Per capita output convergence: the Dickey-Fuller test under the simultaneous presence of stochastic and deterministic trends

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1 Per capita output convergence: the Dickey-Fuller test under the simultaneous presence of stochastic and deterministic trends Manuel Gómez-Zaldívar Daniel Ventosa-Santaulària Universidad de Guanajuato January 13, 010 Abstract We reconsider previous studies that analyze the convergence hypothesis in a time series framework. In doing so, we first describe two possible outcomes overlooked in this literature, namely loose catching-up and loose lagging-behind, these cases are in-between divergence and catchingup; then, we provide evidence of the proficiency of the Dickey-Fuller (DF) test to identify the new outcomes by means of asymptotic theory as well as Monte Carlo experiments. Finally, in the empirical section we illustrate that the excessive evidence in favor of divergence may be due to lack of attention to the cases loose catching-up and loose laggingbehind. Keywords: Convergence, Divergence, Loose Catching-up/Lagging-behind, Deterministic and Stochastic Trends. JEL classification: C3; O40 Departamento de Economía y Finanzas, Universidad de Guanajuato. manuel.gomez@ugto.org Corresponding Author: Departamento de Economía y Finanzas, Universidad de Guanajuato. DCEA-Sede Marfil Fracc. I, El Establo, Guanajuato, Gto. C.P. 3650, México. Phone and Fax (+5) (473) ext. 95, daniel@ventosa-santaularia.com 1

2 1 Introduction Bernard and Durlauf (1996) demonstrated that the traditional cross-sectional notion of convergence is weaker than the time series notion of convergence. Since then, most of the empirical literature studying output convergence has relied on unit root and cointegration tests. These studies have focused only on three cases out of all the possible results that the different testing procedures identify: catching-up, convergence and divergence; whereas catching-up or stochastic convergence (the weaker definition of convergence) is the case where the logarithmic difference in per capita output between two economies is related to a trend stationary process, convergence or deterministic convergence (the stronger definition of convergence) is associated to a constant mean stationary process. Finally, divergence is linked to a process that contains a unit root. Time series evidence has not been completely supportive of the convergence hypothesis. Bernard and Durlauf (1995) investigate this issue using cointegration techniques; they search for similar long-run trends in per capita output either stochastic or deterministic, finding no evidence in favor of convergence in 15 OECD economies. Studies that used the DF test have found it difficult to reject the null hypothesis of unit root (see, for example, Carlino and Mills (1993), Oxley and Greasley (1995), Loewy and Papell (1996), and Li and Papell (1999) among others). As shown by Perron (1989), the effectiveness of a unit root test decreases significantly in the presence of structural breaks. Hence, in order to find evidence in favor of this hypothesis, researchers have employed tests that allow for structural changes at the intercept, on the slope of the deterministic trend, or both. Nevertheless, these tests also yield mixed results [see Lee, Lim, and Azali (005)]. Evidence against the null hypothesis of divergence has been found when relaxing the assumption of structural stability. Datta (003) applies a time-varying parameter method (Kalman Filter) and finds evidence of catching-up for a sample of OECD countries in relation to the U.S. Similarly, Bentzen (005) modifies the methodology proposed by Nahar and Inder (00) by allowing for shifts in the rate of convergence. He obtains support to affirm that, during the period of , OECD countries have been closing the income gap with regard to the U.S. at varying speeds. Other studies have used non-linear unit root tests to evaluate the hypothesis

3 of convergence. 1 With this methodology, it is possible to find evidence of income convergence for Hong Kong, Korea, Taiwan and Singapore with respect to Japan (Liew and Lim 005). Also, Liew and Ahmad (009) find strong evidence in favor of convergence in Nordic economies. In particular, they claim that Finland, Norway and Sweden converged with Denmark. Finally, Chong, Hinich, Liew, and Lim (008) apply this test to twelve OECD countries and find evidence that the income gaps behave non-linearly; they find two cases of long-run convergence and four cases of catching up. As for the other six countries, they conclude divergence with respect to the U.S.. We assert that the lack of support for the convergence hypothesis may be due to two factors: firstly, the range of outcomes so far considered is incomplete; given the empirical evidence, it is necessary take into account an outcome occurring frequently in this literature, namely the simultaneous presence of stochastic and deterministic trends in the series under analysis. Previous studies assess that deterministic trends play an unimportant role, therefore, are seldom mentioned. Secondly, it is essential to use a proficient testing framework able to detect unit roots whilst simultaneously identifying the presence of a deterministic trend. We argue that the DF methodology fulfils these requirements; asymptotic and finite sample evidence is provided in this sense. The article is organized as follows: Section briefly reviews the definitions of convergence in a time series setting. Section 3 lists the relevant Data Generating Processes (DGP) that are linked to this literature. Section 4 analyzes the asymptotic efficiency of the DF test in estimating both the sign and estimated value of the parameter associated with the constant and the deterministic trend. Section 5 presents a Monte Carlo exercise to evaluate the performance of this test in finite samples. Section 6 shows an empirical application; the results suggest that there is evidence of stochastic and deterministic trends in two out of the five series analyzed, an outcome not considered in previous studies. Finally, the main conclusions are presented in Section 7. Definitions of convergence in a time series setting Following Bernard and Durlauf (1996), the literature that studies convergence in a time series setting defines convergence according to the implications of 1 Specifically the KSS tests of Kapetanios, Shin, and Snell (003). 3

4 the neoclassical growth model. Let x i,t and x j,t represent the log of real GDP per capita in country i and country j at time t, respectively. Define y t as the logarithmic difference in real GDP per capita between i and j at time t. The information available at t is denoted I t. Bernard and Durlauf (1996), pp. 165 define convergence as follows: Convergence as catching up. The per capita output gap between country i and j decreases between dates t and t + T. If x i,t > x j,t. E(y t+t I t ) < y t Convergence as equality of long-term forecasts at a fixed time. Countries i and j converge if the long-term forecast of (log) per capita output for both countries are equal at a fixed time t. lim k E(y t+k I t ) = 0 Following these definitions, the analysis of the relative growth dynamic between economies i and j consists in determining the most satisfactory DGP for representing the output difference between the two economies, y t. The next sections show that the DF methodology is a reliable tool to draw inference in this sense. 3 Convergence hypothesis and relevant associated DGP s When testing for convergence with respect to a leader, i.e., the country with the highest per capita output level throughout the period, the main issue is to determine the statistical properties of the logarithmic differences of real per capita output between the leader and country j previously denoted y t. The relevant DGPs to work with are summarized in Table 1. DGP 1 is associated with divergence, the case where the logarithmic difference in per capita output between the two economies follows a random walk; i.e. This definition could be further extended as follows: lim k E(y t+k I t ) = ψ, where ψ=constant. This implies that the per capita output gap is expected to be invariant through time (constant but not necessarily zero). The output gap tends to zero if the two economies have the same steady state. In the other case, the output gap tends to a non-null constant when the steady states are different. 4

5 DGP Interpretation Representation 1 Divergence y t = y t 1 + u yt Convergence y t = µ y + u yt 3 Catching-up or Lagging-behind* y t = µ y + β y t + u yt 4 Loose Catching-up or Loose Lagging-behind** y t = Y 0 + µ y t + t i=1 u yi * Catching-up: where β y < 0 and Lagging-behind: where β y > 0. ** Loose Catching-up: where µ y < 0 and Loose Lagging-behind: where µ y > 0. The process u yt satisfies the conditions stated by Phillip s(1986) lemma, p.369 with variance σy, long-run variance λ, and autocovariances ρ yi for i = 1,,... Table 1: Relevant DGPs. it contains a stochastic trend only. DGP is interpreted as deterministic convergence, the case where the output gap is mean stationary; i.e. it does not contain neither a deterministic trend nor a stochastic trend. DGP 3 is interpreted as stochastic convergence. This entails a systematic narrowing (widening) of the per capita output gap if the sign of the deterministic trend estimator is negative (positive). We call these results catching-up and laggingbehind respectively; i.e. the series under analysis contains a deterministic trend but not a stochastic one. The cases implied by DGP 4 when the series displays a stochastic and a deterministic trend simultaneously have not been mentioned in the existing literature. 3 Such process represents a weaker notion of catching-up or lagging-behind. Indeed, loose catching-up (loose lagging-behind) suggests that the poorer economy is erratically, though inexorably catching up (lagging behind) if the sign of the deterministic trend estimator is negative (positive). The dominance of a deterministic trend over a stochastic is a well-established fact; 4 hence, finding evidence of both trends indicates an inevitable reduction (increase) in output differences in the long-run. 3 This may explain the little evidence against the null of divergence in this literature. Given that this case had not previously considered, the existence of a unit root led directly to the conclusion of divergence. 4 See for example Hasseler (000). 5

6 4 DF test s asymptotic effectiveness in distinguishing the relevant DGPs Unit root tests focus primarily on the statistical properties of autoregressive term s parameter, leaving aside the analysis of the properties of the constant and deterministic trend estimates. As explained in Table 1, the existence of a deterministic trend plays a key role in determining the relative growth dynamic between two economies. Therefore, it is not only essential to use a proficient testing framework to detect a unit root but also a useful one that correctly identifies the presence of the deterministic trend, if it exists. Moreover, it is important to interpret appropriately the result from such a test. 5 The basic methodology to test for convergence using time series is the DF framework. In this case, the relevant auxiliary regression may include only a constant or a constant and a deterministic trend. 6 It has been shown 7 that the presence of a unit root [as in DGP 1 and 4 in Table (1)] is accurately identified by the DF. Nevertheless, its effectiveness in estimating the deterministic trend has not been sufficiently considered in the literature related to the convergence hypothesis. The following two subsections show the asymptotic effectiveness of this procedure in assisting to distinguish among all the relevant cases described in Table 1. For this discussion, it is important to keep in mind that the right assessment of the deterministic trend (drift) must be done through the constant s estimate whenever there is a unit root. 4.1 DF auxiliary regression with a constant and a deterministic trend The DF test can be used to discriminate among the DGPs stated in the previous section. In particular, we can employ regression (1), denoted DF(1) hereinafter, to investigate the statistical properties of series y t, 5 In the empirical section we mention some previous studies that misinterpreted the test s results, therefore, reached arguable conclusions. 6 See, for example: Carlino and Mills (1993), Oxley and Greasley (1995), Bernard and Durlauf (1995), Loewy and Papell (1996), Greasley and Oxley (1997), Li and Papell (1999) and Lee, Lim, and Azali (005). 7 See Dickey and Fuller (1979) and Said and Dickey (1984). 6

7 y t = α + δy t 1 + βt + u t (1) The asymptotic properties of the relevant parameters associated to this auxiliary regression are explained below, Remark 1 Let y t be generated by DGP 1 (divergence), and be used to estimate regression (1). Hence, the estimated parameter, ˆα, diverges asymptotically and its corresponding t-ratio follows a non-standard distribution: [ ω (ω) 1 ] T 1 ˆα d ω(1) ω 1 σ y ω ( ω ) [ ω (ω) 1 ] tˆα d ω(1) ω 1 ( ω ( ω ) ) ω where ω( ) is a standard brownian motion, and ω means 1 0 ω. Proof: see Appendix A..1 Remark (1) implies that it is possible to draw inference concerning the significance of the deterministic trend, and therefore, to correctly identify when a series associated to a pattern of divergence between two economies (DGP 1). Figure (1) shows the asymptotic and simulated distribution of tˆα and table () displays the critical values needed to test the statistical significance of this parameter. They were computed using 100, 000 replications. DF 10% 5% 1% (1) ±.83 ±3.16 ±3.80 Table : Asymptotic Critical Values for tˆα under the null of random walk; auxiliary regression: DF(1). 7

8 Usual Acceptance zone of the Null Figure 1: Sample non-parametrically estimated distribution of tˆα under the null of random walk; R = 100, 000; T = 0, 000; auxiliary regression: DF(1). Remark Let y t be generated by DGP (convergence), and be used to estimate regression (1). Hence, the estimated parameters, ˆα and ˆβ, and their associated t-ratios behave asymptotically as follows: ˆα p µ y T 1 tˆα = O p (1) T 3 ˆβ = Op (1) tˆβ d 3 ( ω(1) ) ω Proof: see Appendix A.. Remark () implies that it is possible to differentiate between the null of unit root (divergence) and the alternative of stationarity around a constant mean (convergence). On the one hand, the estimate parameter ˆα converges to its true value µ y, whilst its corresponding t-ratio diverges. On the other hand, the estimate parameter β collapses, and its corresponding t-ratio converges to a non-standard distribution. 8 Remark 3 Let y t be generated by DGP 3 (catching-up or lagging-behind), and be used to estimate regression (1). Hence, the estimated parameters, ˆα and ˆβ, 8 Actually, the critical values to draw inference regarding this parameter roughly correspond to those of the standard normal distribution. 8

9 converge to their true value, µ y and β y, whilst their associated t-ratios diverge: ˆα p µ y T 1 tˆα = O p (1) ˆβ p β y p 1 T 1 tˆβ Proof: see Appendix A..3 Remark (3) implies the DF procedure allows the practitioner to identify DGP 3 (Catching-up or Lagging-behind). The constant ( ˆα) and the deterministic trend (ˆβ) estimates converge to their true value, µ y and β y, respectively. In addition, their corresponding t-ratios diverge. Hence, the null of no significance will be asymptotically rejected in both cases. Finally, we study the case where y t contains both, deterministic and stochastic trend. Proposition 1 Let y t be generated by DGP 4 (loose catching-up or loose lagging-behind), and be used to estimate regression (1). Hence, the estimated parameter, ˆα, and its associated t-ratio behave asymptotically as follows: ˆα p µ y T 1 tˆα = O p (1) Proof: see Appendix A.1 As mentioned before, whenever there is a unit root, the inference about deterministic trend must be done using ˆα. Proposition 1 shows that ˆα provides a consistent asymptotic estimate of the deterministic trend, µ y ; since its associated t-statistic diverges at rate T, it ensures, asymptotically, the rejection of the null hypothesis. Consequently, Proposition 1 proves that the DF procedure allows the practitioner to discriminate, at least asymptotically, between Loose Catching up/loose Lagging behind (DGP 4) and Divergence (DGP 1). 4. DF auxiliary regression with only a constant In this subsection we investigate the asymptotic behavior of the DF test whose auxiliary regression includes only a constant, denoted as DF(). It is shown 9

10 that such specification remains a useful tool to distinguish among the different cases of the convergence hypothesis. y t = α + δy t 1 + u t () Remark 4 Let y t be generated by DGP 1 (divergence), and be used to estimate regression (). Hence, the estimated parameter, ˆα, and its associated t-ratio behave asymptotically as follows: [ ω T 1 ˆα d 3 ω 1 σ ω(1)] + 3π + 1 [ω(1) 1] [ ] 3 rω ω y π [ 1 ω tˆα d 3 ω 1 ω(1)] + 3π + 1 [ω(1) 1] [ ] 3 rω ω ( π 1 ω 3 rω ) where: π 1 = π = ( ω ω) + 3 [( rω ω(1) ω ( rω ) ω ω ω(1) ) rω ] rω Proof: see Appendix A..4 In this case, tˆα follows, under the null hypothesis, a non-standard distribution that has considerable departure from the standard normal (it is bimodal and has heavier tails). Figure () shows the simulated distribution and table (3) displays the critical values needed to test the statistical significance of this parameter. These new critical values were computed using 100, 000 replications. Using specification DF() would also allow to draw correct inference concerning a possible deterministic trend, in addition to the unit root (that is, looselagging behind/loose catching-up). Remark 5 Let y t be generated by DGP 4 (loose-lagging behind or loose catchingup), and be used to estimate regression (). Hence, the estimated parameter, ˆα, and its associated t-ratio behave asymptotically as follows: ˆα p µ y T 1 tˆα = O p (1) 10

11 Usual acceptance zone of the Null Figure : Sample non-parametrically estimated distribution of tˆα under the null of random walk; R = 100, 000; T = 0, 000; auxiliary regression: DF () DF 10% 5% 1% () ±.5 ±.8 ±3.4 Table 3: Asymptotic Critical Values for tˆα under the null of random walk; auxiliary regression: DF (). Proof: see Appendix A..5 Analogous to Proposition 1, Remark 5 shows that the DF() allows to distinguish the case where y t contains both, deterministic and stochastic trends. 5 Monte Carlo evidence Using Monte Carlo experiments, we study the finite sample behavior of the DF test (with only a constant and with constant and deterministic trend) in the case where the variable under examination contains both, deterministic and stochastic trends. The experiment uses sample sizes, T, that ranges from 5 to 50; the noise is assumed to be either standard normal or autocorrelated, AR(1) with ρ 1y = 0.7; the number of replications is R = 10, 000 and the significance level is 5%. Table (4) shows the proportion of times in which the test, DF(1), correctly identifies DGP 4 (the deterministic trend and its sign as well as the stochastic 11

12 trend), using ˆβ and tˆβ to draw inference about the deterministic trend. 9 Results in Table (4) suggest a poor performance of the test when it is misinterpreted; its effectiveness increases the larger the absolute value of parameter µ y, the larger the sample size and the smaller the autocorrelation. This Monte Carlo experiment may explain the moderate evidence against the null of divergence this literature. Since the test detects the unit root and the incorrect interpretation leads to infer that the deterministic trend does not exist, one may conclude divergence. Sample Size µ y No autocorrelation, ρ = 0.0 Autocorrelation, ρ = Table 4: Efficiency of DF (1) in identifying DGP 4 using the deterministic trend parameter estimate to draw inference about the deterministic trend. Table (5) show the proportion of times in which the test, DF(), correctly identifies the DGP 4, using ˆα and tˆα to draw inference about the deterministic trend. The results are more than encouraging in favor of the DF as a testing framework. Although its efficiency slightly decreases in the presence of autocorrelation, in particular for small samples (less than 50 observations). Table (6) shows the proportion of times in which the test, DF(1), correctly identifies DGP 4, using ˆα and tˆα to draw inference about the deterministic 9 This implies that the practitioner is interpreting improperly the results of the tests because he is looking at the parameter that accompanies the deterministic trend instead of the constant term estimate. 1

13 Sample Size µ y No autocorrelation, ρ = 0.0 Autocorrelation, ρ = Table 5: Efficiency of DF () in identifying DGP 4 using the constant parameter estimate to draw inference about the deterministic trend. trend. The results indicate that, when a deterministic trend is included in the specification, the ability of the test to detect DGP 4 is not as good as with the simpler DF specification. The number of failures to reject the null hypothesis is indeed greater than those obtained with DF(). As stated in the previous section, including a deterministic trend in the DF test does not impel from making a rightful inference concerning the drift; yet, there is a loss of power. 10 This paper is mainly concerned with the simultaneous presence of a unit root and a deterministic trend (the drift), previously labelled in this work as loose catching-up/lagging behind. The joint test proposed by Dickey and Fuller (1981) (hereinafter DF81) where the null hypothesis is H 0 : α = δ = 0 in equation () 11 could be seen as a natural test candidate to identify DGP 1. We therefore study the finite sample properties of the DF81 test statistic. It is noteworthy to mention that the DF81 tests the joint null hypothesis of unit root and the non-significance of the deterministic regressor (the drift). Table 10 Nevertheless, including a deterministic trend permits inference concerning such trend in the absence of a unit root. 11 See τ αµ in DF81 (p. 106). 13

14 Sample Size µ y No autocorrelation, ρ = 0.0 Autocorrelation, ρ = Table 6: Efficiency of DF(1) in identifying DGP 4 using the constant parameter estimate to draw inference about the deterministic trend. 7 shows the results: When the underlying process is a unit root with drift (all rows except for the one with a drift equal to zero), DF81 systematically rejects the null hypothesis because it is half false. Furthermore, the Monte Carlo experiment reveals that the level distortions caused by the presence of autocorrelation are considerable in the DF81 test. 1 We should expect the DF81 to reject the null hypothesis if half the null hypothesis is not correct (this is exactly what the Monte Carlo shows), that is, the alternative hypothesis is selected if the null hypothesis (all or part of it) is not true. This property represents, however, a drawback in the testing of the convergence hypothesis. The outcome proposed here (loose catching-up) is right in between the null hypothesis of the DF81 test and its alternative hypothesis (existence of a deterministic trend). 1 Of course, Dickey-Fuller s auxiliary regression can be adapted to control for autocorrelation; however, there is the issue of selecting the number of lags to consider. We therefore applied Ng and Perron s (001) lag s selection strategy (results available upon request to the authors). Controlling for autocorrelation definitively increases the power of the test but also severely distorts its size. 14

15 Sample Size µ y No autocorrelation, ρ = 0.0 Autocorrelation, ρ = Table 7: Rejection rate of the DF-F test when the data is generated by DGPs 1 (seventh row) and 4. 6 Empirical evidence To illustrate that the excess of evidence in favor of divergence in previous studies may be due to the fact that loose-lagging behind/loose catching-up were not considered as relevant possibilities, we present an empirical illustration that re-evaluates Lee, Lim, and Azali s (005) results, denoted hereinafter LLA. We actually used the same data set of LLA to be as unambiguous as possible, and analyzed the convergence hypothesis pertaining to a leader economy, Japan. The real GDP per capita of Japan, Indonesia, Malaysia, Philippines, Singapore and Thailand were retrieved from the Penn World Table. As in the original study, the data span is from 1960 to LLA did not include 1998 and further because the Asian financial crisis may had interrupted the long-run convergence dynamic. The results of the DF test with constant and deterministic trend are shown in Table (8). They can be interpreted as follows: for Indonesia and Singapore the test rejects the null of unit-root, (δ 0), and accepts the existence of a negative deterministic trend, (β < 0). This implies that these two economies 15

16 are catching up with Japan. LLA also reject the null of unit-root with the Andrews and Zivot s test 13 for Singapore but not for Indonesia. Nevertheless, they do not make any comment regarding the deterministic trend, in fact, they do not report the estimate value of this parameter. Therefore, they conclude that there is long-run income convergence between the Japanese and Singaporean economies and that Japan diverges from Indonesia. For Philippines and Malaysia we can not reject the null of unit-root, (δ = 0); but we find evidence of a positive drift, (α > 0). Therefore, this test indicates that Philippines and Malaysia are loose lagging behind Japan. These results are similar to those of LLA. Nevertheless, they do not mention the importance of interpreting the deterministic trend. Therefore, they conclude that these two countries diverge from Japan Finally, we cannot reject the null of unit-root for Thailand nor accept the existence of a deterministic trend. Therefore, we interpret the presence of a unit root and the nonexistence of a deterministic trend in the output gap series as divergence. Country α a δ β a Philippines 0.15** (3.69) (-.59) (1.35) Indonesia 0.407*** * *** (3.84) (3.36) (-5.05) Malaysia 0.63** *** (3.17) (.54) (3.91) Singapore 0.15*** * *** (4.7) (3.8) (5.19) Thailand *** (.63) (1.98) (-5.0) Table 8: Test of convergence with respect to Japan. The symbols *, **, and *** denote rejection of the null at 10%, 5%, and 1% level, respectively. a Inference regarding the deterministic trend must be done with the parameter estimates marked in bold numbers (given the evidence about the stochastic trend). The empirical application highlights the importance of DGP 4 (loose lagging behind/loose catching-up) in this literature. When this outcome is taken into 13 This test estimates jointly structural changes in the intercept and in the deterministic trend. See Zivot and Andrews (199). 16

17 consideration, the rejection of the null hypothesis of divergence occurs more frequently. We reject the null of divergence in four out of five countries whilst LLA reject it just for one of them Conclusions We argue that the lack of empirical support for the convergence hypothesis may be due to two factors: 1) the failure of previous studies to consider the case where the difference in per capita output contains both a deterministic and a stochastic trend defined in this study as loose catching-up or loose laggingbehind; ) the misinterpretation of the DF test when analyzing a series with these traits. Inference about the deterministic trend must be done with the constant s estimate of the DF whenever there is evidence of a unit root. Empirical studies pay little attention to the proper estimation of the deterministic trend parameter (absolute value and sign). Therefore, they restrict themselves from the correct interpretation of the relative growth dynamic between the two economies. We prove that the Dickey-Fuller test remains useful when studying the convergence hypothesis, although drawing inference from it requires additional attention to some of the parameters previously overlooked, such as the deterministic trend. A Appendix A.1 Proof of Proposition 1 The expressions needed to compute the asymptotic value of tˆβ are: 14 Though LLA used a test that also contemplates structural breaks and we did not. 17

18 yt 1 = Y 0 T + µ y t µy T + ξ y,t 1 }{{} O p (T 3 ) yt 1 t = Y 0 t + µy t µ y t + ξy,t 1 t }{{} O p (T 5 ) yt = µ y T + u y,t }{{} O p (T 1 ) yt t = µ y t + uy,t t }{{} O p (T 3 ) y t 1 = Y0 T + µ y t + µ yt + ξy,t 1 +Y 0 µ y t Y0 µ y T +... }{{} O p(t ) Y 0 ξy,t 1 µ y t + µy ξy,t 1 t µ y ξy,t 1 yt y t 1 = Y 0 µ y T + µ y t µ y T + µ y ξy,t + Y 0 uy,t +... µ y uy,t t µ y uy,t + u y,t ξ y,t 1 }{{} O p(t) y t = µ y T + u y,t +µ y uy,t }{{} O p(t) where ξ y,t = t i=1 u y,i and all the other summations range from 1 to T. The orders in probability can be found in Phillips (1986), Phillips and Durlauf (1986) and Hamilton (1994). These expressions can be written in Mathematica 4.1 code; the software computes the asymptotics of the classical OLS formula (X X) 1 X Y as well as the asymptotic value of the variance estimator: σ u = T 1 T t=1 û t where T yt 1 t X X = yt 1 yt 1 yt 1 t t yt 1 t t 18

19 and, yt Y = yt y t 1 yt t As indicated previously, the proof was achieved with the aid of Mathematica 4.1 software. 15 A. Proof of Remarks 1-5 The procedure to obtain the results appearing in Remarks 1-5 is analogous to the one employed in Proposition 1. The main difference lies in the expressions needed to compute the asymptotic values: A..1 Remark 1 yt 1 = Y 0 T + ξ y,t 1 yt 1 t = Y 0 t + ξy,t 1 t yt = u y,t yt t = u y,t t y t 1 yt y t 1 = Y0 ξy,t 1 + Y 0 ξy,t 1 = Y 0 uy,t + u y,t ξ y,t 1 y t = u y,t 15 The corresponding code is available at: 07a.zip 19

20 A.. Remark For simplicity, O p (1) terms have been ommitted since their are asymptotically irrelevant. yt 1 = µ y T + u y,t yt 1 t = µ y t + uy,t t + u y,t T u y,t yt = u y,t }{{} O p(1) yt t = T u y,t u y,t y t 1 = µ y T + u y,t + µ y uy,t yt y t 1 y t = A..3 Remark 3 = u y,t u y,t 1 u y,t }{{} O p(t) [ u y,t u y,t u y,t 1 ] For simplicity, O p (1) terms have been ommitted since their are asymptotically irrelevant. yt 1 = µ y T + β y t βy T + u y,t yt 1 t = µ y t + βy t β y t + uy,t t + u y,t T u y,t yt = β y T yt t = β y t + T uy,t u y,t 0

21 y t 1 = µ y T + β y t + β y T + u y,t +... µ y β y t µy β y T + µ y uy,t... ( t + βy uy,t t + ) u y,t T u y,t... β y β y uy,t yt y t 1 = (µ y β y ) y t + β y yt t +... β y uy,t + u y,t u y,t 1 u y,t [ y t = βyt + u y,t ] u y,t u y,t 1 A..4 Remark 4 The expressions needed to compute the asymptotics appear in the demonstration of Remark 1, only the OLS matrices are smaller: X X = ( T yt 1 yt 1 yt 1 ) and, Y = ( ) yt yt y t 1 A..5 Remark 5 The expressions needed to compute the asymptotics appear in the demonstration of Proposition 1, only the OLS matrices are are those employed in the demonstration of Remark 4. References Bentzen, J. (005): Testing for catching-up periods in time-series convergence, Economics Letters, 88(3),

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