metadata: authors: 0: author: Fujii,Tomoki
1: author: Van Der Weide,Roy
count: World
docna: 0: docna: Cost-effective estimation of the population
            mean using prediction estimators
docty: Policy Research Working Paper
subtopic: Inequality
abstracts: cdata!: This paper considers the prediction
            estimator as an efficient estimator for the population mean.
            The study may be viewed as an earlier study that proved that
            the prediction estimator based on the iteratively weighted
            least squares estimator outperforms the sample mean. The
            analysis finds that a certain moment condition must hold in
            general for the prediction estimator based on a
            Generalized-Method-of-Moment estimator to be at least as
            efficient as the sample mean. In an application to
            cost-effective double sampling, the authors show how
            prediction estimators may be adopted to maximize statistical
            precision (minimize financial costs) under a budget
            constraint (statistical precision constraint). This approach
            is particularly useful when the outcome variable of interest
            is expensive to observe relative to observing its covariates.
repnme: repnme: Cost-effective estimation of the population
            mean using prediction estimators
display_title: Cost-effective estimation of the
            population mean using prediction estimators
listing_relative_url: /research/2013/06/17928599/cost-effective-estimation-population-mean-using-prediction-estimators
topicv3: Poverty m1326687 695
publishtoextweb_dt: 2013-06-01T00:00:00Z
no_of_pages: 38
author_keys: 0: author_key: 605729
1: author_key: 677092
model_extractions: WPS6509 Policy Research Working Paper 6509 Cost-Effective Estimation of the Population Mean Using Prediction Estimators _Tomoki Fujii Roy van der Weide_ The World Bank Development Research Group Poverty and Inequality Team June 2013
Policy Research Working Paper 6509

# **Abstract**

This paper considers the prediction estimator as an efficient estimator for the population mean. The study may be viewed as an earlier study that proved that the prediction estimator based on the iteratively weighted least squares estimator outperforms the sample mean. The analysis finds that a certain moment condition must hold in general for the prediction estimator based on a Generalized-Method-of-Moment estimator to be at least as efficient as the sample mean. In an application to cost-effective double sampling, the authors show how prediction estimators may be adopted to maximize statistical precision (minimize financial costs) under a budget constraint (statistical precision constraint). This approach is particularly useful when the outcome variable of interest is expensive to observe relative to observing its covariates.

This paper is a product of the Poverty and Inequality Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at rvanderweide@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Cost-effective estimation of the population mean using prediction estimators Tomoki Fujii! Roy van der Weide!

JEL classification codes: C20, C53, 132.

Keywords: Prediction; Double sampling; Maximum likelihood; Generalized method of moment; Regression estimator.

*This study has been funded by the World Bank’s Knowledge for Change Program for which the authors are grateful.

Singapore Management Uni ersity. Email: tfujii@smu.edu.sg * World Bank. Email: rvanderweide @worldbank.org
<!-- Start of picture text --> 1 Introduction<br>In economics, health sciences, and other disciplines, it is common to ha e a situation where the<br>outcome ariable of interest is costly to observe while its co ariates are relati ely inexpensi e<br>to observe. or example, measurement of consumption po erty is expensi e because it in olves<br>a long questionnaire often administered 0 er an extended period of time. Howe er, its co ari-<br>ates, such as asset holdings, ater and lighting sources, and housing materials, are relati ely<br>inexpensi e to observe. In health sciences, simple oral questions and anthropometric data taken<br>by anon-in asi e device often serve as a predictor of the outcome that is expensi e to measure.<br>In industries, certain non-destructi e testing may serve as a cheaper but less accurate alternati e<br>of destructi e testing.<br>In these cases, prediction estimators offer a useful alternati e. Prediction estimators es-<br>timate the mean outcome for the population by e aluating the mean of predicted outcomes,<br>which exploits the information contained in the a ailable co ariates. The data on co ariates is<br>not utilized when estimating the population mean by the mean of observed outcomes. or appli-<br>cations of this type of estimator to po erty measurement see e.g. Elbers et al. (2003), Stifel and<br>Christiaensen (2007), Christiaensen et al. (2012), and Douidich et al. (2013), where po erty rate<br>estimates are deri ed from predicted household consumption data. In an application to health<br>measurement, Fujii (2010) estimates the pre alence of stunting and underweight of children<br>using predicted data.<br>As is shown by Matlof (1981), prediction estimators may be useful e en when the outcome<br>ariable is in act observed for all subjects, simply because the mean of predicted outcomes can<br>be a more efficient estimator of the population mean than the sample mean of the observed out-<br>come ariable. — extension, this suggests that prediction estimators might also be incorporated<br>into the design of surveys where the outcome ariable is collected for a sub-sample of subjects<br>only.<br>This study can be viewed as a generalization of Matlof (1981), who uses a weighted-least<br>square estimator for prediction, in se eral directions. First, we establish the properties of the<br>prediction estimator when prediction is based on a general class of consistent and asymptoti-<br>cally normal estimators. Second, we deri e the condition under which the prediction estimator<br>2<br><!-- End of picture text -->
based on Generalized-Method-of-Moment (GMM) estimation is no less efficient than the sample mean and any linear combination of the sample mean and the prediction estimator itself. In the special cases where the estimator is a least-squares estimator or maximum likelihood (ML) estimator, this condition is guaranteed to hold under suitable regularity conditions. Third, we further specialize in a case where the outcome ariable of interest is binary. This is an interesting case as it is often encountered in empirical applications. It also offers an opportunity to compare a number of different prediction strategies that are moti ated by different assumptions. —e consider two ML estimators; one where the continuous state ariable that generates the outcome ariable serves as the dependent ariable (MLC) and the other where the binary outcome ariable serves as the dependent ariable (MLB). The estimator is included as a third candidate, which coincides with MLC when the errors are normally distributed. or general errors, may be viewed as pseudo-ML. As expected, prediction based on MLC is most efficient.

Finally, we apply the prediction estimator to cost-effecti e double-sampling, where the outcome _ ariable is collected only for a subset of subjects, while the co ariates are observed for all subjects. This allows us to maximize statistical precision [minimize financial costs] under a budget constraint [statistical precision constraint]. Such a sampling scheme is particularly useful when it is expensi e to collect data on the outcome ariable while the budget for data collection is limited.

Therefore, besides generalizing Matlof (1981), this study also contributes to the literature on double sampling (also called two-phase sampling), which dates back to Neyman (1938) and Bose (1943). considering a prediction estimator in the context of GMM estimation, we extend earlier studies of double sampling based on a linear model such as Cochran (1977), amhane (1978), Palmgren (1987), Davido and Haitovsk (2000), and arndal et al. (2003).! Among these studies, this study is particularly closely related to arndal et al. (2003), who applies double sampling to stratification to economize on the cost of estimation. Howe er, our study is different because we allo for non-linear models and only use predictions for estimation.

ith notable exceptions of Cochran and arndal et al. (2003), the discussion of cost| oran empirical application of double sampling, see Hansen and Tepping (1990).

3
<!-- Start of picture text --> effecti eness is only implicit in most double-sampling studies. _ bringing the prediction esti-<br>mator to a general GMM context and explicitly discussing optimal cost-effecti e estimation of<br>the population mean, we hope to expand the scope of possible applications of double-sampling.<br>recent example where double-sampling (without optimization) is explored as an option to<br>reduce the costs of po erty measurement in Bangladesh can be found in Ahmed et al. (2013).<br>2 Prediction estimator<br>Suppose that the potentially limited outcome ariable Y; for subject 7 is related to the continuous<br>state ariable y; by Y; = h(y;), where h is a function of y;. Further, the state ariable y; is related<br>toa ector of co ariates x; and a disturbance term u; by y; = q(2;, ui; O), where O is a _ ector<br>of parameters. e assume for no that the state ariable y; is unobservable. — e define the<br>expected alue of Y; gi en 2; by g(z;,0) = E,[Y;|x;], where 0 is a kK ector of identifiable<br>model parameters with kK < dim(Q) and g is assumed differentiable with respect to 6. _e also<br>define ¢; = Y; — g(2;, 0) and make the following assumptions about x; and u;<br>Assumption 1 The pair (x;, u;) is independently and identically distributed across i.<br>This assumption is relaxed in Section 6.<br>Assumption 2 The variables x; and u; are independent for all 1.<br>Notice that the parameter of interest in this study is 1 = E[Y;] = E,[g(x;,@)] and not O<br>or §. The standard ay to estimate ju is to take the sample mean Y = N-!S> ; Yi, where N<br>denotes the sample size. Howe er, Y is not the most efficient estimator in general. Using a<br>weighted least-square estimator 6” for 6, Matlof has shown that the sample<br>mean fi'”"5 = N-! S~, g(a;, 65) of predicted alues is asymptotically no less efficient than<br>Y.<br>It is not obvious whether the results of Matlof extend to other estimators. Therefore,<br>we first extend Matlof to a general case where predictions are made using a consistent<br>and asymptotically normal estimator for #. o this end, we make the following assumption:<br><!-- End of picture text -->
Assumption 3 The estimator 0 of the model parameters 6 satisfies the following properties: 67,6h and VN(O—0)5N(0,Q7!)h d _ as Noo, where QQ is a symmetric positive-definite K x K matrix.

Hereafter, we assume that suitable regularity conditions al ays hold. In particular, we assume the almost-sure existence and non-singularity of rele ant moments, which typically poses no problem in empirical applications. Using 6, we can no construct the prediction estimator /i(@) = N~! >, g(a, 6), which satisfies the following properties:

> Theorem4 Let M, = E|0g(«;,0)/00|(¢ 0x) and V, = ar|g(x;,0)|, where 0% is a column

kK -vector of zeros. Then, under Assumptions 1, 2, and 3, we have: “ <7 jp and VN(ji(0)h — p) >d N(0,V,+ _ MjQ°'M,) as Noo. Theorem is a direct extension of Matlof to a_ ery general case where the estimator for the model parameter ector @ used for prediction is consistent and asymptotically normally distributed. Note that the asymptotic ariance of (i(0) can be consistently estimated by replacing V,, M, and {2 with their consistent estimators. or ease of presentation, we drop the argument 6 and simply write /z until Section 4. eno specialize in a case where the estimator for prediction is the optimal GMM estimator. This is still airly general and rele ant to empirical applications since man _ of the estimators widely used in practice, including the least-squares and ML estimators, can be viewed as an optimal GMM estimator, e en though the concept of “optimality” is irrele ant when @ is exactly identified.

Henceforward, we maintain the following assumptions: Assumption 5 An L-vector of moment functions m(6, x;, Y;) satisfies E,,|m(0, x;, Y;)|xi] = Or, where m is differentiable with respect to 0, L > K, and V,, = ar|m(0, x;, Y;)]. *We a oid using the term “regression estimator” in this study because it typically refers to the prediction estimator under the assumption of linearity (and often a single co ariate). All proofs are provided in Appendix B.

5
<!-- Start of picture text --> Assumption 6 The estimator 0 of the model parameters is given by:<br><!-- End of picture text --> = § = arg min0 =N Som" (0 94a,ai, Y;)|42 WN —S7m(6N 9%,ri, Yi)42 ’ (2) :

where Wy is a positive-definite symmetric weighting L x L-matrix satisfying Wy > V,! as N->o.

It is con enient to define M,,, = E[Om(0, x;, Y;)/00"], which is the expected gradient of the moment functions. The following results are well-known (Hansen, 1982; Cameron and ri edi, 2005): Theorem Assumptions 1, 2,5, and imply Assumption withQ = MZV>!Mmn. Remark — A conventional choice of Ww is[N'S?, m(0, Xi, Y;)m! (6, r;, Y;)|~', where Aisa consistent estimatorfor 0.

In what follows, we erify whether the following condition holds: aco [Y, ji] =a arf/i).

(3) 0 see the significance of eq. (3), the following lemma is useful: Lemma = Suppose that Assumptions 1, 2, and hold. Then, if eq. holds, the asymptotic variance of the linear combination jt = aY + (1 — a) {i of the sample average Y and prediction estimator jtfor a € R is minimized when a = 0 (i.e., when ft = ft). Furthermore, eq.

implies: a ar|fi] <a ar[Y],

where the equality holds if and only if ji = Y almost surely.

> Lemma__ shows that the prediction estimator /i is asymptotically at least as efficient as the sample a erage Y and an linear combination of / and Y, provided that eq. (3) is satisfied. It thus follows that eq. (3) is crucial for establishing the efficienc of ji. Note that eq. (3) is exactly what Matlof has shown to hold for the prediction estimator (based on estimation

of the model parameters) to establish its efficienc relati eto Y.
<!-- Start of picture text --> The question no is whether eq. (3) holds in general. It turns out that a moment condition<br>has to be satisfied for eq. (3) to hold:<br>Theorem 10 Let m; = m(6,2;,Y;). Then, under Assumptions 1, 2, 5, and 6, eq. holds<br>regardless of the distribution of x if and only if the following equation holds:<br>M, = —MiV,,'E[myYi]. (5)<br>When @ is exactly identified, eq. is equivalent to:<br>E(mYi] = —Mj," VinMg.<br>Theorem 10 shows that the moment condition eq. (5) must be satisfied for the prediction es-<br>timator to be at least as efficient as the sample mean. One important special case of Theorem 10<br>is when 6 is an estimator, in which case eq. (5) is automatically satisfied.<br>Assumption 11 The estimator 0 of 8 is an LS estimator such that it satisfies:<br>0) = arg mininN'S >» (¥;i —— g(x,g(a, ))9))’.<br>Further, for m; = (Y; — 9i)0g;/00, Assumption _ is satisfied.<br>Under Assumption 11, Theorem 10 leads to the following corollary:<br>Corollary 12 Suppose Assumptions 1, 2, and are satisfied. Then, Assumption and eq.<br>hold.<br>Note that the results abo edo not change e en when the expectations are taken with weights.<br>Therefore, Corollary 12 is simply a restatement of the results pro en by Matlof (1981). It<br>should also be apparent from Theorem 10 and Corollary 12 that the main finding of Matlof<br>that the WLS-based prediction estimator is at least as efficient as the sample mean does<br>not hold for GMM-based prediction estimators unless the moment condition gi en in eq. (5)<br>holds.<br>Another important special case is where 6 is an ML estimator.<br><!-- End of picture text -->
<!-- Start of picture text --> Assumption 13 Suppose that the estimator 6 of 2 is an ML estimator such that it satisfies the<br><!-- End of picture text --> following first-order condition: Y Ol; i Fl,,="0=6 where 1; (0, 2;, Y;) is the individual log-likelihoodfunction. Further, for m; = 01;/00, Assumption is satisfied.

Assumption 14 Suppose that ¢; has a once-differentiable conditional probability density function f.(¢;|x;). Further, let l; = \n f.(Y; — g(x, 9)|a;). Then, the standard ML regularity conditions are satisfied for the ML estimator in Assumption 13.

ith these assumptions, we ha e the following corollary: Corollary 15 Suppose that Assumptions 1, 2, 13, and are satisfied. Then, eq. holds. So ar, we ha e not specified the functional form of q(x;, u;;O). Hereafter, we make the following linearity assumption, which is commonly used: Assumption 16 The variables y; and x; are related by the following equation: Yi = Uti, U4; O) = 278 + Ui, where O = {6,0}, the disturbance term u; has a twice-differentiable cumulative distribution function Fy and a probability density function fo(= F}) with Elu;|x;] = 0 and o? = ar[uj]. It is useful to define the normalized disturbance term: i = u/o,, where its probability density function f and cumulati e distribution function F satisfy f(a) = o,fo(u) and F(a) =

Fo(u) for all a € R, respecti ely.

One model that satisfies Assumption isthe obit model. This is an important model as it is widely used and provides a concrete example where the prediction estimator is in general not efficient.

Example 17 Assume that Assumptions 1, 2, 13, and hold. Further, u has a standard normal distribution and Y; = h(y;) = max(0, y;). Then, it can be shown that eq. does not hold in
general. Therefore, there exists a linear combination of ju and Y such that it is different from and more efficiency than {u.

# 3 3 Binary outcome variable

In this section we further specialize in a case where the outcome ariable is binary. This is an important case for two reasons. First, binary outcomes are often encountered in empirical applications which makes the binary model empirically rele ant. Second, eq.(3) is seen to hold regardless of the underlying distribution of u in this case, provided that the ML estimator is used for 6. 0 be more specific, we make the following assumption.

Assumption 18 The variables Y; and y; are related by Y; = Ind(y; > z), where z is an unknown constant. Further, the first component of x; is a constant and {3 is a column K -vector. Note that under Assumption 18, we cannot identify o,, 3, and z separately. _e can only identify 9 = B= (61 — 2)/Ou, B2/u, P3/Ou,+++,Px/ou). — ith this notation, we ha e the following result: Theorem 19 Under Assumptions 1, 2, 13, 16, and 18, eq. holds.

Matlof found that j: = Y holds almost surely for logistic regressions. It turns out that this is the only case where js = Y holds regardless of the distribution of x; when the outcome ariable is binary.

Theorem 20 Suppose that Assumptions 1, 2, 13, 16, and hold. Then, ji = Y and hence a ar|ji] = a ar[Y] almost surely for any distribution of x; if and only if u has the following cumulative distribution function F:

~ 1 F(u)(w) = ———.1l+e% 3It can be shown ji = N~! 0, f(—a72B) + a7 B/[1 — F(—27B)] where f is an ML estimate of 3. See also Appendix C for further discussion on Example 17.
<!-- Start of picture text --> 4 Comparison of prediction estimators<br><!-- End of picture text --> No_, suppose that z is known and both the continuous state ariable y; and the binary outcome ariable Y; are observable. In this case, we can still opt to estimate 6 by ML taking Y; as the (binary) dependent ariable (MLB). Alternati ely, we can also ork with y; as the (continuous) dependent ariable and estimate @ either by ML estimation (MLC) or ordinary least squares estimation. 0 clearly distinguish the prediction estimators based on these different estimators of 6, we introduce the following notations:* pols = B2OLS GOLs gMIB = arg max N7! S- Yj; In{1 — F(a2}'8)] + (1 — Yj) In F(a?<sup>8)</sup> gMiC = arg max Nt S° In[f (yi — 22 B)/ou)] — nou, Tu F where BOYS = arg min Nt du —x} 8)? and Gh = a De — or30tsy| 1/2 Because the estimation of o,, is not rele ant for 6“"8, weha e KM42 = = = dim(ax;) and KOS = KMLC ~ +1, where K* for ¢€ { LS,MLB,MLC}(= €) represents the dimension of 6°. Using §* for © E€ defined abo e, we can obtain the estimate Ba of Be 5 In the case where the outcome ariable is binary, we ha e: ji(0*) = N-! 7, F(—a? 6%)(= ji*)

for €€.

In this section, we make the following assumption: Assumption 21 f is twice differentiable and satisfies f(u) = f(—u) forallu€ R ith this assumption, we deri e the following results: 4To keep the mathematical expressions and proofs simple, we ignore the degree-of-freedom adjustment for GOES,

> *Thatis,* = ((31 — 2)/6u, B3/Gu, B§/Gus-*+ ,BG/Gu) fora € {OLS, MLC} and BMPR = OMEB.

10
Theorem 22 Suppose that Assumptions 1, 2, 13, 16, 18, and hold. Then, the asymptotic variances and covariances of [i" for © E are as follows: A _ ( uo Ou) FB a ar[jio’*] = Vy + vin Dae fe + 4q4 7 a arf] = Vy +078" [FC a) ( dail] Spe a ara] = Vy +B," (Pf @/A@)?| SEASye —= aco ors, MEO) — aco [a eP yMhC) where , = Elu‘], ; = —276, Y= ar[F( ,)], Life = E,[f( ai) “pp = E,[f( a) id: Mor = E, [aja] ],and ( )=fC )/F( JA-FC a)).

The following corollary immediately follows: Corollary 23, Under Assumptions 1, 2, 13, 16, 18, and 21, we have the following relationships concerning the asymptotic variances of [i.

a ar [ji 2") <a ar[fi?"* J, a ar [ji YO] <a ar [ji] Remark 24 a ar[fi“"] < a ar[jfi?"*] can also be proved by directly comparing them. By Assumption 21, E,[f'(%)/f(a)] = 0. Further, by the definition of ti, ar[ti] = 1. Hence, we have: Ef (@/F@)P) = an f(@)/P@) + LP @/F(@] = arlf'(@/F(@)] aralii > co [F(a)/F(a@),al] = [Bula (@/F@]) = 1, Theorem 22 implies that if y; is observed and the model is correctly specified, the MLC prediction estimator is most preferable. Also, e en though the MLB prediction estimator only uses the information on whether y; is abo e z, it is not necessarily less accurate than the prediction estimator. Moreo er, the MLB prediction estimator is robust to a misspecification
<!-- Start of picture text --> that does not affect the sign of y; unlike the and MLC prediction estimators.°<br>Another important point to note is that the MLB prediction estimator can be a useful al-<br>ternati e to the MLC prediction estimator in practice, if Y; is more cheaply observed than y;.<br>o highlight this point, suppose that the budget for data collection is fixed and we are only<br>interested in estimating jz. Further, the cost of collecting Ny observations of (;, Y;) is equal<br>to the cost of collecting N, observations of (x;,y;). Then, if we ha ea ar[ji”’"?]/Nisiep<br>a ar{fi@"°]/Nuzc, collecting Ny observations of (x;, Y;) and using jiyy_p leads to a more<br>accurate estimate of j. than collecting NV, observations of (x;, y;) and using fiyrrc.<br>5 Application to cost-effective double sampling<br>In this section, we apply the prediction estimator to cost-effecti e sampling. If collecting data<br>on x is cheaper than Y, double sampling—where x; is observed for all subjects but Y; is ob-<br>served only for a subset of the sample—may be preferred to the standard single sampling—<br>where both x; and Y; are observed for all subjects in the sample. o highlight the benefits of<br>double sampling, we consider the problem of maximizing statistical precision under a gi en<br>budget constraint and its dual problem of minimizing financial costs gi ena statistical precision<br>constraint. ormally, we make the following assumption in this section:<br>Assumption 25 The covariates x; are observed for all subjects i € {1,2,...,N}= , while<br>Y; is observed only for subjects i € {1,2,...,| N|} = ‘4, where |-| is the floor operator<br>that gives the maximum integer that does not exceed the argument and € (0, 1] denotes the<br>proportion of subjects for which Y; is observed as N + .<br>Let us referto ‘asSample andto ‘’= \ ‘as Sample 2, respecti ely. Because the<br>indexing of the observations may be changed arbitrarily under Assumption 1, we assume that<br>the first N’(= | N]) observations are those that contain Y;.__ e further make the following<br>assumption:<br>"© or example, consider the case where the true relationship between y; and 2; is gi en by y; = (278 + ui)?<br>but we use the following misspecified model: y; = a! 8 + uj. This does not affect p”@"* but it affects aM“.<br>12<br><!-- End of picture text -->
Assumption 26 The distribution of (x;, u;) is independent of the sample selection. That is: This requires that the selection into either sample carries no information about x; or u;. This is a reasonable assumption when the researcher can decide whether to observe Y;. In this setup, we use / to compute the estimator 6! of @ and predict g(x;, 6") for all obser ationsin . Therefore, the prediction estimator for jz under Assumption 25 is gi en by: pe? = N 1 So g(a, 0"). ies It is straightforward to deri e the following results: Theorem 27 Suppose that Assumptions 1, 2, 25,and holds and 6! satisfies Assumption 3. Then, [i?® satisfies the following properties: pre? and VN(jfi—p)SN(0,V,)+ “'M7Q71M,) as Noo. (10) Theorem 27 is particularly useful for estimating the population mean yp in a cost effecti e manner, which can be done by choosing appropriately. osho this, we make the following assumption: Assumption 28 The ratio of the cost of observing only x; to that of observing both x; and Y; for a given subjectiis (1).

Hereafter, we normalize the cost of observing both x; and Y; to be unity and ignore the act that N and N’! are integers. The following theorem is rele ant if there is a binding budget constraint (i.e., the budget is just enough to collect observations of both x; and Y; Theorem 29 Given the binding budget constraint, the variance of ju°? is minimized when N and _ satisfy: (VN, ) = arg min|VgV,p + 1M OQ} M,]/v st. [ + (- )| = = (/[*+ G- *)], *),

13
<!-- Start of picture text --> where * = min (1, ,/ (= MPO“ MV") and the minimized variance is given by:<br><!-- End of picture text --> In other cases, we may ant to minimize the cost of data collection for a gi en accurac . In this case, the dual ersion of Theorem 29 is rele ant. Corollary 30 Suppose that the maximum acceptable variance of ju°? is given by. Then, the cost of achieving this accuracy is minimized when N and _ satisfy the following equation: (Ny) = argminl + (1 J) ste [Vp + MPO"'M,/p|/v =~ = (Yt M,2°'M,/p*)/A, *),

<!-- Start of picture text --> where * = min (1, ,/ - )IMPO-M,V,-") and the minimized cost is given by<br><!-- End of picture text --> Note that when -—> 0,wehae: Noo, —0,andN’ = N-- . Inother ords, in the limit where the co ariates x; can be collected atnocost — Q), itis cost-effecti e to collect x, for an infinite number of subjects (or the entire population) without spending an_ resources (i.e., N —> oo and spend all budget on collecting the outcome ariable Y; (i.e., N’ >). Under this scenario, the ariance will be entirely dri en by the model error; with N — ov, the sample ariance component will tend to zero. Note also that under these simplifying assumptions concerning the cost function, the ratio of the minimized ariance under double-sampling relati e to the ariance under single-sampling coincides with the ratio of the minimized budget under double-sampling relati e to the budget under single-sampling: (x,a) = (va +VU=ajl— )) 2 <!-- Start of picture text --> (12)<br><!-- End of picture text -->
where a = V,/(V,+M.; (Q~' M,), which is the proportion of the ariance due to sampling error to the total ariance of the double-sampling of the single sample estimator.

<!-- Start of picture text --> a<br>fo - i.<br>of<br>a<br>a<br>7<br>7<br>7<br>7<br>7<br>7<br>7 —<br>7<br>4<br>4<br>/<br>/<br>!<br><!-- End of picture text --> Figure («,a)asafunctionof for: a= 0.5 anda = 0.8.

Figure plots the ariance ratio (or budget ratio) as a function of fora = 0.5 anda = 0.8. The figure shows that the benefits of double-sampling are larger when the model error component is small relati e to the original sampling error component. The figure also confirms that there is athreshold aluefor abo e which there will be no gains in using double-sampling relati e to single-sampling.

6 Extensions In the discussion so ar, we maintained Assumption _ to keep the presentation simple. Howe er, this assumption can be relaxed in at least two important ays. First, we will consider cluster sampling which is a common feature of surveys. This can be easily accommodated in Theorem 4. Since we essentially only rely on Assumptions 2 and 3 for the proof of Theorem 4, all we need to do is to compute the estimates of 6, 2, V,, and M, ina clustering-consistent manner.’ 7See Appendix _ for further discussion on this.

15
Second, Assumption 26 implies that Y; is missing completely at random. Because we are concerned about cost-effecti ely estimating the population a erage by choosing not to observe Y; for some subjects, this is a reasonable assumption for our main purpose. Howe er, Assumption 26 may be too strong in other cases for which our prediction estimator is potentially applicable. One example is the case of a censored outcome §ariable.

Example 31 Consider Example — again. This time, assume that Y; is missing when the (unobservable) state variable y; satisfies y; < 0. If @ is estimated only with Sample (i.e., subjects for which both x; and Y; are observed), 6 will be biased and thus ji will also be biased.

Assumption implies that Samples and 2 discussed in Section 5 are generated from the same underlying population. In some cases, the alidity of this assumption is not clear, especially when the two samples are taken from two different data sources. In such cases, one can test the alidity of the prediction estimator using the following statistic: Theorem 32 Suppose that the estimator 6! of 0 is estimated only with ! and satisfies Assumption 3. Further, let Y! = ies! Y,/N! be the sample average of the outcome variablefor ' and ji!" = Yoiegr 9 (ai, 61) /NU be the prediction estimatorfor Sample 2, respectively. Then, under the null hypothesis that Assumptions Assumptions 1, 2, and asymptotically follows a standard normal normal distribution: that Assumptions Assumptions 1, 2, and hold, the test statistic defined below standard normal normal distribution: 7~ M+ “OV, + Y!are)— jill+ —EMPQ-1M,)/N Note that the rejection of the null hypothesis is consistent with arious possibilities. or example, the distribution of 7; may be different between Sample and Sample 2, which can be tested separately. It is also possible that @ is different in the underlying populations of two samples because, for example, the timing of sampling is different. The test described in Theorem 32 alone cannot distinguish these and other possibilities.
<!-- Start of picture text --> Discussion<br>The primary moti ation for the use of prediction in economics, health sciences, and other disci-<br>plines has been to deal with arious forms of missing data problems. e make a case for using<br>a prediction estimator to obtain more efficient estimates of the population mean by extending<br>the results of Matlof (1981). In an application to cost-effecti e double sampling, we sho<br>ho prediction estimators may be adopted to maximize statistical precision [minimize financial<br>costs] under a budget constraint [statistical precision constraint]. The approach is particularly<br>useful when the outcome ariable is relati ely more expensi e to observe than its co ariates.<br>There ha e been a number of cases in which predictions are used ex post to estimate the<br>outcome ariable of interest. or example, Elbers et al. (2003) ha e put forward an approach<br>where the prediction of household consumption per capita is made for each census record, which<br>in turn is used to compute po erty statistics for small areas. Their approach is useful because<br>household consumption is expensi e to observe and thus typically observed only for a small<br>set of households in the population of interest. _ similar approach can be used to estimate the<br>pre alence of stunting and underweight of children for small areas (Fujii, 2010). There are also<br>a number of recent case studies in which a consumption surve is combined with secondary<br>surveys (without a consumption component) to supplement existing po erty estimates (Stifel<br>and Christiaensen, 2007; Douidich et al., 2013).<br>Our study shows ho _ prediction methods may be fruitfully adopted e en ex ante by de-<br>signing surveys appropriately. recent study by Ahmed et al. (2013) on Bangladesh shows<br>that po erty can be estimated with reasonable accurac_ with a relati ely small sample (e.g.,<br>Primary Sampling Units (PSUs) with 10 households per PSU) of the outcome §ariable and its<br>co ariates when combined with a larger sample of the co ariates only (612 PSUs). Their cost<br>estimates suggest that, compared with a full scale consumption surve (complete observations<br>of both outcome ariable and its co ariates for 612 PSUs), the manpower cost for data collec-<br>tion could by cut by more than 90 percent without se erely undermining the accurac_ of the<br>national po erty estimate. If our approach is adopted, the trade-of between financial costs and<br>statistical precision can be explicitly optimized in a situation like theirs.<br>o what extent we can successfully bring down the costs while maintaining statistical pre-<br><!-- End of picture text -->
<!-- Start of picture text --> cision is intrinsically an empirical question. In general, our prediction estimator is most useful<br>when there are co ariates that are both inexpensi ely observable (1.e.,lo alueof and offer<br>a good model for prediction (i.e., lo — alue of [MP Q7!MglV,-'). It should be apparent that<br>prior estimates of these parameters are required to calculate the optimal sample sizes under<br>double sampling. When these are not a ailable, data collection can be done in two stages. That<br>is, one can collect a small sample with both Y; and 2; in the first stage, and use this sample to<br>compute the parameters needed to determine the optimal double sampling structure. In practice,<br>this exercise can be done at the time of a pilot surve , in which case there may be practically<br>no additional logistical costs. While the estimate of obtained from the first-stage sample is<br>likely to be noisy, this does not bias the estimate of 4. Moreo er, the resulting estimates are<br>likely to be more accurate than the estimates created with only one sample under the same bud-<br>get constraint. Therefore, the cost-effecti e estimation approach proposed in Section5 is likely<br>to be a helpful alternati e that will benefit arious institutions, including go ernments, hospi-<br>tals, laboratories, and actories, that are financially constrained or wish to minimize the cost of<br>estimation while achieving a gi en le el of accurac .<br>References<br>Ahmed, ., C. Dorji, S. akamatsu, and N. oshida (2013) ‘Conducting a hybrid surve to<br>impro e the reliability and frequenc of po erty statistics in Bangladesh. mimeo, _ orld<br>Bank<br>Bose, C. (1943) ‘Note on the sampling error in the method of double sampling. Sankhya 6, 329-<br>330<br>Cameron, A.C., and P.K. ri edi (2005) Microeconometrics: Methods and Applications (Cam-<br>bridge Uni ersity Press)<br>Christiaensen, L., P. Lanjouw, J. Luoto, and D. Stifel (2012) ‘Small area estimation-based pre-<br>diction methods to track po erty: alidation and applications. ournal of Economic Inequal-<br>ity 10(2), 267-297<br>Cochran, .G. Sampling Techniques, 3rd edition ed. (John — ile Sons)<br><!-- End of picture text -->
<!-- Start of picture text --> Davido ,O., and = Haitovsk (2000) ‘Optimal design for double sampling with continuous<br>outcomes. ournal of Statistical Planning and Inference 86, 253-263<br>Douidich, M., A. Ezzrari, R. ander eide,andP. erme (2013) ‘Estimating quarterly po erty<br>rates using labor force surveys: primer. orld Bank Polic Research orking Paper 6466,<br>orld Bank<br>Elbers, C., J. Lanjouw, and P. Lanjouw (2003) ‘Micro-le el estimation of po erty and inequal-<br>ity. Econometrica 71, 355-364<br>Fujii, . (2010) ‘Micro-le el estimation of child undernutrition indicators in Cambodia. orld<br>Bank Economic Review 24(3), 520-553<br>Hansen, L.P. (1982) “Large sample properties of generalized method of moments estimators.<br>Econometrica 50(4), 1029-1054<br>Hansen, M.H., and B.J. epping (1990) “Regression estimates in federal welfare quality control<br>programs. ournal of the American Statistical Association 85(411), 856-864<br>Matloff, N. ‘Use of regression functions for impro ed estimation of means. Biometrika<br>68, 685-689<br>Neyman, J. (1938) ‘Contribution to the theory of sampling human populations. ournal of the<br>American Statistical Association 33, 101-116<br>Palmgren, J. ‘Precision of double sampling estimators for comparing two probabilities.<br>Biometrika 74(4),<br>arndal, C.-E., B. Swensson, and J. Wretman (2003) Model Assisted Survey Sampling<br>(Springer)<br>Stifel, D., and L. Christiaensen (2007) racking po ertyo er time in the absence of comparable<br>consumption data. orld Bank Economic Review 21(2), 317-341<br>amhane, A.C. ‘Inference based on regression estimator in double sampling. Biometrika<br>65(2), 419-427<br><!-- End of picture text -->
<!-- Start of picture text --> A Random-effects model<br>It is not uncommon that errors between subjects are correlated , which calls for a relaxation<br>of Assumption 1. popular model that accommodates correlated errors is the random-effects<br>model. oallo for random effects in each cluster, we replace the index for each subject 7 by<br>the combination of “cluster” and “household” h. The disturbance term w, therefore, has the<br>following structure:<br>U= +,<br>where and are, respecti ely, the cluster-specific and household-specific random-effects<br>terms that satisfy E[ ]= E[ |=0. edefineo® = ar[| | ando”?= ar|_ | and denote the<br>size of cluster inthe sample by ( oc) (i.e., there are households in cluster ). Therefore,<br>the total number of households is equalto N= 5S).<br>In what follows, we make the following assumption instead of Assumptions — and 2:<br>Assumption 33 The variables (x , ) are independently and identically distributed across<br>ch. Further, and are independently and identically distributed across . The variables x ,<br>, ,and are independent with each other.<br>Under Assumption 33, the ariance of the household disturbance term u is ar[u |] = 0? +<br>o*(= 02). Note that u is not independent across households because of the cluster-specific<br>random-effects term .Thecorrelationofu inagi encluster is gi enby: , =07/o?.<br>While the estimator of @ is still consistent under Assumption 33, it is in general inef-<br>ficient. Therefore, we consider the case when @ is estimated by a (feasible) Generalized-Least-<br>Squares estimator 0°", 0 acilitate the discussion, we further make two additional<br>assumptions:<br>Assumption 34 The number of clusters tends to infinity as the total number of households N<br>tends to infinity.<br>e also assume that consistent estimators of the ariance parameters are a ailable.<br>20<br><!-- End of picture text -->
<!-- Start of picture text --> Assumption 35 or each , we can compute G? and G? with the sample. Further, the following<br>asymptotic properties are satisfiedas — ov:<br>a Fo?<br>a FG?<br>In the discussion belo _, it is con enient to introduce matrix notation. Plugging i = u/o,<br>into eq. (7), and writing the result in matrix form, we obtain the following equation:<br>y =X$+o0,U (13)<br>where y, X, and U are, respecti ely, the matrix ersionofy ,x ,andu formed by stacking<br>all the households in the sample ertically. or example, y is defined as follows:<br>y== (Yui. *° YL ys Yat U2 att Yi ) T .<br>No ,define =o07E (UU"). Due to the cluster-specific random effects, is a block-diagonal<br>matrix with the number of blocks equal to . The -th block solves: = 071 1" +o .,<br>where 1 and areacolumn- ectorofonesanda x _ -identity matrix, respecti ely. ith<br>these notations, we obtain the following lemma.<br>Lemma 36 Let B¢!5 = ( T° -! )-! T° -1ly denote the feasible GLS estimatofo r 8,<br>where — is a consistent estimator for. Furthermore, fori = y— BCS, let 62'S =<br>|a7a/(N — 1)| "? be the corresponding estimator for o,,. Further, define 0G" = ((BG45 —<br>2) (GEES, BEES /GGLS | BEES /GGLS) SY = Elx a], = o?/( o? +07), and =<br>(l— )( —1)/k. Then, under Assumptions 33, 34, and 35, 0°" satisfies the following<br>properties as —> ©:<br>gous Bg<br>JN(0%5 —0) S N(0,— i) [((1— EL ])See+ El J are JJ‘),<br>where the expectations E| | and E| |aretakenover .<br>21<br><!-- End of picture text -->
Let ju?“* denote the estimator for j: that is based on 6°45, Because Lemma 36 shows that 9°45 satisfies Assumption 3, the asymptotic ariance of ji@“5 has the same form as in the model under Assumption |. The only difference lies in the act that the asymptotic ariance matrix

(.~' must be adjusted to account for clustering.

B_ Proofs <!-- Start of picture text --> Proof of Theorem 4 an exact first-order aylor expansion of 6 around 6, the La of Large<br>Numbers, and Assumption 3, we ha e:<br>won 1 “ 1 Og(xj,97) ,- p<br>fi(9) = NDI) + No or? — 0) > pb,<br>where 6+ is between 6 and 6.<br>the Central Limit Theorem and Assumptions 2 and 3, we ha e:<br>. 1 . L yn 0g(xi, 07) j<br>N(p- = — i, 9) — — ———VN(0-90<br>VN HW) = Te Dale) wt DI gr VN 8)<br>*, N(0,V, + M7Q'M,),<br>which completes the proof. L<br>Proof of Theorem Because this theorem is well known, we only provide a sketch of proof<br>that is rele ant to the rest of the paper. aking the first order condition of the minimization<br>problem in eq. (2) and dividing by two, we ha e:<br>1 Am? (0,23, yi) | os 1 “<br>aking an exact first-order aylor expansion around § = 0, there exist 0**+ between @ and 6<br>such that:<br>1 dm! (6,x;,y:) | 2, | 1 1 Om(0**, vi, Yi)»<br>QT EM te Yi | 0,2;,y) +—y OX 6 — 6)| =0.<br>Ed a ied > aeee?ea ee<br>22<br><!-- End of picture text -->
<!-- Start of picture text --> Solving for # and applying the La of Large Numbers, we obtain:<br>h 1 Om?(4,24, y:) L dm(6+*, 21,4) |]<br>6 mm » Ui, Vi 7 m » Li, Yi<br>= @-||— ——_——| Wy |— ———<br>oe1 dm! (6,2;,y;)eee | Wy yD1 (OHH) (15)<br>* 0.<br>This in turn implies 67+ ?. 9g, this, Wy 2s V,,', and the Central Limit Theorem, we ha e:<br>h L Om!(4,2;, ys) Lc dm(6++,2:,y) 1]<br>|N(@-@) = —-||— eem » Li, Yi | Wy; | leem A » ReUi,EATVi<br>> eae da<br>1 Am! (6,2, yi) | vs 1<br>— ae | Wa | 0,.2;, Yi<br>ee ts [em<br>4 N(0,(MEV2!Mp]7').<br><!-- End of picture text -->

<!-- Start of picture text --> (15)<br><!-- End of picture text --> O Proof of Lemma eq. (3), we obtain eq. since: a ar[Y — fl] =a ar[Y] +a ar[fi] — 2aco [Y, fA] =a ar[Y] —a ar[fi] > 0. The equation abo e is satisfied with equality if and only if a ar[Y — fj] = 0, which in turn is equi alent to 2 = Y almost surely because ji and Y are both consistent estimator of ju. It is clear that / is consistent and its asymptotic ariance is: a ar[ji] = 7a ar[Y] + (1 — a)?a arf] + 2a(1 — ajaco [Y, fi. If , = Y holds almost surely, the asymptotic ariance of /1 is constant and therefore trivially minimized when a = 0 (or an other alue of a). Thus, we assume belo _ that i 4 Y almost surely. taking the first-order condition with respect to a, it can be seen that a ar[/i| is minimized

23
when: <!-- Start of picture text --> Q = Sea ar(ji] — aco [YA<br>a ar[Y] +a ar[/i] — 2aco [Y, ji]<br><!-- End of picture text --> <!-- Start of picture text --> Hence, if eq. (3) holds, a ar| ji] is minimized when a = 0, in which case we ha e ji = ju. O<br><!-- End of picture text -->

Proof of Theorem 10 Theorems and7,wehae a ar[/i] = Vy + MP(MVMin ‘My. Let us define 9; = 9(x;,9), 9; =9(xi, 97), m{* =m(0"*, x;, Y;), and mh; =m(0t*, x, Yj). Then, using eqs. and (15), we obtain: <!-- Start of picture text --> 2 Vy — MP [MEV in! Mm) 'M;,V~'co [mi, €](= aco [Y, fl).<br><!-- End of picture text --> the la of total co ariance and E,[m;|x;] = 0, we ha e: co [m6] = co »[E,[mi|xi], Ey[ei|vi]] + E,[co y[mi, |x] = E,{co ylm, Yi|zi]]

Therefore, by eqs. (17), (18), and (19), eq. (3) holds if and only if we ha e: MAMVin! Mm) "My = —MP (MEV Mn) (MiVn' ElmiYi). or this to hold regardless of the distribution of x, it is necessary and sufficient to ha e eq. (5). In the exactly-identified case, premultiplying eq. (5) by M7, V,,, we obtain eq. (6). O 24
<!-- Start of picture text --> Vin = B,[ ary[m;]] = B, oo ar{e?]. (22)<br><!-- End of picture text --> <!-- Start of picture text --> Ey AO) |= Le(ci|vi) G | = erFleilriMalnJedeede = [fe(elrile:[ilohede —— Fe(eilvi)|e-",Flan)€;=00 = —1.<br><!-- End of picture text -->
Theorem 10, eq. (3) holds.

L Proof of Theorem 19 First, notice that this is also an exactly identified case. Note also that the following relationship holds: Yir>z & a«a4P-z+u>0 © a; > —a 6B = al 9 = ie Thus, we ha e g(a;,0) = E[Ind(a; > ;)] = 1—F(_ ;). The individual log-likelihood function is: 1]; = Y;ln[1— F( i)|+(-Y;) In F( i). Hence, solving the first order condition, we obtain: Ali -Y; 1-Y¥)0; FC i)+¥%-1 ; oO |1-F( i) FC a) 08) FC i)A-FC 3)

Because Y; is binary, Y? = Y;. Using this, we ha e: <!-- Start of picture text --> F( i)A— FC 3) 1—F( ;)<br>aking expectations 0 er x yields E[Y;m,] = M,. Furthermore, we ha e:<br>mM, - —~p|/(——%Y; _ , 1-Y;45") \ xp Ga|O ;O ;<br>(at—Y; 1-Y;\a) Pon- Oae;<br>= —Vp.<br>Because V,,, is symmetric, eq. holds. Theorem 10, eq. (3) holds. |<br>Proof of Theorem 20 eq. (23) and Assumption 13, @ satisfies:<br>— 3. (F(-270) +¥5-1) ee ei) x, —0 (24)<br>N« oO F(—a70)(1— F(-af6)) }<br>Because eq. (24) holds for each component of x;, it also holds for the first component, or the<br>constant term. Thus, letting {= —«? 6, the following equation must hold to ha e fi = Y<br><!-- End of picture text -->

26
almost surely: or this to hold for an distribution of x;, the following equation must hold for all 7 F( )Q—FC( i) Rearranging terms, we ha e: i= = a = in i) — in _ i C, : | F(1 ij) 4+—41-F 3)|r InF(°;) —In(1— F( ,))<sup>+</sup>

where is aconstant. Solving for F( “is we obtain: F(°) =[b4e0@- 7-4. (27) The mean of ti; in this case is: oo 7 1 . . 7 E\a;| = uf(ujdu= (InF-—In(l-—F)+ )dF=c. (28) —oo 0 Because E[u;] = 0, we musthae = 0. Plugging this into eq. (27), we ha e eq. (8). Con ersely, if eq. holds, eq. (26) holds. Hence, eq. (24) reduces to Y = ju. | Proof of Theorem 22 0 distinguish the moments for different prediction estimators, we use the superscripts LS, MLB, and MLC. Letting ; = (y; — x7 8)/o,, and solving the first

27
order conditions yields: mols = —(¥; — a7 B)a; on — (Yi 2} 8)? MLB = Fi )+Vi-1l F( _)xr. mMiC = ft i) C/O Ox ‘ fC i) i/Ou 1/ou o calculate the rele ant moments belo_, the following formulae are useful: E, |<sup>———"—-|</sup> f(a) f" (a) = - ————f(a) f" (a) eI-\Axf(u)du ~ pili ~\ a~ pl i~ u=00 | Fa) | ag [a= = Fwu)du = fw]u = =0 B, or,f(a) f" (a) - = Mananpili ~\~ a~ = [7pl @u-(~ a]~ u=00 =0 Ey f(a) f"(@) -» _ nny x2 ~| glia) a2 ~ 6 ~ ed _ or, | = _/ (a)u“di = li (a)u“ — 2uf (au) + 2F (ii) =2 f'(a) fH) a ay Lz. 00 E,|\=—| = —— f(u)du= u)du val Fay si= , P@ai= = [fo],U = =9 Ey, 2.f(a) = _/eye (a)udiae = foreI \ - P(a)|~\])u=~ewe —1 E, | | 7———_| | = , u)dum **a** == [FO],U == FOdij

Assumption 21, we also ha e: fu) 28
Using these andu; = ;, we ha e: <!-- Start of picture text --> MOLS — Mer 0<br>0 2o,,<br>yols _ pF (yi — aj Paya} —(a% — (yi — 27 8)?)(yi — @F 8) xy<br>—(a% — (yi — 278)?)(yi — 27 B) at (a0 — (yi — a7 B)?)?<br>O27 ee 01x<br>0 x1 uu o2<br>MMIB — _B F(1—F) —(F+¥i- L)(L= 2F) po FLYi-1 wat<br>m F2(1—F) F(1—F) a<br>— _p, ;F( i)(l—fP( i)FC i)) vat = _yMLB<br>rary ry vit? aya; ry Li<br>7<br>punem _ 7 ff" f?— (fy aaiar “oea? Lff 0.al oeQu; i" 0 x 0 1x4<br>ia. 2<br>Ors By E ( F(a) 2+<br>_1 (5)i(ai))?| Be [nse]T ys _ _ye<br>Oy 0 7 (Zan)? om<br><!-- End of picture text -->

Also, note the following: 9°" (at 0) = gC (at, 0) =1— FC i) = 9"? (a7, 0).

Therefore, we ha e: OLS _= YM pe/o M, fa/@u Upp /Cu My? = Mfe MLC Lpe/Ou M, = .

0 29
Using these, we obtain: . _ (ur On) FB a ar[fi0’’]) = Vo + whe mae Ufa += m7 a arf?) = V,+%),E5' |<sup>==></sup> FC )G—-FC(PX i) i) xix; | “po ~2 a ara’) = v,4pz1 (Lee!a). Dhaene Uf:

or the asymptotic co ariance between ji° and ji witha, € €, the following follows from eq.

Ni = pw) —p) Vet “MPC "(= aco [fi*,jv ]), (29) where = M)|MIV.'Mn|'MmV,,' for each of the superscripts and , and Mw) = Bim} (m,)"].

OLS _— i yt yl Lys o, | J 203 ae FC )G—FC( i) Ph. 2 MLC _ —o.S4,E;! Pui) [S32 0) sal.

Further, we ha e: 30
<!-- Start of picture text --> Plugging these moments in eq. (29), we obtain:<br><!-- End of picture text --> which completes the proof.

O Proof of Corollary 23 Because a ar[ji@"°] = aco [f°%%, f!@"°] = aco [aTM@"®, jM@"], the same argument as Lemma __is applicable to the comparison between ji?"* and fi@"© and between ju@"? and f°. oO Proof of Theorem 27 an exact first-order aylor expansion of 6! around 6, the La of Large Numbers, and Assumption 3, we ha e: ~DS 1 ~ 1 Og(xj, OT) Lh __~ LI + HD aGr—Oie) I — 6) Pp> B, (30) where 6+ is between 0 and 6.

31
<!-- Start of picture text --> the Central Limit Theorem and Assumption 3, we obtain:<br>YNGS—W)N(jiP> — == —Te1 LaleoBi,9)“ — wt N—say—1 aN0g(x;,07)Lage NIVN (6!~ —— 8 8)<br>*, N(0,V,+ ~'MPOQ'M,),<br>which completes the proof. LO<br>Proof of Theorem 29 Plugging the constraint into the objecti e function from eq. (11), the<br>objecti e function is seen to solve:<br>[V, + MPOM,(1— )+V,(1— ) +MZOM, /l/C.<br>This function is con ex with respect to . The theorem follows from this and € (0, 1]. O<br>Proof of Corollary 30 The proof is similar to the proof of Theorem 29. L<br>Proof of Theorem 32 It is straightforward to sho _ that:<br>VNI(Y!= — p) +d N(0,Vy + arfe)). (31)<br><!-- End of picture text --> Because Y! and ji!’ are independent under Assumption 26, Theorem 32 follows from eq. (31) and Theorem 4. LO Proof of Lemma 36 Under Assumptions 33, 34, and 35, 3°” is consistent and asymptotically normal distributed as — oo gers Bg VN(BMS — 8) S NOC TY).

e are interested in 9°45 = ((BE45 — 2)/GEPS, BEES GEES... BEES /GG"S). The consistenc of 6¢“5 immediately follows from the consistence of 3¢/5 and 0°45, What remains to be established is the asymptotic ariance and normality of OCLs Since the constant z merely introduces a deterministic shift in the intercept it does not affect 32
the asymptotic ariance and normality. This means that our interest is in asymptotic distribution of B¢45 /G¢45. An application of Slutsk s theorem yields: VN (0945 — 0) SN (0,951) as 0, where 2G). Satisfies: ( 7 OEX/N) "op = (0, * OLX/N)"! > Oars: o obtain the analytic expression of QGj.g, let us first e aluate the matrix in erse of . As is a block-diagonal matrix, its in erse too is a block-diagonal matrix with blocks ~'. The <!-- Start of picture text --> in erse of is:<br>2<br>——— i _ 7 1?<br>o> \ © o%+ oF © ef}-<br>Substituting thisinto “7 ~! yields:<br>Tol S- T -1<br>_ i T o T<br>7 oe cot get<br>=<br>_ ti St = aa<br>3 EE |,<br>where = (’\,---,a7)" denotes the matrix of x stacked forcluster andz = 11 ,/k<br>is the clustera erageofax .<br>Therefore, 0°" is gi en by:<br>QeLs _— a2 ( T -1 )/N<br>Oi 1 T = oT<br>-Gx[D" os<br>Oy 1 GQ- JC =)<br>[x]<br>= oF re ja at +<br>S (l= a) [LBL Eee+ EL] arfe j],<br><!-- End of picture text -->

33
by the independence between x and andLa_ of Large Numbers, which completes the proof.

O C Further discussion on Example 17 oclearly sho _ the violation of eq. (5), we explicitly compute each moment. Let 0 = (87, 0,,)" and denote the probability density function and the cumulati e distribution function for the standard normal distribution by and ®, respecti ely. Then, we ha e: g(z,8) = ElyiInd(y; > 0)|a] = El(at6 +o,i;)Ind(a; > —x} B/o,)|a;] —2F Blow Ou

where ; = —2/ 8/o,.

wee, 0g;/OB2 OBC1—®( ;))x/o,De) or the obit model, the individual log-likelihood function /; can be written as: |; = Ind(Y; = 0) In ®( ;) + Ind(Y; > 0) In{ —1 |———}Y, — 27B }. om Ou the moment function is: = Mm;<sup>=</sup> 4 |—Ind(¥; =T 0)bg“ i. + Ind(y; > 0) ee]u i . st Ind(¥; = 0)x4 + Ina(; > 0) [-2 + SSE._@T B)2 |

Then, the moment function is: 34
The following relationships are useful: E\¥,>0]) = Elu> i= (a)di= 1— ®( ;) Ela|¥i > 0) = ue(ujda = —[ (w);=S = (a) Blu"|¥, > 0) = w (ude = (O(a) —ad(@)JIEX = ¢ ( )+1-O( 3) Blw'|¥; > 0) = w (ajdt = —[(@ +2) (WI=~ =( 7 +2) (a) Blw'|¥; > 0) = uw! (u)dt = [3@(a) — (a) (w + 3a))7= = 31-8 ))+ (JC 7+3 9)

Therefore, letting ; = (y; — 2/8) /o,,(= @) and taking expectations, we ha e: Mn<sup>=E</sup> |=om a 0 rj + 2 ayatay ; = £ Ind(y=0) + * ite +E 1 art -2 ja; Ind(¥; > 0) Cu = =-2;, 1-3 ? = - [L—( iit 7 41)8( i)leia?t —% i ( 4) > 2 ) eu —i ( aay —i i ( i)+20-( 3)

35
where ;= ( ;)/®( ;)and ;=[( ;+ 4) ;4+1]. Also, we ha e: Vin = Elm] = E Ind(¥;=0)4 + Pu itp ; E 1 peep (> at Pai Ind(¥;> 0) Cu (= Gt Pah ( F-1P = £, +~—— C3) + Pu itp H pn.v22 (i( i) - OC i) +)aiay (7+) ( idx %u ( 7+1) ( iat 21-®( i))+( 7+ i) Ca) _i, (Cit a) ( )+1-O( j))aiat ( i®C i)<sup>+ida</sup> = sE, Os ( @( i+ i)az 2(.1—®( j))+ iCi (i) it 4), where ;=( ?+4+1) ( 4).

> Ind(i;> 4)-[— ¢+%j][-1+ 07] _i, H-CiCd +O d)G ( aa ou —(( d+ ¢ ( )-CF+2)(a) op 1—© EHR 4) drilou)Xi/ Oy ( i)/Ou Using the moments calculated abo e, we can erify that eq. (5) do not hold for the obit model.

36