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R + \

for u = 1,2,...,— ---1. We should specify Q = V, where V is the sample variance matrix;■ix; that is, V = —1 / - z j \' / - z j \ with z = —1 , • Thus, for

^

det^K j

We will now focus on the problem o f bandwidth selection. We need the convergence rate o f h to be faster enough. To build up such convergence rates we will use the uniform convergence rates o f Bierens (1987). From the point of view o f uniform consistency and under conditions satisfied by the high-order bias reducing kernels o f Bierens,

(3.6)

is stochastically bounded. Clearly, the best uniform consistency rate is obtained for such that min^c^^ j is maximal. This is the case if

We then have m in(c/'-'V ÏV ,c„-(''*'>) oc

C H A PTER S. A NEW ESTIMATOR 6 9

= c - a value of R that should satisfy the inequality

R + \ > T - f .

Now, the bandwidth selection problem is reduced to choose the constant c. A natural way to proceed (Hardie and Linton (1994), Hardie (1990) and Silverman (1986)) is to choose c so as to minimise some kind o f measure o f the “distance” o f the estimator from the true value (according to some performance criterion). If we are, for example, interested in the quadratic loss o f the estimator at a single point z ,

which is measured by the mean squared error, MSe |^^ (z) | , then we will minimise

the MSB over c in order to get an approximately optimal value for c.

By following this minimisation method the optimal bandwidth depends on elements that we do not know unless we know the optimal bandwidth. In practice, these quantities have to be estimated on the basis o f some preliminary smoothing process which raises a second-order bandwidth selection problem. A first step bandwidth c should be chosen to estimate the quantities that determine the optimal bandwidth c*.

3.4 Single Estimates for the Whole Panel

In section 2 and 3 we have introduced the proposed second and first stage estimators. There, the analysis was based on two periods. In this section we will illustrate how to

CH A PTER S. A NEW ESTIMATOR 7 0

combine the estimators coming from each two waves o f the panel to come up with a single estimate.

The estimators used in the first step are obtained by maximising the likelihood

function o f ,

12V bivariate probits separately, each o f them based in a different combination o f two time periods. Once estimates o f the correction terms are included, equation (2.7) can be estimated using least squares for each combination o f panel

waves (/, 5^, / ^ 5^'; this gives a total o f pairs for a panel o f length T . A

minimum distance procedure, with the corresponding weighting matrix, can then be applied to combine these estimates. To estimate the weighting matrix an estimate for the covariance matrix o f the estimators for the different time periods is required. The block diagonal matrices are simply the corresponding covariance matrices estimates for each pair. To get the block off-diagonal matrices o f the weighting matrix we just need to combine the corresponding two influence functions for each combination o f two pairs. These covariances among pairs do not converge to zero. In the minimum distance step we restrict the estimates for fl to be the same for each combination

(

(t, 5) and we estimate x 2 coefficients for the correction terms in all the pairs

(two correction terms per pair)^^. The latter group o f parameters is left unconstrained in the minimum distance step. The number o f parameters associated to the correction

N o tice that for sam ple selection m odels w e gain in efficiency by considering all p ossible pairs in place o f just first differences. Different combinations o f individuals appear in different pairs because o f the observability rule

driven by = 1 or =0 .

In the minimum distance step w e can recover a time trend coefficient or time dum m ies coefficients (reflecting changing econom y conditions com mon to all individuals).

CHAPTER 3. A NEW ESTIM ATOR 71

terms is a function o f T and it grows faster than T. This is so because the estimator allows the variance o f the time varying errors in both equations to vary over time and it does not restrict the correlations between the time-differenced errors in the main equation and the errors in the selection rule to be time-invariant. As we focus on the case where the data consist o f a large number o f individuals observed through a small (fixed) number o f time periods and look at asymptotics as the number o f individuals approaches infinity the growth in parameters does not impose a problem, in principle.

Alternatively, for the more parametric first step, we can use a strategy that might asymptotically be more efficient. A minimum distance estimator can be

( A obtained from the

V27 sets o f bivariate probits estimates that we can form with a panel o f length T. Lambda terms based on the resulting estimates can be plugged into equation (2.7) that is again estimated by pairs. A second minimum distance step is computed in the same way it was applied for the estimator based on the results o f bivariate probits estimated separately. However, although this strategy might asymptotically be more efficient, the other one is easier from a practical point o f view and it still provides consistent estimates.

z'

To test for the assumption o f the x 2 correction terms being jointly

\2 J

significant is easily carried out by constructing a Wald statistic. This can be used as a test for selection bias. A test o f overidentifying restrictions in the minimum distance step can be also performed. The latter, in fact, implies testing whether the imposed restrictions (J3 being constant over time) cannot be rejected.