CHAPTER-IV PRE-TEST ESTIMATOR OF REGRESSION COEFFICIENTS: PERFORMANCE UNDER LINEX LOSS FUNCTION

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1 CHAPTER-IV PRE-TEST ESTIMATOR OF REGRESSION COEFFICIENTS: PERFORMANCE UNDER LINEX LOSS FUNCTION 4.1 INTRODUCTION It hs lredy been demonstrted tht the restricted lest squres estimtor is more efficient thn the ordinry lest squres estimtor in terms of vrince. This result is true even if the imposed restrictions re not correct. Therefore, the decision whether to use the restricted estimtor or not involves trdeoff between bis nd vrince reduction. One wy to evlute this trdeoff is using loss function. In generl, loss function reflects the loss incurred by incorrectly guessing n estimtor by using its unbised estimtor. The most commonly used loss function is qudrtic error loss function, but in mny situtions the use of qudrtic error loss function is inpproprite. If such loss function is tken s mesure of inccurcy, then the resulting risk is often too sensitive to the ssumption bout the behvior of the til of the probbility distribution. The choice of the qudrtic error loss function is more undesirble if it is supposed to represent rel finncil loss. When over estimtion nd under estimtion re not equl, the symmetry in qudrtic error loss function becomes trouble. In prctice, overestimtion nd underestimtion of the sme mgnitude often hve different economic consequences nd the ctul loss function is symmetric. There re numerous such exmples vilble in this direction. Vrin(1975) pointed out tht, in rel estte vlutions, underssessment results in n pproximtely liner loss of revenue wheres overssessment often results in ppels with ttendnt substntil litigtion nd other costs. In food-processing industries it is undesirble to overfill continers, since there is no cost recovery for the overfill. If the continers re underfilled, however, it is possible to incur much more severe penlty rising from misrepresenttion of the product s ctul weight or volume. In dm construction n underestimtion of the pek wter 83

2 level is usully much more serious thn overestimtion.[see; e.g. Zellner (1986)] The use of LINEX loss function in estimtion procedures hs been considered in different context by Chin nd Jnssen (1995), Ohtni (1995), Gles nd Giles (1996), Ohtni (1999), Tkd(000). In seminl work Bncroft (1944) explored the ide of the pretest estimtor nd studied its performnce properties under qudrtic error loss function. Such estimtor incorportes the smple informtion with the nonsmple informtion bout the vlue of the prmeter. Much reserch hve been completed concerning risk properties ssocited with vrious estimtors of liner model tht explicitly utilize prior liner constrints on the model prmeters, including restricted lest squres nd Pre-test estimtor [see; Bock et. l. (1973), Judge nd Bock (1978)]. Clrke et. l (1987) derive exct finite smple expression for the bises nd risk of severl common pre-test estimtors ssocited with lest squres, mximum likelihood nd minimum men squred error of pre-test estimtor in liner regression model. In the field of pre-test estimtor for vrious model prmeter Sleh(006), Khn et.l. (005) studied the performnce properties of four lterntive estimtors, including the pre-test estimtor of the intercept prmeter of simple liner regression model. Their study shows tht under certin conditions pre-test estimtor domintes the lest squres estimtor. However, ll these studies hve been crried out under qudrtic error loss function. As the LINEX loss function is more generl thn the qudrtic error loss function, therefore, the im of this chpter is to study the performnce property pre test estimtor under the LINEX loss function hve been crried out. The orgniztion of this chpter is s follows: Section 3. describes the model nd the estimtors, the performnce properties of the estimtors under LINEX loss function hve been studied in Section 3.3. In order to study nd compre the performnce properties of the estimtors empiriclly, progrmme in MATLAB (013) is developed. Lstly, in section 3.4 proofs of theorems re provided. 84

3 4. THE MODEL AND THE ESTIMATORS Let us consider the orthonorml liner regression model (4.1) y = Xβ + ε where y is n n 1 vector of observtions on the response vrible, β is p 1 vector of unknown prmeters ssocited with the p regressors nd ε is n n 1 vector disturbnces. The elements of the disturbnces vector ε re ssumed to be independently nd identiclly distributed ech following norml distribution with men zero nd vrince σ, so tht E(ε) = 0 E(εε ) = σ I n where σ is finite but unknown. Under orthonorml model the lest squres estimtor from model (4.1) becomes (4.) b = X y ; b~(β, σ I p ) Let us now ssume tht if some prior informtion in the form of restrictions on β re vilble, which is given by (4.3) r Rβ = δ where R is q p full row rnk mtrix of known elements, q being the number of restrictions imposed on the coefficients, r is q 1 vector of known elements nd δ is q 1 vector representing the errors in the restrictions. Appliction of lest squres to (4.1) subjected to (4.3) yields the restricted lest squres estimtor of β nd is given by (4.4) b R = b + R [RR ] 1 (r Rb) As discussed in Chpter-II tht b R follows norml distribution with men E(b R ) = β + Aδ ; A = R (RR ) 1 nd vrince covrince mtrix 85

4 V(b R ) = σ [I R (RR ) 1 R] In order to test the comptibility of smple informtion (4.1) nd the nonsmple informtion (4.3), Pre-test estimtor of regression coefficient my be defined s discussed previously, is given by (4.5) b PT = b I [0,c) (u)(b b R ) let us write the k th elements of p 1 vector b PT s (4.6) (b PT ) k = b k I ( u 1 u < c ) (b k b R k ) where c = q c v, nd k = 1,, 3,., p 4.3 PROPERTIES OF ESTIMATORS UNDER LINEX LOSS FUNCTION Properties of estimtor of the coefficients re generlly investigted under symmetric loss structures such s bsolute error loss function [see; e.g. Ohtni nd Giles (1996)]. Such loss structure ssigns equl weightge to positive nd negtive estimtion errors of the sme mgnitude. However in such situtions when under estimtion is more serious thn overestimtion nd vice-vers, it my be pproprite to use symmetric loss function. Over the yers, greter emphsis is being plced on LINEX loss function which is symmetric nd gives unequl weights to over nd under estimtion errors. Therefore, we consider the following LINEX loss function s proposed by Vrin (1975). (4.7) L( ) = b[e 1] Where = (β β) β is estimtion error, < < ( 0), b (> 0) re the sclrs specifying the loss function. The constnt b serves to scle the loss function (4.7) nd the constnt determines shpe of the loss function (4.7). 86

5 Without loss of generlity choose b to be one so tht the loss function (4.7) reduced to (4.8) L( ) = e 1 Therefore, the loss function of k th elements of p 1 vector b, under (4.8) is given by (4.9) L(b k, ) = e b k. b k 1 nd risk of k th elements of ordinry lest squres estimtor under LINEX loss function is given by (4.10) R(b k ) = E[L(b k, )] = E (e b k β b k ) E ( k ) 1 Thus, the following theorem gives the risk of ordinry lest squres estimtor under LINEX loss function. Theorem 4.1: When the errors in the model (4.1) re normlly distributed, the risk of k th element of the ordinry lest squres estimtor (4.) under LINEX loss function is given by (4.11) R(b k ) = e 1 σ.( ) 1 Proof : See Proof of the Theorem. Clerly, the risk function of the ordinry lest squres estimtor depends on the mgnitude of but not on its sign. Therefore, the functionl form of risk of the ordinry lest squres estimtor, it is evident tht s grows lrger, the risk of the ordinry lest squres estimtor lso grows lrger. While for smll is risk of ordinry lest squres estimtor smll. Using (4.8) the loss function of k th elements of p 1 vector b R,my be written s 87

6 (4.1) L(b R k, ) = e b Rk. b Rk 1 nd risk of k th elements of restricted lest squres estimtor under LINEX loss function is given by (4.13) R(b Rk ) = E (e b Rk ) E ( b Rk ) 1 The following theorem gives the risk of restricted lest squres estimtor under LINEX loss function. Theorem 4.: When the errors in the model (4.1) re normlly distributed, the risk of k th element of the restricted lest squres estimtor (4.4) under LINEX loss function is given by (4.14) R(b Rk ) = e δ k + 1.( σ ) + δ k 1 Proof: See Proof of the Theorems. When restriction is true (4.14) reduced to 1 (4.15) R(b Rk ) = e.( σ ) 1 In order to compring the risks of ordinry lest squres estimtor nd restricted lest squres estimtor from (4.11) nd (4.15), we my write (4.16) R(b k ) R(b Rk ) = e 1 σ.( ) 1 e.( σ ) It is pertin to mention tht the vrince of restricted lest squres estimtor is lwys smller tht the ordinry lest squres estimtor s discussed in Chpter-I, therefore, from (4.16) it is evident tht if prior informtion is correct the risk of restricted lest squre estimtor under LINEX loss is less tht the risk of ordinry lest squres estimtor. Our empiricl study grees this theoreticl result which is presented in Tble 4.1. for σ = 1, = 1 nd different vlues of σ. The vlues obtined Tble 4.1 for the risk re then presented grphiclly. 88

7 Tble 4.1 Risk of estimtors for σ = 1, = 1 nd selected vlues of σ δ k =0 δ k =.5 b b R k t σ_*=.1 b Rk t _*=.5 b Rk t σ_*=.9 b Rk t σ_*=.1 b Rk t σ_*=.5 b t σ_*=.9 Rk

8 Risk The Figure 4.1 presents the grph tht gives comprtive performnce of the ordinry lest squres nd restricted lest squres estimtor when prior informtion is correct. From Figure 4.1 it is evident tht the risk of restricted lest squres estimtor is less thn risk of restricted lest squres estimtor. It is importnt to noticing tht when the vlue of σ increse the risk of restricted lest squres estimtor lso increses. Figure 4.1 Risk of Estimtors for = 1, δ k = 0, σ = 1 nd selected vlues of σ Risk Comprison of OLS nd RLS b br σ_*=.1 br σ_*=.5 br σ_*= If prior informtion is not correct, Figure 4.1(A) present the risk of restricted lest squres estimtor for fixed vlue of σ nd different vlues of δ k. From Figure 4.1(A) it is esy to see tht when restriction specifiction error increse the risk of restricted lest squres estimtor increses. 90

9 Risk Figure 4.1 (A) Risk of Restricted Lest Squres Estimtor for = 1 nd selected vlues of δ k nd σ 3.5 Risk of RLS Estimtor br σ_*=.1,δ*=0 br σ_*=.1,δ*=.1 br σ_*=.1,δ*= Now let us find the risk of k th elements of the Pre-test estimtor, for this purpose using (4.8), we cn write (4.17) R[(b PT ) k ] = E (e (b PT) k ) E ( (b PT ) k ) 1 Using it we get the following theorem gives the risk of Pre-test estimtor of regression coefficient under LINEX loss function. Theorem 4.3: When the errors in the model (4.1) re normlly distributed, the risk of Pre-test estimtor (4.6) under LINEX loss function is given by (4.18) R[(b PT ) k ] = e 1 σ.( ) 1 1 {e.( σ ). e (δ k 1) δk } i=0 w i (λ)ι qc v+qc ( q + i, v ) Proof: See Proof of the Theorems. From the bove theorem it is esy to see tht when prior informtion is correct, the risk of Pre-test estimtor is 91

10 1 R[(b PT ) k ] = R(b k ) {e.( σ ) } Ι qc ( q v+qc, v ) As erlier discussed the vrince of restricted lest squres estimtor is lwys less thn the vrince of lest squres estimtor nd Ι qc ( q, v ) 1, therefore v+qc when prior informtion is correct the risk of Pre-test estimtor is less thn the risk of ordinry lest squres estimtor. In order to justify this theoreticl result n empiricl study hs been mde, which presented in Tble 4.. The prmeter vlues for numericl computtions for compring risks re v = 0, q = 1, = 1, δ k = 0, σ = 1, σ =. The vlues obtined for the risks re then presented grphiclly. The Figure 4. presents the grph tht gives comprtive performnce of the ordinry lest squres, restricted lest squres nd Pre-test estimtors bsed on their risk functions when the size of the criticl vlue is vried. From the Figure 4., it is observed tht risk of Pre-test estimtor is smller thn of ordinry lest squres estimtor but lrger thn the risk of restricted lest squres estimtor. Therefore, when prior informtion is correct the restricted lest squres perform better thn the ordinry lest squres estimtor nd gree with theoreticl results. Tble 4. Risk of Estimtors for = 1, δ k = 0, σ = 1 nd σ =. 5 b br b_pt c=1 b_pt c= b_pt c=

11 Risk Figure 4. Risk of Estimtors for = 1, δ k = 0, σ = 1 nd σ =. 5 Risk Comprison b br b_pt Now in order to understnd the behviour of risk properties of the Pre-test estimtor under LINEX loss with the chnge in the mount of non-centrlity prmeter nd vlue of shpe prmeter, the risks vlues hve been tbulted for vrious vlues of v, λ nd c. The Tbles 4.3 to 4.6 present the results on risk performnces of the Pre-test estimtor. Tble 4.3 Risk of Pre-test estimtor for selected vlues of σ = 1, v = 10, q = 1, = 1, δ k =. 5 nd σ =. 1 λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ=

12 Tble 4.4 Risk of Pre-test estimtor for selected vlues of σ = 1, v = 10, q = 1, = 1, δ k =. 5 nd σ =. 5 λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ= Firstly the grphs hve been plotted for ll the tbulted vlues of λ which re shown in Figure 4.3 which presents comprehensive view of ll possible vlues of λ. As the grph does not provide cler picture of the ptterns obtined, the grphs hve gin been plotted for smller nd then for few selected vlues of λ. Figures 4.3(A) shows some specific smller vlues of λ. It is evident from the Figure 4.3(A) tht for smll vlues of the non-centrlity prmeter, in the rnge 0< λ < 0.1, the risk of the Pre-test estimtor under LINEX loss hve lmost sme vlues. Figure 4.3(B) nlyze the risk of Pre-test estimtor for vlue of noncentrlity prmeter λ>.1. From Figure 4.3(B) it is importnt to mention tht when the vlue of non-centrlity prmeter increse the risk of the Pre-test estimtor decreses for negtive while the risk increse for positive. nd the mgnitude of risk difference increses with increse in λ. When the vlue of shpe prmeter is ner zero, the risk of the Pre-test estimtor is lmost sme nd minimum, this is due the fct tht for smll vlues of the LINEX loss function is equl to qudrtic error loss function. 94

13 Risk Risk Figure 4.3 Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 1 Risk of Pre-test Estimtor λ=0 λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ=10 Figure 4.3(A) Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 1 Risk of Pre-test Estimtor λ=0 λ=0.01 λ=0.05 λ=

14 Risk Risk Figure 4.3(B) Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 1 Risk of Pre-test Estimtor λ= λ=.5 λ=1 λ=5 λ= Coming to the vlues computed in Tble 4.4, it cn be seen tht n increse in the vlue of σ, mkes n increse in the risk of Pre-test estimtor. These vlues re grphiclly represented in Figures 4.4(A) nd 4.4 (B). Figure 4.4 Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 5 Risk of Pre-test Estimtor E λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ=10 96

15 Risk Risk Lstly let us study the effect of chnge in the degrees of freedom on the performnce properties of Pre-test estimtor. In the Tbles 4.3- nd 4.6 for fixed σ =.1 nd in the Tbles 4.4. nd 4.6 for fixedσ =.5 the vlue of the degrees of freedom is chnged from 10 to 50. It is importnt to notice tht the behviour of risk of the Pre-test estimtor does not chnge if we increse degrees of freedom i.e v, this cn be verified from Tbles nd Tbles 4.4 nd 4.6. Figure 4.4(A) Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 5 Risk of Pre-test Estimtor E λ=0 λ=0.01 λ=0.05 λ=0.1 λ= Figure 4.4(B) Risk of Pre-test Estimtor for v = 10, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 5 Risk of Pre-test Estimtor λ=.5 λ=1 λ=5 λ=10 6E

16 Tble 4.5 Risk of Pre-test estimtor for selected vlues of σ = 1, v = 50, q = 1, = 1, δ k =. 5 nd σ =. 1 λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ= Tble 4.6 Risk of Pre-test estimtor for selected vlues of σ = 1, v = 50, q = 1, = 1, δ k =. 5 nd σ =. 5 λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ=

17 Risk Risk Figure 4.5 Risk of Pre-test Estimtor for v = 50, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 1 Risk of Pre-test Estimtor λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ=10 Figure 4.6 Risk of Pre-test Estimtor for v = 50, q = 1, = 1, δ k = 0. 5, σ = 1, σ = 0. 5 Risk of Pre-test Estimtor λ=0.01 λ=0.05 λ=0.1 λ= λ=.5 λ=1 λ=5 λ= Proof of lemm nd Theorems Lemm 4.1 If w is normlly distributed with men zero nd vrince one, nd w nd S re independent then for ny Borel mesurble functionφ: R (0, ) R, nd for ny ξ R then E[e ξ w φ(w, S)] = e ξ E[φ(w + ξ, S)] 99

18 provided [e ξ w φ(w, S)] is integrble. Proof: By definition E[e ξ w φ(w, S)] = E{E[e ξ w φ(w, S)/S]} = E { 1 π w φ(w, S)eξw R dw} = e ξ E { 1 π 1 φ(w, S)e (w ξ) Now tking t = w S the bove expression reduced to R dw} E[e ξ w φ(w, S)] = e ξ E { 1 Which complete the proof of lemm. Proof of the Theorems Proof of Theorem 4.1 π 1 φ(t + ξ, S)e t R = e ξ E[φ(w + ξ, S)] dw} Using (4.8) the risk of k th elements of ordinry lest squres estimtor under LINEX loss function my be written s (A.1) R(b k ) = E (e b k β b k ) E ( k ) 1 In order to find the risk function (A.1), let us first solve the first term of the right hnd side in (A.1) (A.) where w = b k ~N(0,1) σ E (e b k σ w ) = E [e ] Hence pplying the Lemm 3.1, we obtin 100

19 (A.3) E (e b k β 1 k ) = e.σ β k Since b k is unbised estimtor of, therefore, the second term of the right hnd side in (A.1) is zero. Hence substituting the vlue of (A.3) in (A.1) we obtin the risk of ordinry lest squres estimtor s given in Theorem 4.1. Proof of Theorem 4. The risk of k th elements of restricted lest squres estimtor under LINEX loss function is given by (A.4) R(b Rk ) = E (e b Rk ) E ( b Rk ) 1 As b R is lso follows norml distribution, therefore, the proof of the Theorem is k stright forwrd on the pttern of Theorem 4.1. Proof of Theorem 4.3 Using (4.8) the risk of k th element of Pre-test (b PT ) k under LINEX loss function cn be written s (A.5) PT) R[(b PT ) k ] = E [e (b k ] E [ (b PT ) k ] 1 For the purpose to solve the risk function (A.5), let us first solve the first term of the right hnd side in (A.5) (A.6) PT) E [e (b k ] = e (b PT ) k f((b PT ) k )d(b PT ) k Therefore, using the density function of the Pre-test estimtor (b PT ) k defined in Chpter-II s f(τ) = 1 π [1 σ e 1 (τ σ ) φ 1 e 1 (τ δ k) σ ] σ the eqution (A.6) my be written s 101

20 E [e (b PT ) k ] = e τ f(τ)dτ (A.7) PT) E [e (b k ] = 1 τ β { k π σ e e 1 (τ σ ) dτ w i (λ) Ι pc i=0 v+pc ( p + i, v ) e τ e 1 (τ δ k) σ dτ} Now using the trnsformtions τ σ = t 1 in fist integrl in right hnd side of (A.6) nd τ δ k = t σ in second integrl in right hnd side of (A.7), the eqution (A.7) reduced to (A.8) PT) E [e (b k ] = 1 π { e σ t 1 e t 1 dt 1 w i (λ) Ι qc i=0 v+qc ( q + i, v ) e (δ k 1) e σ t e t dt } Now using the formul e x x PT) (A.9) E [e (b k ] = e 1 σ.( ) 1 e.( σ ) dx = π e 4 in (A.8) we obtin. e (δ k 1) i=0 wi (λ) Ι qc ( q + i, v ) v+qc since the second term involve the first order moment of the Pre-test estimtor, which is obtined in Chpter-II, therefore, using the first order moment of the Pre-test estimtor directly s i=0 (A.10) E[(b PT ) k ] = δ k w i (λ) Ι qc v+qc ( q + i, v ) Thus, using (A.9) nd (A.10) in (A.5) we get the risk of k th element of the Pre-test estimtor (b PT ) k under LINEX loss s given in Theroem