

0.194

Note: ** indicates significance at the 1% level.

22 Table 3 (Continued) n T

120

10

20

50

100

150

200

250

300

350

X 22

104.61**

61.67**

38.17**

8.91

9.59

7.75

11.51

5.75

10.87

β2

0.184

0.190

0.194

0.203

0.196

0.194

0.199

0.204

0.200

0.114

0.110

0.101

0.106

0.103

0.106

0.100

0.103

0.105

2.009

1.822

1.557

1.443

1.406

1.385

1.372

1.364

1.352

0.230

0.246

0.231

0.198

0.194

0.191

0.183

0.186

0.180

89.30**

87.01**

34.13**

5.92

3.39

9.05

0.183

0.186

0.190

0.198

0.200

0.199

0.196

0.197

0.200

0.112

0.102

0.100

0.098

0.102

0.102

0.100

0.105

0.101

2.013

1.836

1.559

1.441

1.410

1.373

1.369

1.366

1.355

0.226

0.245

0.225

0.191

0.179

0.173

0.175

0.173

0.163

84.93**

38.57**

41.63**

0.193

0.193

0.189

0.198

0.198

0.199

0.197

0.196

0.199

0.108

0.105

0.096

0.097

0.098

0.101

0.096

0.098

0.093

2.019

1.818

1.560

1.443

1.401

1.387

1.367

1.363

1.354

0.219

0.225

0.214

0.190

0.189

0.168

0.172

0.173

0.166

84.91**

64.37**

48.38**

23.81**

7.73

0.187

0.192

0.189

0.195

0.202

0.200

0.200

0.203

0.197

0.103

0.095

0.089

0.091

0.091

0.088

0.091

0.090

0.088

2.009

1.830

1.554

1.449

1.402

1.379

1.382

1.366

1.359

0.200

0.217

0.193

0.175

0.166

0.155

0.157

0.148

0.159

121.17**

69.03**

27.68**

7.62

8.51

0.184

0.188

0.195

0.199

0.197

0.200

0.200

0.197

0.203

0.091

0.088

0.088

0.089

0.084

0.086

0.085

0.085

0.081

2.024

1.818

1.564

1.445

1.409

1.383

1.373

1.360

1.357

0.191

0.196

0.186

0.167

0.152

0.147

0.142

0.150

0.154

110.34**

56.65**

41.55**

23.34**

7.50

4.79

7.38

0.185

0.188

0.196

0.192

0.195

0.200

0.200

0.202

0.196

sd β2

σ22 2 sd σ2 X 22

β2

130

sd β2

σ22 2 sd σ2 X 22

β2

140

sd β2

σ22 2 sd σ2 X 22

β2

160

sd β2

σ22 2 sd σ2 X 22

β2

180

sd β2

σ22 2 sd σ2 X 22

β2

200

sd β2

15.40

15.90

7.21

5.26

19.40

10.55

14.49

12.79

10.04

19.01

4.39

4.17

12.42

18.10

11.98

13.85

16.90

12.95

20.29

0.088

0.087

0.081

0.081

0.080

0.078

0.081

0.084

0.080

σ22

2.024

1.824

1.566

1.444

1.401

1.391

1.377

1.364

1.353

2 sd σ2

0.179

0.183

0.171

0.149

0.143

0.137

0.135

0.139

0.136

Note: ** indicates significance at the 1% level.