Testing for a Unit Root in a Random Coefficient Panel Data Model

Transcription

1 esting for a Unit Root in a Random Coefficient Panel Data Model Joakim Westerlund Lund University Sweden Rolf Larsson Uppsala University Sweden February, 009 Abstract his paper proposes a new unit root test in the context of a random autoregressive coefficient panel data model, in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance. he asymptotic distribution of the test statistic is derived and simulation results are provided to suggest that it performs very well in small samples. JEL Classification: C3; C33. Keywords: Panel unit root test; Random coefficient autoregressive model. Introduction Consider the panel data variable y it, observable for t,..., time series and i,..., N cross-sectional units. he analysis of such variables has been a growing field of econometric research in recent years, with a majority of the work focusing on the issue of unit root testing. he main reason for this being the well-known power problem of univariate tests in cases when is small, and the potential gain that can be made by pooling across a cross-section of similar units. he most common approach, pioneered by Levin et al. 00), is to assume that y it admit to a first-order autoregressive AR) representation with a common slope coefficient, y it ρy it + u it, Financial support from the Jan Wallander and om Hedelius Foundation under research grant numbers P005-07: and W : is gratefully acknowledged. Corresponding author: Department of Economics, Lund University, P. O. Box 708, S-0 07 Lund, Sweden. elephone: , Fax: , address: joakim.westerlund@nek.lu.se.

2 where u it is a stationary mean zero disturbance. A pooled least squares t-statistic is then computed, and the null hypothesis that ρ is tested against the alternative that ρ <. he major limitation of this approach is that ρ is restricted to be the same for all units. he null makes sense under some circumstances, but the alternative is too strong to be held in any interesting empirical cases. For example, when testing for growth convergence, one can formulate the null as implying that none of the countries under study converges. But it does not make any sense to assume that all the countries will converge at the same rate if they do converge. Im et al. 003) relax the assumption of a common AR coefficient under the alternative. he idea is very simple. ake the above model and substitute ρ i for ρ, which in the usual formulation where ρ i is fixed results in N separate AR models, one for each unit. hus, instead of looking at a single pooled t-statistic, we now look at N individual t-statistics, which can be combined by taking for example the average. he resulting average statistic tests the null that ρ i ρ for all i against the alternative that ρ i < for a positive fraction of N. But this is basically the same as saying that the null should be rejected if at least one of the individual tests end up in a rejection at the appropriate significance level, which brings us back to the original problem, namely that has to be large. Put in another way, to be able to determine the fraction of units for which the null is rejected, needs to be large. But if is large enough for valid inference at the individual level, then there is hardly no point in pooling. his leaves us with an intricate dilemma. On the one hand, we would like to exploit the additional power that becomes available when we pool, and when we do this we would like to allow for some heterogeneity in ρ i. On the other hand, this allowance requires to be large, in which case we can just as well go back to doing unit-by-unit inference. he appropriate response here depends on the relative size of N and. But if only N is large enough, then it should be possible to devise powerful tests that are informative in an average sense, even if is small. his leads naturally to the consideration of a random specification for ρ i. Suppose for example that ρ i + c i, where c i is an independently distributed random variable with mean µ c and variance ωc. hen the null of a unit root corresponds to the joint restriction that µ c ωc 0, while the alternative is that µ c < 0 or ωc > 0, or both. his random specification of c i has many advantages in comparison to the traditional

3 fixed specification. Firstly, working with incompletely specified models inevitably leads to a loss of efficiency. he random specification reduces the number of parameters that needs to be estimated, and is therefore expected to lead to more powerful tests. Secondly, the random specification is more general, because fixed coefficients are special random variables. Whether something is random or not should be decided by considering what would happen if we were to replicate the experiment. Is it be realistic to assume that c i stays the same under replication? If not, then the random specification is more appropriate. hirdly, by considering not only the mean of c i but also the variance, random coefficient tests account for more information, and are therefore expected to be more powerful. aking this random coefficient model as our starting point, the goal of this paper is to design a procedure to test the null hypothesis that µ c ωc 0, which has not received much attention in the previous literature. In fact, the only attempt that we are aware of is that of Ng 008), who uses a random coefficient model as a basis for proposing an estimator of the fraction of units with a unit root. However, this procedure does not exploit the fact that under the null hypothesis the variance of c i is zero, which makes it suboptimal from a power point of view. It is also rather restrictive in nature, and cannot be easily generalized to accommodate for example high-order serial correlation. Our testing methodology is rooted in the Lagrange multiplier LM) principle, and can be seen as a generalization of the recent time series work of Distaso 008) and Ling 004), who consider the problem of testing for a unit root when the AR coefficient is time-varying. It is also very similar to the seminal approach of Schmidt and Phillips 99), from which it inherits many of its distinctive features. he test is for example based on an optimal detrending procedure that imposes the null hypothesis, and if a linear trend is included the test is asymptotically invariant with respect to the presence of a level break. It is also very straightforward and easy to implement. he asymptotic analysis reveals that the LM test statistic has a limiting chi-squared distribution that is free of nuisance parameters under the null hypothesis. In particular, it is shown that the asymptotic null distribution is the same regardless of whether there is a constant or a constant and trend in the model. hus, unlike most test procedures around, with this test there is only one set of critical values that applies in both cases. We also study the limiting behavior of our test statistic under both random and local 3

4 alternative hypotheses. On the one hand, when c i is distributed independently of N and we show that the test is consistent, and that the rate of the divergence is higher than for the conventional t-test of Levin et al. 00). On the other hand, when c i is allowed to shrink towards zero at the rate N, we show that while unbiased in the case with a constant, in the case with a constant and trend the test has power equal to size, which is a reflection of the so-called incidental trends problem. A small simulation study is also undertaken to evaluate the small-sample properties of the test, and the results show that the asymptotic properties are borne out well, even in very small samples. A word on notation. he symbols, p and : will be used to signify weak convergence, convergence in probability and definitional equality, respectively. As usual, y O p r ) will be used to signify that y is at most order r in probability, while y o p r ) will be used in case y is of smaller order in probability than r. In the case of a double indexed sequence y N,, N will be used to signify that the limit has been taken while passing both indices to infinity jointly. Restrictions, if any, on the relative expansion rate of and N will be specified separately. Model and assumptions he data generating process of y it is given by y it d it + yit, s ) where d it is the deterministic part of y it, while yit s is the stochastic part. he typical elements of d it include a constant and a linear time trend, and this is also the specification considered here. Specifically, using p to denote the lag length, then d it α i + β i t p), which nests two cases. In the first, there is no trend, while in the second, there is both an intercept and trend. he parameters α i and β i can be either known or unknown to be estimated along with the other parameters of the model. he stochastic part is assumed to evolve according to a first-order AR process, yit s + c i )yit s + u it ρ i yit s + u it, ) If y is deterministic, then O p r ) and o p r ) are replaced by O r ) and o r ), respectively. 4

5 or, equivalently, y s it c i y s it + u it, where c i satisfies the following random coefficient assumption. Assumption. c i [ M, 0 ] is i.i.d. with mean µ c and variance ω c for all i, where M 0. he error u it is assumed to follow a stationary and invertible AR process of order p, φ i L)u it ε it, 3) where φ i L) : p j φ ijl j is a polynomial in the lag operator L and ε it is an error term that satisfies the following assumption. Assumption. i) ε it is i.i.d. across both i and t, and has mean zero and variance σ i, ii) φ i L) is non-random with all roots falling outside the unit circle, iii) ε it is independent of ρ i, iv) y s i0,..., ys ip are O p). At times we will also assume that ε it is normal. Assumption 3. ε it is normally distributed. Define e it : φ i L)y it d it ) as well as the scaled variables e it : σ i e it and ε it : σ i ε it, then under the above conditions, e it ρ i )e it + ε it c i e it + ε it 4) or, in terms of the observed variable, y it y it e it + c i e it + ε it, where the stochastic part of y it zit ) lies entirely in the past. hus, letting F t denote the information set available at time t, vary it F t ) var c i e it + ε ) it F t ω c e it ) +, 5) Ey it F t ) y it e it + µ c e it, 6) 5

6 which can be used to obtain the log-likelihood function L of y ip+,..., y i. suppose that Assumption 3 holds so that ε it is normal, then apart from constants, L i tp+ i tp+ ln vary it F t ) ) ln ω c e it ) + ) i tp+ i tp+ his is the starting point for the test statistic that we will consider. he null hypothesis of interest is formulated as H 0 : µ c ω c 0, yit Ey it F t ) ) In particular, vary it F t ) ci µ c )e it + it) ε ωc e. 7) it ) + while for the alternative hypothesis, we will consider two candidates, H and H. he first is formulated as H : µ c < 0 or ω c > 0 or both, which is similar to the conventional heterogeneous specification of Im at al. 003). Of course, with such a specification, we only learn whether the test is consistent and if so at what rate. herefore, to be able to evaluate the power analytically, we will also consider an alternative, denoted H, in which ρ i is local-to-unity as N,. he following near unit root model handles both unit roots and such local alternatives ρ i + c i N. 8) Note that under this specification, H 0 is the same as before, while the local alternative that we will consider is given by H : µ c < 0, corresponding to an AR coefficient that approaches one at rate N. Coincidentally, N is also the rate of consistency of the pooled least squares estimator of ρ i under H 0, which is going to turn out to form the basis of our test statistic. Being an estimate of the slope of the mean function, it is logical to expect that the main effect of 8) is to induce via µ c a non-centrality of the asymptotic distribution of the test. his is the basic reason for why H is stated in terms of µ c, and not ω c. he test simply does not have any power against alternatives of the type ω c > 0. 6

7 Assumptions and are used throughout the analysis. he assumption that Ec i ) and varc i ) are equal across i is made for convenience, and could be relaxed as long as the crosssectional averages of these moments have limits such as µ c and ωc, respectively. he assumed independence across i is restrictive but is made here in order to make analysis manageable. Some possibilities for how to relax this condition are discussed in Section 3. Assumption 3 is only needed for deriving the true LM test statistic and is not necessary for developing the asymptotic theory. 3 he test procedure In this section, we first derive the LM test statistic, and then we establish its asymptotic properties under both the null and alternative hypotheses. Finally, we discuss some generalizations. 3. he LM test statistic Consider the model with trend, in which case s it can be written as e it φ i L) y it α i β i t p) ) y it Φ iy it αi β i φ i L)t p) 9) with e it φ i L) y it β i ) y it Φ i y it βi, 0) where αi : φ i)α i + φ i L)yip s, β i : φ i)β i and y it y it,..., y it p ) is the vector of lags with Φ i φ i,..., φ ip ) being the associated vector of slope coefficients. If there is no trend, then β i 0 and so e it y it Φ i y it αi. In any case, if all parameters but σ i are all known, then we denote the resulting LM test statistic by LM, which at times will be subscripted by c or t to indicate whether there is a constant or a constant and trend in the model. he formulae for LM is given in the following theorem. heorem. Under H 0 and Assumptions to 3 the LM statistic is given by LM i tp+ it ) ěitě i tp+ + ě it ) )ě it )) i tp+ ě it ) i tp+ ě it ) )ě, it )4 where ě it : ˇσ i e it with ˇσ i p tp+ e it). Remarks. 7

8 a) he equality in heorem is not really strict as the right-hand side expression is based on a block-diagonal Hessian matrix, which is an asymptotic result, see Appendix A. In this sense LM is only an approximate LM test statistic. However, given the blockdiagonal structure, the formulae is very simple and intuitive. In fact, a careful inspection reveal that the first part is nothing but the LM test statistic for testing the null that µ c 0 given ω c 0. hat is, the first part is the LM unit root statistic based on a homogenous ρ i. he second part is the LM statistic for testing the null that ω c 0 given µ c 0. b) he formulae for LM reveals some interesting similarities with results obtained previously in the literature. In particular, note how the first part is the squared equivalent of the panel unit root test considered by Levin et al. 00). 3 he second has no direct resemblance of anything that has been proposed earlier in the panel unit root literature. However, it can be seen as a panel version of the test statistic of Leybourne et al. 996), who considers the problem of testing the null of a fixed unit root against the randomized alternative in the context of a single time series. he test statistic as a whole can be regarded as a panel extension of the time series statistics discussed in Distaso 008) and Ling 004). c) Note that φ i L)t p) φ i )t p) + p j jφ ij, where the sum is O) as long as p is fixed. It follows that e it yit Φ iy it αi βi t p) ) + O ) hus, LM is asymptotically equivalent to the test statistic obtained by replacing e it with y it Φ i y it αi β i t p). We will make use of this in the next subsection when we consider a feasible version of LM, which is appropriate in general when Φ i, α i and β i are unknown. 3. Asymptotic null distribution o perform the new tests, the exact distribution of LM must be known a priori. Unfortunately, the exact distribution of this statistic is untractable even when Assumption 3 holds so that ε it is normal. As a response to this, in this section we use asymptotic theory to obtain 3 he first part of LM can also be regarded as a panel version of the LM unit root tests proposed in the time series literature by for example Ahn 993) and Schmidt and Phillips 99). 8

9 the limiting distribution of LM as N,. Although this means that N and must be large for the test to be accurate, it also means that in principle there is no need for any distributional assumptions like normality. However, for some of the proofs normality greatly simplifies the calculations, and we therefore keep Assumption 3 even in this section. he asymptotic null distribution of LM is given in the following theorem. heorem. Under H 0 and Assumptions to 3, as N, with N 0 LM χ ) + 5 χ ), where χ ) and χ ) are two independent chi-square distributed variables with one degree of freedom each. All results reported so far are based on the assumption that Φ i, α i and β i are all known, which is of course not very realistic. Let us therefore consider replacing e it with u it : y it ˆΦ iy it ˆα i ˆβ i t p), ) where ˆα i y ip+ ˆΦ i y ip ˆβ i with ˆβ i and ˆΦ i being the least squares estimators of β i and Φ i, respectively, in the first differenced regression y it β i + Φ i y it + ε it. ) If there is no trend, then we just remove the intercept, and compute ˆα i y ip+ ˆΦ i y ip and u it y it ˆΦ i y it ˆα i.4 he feasible LM statistic is defined as LM + : i it ) ûitû i + û it ) )û it )) i û it ) i û it ) ) û, it )4 where û it : ˆσ i u it with ˆσ i : u it). he asymptotic distribution of LM + is given in the following theorem. heorem 3. Under the conditions of heorem, as N, with N 0 a) LM + c χ ) + 5 χ ), b) LM + t 3 lim N N + χ ). Remarks. 4 If in addition there is no serial correlation, then ˆα i y i and u it y it ˆα i. 9

10 a) heorem 3 a) shows that the asymptotic distribution of LM c + is the same as that of LM c. his is very interesting because typically demeaning leads to an asymptotic bias that has to be removed in order to prevent the statistic from diverging, see for example Levin et al. 00) and Im et al. 003). However, while unbiased in the case with a constant, as shown in b), the presence of a trend that needs to be estimated makes LM t + divergent. he source of this divergence is the first term in the formulae for LM +. Interestingly, the usual practice of mean-adjusting the statistic does not work here in the sense that the resulting distribution of the first term is degenerate. b) As shown in Lemma A. of Appendix A, ˆσ, ˆα i, ˆβ i and ˆΦ i are the feasible restricted maximum likelihood estimators of σ, αi, β i and Φ i, respectively. Although already relatively uncomplicated, as shown in Appendix B the implementation of LM + c can be made even simpler by using the following modification LM ++ c : i it ) ûitû + 6 i û it ) )û it )) i û it ) 5, i û it ) û it )4 which converges in distribution to χ ). An obvious way to rescue LM + t to remove the part that diverges, and to use from divergence is LM ++ t : N i û it ) )û it )) N i û it ) û it )4, whose asymptotic distribution is given by χ ). 3.3 Asymptotic power If the alternative is random in the sense that ρ i does not depend on N and, then we have the following result. heorem 4. Under H and Assumptions and, LM, LM + O p N ). heorem 4 shows that the tests are consistent, and that their power therefore goes to one as N,. he rate of the divergence is O p N ), which is twice as fast as for the Levin et al. 00) test, see Appendix C for a formal proof. Consider next the results obtained under the local alternative H, which are summarized in heorem 5. 0

11 heorem 5. Under H and Assumptions to 3, as N, with N 0 a) LM, LM c + µ c + µ c N0, ) + χ ) + 5 χ ), b) LM + t 3 lim N N + χ ). Remarks. a) he first thing to note is that ω c does not enter the asymptotic distribution of the test. he reason is that the rate of shrinking of the local alternative is just enough to manifest µ c as a nuisance parameter in the asymptotic distribution of the first part of the test statistic. he normalizing order of the second part, which represents the test of ω c 0, is higher and ω c is therefore kicked out. Consider for example the asymptotic distribution of LM. he first three terms, which contain the dependence on µ c, originate with the first part of the statistic, while 5 χ ) originates with the second part, see Appendix C. b) he specification in 8) has two effects. he first is to shift the mean of the limiting distribution of the test. In particular, since µ c > 0, this means that the mean shifts to the left as we move away from H 0, suggesting that the test is unbiased and that its asymptotic local power therefore is greater than the size. he second effect is to increase the variance of the limiting distribution. his effect is especially noteworthy as usually there is only the mean effect. ake as an example the Levin et al. 00) test statistic, which is standard normal under the null that µ c 0. As Moon and Perron 008) show, under H this statistic converges in distribution to µ c + N0, ), which entails only a mean effect. c) Note that the local power of LM + c is the same as that of LM c, so the demeaning does not affect the local power. his result is in agreement with the work of Moon et al. 007), who develop a point optimal test statistic for the null that µ c 0. According to their results estimation of intercepts does not affect maximal achievable power. 5 5 Unfortunately, the optimality property of the single parameter case does not translate directly to the present multiparameter case. he problem lies in that optimality for the single parameter case follows from maximizing power in the only direction available under the alternative hypothesis. In our case we have a power surface defined over all possible values of µ c and ω c, and hence there is no obvious direction that should be used to maximize power.

12 d) Since the asymptotic distribution under H is the same as the one that applies under H 0, the local asymptotic power of LM + t is equal to the size. 6 his stands in sharp contrast to the results obtained for LM and LM + c, which have significant asymptotic power against H. his difference is a manifestation of the difficulty in detecting unit roots in the presence of heterogeneous trends, commonly referred to as the incidental trend problem, see Moon and Phillips 999). 3.4 Generalizations 3.4. Cross-section dependence One drawback with the above analysis is that it supposes that the cross-sectional units are independent, an assumption that is perhaps too strong to be held in many applications. Accordingly, more recent panel unit root tests such as those of Bai and Ng 004), Moon and Perron 004), Phillips and Sul 003), and Pesaran 007) relax this assumption by assuming that the dependence can represented by a common factor model. his approach fits very well with the parametric flavor of our LM framework, and it will therefore be used also in this paper. Suppose that ε it in 3) has the factor structure ε it θ i f t + w it, 3) where we assume for simplicity that f t is a known common factor with loading θ i, while w it is completely idiosyncratic. Both variables are assumed to satisfy Assumption. Under these conditions, 4) can be written as e it θ i ft c i e it + ε it. 4) By using the results provided in Appendix A, it is not difficult to see that ˆθ i, the maximum likelihood estimator of θ i, can be obtained by running the following least squares regression y it β i + Φ i y it + θ i f t + ε it. 5) 6 Despite this, LM + t is consistent against a random alternative in the sense that the probability of a rejection goes to one as N, for a set of AR parameters that does not depend on N or.

13 In view of 4) it is also not difficult to see that in this case LM + is given by LM + + i û it ˆθ i ˆf t )û it ) i û it ) i û it ˆθ i ˆf t ) )û it )) i û it ˆθ i ˆf t ) ) û, it )4 where ˆf t : ˆσ i f t with an obvious definition of LM ++. Here û it is as in ) but with ˆα i given by ˆα i y ip+ ˆΦ i y ip ˆβ i ˆθ i f p+, while ˆβ i and ˆΦ i come from fitting 5) by least squares. ˆσ i is obtained as before but with û it ˆθ i f t ) in place of û it. he asymptotic distribution of the resulting statistic is the same as the one given in heorem 3 for the case with cross-section independence. If f t is also unknown, then we proceed as in Pesaran 007), using the cross-sectional averages of y it as an approximation. Assuming for simplicity that Φ i Φ for all i, and by using a bar denote cross-sectional averaging, then we have from 5) that y t β + Φ y t + θf t + ε t, which in turn yields f t θ y t β Φ y t ε t ) ) θ y t β Φ y t ) + O p. N he implication is that f t can be approximated by a linear combination of y t and y t, which can be found by applying least squares to y it β i + Φ i y it + π i y t + Π i y t + ε it, where ε it absorbs both ε it and the approximation error. We then use ˆβ i + ˆπ i y t + ˆΠ i y t in place of ˆθ i f t when computing the test statistic. Once this approximation has been made, there is of course no claim of validity of the resulting test, and we do not prove here that this approach is asymptotically valid. However, intuition suggests that it should perform well in practice, and unreported simulation evidence confirms this Structural breaks Analogous with the time series statistic studied by Amsler and Lee 995), the asymptotic null distribution of LM t + computed under the assumption of a linear trend but no structural break is unaffected by the presence of a break in the level of y it. 3

14 Let D t t > t ), where x) is the indicator function and t indicates the timing of the break, which may be unit specific. he intuition behind the above result follows from the fact that D t 0 as, suggesting that the break has no effect on u it. Moreover, since D t 0 for all t except when t t, the effect on u it is eliminated when subtracting the mean. he asymptotic null distribution of the LM + t is therefore unaffected. he problem is that exclusion of the break makes the test biased towards accepting H 0. hus, although the break does not affect the asymptotic null distribution of the test statistic, it does reduce its power. o avoid this, D t can be included in the analysis as an additional deterministic regressor, forming d it α i + β i t p) + δ i D t, where δ i measures the magnitude of the break. he analysis can now be conducted exactly as before, augmenting ) with ˆδ i D t, where ˆδ i is the least squares estimate of δ i : φ i)δ i in y it β i + Φ i y it + δ i D t + ε it. he development of procedures that accommodate breaks that are unknown is of interest but beyond the scope of the present contribution. 4 Simulations o be completed. 5 Concluding remarks o be completed. 4

15 References Ahn, S. K. 993). Some ests for Unit Roots in Autoregressive-Integrated-Moving Average Models with Deterministic rends. Biometrica 80, Amsler, C., and J. Lee 995). An LM est for a Unit Root in the Presence of a Structural Break. Econometric heory, Bai, J., and S. Ng 004). A Panic Attack on Unit Roots and Cointegration. Econometrica 7, Billingsley, P. 986). Convergence of Probability Measures. Wiley, New York. Breitung, J., and M. H. Pesaran 005). Unit Roots and Cointegration in Panels. Forthcoming in Mátyás, L., and P. Sevestre Eds.), he Econometrics of Panel Data. Klüwer Academic Publishers. Distaso, W. 008). esting for Unit Root Processes in Random Coefficient Autoregressive Models. Journal of Econometrics 4, Leybourne, S. J., B. M. P. McCabe, and A. R. remayne 996). Can Economic ime Series be Differenced to Stationarity? Journal of Business and Economic Statistics 4, Levin, A., C. Lin, and C.-J. Chu 00). Unit Root ests in Panel Data: Asymptotic and Finite-sample Properties. Journal of Econometrics 08, 4. Ling, S. 004). Estimation and esting Stationarity for Double-Autoregressive Models. Journal of the Royal Statistical Society: Series B 66, Moon, H. R., and B. Perron 004). esting for Unit Root in Panels with Dynamic Factors. Journal of Econometrics, 8 6. Moon, H. R., and B. Perron 008). Asymptotic Local Power of Pooled t-ratio ests for Unit Roots in Panels with Fixed Effects. Econometrics Journal, Moon, H. R., and P. C. B. Phillips 999). Maximum Likelihood Estimation in Panels with Incidental rends. Oxford Bulletin of Economics and Statistics 6,

16 Moon, H. R., B. Perron and P. C. B. Phillips 007). Incidental rends and the Power of Panel Unit Root ests. Journal of Econometrics 4, Pesaran, M. H. 005). A Simple Panel Unit Root est in the Presence of Cross Section Dependence. Journal of Applied Econometrics, Phillips, P. C. B., and H. R. Moon 999). Linear Regression Limit heory of Nonstationary Panel Data. Econometrica 67, 057. Phillips, P. C. B. and D. Sul 003). Dynamic Panel Estimation and Homogeneity esting Under Cross Section Dependence. Econometrics Journal 6, Schmidt, P., and P. C. B. Phillips 99). LM ests for a Unit Root in the Presence of Deterministic rends. Oxford Bulletin of Economics and Statistics 54,

17 Appendix A: Proofs for Section 3. In this appendix we prove heorem. For brevity, the results are only provided for the case with a trend, which is the most general deterministic specification considered. Lemma A.. Under the conditions of heorem and in the case with a trend the maximum likelihood estimators of σi, α i, β i and Φ i are given by a) ˇσ i ε p it, tp+ b) ˇα i y ip+ Φ iy ip β i, c) ˇβ i y i Φ i y i, d) ˆΦi y it y i ) y it y i ) y it y i ) y it y i ), where y i : p y it with an analogous definition of y i. Proof of Lemma A.. Consider a). With the trend specification of d it, 4) becomes for t p +, and for t p +,...,, φ i L)y ip+ φ i )α i + β i ) + ρ i φ i L)y s ip + ε ip+ A) φ i L)y it ρ i )φ i L) α i + β i t p) ) + ρ i φ i L)y it + β i ) + ε it. A) Under H 0 these equations reduce to respectively. φ i L)y ip+ φ i L)α i + β i + y s ip) + ε ip+ α i + β i + ε ip+, A3) φ i L)y it φ i L)y it + β i ) + ε it φ i L)y it + β i + ε it, A4) Moreover, by using 7), we have that under H 0 the log-likelihood function reduces to L p lnσi ) i σ i i tp+ ε it. A5) Clearly, L σ i p σ i + σ 4 i tp+ ε it, 7

18 which can be put equal to zero, and then solved for σ i, giving ˇσ i p tp+ ε it. his establishes a). Consider b). By using ˇσ i in place of σi, we obtain the following concentrated loglikelihood function L p lnˇσ i ) i where, making use of A3) and A4), p ln p i tp+ ε it, A6) ε it ε ip+ + tp+ ε it φ i L)y ip+ αi βi ) + φi L) y it β ). i It follows that L α i ˇσ i φi L)y ip+ αi βi ), implying ˇα i φ i L)y ip+ β i y ip+ Φ iy ip β i. Moreover, since ε ip+, and therefore also ε ip+, is zero when evaluated at ˆα i, L β i ˇσ i φi L) y it βi ), which can be used to establish c), as seen by writhing ˇβ i p φ i L) y it y i Φ i y i. Similarly, by using ˇβ i in place of β i, tp+ ε it φi L) y it ˆβ i ) yit y i ) Φ i y it y i ) ), and so L Φ i ˇσ i yit y i ) Φ i y it y i ) ) y it y i ), 8

19 from which we deduce that ˆΦ i y it y i ) y it y i ) y it y i ) y it y i ). his establishes d) and hence the proof of the lemma is complete. Lemma A.. Under the conditions of heorem, e it ε it and e it s it, where s it : t sp+ ε is. Proof of Lemma A.. From A4), e it φ i L) y it β i ) φ i L)y it φ i L)y it β i ε it, implying that t φ i L)y it φ i )α i + β i φ i L)t p) + φ i L)yit s αi + β i φ i L)t p) + ε is, and so we get t e it φ i L)y it α i β i t p)) ε is s it. sp+ sp+ Lemma A.3. Under the conditions of heorem, L a) ě µ itě it, c b) c) d) e) i tp+ L N µ c ) L ω c i tp+ i tp+ L ω c ) L µ c ω c ě it ), ě it ) ) ě it ), i tp+ i tp+ ě it ) ) ě it ) 4, ě itě it ) 3. 9

20 Proof of Lemma A.3. We prove a). he proofs of b) to e) follow by similar arguments. Making use of 7), L µ c i tp+ ci µ c )e it + ) ε it e it ωc e it ) + i tp+ ε ite it, where the last equality imposes H 0 by setting ωc c i µ c 0. he required result is obtained by using Lemma A. to substitute for ε it, and then to concentrate the resulting expression with respect to σi. Proof of heorem. Define g : H : L ) µ c L ωc L µ c ) L µ c ω c L ω c ) g g ) ), H H H ), where the elements of g and H are given in Lemma A.3. he LM test statistic is defined as LM t : g H) g. We now show that H is diagonal, which yields the desired result after substituting for g and H. Note that LM t G g) G HG ) G g), where N 0 ) G : N 3/. Making use of Lemma A. the off-diagonal element of G HG can be written N 5/ H N 5/ N 5/ i tp+ i tp+ ě itě it ) 3 N 5/ w i ε its it ) 3, 0 i tp+ ˇε itš it ) 3

21 where w i : σ i. As in the main text, a circle superscript denotes division by σ ˇσ i i, while a circle and and a check denote division by ˇσ i. Consider H i tp+ ε it s it )3. Since ε it is independent across both i and t, N 5/ EH ) N 5/ i tp+ Eε it)es it ) 3 ) 0. Moreover, by a functional central limit theorem, s it O p), which in turn suggests that a central limit theorem should apply to N i tp+ it ε s 3. it ) Hence, N 5/ H and by the Cauchy Schwarz inequality, N 5/ H H ) N 5/ i tp+ O p ) N wi )ε its it ) 3 ) / wi ) N i ) ) O p O p, where we have made used of the fact that N 5 5/ i tp+ ) ε its it ) 3 ) w i O p, A7) as follows from a first-order aylor expansion of the inverse of ˇσ i, which in turn is such that ˇσ i p s it ) tp+ p tp+ see for example Levin et al. 00). In fact, we even have / ) ε it σi + O p, A8) Eˇσ i ) p tp+ Eε it) σ i, A9) meaning that ˇσ i is not only consistent but also unbiased. It follows that N 5/ H O p ) ) + O p, N

22 which is O p / N ) if we assume that N G g O p ), yields LM t g H 0 H ) i tp+ it ) ěitě i tp+ ě it ) 0 as N,. his result, together with ) g + O p N ) g g + O p H H N + N i tp+ ě it ) )ě it )) i tp+ ě it ) )ě it )4 + O p ), N where the last equality follows from using Lemma A.3 to substitute for g, g, H and H. Appendix B: Proofs for Section 3. In this section we derive the asymptotic results that hold under H 0. Proof of heorem. From heorem we have that LM t + i tp+ it ) ěitě i tp+ ě it ) i tp+ ě it ) )ě it )) i tp+ ě it ) )ě it )4 i tp+ w iε it it ) s i tp+ w is it ) + O p ) N + i tp+ w i ε it ) ) w i s it ) ) i tp+ w3 i ε it ) ) w i s it ) + O 4 p ) I + II + O p. N Consider I, which can be written as ) N I N i tp+ w iε it it ) s tp+ w is it ) N N i I a I b.

23 Note that if I a N N i tp+ ε it s it, then Ia I a N N i tp+ i w i )ε its it ) / w i ) O p N /4 )O p N /4 ), N i tp+ ε its it ) / which goes to zero if N to I a, whose expectation is given by 0 as N,. It follows that I a is asymptotically equivalent E I a ) N i tp+ Eε its it ) N t i tp+ sp+ Eε itε is) 0. As for the variance, it holds that var I a ) N N N N i j tp+ sp+ i tp+ i tp+ i tp+ E ε it) s it ) ) E t ε ) it) E t ε ) it) t p) tp+ Eε itε js)es it s js ) t sp+ kp+ sp+ tp+ E ε is) ) t + O Eε isε ik ) ) 0 rdr as, where r [0, ]. Define G i : N tp+ ε it s it, which is independent with mean zero and varg i) O/N). herefore, according to condition 3.0) of Phillips and Moon 999), if we can show that for all δ > 0, lim N, E G i G i > δ) ) 0, i where x) is the indicator function, then N i G i N0, ) as N,. 3

24 yields In order to verify this condition we make use of the Cauchy Schwarz inequality, which E G i G i > δ) ) EG 4 i )E G i > δ)) and by further application of the Markov inequality, E G i > δ)) EG δ i ). hus, E G i G i > δ) ) δ i O EG 4 i )EG i ) N ) / N EG 4 i ) EG i ) δ i N ) O). i i ) / herefore, since the above condition holds, if we assume that N Ia ) N Ia + O p N ) N G i + O p i 0 as N,, N0, ), A0) which, via the continuous mapping theorem, gives I a χ ). Consider I b N N i tp+ s it ). As, EI b ) N i tp+ t p). tp+ E s it ) ) N t i tp+ sp+ E ε is) ) hus, by Corollary of Phillips and Moon 999), if tp+ s it ) is uniformly integrable in, then I b p as N,. o verify this condition we make use of heorem 5.4 in Billingsley 968), which states that since s it ) tp+ 0 W i r)) dr as, where W i r) is a standard Brownian motion on r [0, ], uniform integrability is equivalent to requiring E which is obviously true. tp+ s it ) ) E W i r)) dr, 0 4

25 Hence, because I b I b N i tp+ w i )s it ) ) / w i ) N N i O p ) O p ), i ) s it ) tp+ / we have that I b Ib + O p/ ) p, which in turn implies I I a I b I a I b ) N + O p χ ). A) Next, consider II, which can be written as II N 3/ i tp+ w i ε it ) ) ) w i s it ) ε it ) ) II a. w i s it ) 4 II b N 3 N i tp+ w3 i Let IIa N 3/ i tp+ ε it ) ) w i s it ). By the same steps used before when evaluating I a we get IIa II a + O p N ), implying that II a is asymptotically equivalent to II a as N, with N the unbiasedness of ˇσ i, Eε it ˇσ i ) 0, which in turn implies that. By E II a ) N 3/ σ i tp+ i Eε it ˇσ i )E s it ) ) 0 he variance of II a is given in the following lemma. Lemma B.. Under the conditions of heorem, var II a ) O p/ ). Proof of Lemma B.. We omit this proof in this paper. All details can be found in Larsson and Westerlund 009). hese results, together with the joint limit theory of Phillips and Moon 999), yield IIa ) N 5 IIa + O p N0, ) 6 5

26 as N, with N, implying II a 5 6 χ ). As for II b, note that II b N 3 N 3 N 3 i tp+ i tp+ i tp+ w 3 i w 3 i ε it) wi ) s it ) 4 ε it) wi ) s it ) 4 + N 3 ) wi 3 ε it) s it ) 4 + O p, N where the last equality follows from the fact that II a O p ). i tp+ w 3 i ε it) s it ) 4 Let II b N 3 N i tp+ ε it ) s it )4. By a law of large numbers as N, IIb p lim 3 lim 3 tp+ tp+ E ε it) ) E s it ) 4) lim 3 ε E[ i ) B t ε ) ] i, tp+ E s it ) 4) where ε i is a p) dimensional vector of stacked observations on ε it, B t : b t b t, b t : t sp+ δ s and δ s is a p) dimensional vector of zeros except for a one at position t. By using the technique of Magnus 978), we obtain ε E[ i ) B t ε ) ] i trb t ) ) + trb t ) 3t p), from which it follows that IIb p lim 3 tp+ ε E[ i ) B t ε ) ] i 6 lim 3 tp+ t + O p ). But II b II b + O p/ ) and so we get II II a II b II a II b ) N + O p 5 χ ) A) as N, with N 0, which completes the proof. he following lemma replaces Lemma A. in providing a representation of u it and u it in terms of ε it. Lemma B.. Under the conditions of heorem 3 and in the case with a trend, as 6

27 a) u it ε it + O p ), b) u it g it + O p ), where g it : t sp+ ε is ε i ) and ε i : p ε it. If there is no trend then g it s it. Proof of Lemma B.. We begin with b). From Lemma A., u it y it ˆΦ iy it ˆα i ˆβ i t p) s it ˆΦ i Φ i ) y it ˆα i α i ) ˆβ i β i φ i L))t p) s it ˆΦ i Φ i ) y it ˆα i α i ) ˆβ i β i )t p) β i φ i ), where the last equality uses the Beveridge Nelson decomposition of φ i L) as φ i L) φ i ) + φ i L) L). From A4) we have that φ i L) y it β i + ε it, or y it Φ i y it + β i + ε it, A3) which implies that ) ˇβ i y i Φ i y i Φ i y it + βi + ε i Φ i y i βi + ε i βi + O p, and so ) ˆβ i βi ˆβ i ˇβ i + ˇβ i βi ) ˆβ i ˇβ i + O p. Consider ˆβ i. By using A3), y it y i ) Φ i y it y i ) + ε it ε i, which can be used to show that as ˆΦ i Φ i + y it y i ) y it y i ) herefore, Φ i + O p ) O p ). y it y i )ε it ε i ) ˆβ i y i ˆΦ i y i y i Φ i y i ˆΦ i Φ i ) y i ˇβ i + O p ) O p ) A4) 7

28 suggesting that ) ˆβ i βi O p, A5) and by a similar calculation, ˆα i y ip+ ˆΦ iy ip ˆβ i y ip+ Φ iy ip ˇβ i ˆΦ i Φ i ) y ip ˆβ i ˇβ i ) ) αi + O p. A6) he results in A4) to A6) imply u it s it ˆΦ i Φ i ) y it ˆα i α i ) ˆβ i β i )t p) β i φ i ) g it + O p ), A7) where the second equality follows from noting that s it ˆβ i βi )t p) s it t p)ε i t ε is ε i ) g it. his establishes b), and by similar arguments, sp+ which establishes a). u it y it ˆΦ i y it ˆβ i ε it ˆΦ i Φ i ) y it ˆβ i βi ) ) ε it + O p, A8) If there is no trend in d it, then ˆβ i β i )t p) in A7) drops out and therefore so does ε i. Hence, g it reduces to s it. Lemma B.3. Under the conditions of heorem 3 and in the case with a trend, as a) û itû it ˇε itǧit + O p ), b) c) d) 3/ 3 û it ) ǧ it ) + O p ), û it ) ) û it ) 3/ û it) û it ) 4 3 ˇε it ) ) ǧ it ) + O p ), ˇε it) ǧ it ) 4 + O p ). 8

29 If there is no trend then g it s it. Proof of Lemma B.3. By Lemma B., ˆσ i u it ) ) g it ) + O p ) s it ) + O p ) ˇσ i + O p, A9) where we have used the fact that g it s it + t p) t p ))ε i s it + ε i ε it + ε i. hus, by a first-order aylor expansion of the inverse square root of ˆσ i and then application of Lemma B., û it ˆσi u it ) ǧit + O p, ) u it + O p ˇσi ) g it + O p ˇσi and similarly, ) û it ǧit + O p. Parts a) to d) are direct consequences of these results. Proof of heorem 3. Lemma B. shows that the feasible maximum likelihood estimators converge to their unfeasible counterparts. herefore, since ε ip+ is zero when evaluated at the unfeasible estimators, this implies that the first observation, t p +, can be disregarded when forming the feasible LM test statistic. Consider now part a) and the model with a constant, in which case the feasible statistic 9

30 can be written as LM + c + i it ) ûitû i û it ) ) û it + )) i û it ) N i it ) ûitû û it ) N N i i û it ) ) û it )4 N 3/ i û it ) ) ) û it ) ) + O û it ) û p it )4 N N 3 N i I + II + O p N ), where the second equality uses the results provided in the proof of heorem. Consider I I a Ib, where I a and I b are implicitly defined from the above equation. By Lemma B.3, Ia N N i i û itû it N ) N w i ε its it + O p i and by using the same arguments as in the proof of heorem, Ia N i ) N ε its it + O p ) N ˇε itš it + O p I a + O p N ). Similarly, I b N i û it ) N i ) s it ) + O p Ib + O p ). hus, as in A3) we have that as N, with N 0 I I a I b ) N + O p χ ). But the same steps can be applied to show that II 5 χ ), which completes the proof for LM + c. 30

31 Next, consider b). By use of Lemma B.3 LM t + LM t + i it ) ûitû i û it ) may be written as + i û it ) ) û it )) ) + O û it ) û p it )4 N N i i it ) ˇε itǧ + i ǧ it ) i w iε it it ) g i w igit ) i ˇε it ) ) ǧit )) i ˇε it ) ǧit )4 ) N + O p + i tp+ w i ε it ) ) w i g it ) ) i tp+ w3 i ε it ) git )4 + O p N ), which, by the same steps used in the proof of heorem, becomes LM t + N i ε it it ) g g it ) + N N i N N 3/ i N N 3 i I a I b + II a II b + O p tp+ ε it ) w i ) g it ) ) tp+ ε it ) g it )4 ) N I + II + O p N ). ) N + O p Consider I. Note that g it g it + g it ) g it + g it) + g it g it, suggesting that g it g it g i g ip+) g it ) as g i g ip+ 0. But g it ε it + O p / ) and hence Ia N N p i N lim N ε itg it i N i ) N git) + O p g it ), ) N gitg it + O p N i ) N ε it) + O p as N, with N, which uses the fact that ε it ) p as. Moreover, E g it ) ) E s it ) t p ) ε i ) ) ) + O p 3 6

32 as, where we have used that E s it )) and t p ) E ε i ) ) Hence, by a law of large numbers, as N, from which we deduce that I b N i t p ) t sp+ t + O kp+ sp+ E ε is) ) + O git ) p 6, ) 3. Eε ik ε is) ) I p 3N lim N. A0) Next, consider II. Since ˇσ i is unbiased, IIa N 3/ σ i tp+ i Eε it ˇσ i )E git ) ) 0. var II a ) is given in Lemma B.4. Lemma B.4. Under the conditions of heorem 3, var II a ) 0 + O p/ ). Proof of Lemma B.4. We omit this proof in this paper. All details can be found in Larsson and Westerlund 009). It follows that as N, with N Also, E 3 ε it) git ) 4 II a 0 χ ). 3 E ε it) ) E g it ) 4) 3 E g it ) 4). 3

33 Note that ε it ε i δ t p ι ) ε i, where ι is a p ) dimensional vector of ones. Hence, git ) 4 b t ε i ιε i ) ) 4 ε i ) I ) p ιι b t b t I ) ) p ιι ε i, where I is the identity matrix. If we further define C t : I p ιι ) b t b t I p ιι ), then 3 E g it ) 4) 3 ε E[ i ) C t ε ) ] i, where, by following the same steps used in the proof of heorem, ε E[ i ) C t ε ) ] i It follows that 3 trc t ) ) + trc t ) 3t p ) t p ). p E g it ) 4) ε E[ i ) C t ε ) ] i t p) t p ) p 0, suggesting II b p 5, and so II χ ). Appendix C: Proofs for Section 3.3 In this section we derive the results that apply under H and H. We begin with the following lemma, which takes the place of Lemma A. in providing a representation of e it that holds even when we are not under H 0. Lemma C.. e it ρ t iφ i L)y s ip + Proof of Lemma C.. By combining ) and 3) we get t sp+ ρ t s i ε is. φ i L)y s it ρ t iφ i L)y s ip + t sp+ ρ t s i ε is. But φ i L)y s it φ il)y it α i β i t p)) e it, and so the proof is complete. 33

34 Proof of heorem 4. Write N LM t + N i ěitě it ě it ) N N i N N i N N i I a I b + II a II b + O p ) ) ě it ) )ě it ) ) + O ě it ) )ě p it )4 N ) 3/ ) N ) 3/ I + II + O p N ) 3/ By equation ) and a law of large numbers as N, ). A) Ia p N lim i ě itě it N i ci ě it + ˇε )ě it it E c i ě it ) + ě it ˇε it) µc lim E ě it ) ) O), which uses that ˇσ i O p) regardless of whether we are under H 0 or not, and that by Lemma C., ) Eě it ˇε it) E ˇσ i But we also have I b N and hence I O). i E ρ t i φ i L)y s ip + ě it ) p lim t sp+ ρ t s i ε is Eε it ) 0. E ě it ) ) O) Moreover, by another application of a law of large numbers, as N, IIa p N N i lim i ě it) )ě it ) Ec i ) lim c i ě it ) + c i ě it ˇε it ˇε it) ) ) ě it ) E ˇσ i ) E c i e 4 it + c i e 3 it ε it ε it ˇσ i )e ) it ) E ˇσ i E e 4 it ) O), 34

35 where we have used that Ee 3 it ε it) Ee 3 it )Eε it) 0 and E ε it ˇσ )e it ) Eε it ˇσ )Ee it ) 0. By a similar calculation, II b O p ) and so II O p ). By combining these results, ) N LM t I + II + O p N ) 3/ O p ), or LM t O p N ), and so the proof for LM t is complete. he proof for LM t is identical and is therefore omitted. he proofs for LM + c and LM + t follow from û it ě it + O p) and û it ě it + O p ), which imply that the order of these statistics is the same as that of LM c and LM t. Proof of heorem 5. he proof of part a) follows by a simple manipulation of the proof of heorem 4. For simplicity we focus on LM t. he proof for LM c is identical. We begin by inserting 8) into 4), which yields e it ρ i )e it + ε it c i N e it + ε it. Making the usual decomposition of LM t into I Ia I b and II IIa II b we have Ia N i ě itě it N i w i e ite it, where ˇσ i p N 3 e it ) tp+ tp+ ρi )e it + ε it ) ρi ) e it + ρ i )e it ε it + ε it) c i e it + N ) ε it + O p N c i e it ε it + ε it ) σi + O p. A) 35

36 hus, A7) holds even under H. It follows that Ia N N I a + I a. i i w i e ite it N i ) N ρi )e it ) + e it ε is) + Op By using the results provided in the proof of heorem, EI a) N i E c i e it ) ) µ c N i ) N e ite it + O p E e it ) ) µ c, suggesting that I a p which implies µ c as N,. Also, from A9) we have that I a N0, ), I a I a + I a) I a) + I ai a + I a) µ c 4 + µ c N0, ) + σ4 χ ). But we also have that I b p as N, with N 0, and so I µ c + µ c N0, ) + χ ). A3) Consider II, where IIa N 3/ N 3/ i i II a + II a + II a ě it) )ě it ) ρi ) ě it ) + ρ i )ě it ˇε it ˇε it) ) ) ě it ) so that II a II a+ii a +IIa). From the proof of heorem we know that IIa 5 6 N0, ) as N, with N II a II a 0, while N 3/ 7/ N 5/ i i ) c i ě it ) 4 O p, N c i ě it ) 3ˇε ) it O p. N 36

37 Part II b can be written as II b N 3 N 3 i i II b + II b + II b + O p where II b p, while ) ě it) ě it ) 4 + O p N ρi ) ě it ) + ρ i )ě it ˇε it + ˇε it) ) ) ě it ) 4 + O p N ), N II b II b N 5 4 N 3/ 7/ i i ) c i ě it ) 6 O p, N c i ě it ) 5ˇε ) it O p N. hus, by aylor expansion, II II a II a) ) II b IIb + O p N 5 χ ). A4) which, together with A3), establishes the required result for LM t. If there is a constant that needs to be estimated but no trend, û it ě it + O p/ ) and û it ě it + O p). hus, if we assume that N distribution of LM + c is the same as that of LM t. 0 as N,, then asymptotic Next, consider b). In order to isolate the effect of the trend note that from Lemma B., ) u it y it ˆΦ i y it ˆβ i y it Φ i y it ˆβ i + O p ) y it y i Φ i y it y i ) + O p ) φ i L) y it y i ) + O p. By using ) and then ), we get φ i L) y it y i ) φ i L) yit s p sp+ ρ i )φ i L) yit s p y s is sp+ y s is + ε it ε i ), 37

38 and by further use of Lemma C. and 8), φ i L)y s it ρ t iφ i L)y s ip + t sp+ ρ t s i ε is + c i N ) t φ i L)y s ip + t sp+ ) φ i L)yip s + s it + O p, N from which it follows that φ i L) y it y i ) ρ i ) s it p sp+ ) ρ i )A it + g it + O p N, + c i N ) t s ε is s is ) + g it + O p N giving ) u it ρ i )A it + g it + O p. A5) Similarly, u it ρ i )φ i L) ρ i ) t sp+ t sp+ y s is t p p s is t p p ρ i )B it + g it + O p ), sp+ sp+ s is y s is + g it + O p ) + g it + O p ) or u it ) ) ρi )B it + g it + Op. A6) Consider again the decomposition of LM + t into I Ia I b and II IIa II b. Making use of A5) 38

39 and A6) we obtain Ia N N N i i ) N + O p + N 3/ 3 N i i i û itû it I a + I a + I a + O p N ), ) ρi )Â it + ĝit ) ρi ) ˆB N it + ĝit ) + Op ρi ) Ǎ it ˇB it + ρ i ) ǧ it ˇB it + Ǎ itǧ it ) + ǧ itǧ it ) c i Ǎ it ˇB it + N ) N ǧitǧ it + O p i c i ǧ it ˇB it + Ǎ itǧ it ) where as usual a hat denotes division by ˆσ i, while a check denotes division by ˇσ i. he third equality follows from ˆσ i u it ) N 3 ) ) ρi )A it + g it + Op ρi ) A it + ρ i )A it g it + g it ) ) ) + O p c i A it + N ) g it ) + O p c i A it g it + ) ˇσ + O p ) g it ) + O p A7) Note that I a O p / N ) and since ˆσ i σ i +O p/ ), we also have Ia p lim N N. Part I a requires more work. Note first that by a law of large numbers as N, I a N i p µ c lim c i ǧ it ˇB it + Ǎ itǧ it ) E g itb it + A itg it ). A8) 39

40 Consider E git B it ), which can be expanded as E gitb it ) t E ε it ε i ) s is t p s is p sp+ sp+ E ε t it ε it s is t p p p sp+ t E ε it s is t p p E ε it s is sp+ sp+ p E t ε it s is + t p p ) E sp+ sp+ s is ε it s is sp+ where the first term on the right-hand side is zero, while the second is ) t p p E ε it s is t p p E ε it s is t p t). p sp+ Similarly, p E t p p ) E t ε it s is sp+ ε it s is sp+ st p E t t ε it s is sp+ t p )t p 3), p ) t p p ) E ε it s is sp+ t p p ), p ) which yields E gitb it ) t p )t p 3) + t)t p ) p ) ) p )t p ). A9) Next, consider EA it g it ). It holds that EA itgit ) E s it s is s it t p )ε i ) p sp+ E s it ) ) t p )Es it ε i ) p E s it + t p p E ε i s is, sp+ 40 sp+ s is,

41 where E s it )) t p ), implying t p )Es it ε i ) t p p E Also, p E t p p E s it s is sp+ from which we deduce that ε i s is sp+ t p ) p. s it ε is sp+ p E s it t p p E s it ) ) t sp+ s is + st t p ) t p ) + t), p ) t p p ) E t p p ), p ) ε is s is sp+ sp+ s is EA itgit ) p ) t p ) p ) t p ) t p ) ) t) + p ) Equations A8) to A30) imply I a p µ c lim t p ) t). A30) p ) µ c lim p ) ) t p ) t) µ c lim p ) E g itb it + A itg it ) p )t p ) t p )t p 3) t p ) O Hence, I a p lim N N as N, with N in the proof of heorem 5, I b p 6. herefore, I p 3N lim N. But it also holds that II χ ), and so the proof is complete. ). A3) 0, and by following the same steps used 4