面板閾值模型

1、6.3 Threshold Effects in Non-dynamic PanelsAre regression functions identical across all observations in a sample, or do they fall into discrete classes? This question may be addressed using threshold regression techniques. Threshold regression models specify that individual observations can be divi

2、ded into classes based on the value of an observed variable. Hansen (1999) introduces econometric techniques appropriate for threshold regression with panel data.6.3.1 Model SetupThe observed data are from a balanee panel 幾耳忑：1 <i <NJ <t <T . The dependent variable ylt is scalar the thre

3、shold variable >s scalar, and the regressor % is a k vector. The structural equation of interest isYxt =ZA + (d ")+(5 >力+ %©52)where !() is the indicator fu notion. An alternative intuitive way of wnti ng (6.52) is%(kt<7andAnother compact representation of (6.52) is to setso that

4、(6.52) equalsyrt=/4 + Z?t(7)+elt(6 53)The observations are divided into two Tegimes depending on whether the threshold vanable qit is smaller or larger than the threshold 7 . The regimes are distinguished by differing regression slopes,久 and 0?. For the identification of and , it is required that th

5、e elements of are not time invanant. We also assume that thethreshold variable is no t time i nvzria nt.The error elt is assumed to be independentand identically distributed (iid) with mean zero and finite vanance a2 The nd assumption excludes lagged dependent variables from .The analysis is asympto

6、tic with fixed T as N ts.6.3.2 Model EstiamtionOne traditional method to eliminate the individual effect /zx is to remove in dividual-specific means Note that taki ng averages of (6.52) over the time in dex t andtaking the difference of (6.52) and the averages producesy： = "X(7)+e：(6.54)T where

7、 y； = ylt-yx , >C(7)= (/)-(/) . £ =命一可 and %=嚴(yán)工人, t-1TT(7)= T"1Vj(/)i =T-1eit . Using the matrix notation, (6 54) is equivalent tot=it=i(6.55)Y* = X*(/)Z? + e#For any given /, the slope coefficient fl can be estimated by ordinary leastsquares (OLS). That is,(6.56)(/) = (x*(/)r Xe(/) X*(

8、z)rY*The vector of regression residuals ise*(/) = V -X*(/)(/)(6.57)and the sum of squared errors is(6.58)Chan (1993) and Hansen (2000) recommend estimation of / by least squares. This is easiest to achieve by minimization of the concentrated sum of squared errors (6.58).I lence the least squares est

9、imators of / is(6.59)/= argminS1(7)rOnce / is obtained, the slope coefficient estimate is 0 = 0(?) The residualvector is e = e*(/)and residual varianee(6.60)e eN(T-1)6.3.3 Inference(1) Testing for a thresholdIt is important to determine whether the threshold effect is statistically significant The h

10、ypothesis of no threshold effect in (6.52) can be represented by the linear constraint h0：A = 2Under the null hypothesis of no threshold, the model is九二"+ 01冗 + 陷(661)After the fixed-effect transformation is made, we havey； = A冗+ e；(6.62)By OLS , we have px 可 and the sum of squared errors Sq =

11、e*'e*. The likelihood ratio test of HO is based on偲-$)件(663)The asymptotic distribution of F】is non-standard Han sen (1996) shows that a bootstrap procedure attains the first-order asymptotic distribution, so p-values constructed from the bootstrap are asymptotically valid.(2) Asymptotic distrib

12、ution of threshold estimateWhen there is a threshold effect (人豐 0jChan (1993) and Hansen (2000) have shown that / is consistent for /0 ( the true value of / ) and that the asymptotic distribution is highly non-standard Hansen (1999) argues that the best way to form confide nee in tervals for / is to

13、 form the 'no-rejection regi orV using the likelihood ratio statistic for tests on 7. To test the hypothesis Ho:/ = /0, the likelihood ratio test is to reject for large values of LR (/0) whereLKi(7)= (7)-si(7)/©64)Under certain assumptions and Ho:/ = /Ot Hansen shows that LRj(/)asN where g

14、is a random variable with distnbution functionP( <x) = (l-exp(-2)3(6.65)Since the asymptotic distribution is pivotal, i.e.r this limiting distribution does not depend on nuisance parameters, it may be used to form valid asymptotic confidence intervals. Furthermore, the distribution function (6.65

15、) has the inversec(a) = -21og(l->A-a)(6.66)from which it is easy to calculate critical values. For example, the 5% critical value is 7.35. A test of Ho :/ = 70 rejects at the asymptotic level a if LRj (/) exceeds c(a ).To form an asymptotic confidence interval for / , the 4no-rejection regiorV of

16、 confidence level 1-a is the set of values of 7 such that LK, (/) <c(a). This is easiest to find by plotting LRj (/) against 7 and drawing a flat line at c(a).(3) Asymptotic distribution of slope coefficientsAAThe estimator 0 = 0(夕)depends on the threshold estimate /, however, Chan (1993) and Han

17、sen (2000) show that the dependenee on the threshold estimate is not of first-order asymptotic importance, so inference on p can proceed as if the thresholdA estimate 7 were the true value. Hence 0 is asymptotically normal with a covanance matnx which can be /nt,、tv=F< 1=1 t=lyIf the errors are a

18、llowed to be condition ally heteroskedastic, the natural covana nee matnx estimator for 0 is/NT V N Tf oVh=工空（加工工拓（刃拓（刃何J6.3.4 Multiple thresholdsModel (6.52) has a single threshold In some applications there may be multiple thresholds. For example, the double threshold model takes the formYit =A +

19、A M(% ")+ 0忑I(/1 < 5 “2) + 03憑I(7z V qj+ 勺(6 67) where the thresholds are ordered so that 人 < 乙 We will focus on this double threshold model since the methods extend in a straightforward manner to higher-order threshold models.(1) EstimationFor given(7P/2)>(6 67)is linear in the slopes

20、 (/民,03)so OLS estimation is appropnate Thus for given (7P/2) the concentraied sum of squared errors S(/P72) is straightforward to calculate (as in the single threshold model) The joint LS estimates of (/p/2) are by definition the values which jointly minimize S(/p/2) ). While these estimates might

21、seem desirable, they may be quite cumbersome to implement in practice. A grid search overrequires approximately n2 = (NT)- regressions which maybe prohibitively expensive.A remarkable insight allows us to escape this computational burden. It has been found (Chong, 1994; Bai, 1997; Bai and Perron, 19

22、98) in the multiple changepoint model that sequential estimation is consistent. The same logic appears to apply to the multiple threshold model. The method works as follows.Step 1: Let Sj (/) be the single threshold sum of squared errors as defined in (6.58) and let 久 be the threshold estimate which

23、 minimizes q (?). The analysis of Chong a nd Bai suggests that 久 will be consistent for either 為 or /2.Step 2: Fixing the first-stage estimate認(rèn),the second-stage criterion is(6.68)and sec on d-stage threshold estimate is(6.69)/： = arg min S3 (/2)Bai (1997) has shown that is asymptotically efficient,

24、but 久 is not. This is because the estimate 久 was obtained from a sum of squared errors function which was contaminated by the presence of a neglected regime. The asymptotic efficiency of suggests that yx can be improved by a third-stage estimation. Bai (1997) suggests the following reftnement estima

25、tor.Step 3 Fixing 夕；,deftne the refinement critenon(6 70)（方J，if広7i=<and the refinement estimate7； = arg mill(6.71)nBai (1997) shows that the refinement estimator 夕;is asymptotically efficient inchangepoint estimation, and we expect similar results to hold in threshold regression.(2) Determining n

26、umber of thresholdsIn the con text of model (6.67), there are either no thresholds, one threshold, or twothresholds. In Section 6.3.3 we introduced 片 as a test of no thresholds against one threshold, and suggested a bootstrap to approximate the asymptotic p-valuef 片 rejects the null of no threshold, in the context of model (6.67) we need a further test to discnmimate between one and two thresholds.The minimizing sum of squared errors