Estimation and Inference of Discontinuity in Density

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Assumption 1 is on the data structure. Although it is beyond the scope of this paper, it wouldbe interesting to extend the proposed method to weakly dependent data, where we are interestedin the discontinuity of the stationary distribution. For this extension, we would need to introduce ablocking technique to handle the time series dependence in the moment functions (see Kitamura, 1997).Assumption 2 restricts the local shape of the density function around x = c. This assumption allowsdiscontinuity of the density function at x = c. Assumption 3 is on the kernel function K and implies thesecond-order kernel. This assumption is satis…ed by e.g., the triangle kernel K (a) = max f0; 1jajg andEpanechnikov kernel K (a) = 3 4 1 a2 I fjaj 1g. Assumption 4 is on the bandwidth parameter h.The requirement nh 5 ! 0 corresponds to an undersmoothing condition to remove the bias component h i h i(f r f l ) E ^fr E ^fl in the construction of empirical likelihood. The requirement b=h ! 0 ison the bin width b used for the binning estimator. The requirement nh 3 ! 1 is required to obtainthe consistency of the minimizers to solve (12). If we do not have the local linear terms (X ic) and(u c) in (3), this requirement is unnecessary. Assumption 5 is similar to an assumption that wouldbe used in a parametric estimation problem.’Under these assumptions, the asymptotic distributions of the empirical likelihood functions `G ( 0 )and ` ( 0 ) evaluated at the true parameter value 0 are obtained as follows.Theorem. Under Assumption, as n ! 1`G ( 0 ) d ! 2 (1) ;` ( 0 ) d ! 2 (1) :Note that even in this nonparametric hypothesis testing problem, we observe the convergence of theempirical likelihood statistic to the pivotal 2 distribution, i.e., the Wilks phenomenon emerges. Thenull hypothesis H 0 : 0 = against H 1 : 0 6= for some given can be tested by the test statistic`G () or ` () with some 2 (1) critical value. For example, the null of density continuity is tested by`G (0) or ` (0). Also, by inverting these test statistics, the 100 (1 ) % empirical likelihood asymptoticcon…dence sets are obtained asCS G = : `G () 2 1 (1) ;CS = : ` () 2 1 (1) ;where 2 1 (1) is the (1 )-th quantile of the 2 (1) distribution.We now compare our empirical likelihood approach with the Wald approach suggested by McCrary(2008). First, in contrast to the t-test based on (5), the empirical likelihood test based on `G (0) or` (0) does not require any asymptotic variance estimation, which is automatically incorporated in theconstruction of the empirical likelihood functions. Also, while the Wald test requires the derivation ofthe asymptotic variance 2 Kfor each kernel function, the empirical likelihood tests do not require this10
type of derivation. Second, the empirical likelihood con…dence sets CS G and CS do not require thelocal linear (or polynomial) estimators ^f l and ^f r . Thus, even if ^fl or ^f r yields negative estimates in…nite samples, CS G and CS are well de…ned. Third, the empirical likelihood test statistics are invariantto the formulation of the nonlinear null hypotheses. For example, to test the density continuity, wemay specify the null hypothesis as H 0 : log f l = log f r , H0 ~ : f l = f r , H0 f: lf r= 1, etc. For thesehypotheses, the empirical likelihood test statistics are identical (i.e., `G (0) or ` (0)). On the otherhand, the Wald test statistic is not invariant to the formulation of the null hypotheses and may yieldopposite conclusions in …nite samples (see, Gregory and Veal, 1985, for examples).3 SimulationsIn this section we study the …nite-sample behaviors of the aforementioned methods using simulations.First we focus on the point estimators, i.e. the local linear binning estimator ^ G and the local (loglinear) likelihood estimator ^ for 0 .For comparisons, we also consider the local constant binningestimator ~ G and the local (log constant) likelihood estimator ~ . For the kernel function K, we usethe triangle kernel function K (a) = max f0; 1 jajg. For the bandwidth h, we consider both …xedbandwidths h = 1; 2; 3; 4 and data-dependent bandwidths h = h dd , where h dd is the data-dependentbandwidth used by McCrary (2008) and = 1:5 k for k = 1; 0; 1; 2. For the bin size b to implement ^ Gand ~ G , we employ a data-dependent method suggested by McCrary (2008). The data are generatedfrom normal distribution N (12; 3) (following McCrary, 2008) and Student’s t distribution 12 + 3 p5t (5).Both distributions have the same mean and variance. The sample size is n = 1000 and the suspecteddiscontinuity point is c = 13. Since the above densities are continuous, the true value is 0 = 0. Thebiases, variances and mean square errors (MSEs) of the above estimators are reported in Tables 1 and2.Among four estimators, ^ performs best in terms of MSEs. Its MSE is slightly smaller than that ofits competitor, ^ G , when a small bandwidth is used, but is signi…cantly smaller when the bandwidth isrelatively large. The dominance of ^ mainly comes from its superior bias performance on boundaries,while its variance is comparable with that of ^ G . The local constant estimators ~ G and ~ generallyhave smaller variances than ^ G and ^, but have much larger biases and thus larger MSEs. All fourestimators are generally biased downwards. On the other hand, a preliminary simulation indicates thatthese estimators are generally biased upwards if the discontinuity point suspected is on the left side ofthe peak, e.g. c = 11. Typical bias-variance trade-o¤s for the bandwidth selection is also observed: thebiases are larger and the variances are smaller when the bandwidth increases. Data generated from aheavier tailed distribution increase the biases of the four estimators signi…cantly and a¤ect the variancesonly slightly. Again, ^ appears to have smaller MSEs than other estimators.Next we look at the tests for (dis)continuity in the density function. We consider a general set-upof mixture of normal distributions.Suppose that the random variable X is drawn from truncated11
Page 3 and 4: the empirical likelihood con…denc
Page 7 and 8: whereKlj G = K Xj;hG ( JX1 IkGk=1K
Page 9: the binned data, the computational
Page 15 and 16: AMathematical AppendixHereafter “
Page 17 and 18: w.p.a.1. On the other hand, an expa
Page 19 and 20: Let g i;1 ( 0 ) be the …rst eleme
Page 21 and 22: References[1] Angrist, J. D. and V.
Page 23 and 24: Table 1: Finite-sample biases, stan
Page 25 and 26: Table 3: Finite-sample sizes, quant
Page 27 and 28: Frequencies0.0160.0140.0120.010.008
Page 29: Densities0.015Binning EstimatorLoca

Estimation and Inference of Discontinuity in Density

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