Asymptotic Least Squares Theory: Part II

Transcription

1 Chapter 7 Asymptotic Least Squares heory: Part II In the preceding chapter the asymptotic properties of the OLS estimator were derived under standard regularity conditions that require data to obey suitable LLN and CL. Some important consequences of these conditions include root- consistency and asymptotic normality of the OLS estimator. here are, however, various data that do not obey an LLN nor a CL. For instance, the simple average of the random variables that follow a random walk diverges, as shown in Example 5.3. Moreover, the data that behave similarly to random walks are also not governed by LLN or CL. Such data are said to be integrated of order one, denoted as I(; a precise definition of I( series will begiveninsection7.. I( data are practically relevant because, since the seminal work of Nelson and Plosser (982, it has been well documented in the literature that many economic and financial time series are better characterized as I( variables. his chapter is mainly concerned with estimation and hypothesis testing in linear regressions that involve I( variables. When data are I(, the results of Section 6.2 are not applicable, and the asymptotic properties of the OLS estimator must be analyzed differently. As will be shown in subsequent sections, the OLS estimator remains consistent but with a faster convergence rate. Moreover, the normalized OLS estimator has a non-standard distribution in the limit which may be quite different from the normal distribution. his suggests that one should be careful in drawing statistical inferences from regressions with I( variables. As far as economic interpretation is concerned, regressions with I( variables are closely related economic equilibrium relations and hence play an important role in empirical studies. A more detailed analysis of I( variables can be found in, e.g., Hamilton (

2 88 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II 7. I( Variables A time series {y t } is said to be I( if it can be expressed as y t = y t + ɛ t with ɛ t satisfying the following condition. [C] {ɛ t } is a weakly stationary process with mean zero and variance σɛ 2 and obeys an FCL: σ [r] ɛ t = σ y [r] w(r, r, where w is the standard Wiener process, and ( ɛ t, σ 2 = lim var which is known as the long-run variance of ɛ t. his definition is not the most general but is convenient for subsequent analysis. In view of (6.5, we can write ( var ɛ t =var(ɛ t +2 cov(ɛ t,ɛ t j. j= he existence of σ 2 implies that the variance of ɛ t is O(. Moreover, cov(ɛ t,ɛ t j must be summable so that cov(ɛ t,ɛ t j vanishes when j tends to infinity. For simplicity, a weakly stationary process will also be referred to as an I( series. Under our definition, partial sums of an I( series (e.g., t i= ɛ iformani( series, while taking first difference of an I( series (e.g., y t y t yields an I( series. Note that A random walk is I( with i.i.d. ɛ t and σ 2 = σ2 ɛ.when{ɛ t } is a stationary ARMA(p, q process, {y t } is also known as an ARIMA(p,,q process. For example, many empirical evidences showing that stock prices and GDP are I(, and so are their log transformations. Yet, stock returns and GDP growth rates are found to be I(. Analogous to a random walk, an I( series y t has mean zero and variance increasing linearly with t, and its autocovariances cov(y t,y s do not decrease when t s increases, cf. Example 5.3. In contrast, an I( series ɛ t has a bounded variance, and cov(ɛ t,ɛ s decays to zero when t s becomes large. hus, an I( series has increasingly large variations and smooth sample paths, yet an I( series is not as smooth as I( series and has smaller variations. o illustrate, we plot in Figure 7. two sample paths of an ARIMA process: y t = y t + ɛ t,whereɛ t =.3ɛ t + u t.4u t with u t i.i.d. N (,. c Chung-Ming Kuan, 27

3 7.2. AUOREGRESSION OF AN I( VARIABLE 89 5 ARIMA ARMA 25 2 ARIMA ARMA Figure 7.: Sample paths of ARIMA and ARMA series. For comparison, we also include the sample paths of ɛ t in the figure. It can be seen that ARIMA paths (thick lines wander away from the mean level and exhibit large swings over time, whereas the ARMA paths (thin lines are jagged and fluctuate around the mean level. During a given time period, an I( series may look like a series that follows a deterministic time trend: y t = a o + b o t + ɛ t,whereɛ t are I(. Such a series, also known as a trend stationary series, has finite variance and becomes stationary when the trend function is removed. It should be noted that an I( series is fundamentally different from a trend stationary series, in that the former has unbounded variance and its behavior can not be captured by a deterministic trend function. Figure 7.2 illustrates the difference between the sample paths of a random walk y t = y t + ɛ t and a trend stationary series y t =+.t+ɛ t,whereɛ t are i.i.d. N (,. It is clear that the variation of random walk paths is much larger than that of trend stationary paths. 7.2 Autoregression of an I( Variable Given the specification y t = x tβ+e t such that [B2] holds: y t = x tβ o +ɛ t with IE(x t ɛ t =, the OLS estimator of β can be expressed as ( ( ˆβ = β o + x t x t x t ɛ t. c Chung-Ming Kuan, 27

4 9 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II 25 random walk 3 random walk Figure 7.2: Sample paths of random walk and trend stationary series. A generic approach to establishing OLS consistency is to show that, under suitable conditions, the second term on the right-hand side converges to zero in some probabilistic sense. heorem 6. ensures this by imposing [B] on data. With [B] and [B2], we have x tx t = O IP ( and x tɛ t = o IP (, so that ˆβ = β o + o IP (, showing weak consistency of ˆβ. When [B3] also holds, the CL effect yields a more precise order of x tɛ t,namely,o IP ( /2. It follows that ˆβ = β o + O IP ( /2, which establishes root- consistency of ˆβ. On the other hand, the asymptotic properties of the OLS estimator must be derived without resorting to LLN and CL when y t and x t are I( Asymptotic Properties of the OLS Estimator o illustrate, we first consider the simplest AR( specification: y t = αy t + e t. (7. Suppose that {y t } is a random walk such that y t = α o y t + ɛ t with α o =andɛ t i.i.d. random variables with mean zero and variance σ 2 ɛ. From Examples 5.3 we know c Chung-Ming Kuan, 27

5 7.2. AUOREGRESSION OF AN I( VARIABLE 9 that {y t } does not obey a LLN. Moreover, t=2 y t ɛ t = O IP ( by Example 5.32, and t=2 y2 t = O IP ( 2 by Example he OLS estimator of α is thus t=2 ˆα =+ y t ɛ t t=2 y2 t =+O IP (. (7.2 his shows that ˆα converges to α o = at the rate, which is in sharp contrast with the convergence rate of the OLS estimator discussed in Chapter 6. hus, the estimator ˆα is a -consistent estimator and also known as a super consistent estimator. When y t is an I( series, it is straightforward to derive the following asymptotic results; all proofs are deferred to the Appendix. Lemma 7. Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. hen, (i 3/2 y t σ (ii 2 y2 t σ2 w(rdr; w(r 2 dr; (iii y t ɛ t 2 [σ2 w( 2 σ 2 ɛ ]=σ 2 w(rdw(r+ 2 (σ2 σ 2 ɛ. As Lemma 7. holds for general I( processes, the assertions (i and (ii are generalizations of the results in Example he assertion (iii is also a more general result than Example 5.32 and gives a precise limit of t=2 y t ɛ t /. he weak limit of the normalized OLS estimator of α, t=2 (ˆα = y t ɛ t / t=2 y2 t /, 2 now can be derived from Lemma 7.. heorem 7.2 Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. Given the specification y t = αy t + e t, the normalized OLS estimator of α is such that (ˆα 2[ w( 2 σɛ 2 ] /σ2. w(r2 dr where w is the standard Wiener process. In particular, when y t is a random walk, [ 2 w( 2 ] (ˆα w(r2 dr, which does not depend on σ 2 ɛ and σ 2. c Chung-Ming Kuan, 27

6 92 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II heorem 7.2 shows that for an autoregression of I( variables, the OLS estimator is -consistent and has a non-standard distribution in the limit. It is worth mentioning that (ˆα has a weak limit depending on the (nuisance parameters σɛ 2 and σ 2 in general and hence is not asymptotically pivotal, unless ɛ t are i.i.d. It is also interesting to observe from heorem 7.2 that OLS consistency is not affected even when y t and correlated ɛ t are both present, in contrast with Example 6.5. his is the case because t=2 y2 t grows much too fast (at the rate 2 and hence is able to wipe out the effect of t=2 y t ɛ t (which grows at the rate when becomes large. Consider now the specification with a constant term: y t = c + αy t + e t, (7.3 and the OLS estimators ĉ and ˆα. Define ȳ = y t/(. he lemma below is analogous to Lemma 7.. Lemma 7.3 Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. hen, (i 2 (y t ȳ 2 σ 2 (ii (y t ȳ ɛ t σ 2 w (r 2 dr; w (rdw(r+ 2 (σ2 σ 2 ɛ, where w is the standard Wiener process and w (t =w(t w(r dr. Lemma 7.3 is concerned with de-meaned y t, i.e., y t ȳ. In analogy with this term, the process w is also known as the de-meaned Wiener process. It can be seen that the sum of squares of de-meaned y t also grows at the rate 2 and that the sum of the products of de-meaned y t and ɛ t grows at the rate. hese rates are the same as those based on y t, as shown in Lemma 7.. he consistency of the OLS estimator now can be easily established, as in heorem 7.2. heorem 7.4 Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. Given the specification y t = c + αy t + e t, the normalized OLS estimators of α and c are such that (ˆα w (rdw(r+ 2 ( σ2 ɛ /σ 2 =: A, w (r 2 dr ĉ A (σ w(rdr + σ w(. c Chung-Ming Kuan, 27

7 7.2. AUOREGRESSION OF AN I( VARIABLE 93 where w is the standard Wiener process and w (t =w(t w(r dr. In particular, when y t is a random walk, (ˆα w (rdw(r w (r 2 dr. heorem 7.4 shows again that, for the autoregression of an I( variable that contains a constant term, the normalized OLS estimators are not asymptotically pivotal unless ɛ t are i.i.d. It should be emphasized that the OLS estimators of the intercept and slope coefficients have different rates of convergence. he estimator of the latter is -consistent, whereas the estimator of the former remains root- consistent but does not have a limiting normal distribution. Remarks:. While the asymptotic normality of the OLS estimator obtained under standard conditions is invariant with respect to model specifications, heorems 7.4 and 7.2 indicate that the limiting results for autoregressions with an I( variable are not. his is a great disadvantage because the asymptotic analysis need to be carried out for different specifications when the data are I( series. 2. All the results in this sub-section are based on the data y t = y t + ɛ t which does not involve an intercept. hese results would break down if y t = c o + y t + ɛ t with a non-zero c o ; such series are said to be I( with drift. It is easily seen that, when a drift is present, y t = c o t + t ɛ i, i= which contains a deterministic trend and an I( series without drift. Such series exhibits large swings around the trend function and has much larger variation than I( series without drift. See Exercise 7.2 for some properties of this series ests of Unit Root What we have learnt from the preceding subsections are: ( he behavior of an I( series is quite different from that of an I( series, and (2 the presence of an I( variable in a regression renders standard asymptotic results invalid. It is thus practically important to determine whether the data are in fact I(. Given the specifications (7. c Chung-Ming Kuan, 27

8 94 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II and (7.3, the hypothesis of interest is α o referred to as tests of unit root. = ; tests of this hypothesis are usually A leading unit-root test is the t statistic of α o = in the specification (7. or (7.3. For the former, the t statistic is τ = ( t=2 y2 t /2(ˆα, ˆσ where ˆσ 2 = t=2 (y t ˆα y t 2 /( 2 is the standard OLS variance estimator; for the latter, the t statistic is τ c = [ t=2 (y t ȳ 2] /2 (ˆα ˆσ, where ˆσ 2 = t=2 (y t ĉ ˆα y t 2 /( 3. In view of heorem 7.2 and heorem 7.4, it is easy to derive the weak limits of these statistics under the null hypothesis of α o =. heorem 7.5 Let y t be a random walk. Given the specifications (7. and (7.3, we have, respectively, τ τ c 2 [w(2 ] [ w(r2 dr ] /2, w (rdw(r [ w (r 2 dr ] /2, (7.4 where w is the standard Wiener process and w (t =w(t w(r dr. hese statistics were first analyzed by Dickey and Fuller (979 and their weak limits were derived in Phillips (987. In addition to the specifications (7. and (7.3, Dickey and Fuller (979 also considered the specification with the intercept and a time trend: ( y t = c + αy t + β t + e 2 t ; (7.5 the t-statistic of α o = is denoted as τ t. he weak limit of τ t is different from those in heorem 7.5 but can be derived similarly; we omit the detail. he t-statistics τ, τ c, and τ t and the F tests considered in Dickey and Fuller (98 are now known as the Dickey-Fuller tests. he limiting distributions of the Dickey-Fuller tests are all non-standard but can be easily simulated; see e.g., Fuller (996, p. 642 and Davidson and MacKinnon (993, c Chung-Ming Kuan, 27

9 7.2. AUOREGRESSION OF AN I( VARIABLE 95 able 7.: Some percentiles of the Dickey-Fuller distributions. est % 2.5% 5% % 5% 9% 95% 97.5% 99% τ τ c τ t p. 78. hese distributions will be referred to as the Dickey-Fuller distributions; some of their percentiles reported in Fuller (996 are summarized in able 7.. We can see that these distributions are not symmetric about zero and are all skewed to the left. In particular, τ c assumes negatives values about 95% of times, and τ t is virtually a non-positive random variable. o implement the Dickey-Fuller tests, we may, corresponding to the specifications (7., (7.3 and (7.5, estimate one of the following specifications: Δy t = θy t + e t, Δy t = c + θy t + e t, Δy t = c + θy t + β ( t 2 + e t, (7.6 where Δy t = y t y t. Clearly, the hypothesis θ o = for these specifications is equivalent to α o = for (7., (7.3 and (7.5. It is also easy to verify that the weak limits of the normalized estimators in (7.6, ˆθ, are the same as the respective limits of (ˆα under the null hypothesis. Consequently, the t-ratios of θ o =alsohavethe Dickey-Fuller distributions with the critical values given in able 7.. Using the t-ratios of (7.6 as unit-root tests is convenient in practice because they are routinely reported by econometrics packages. A major drawback of the Dickey-Fuller tests is that they can only check if the data series is a random walk. When {y t } is a general I( process, the dependence of ɛ t renders the limits of heorem 7.5 invalid, as shown in the result below. c Chung-Ming Kuan, 27

10 96 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II heorem 7.6 Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. hen, ( τ σ 2 [w(2 σɛ 2 /σ ] 2 σ [ ɛ w(r2 dr ], /2 ( τ c σ w (rdw(r+ 2 ( σ2 ɛ /σ2 σ [ ɛ w (r 2 dr ], /2 (7.7 where w is the standard Wiener process and w (t =w(t w(r dr. heorem 7.6 includes heorem 7.5 as a special case because the limits in (7.7 would reduce to those in (7.4 when ɛ t are i.i.d. (so that σ 2 = σɛ 2. hese results also suggest that the nuisance parameters σɛ 2 and σ 2 may be eliminated by proper corrections of the statistics τ and τ c. Let ê t denote the OLS residuals of the specification (7. or (7.3 and s 2 n denote a Newey-West type estimator of σ 2 based on ê t : s 2 n = t=2 ê 2 t ( s κ n s= t=s+2 ê t ê t s, with κ a kernel function and n = n( its bandwidth; see Section Phillips (987 proposed the following modified τ and τ c statistics: Z(τ = ˆσ 2 τ s (s2 n ˆσ2 ( n s n t=2 y2 t / 2 /2, Z(τ c = ˆσ 2 τ s c (s2 ˆσ2 [ n s n t=2 (y t ȳ 2] /2 ; see also Phillips and Perron (988 for the modifications of other unit-root tests. he Z-type tests are now known as the Phillips-Perron tests. It is quite easy to verify that the limits of Z(τ andz(τ c are those in (7.4 and do not depend on the nuisance parameters. hus, the Phillips-Perron tests are asymptotically pivotal and capable of testing whether {y t } is a general I( series. Corollary 7.7 Let y t = y t + ɛ t be an I( series with ɛ t satisfying [C]. hen, [ 2 w( 2 ] Z(τ [ w(r2 dr ] /2, Z(τ c w (rdw(r [ w (r 2 dr ] /2. where w is the standard Wiener process and w (t =w(t w(r dr. c Chung-Ming Kuan, 27

11 7.3. ESS OF SAIONARIY AGAINS I( 97 Said and Dickey (984 introduced a different approach to circumventing the nuisance parameters in the limit. Note that the correlations in a weakly stationary process may be filtered out by a linear AR model with a proper order, say, k. For example, when {ɛ t } is a weakly stationary ARMA process, ɛ t γ ɛ t γ k ɛ t k = u t are serially uncorrelated for some k and some parameters γ,...,γ k. Basing on this idea Said and Dickey (984 suggested the following augmented specifications: Δy t = θy t + k γ j Δy t j + e t, j= Δy t = c + θy t + k γ j Δy t j + e t, j= ( Δy t = c + θy t + β t + 2 k γ j Δy t j + e t, j= (7.8 where γ j are unknown parameters. Compared with the specifications in (7.6, the augmented regressions in (7.8 contain k lagged differences Δy t j, j =,...,k,which are ɛ t j under the null hypothesis. hese differences are included to capture possible correlations among ɛ t. After controlling these correlations, the resulting t-ratios of θ o = turn out to have the Dickey-Fuller distributions, as the t-ratios for (7.6, and are known as the augmented Dickey-Fuller tests. Compared with the Phillips-Perron tests, these tests are capable of testing whether {y t } is a general I( series without non-parametric kernel estimation for σ. 2 Yet, one must choose a proper lag order k for the augmented specifications in ( ests of Stationarity against I( Instead of testing I( series directly, Kwiatkowski, Phillips, Schmidt, and Shin (992 proposed testing the property of stationarity against I( seires. heir tests, obtained along the line in Nabeya and anaka (988, are now known as the KPSS test. Recall that the process {y t } is said to be trend stationary if y t = a o + b o t + ɛ t, where ɛ t satisfy [C]. hat is, y t fluctuates around a deterministic trend function. When b o =, the resulting process is a level stationary process, in the sense that it moves c Chung-Ming Kuan, 27

12 98 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II around the mean level a o. A trend stationary process achieves stationarity by removing the deterministic trend, whereas a level stationary process is itself stationary and hence an I( series. he KPSS test is of the following form: ( t 2 η = 2 s 2 ê i, n i= where s 2 n is, again, a Newey-West estimator of σ2, computed using the model residuals ê t. o test the null of trend stationarity, ê t = y t â ˆb t are the residuals of regressing y t on the constant one and the time trend t. For the null of level stationarity, ê t = y t ȳ are the residuals of regressing y t on the constant one. o see the null limit of η, consider first the level stationary process y t = a o + ɛ t. he partial sums of ê t = y t ȳ are such that [r] [r] [r] ê t = (ɛ t ɛ = ɛ t [r] hen by a suitable FCL, [r] ê t = σ σ [r] ɛ t [r] ɛ t, r (, ]. ( σ w(r rw(, r (, ]. ɛ t hat is, properly normalized partial sums of the residuals behave like a Brownian bridge with w (r =w(r rw(. Similarly, given the trend stationary process y t = a +b t+ ɛ t,letê t = y t â ˆb t denote the OLS residuals. hen, it can be shown that [r] ê t w(r+(2r 3r 2 w( (6r 6r 2 σ w(sds, r (, ]; see Exercise 7.4. Note that the limit on the right is a functional of the standard Wiener process which, similar to a Brownian bridge, is a tide-down process in the sense that it is zero with probability one at r =. he limits of η under the null hypothesis are summarized in the following theorem; some percentiles of their distributions are collected in able 7.2. heorem 7.8 let y t = a o +ɛ t be a level stationary process with ɛ t satisfying [C]. hen, η computed from ê t = y t ȳ is such that η w (r 2 dr, c Chung-Ming Kuan, 27

13 7.4. REGRESSIONS OF I( VARIABLES 99 able 7.2: Some percentiles of the distributions of the KPSS test. est % 2.5% 5% % level stationarity trend stationarity where w is the Brownian bridge. let y t = a o +b o t+ɛ t be a trend stationary process with ɛ t satisfying [C]. hen, η computed from from the OLS residuals ê t = y t â ˆb t is such that η f(r 2 dr, where f(r =w(r+(2r 3r 2 w( (6r 6r 2 w(sds and w is the standard Wiener process. hese tests have power against I( series because η would diverge under I( alternatives. his is the case because 2 in η is not be a proper normalizing factor when the data are I(. It is worth mentioning that the KPSS tests also have power against other alternatives, such as stationarity with mean changes and trend stationarity with trend breaks. hus, rejecting the null of stationarity does not imply that the series being tested must be I(. 7.4 Regressions of I( Variables From the preceding section we have seen that the asymptotic behavior of the OLS estimator in an auotregression changes dramatically when {y t } is an I( series. It is then reasonable to expect that the OLS asymptotics would also be quite different from that in Chapter 6 if the dependent variable and regressors of a regression model are both I( series Spurious Regressions In a classical simulation study, Granger and Newbold (974 found that, while two independent random walks should have no relationship whatsoever, regressing one random walk on the other typically yields a significant t-ratio. hus, one would falsely reject the null hypothesis of no relationship between two independent random walks. his is c Chung-Ming Kuan, 27

14 2 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II known as the problem of spurious regression. Phillips (986 provided analytic results showing why such a spurious inference may arise. o illustrate, we consider a simple linear specification: y t = α + βx t + e t. Let ˆα and ˆβ denote the OLS estimators for α and β, respectively. Also denote their t-ratios as t α =ˆα /s α and t β = ˆβ /s β,wheres α and s β are the OLS standard errors for ˆα and ˆβ. We are interested in the case that {y t } and {x t } are I( series: y t = y t +u t and x t = x t + v t,where{u t } and {v t } are mutually independent processes satisfying the following condition. [C2] {u t } and {v t } are two weakly stationary processes have mean zero and respective variances σu 2 and σ2 v and both obey an FCL with respective long-run variances: ( 2 σy 2 = lim IE u t, ( 2 σx 2 = lim IE v t. In the light of Lemma 7., the limits below are immediate: 3/2 2 y t σ y w y (r dr, yt 2 σ2 y w y (r 2 dr, where w y is a standard Wiener processes. Similarly, 3/2 2 x t σ x w x (rdr, x 2 t σ2 x w x (r 2 dr, where w x is also a standard Wiener process which, due to mutual independence between {u t } and {v t }, is independent of w y. As in Lemma 7.3, we also have 2 2 (y t ȳ 2 σy 2 (x t x 2 σx 2 c Chung-Ming Kuan, 27 w y (r 2 dr σ 2 y w x (r 2 dr σ 2 x ( ( 2 w y (r dr =: σym 2 y, 2 w x (r dr =: σx 2 m x,

15 7.4. REGRESSIONS OF I( VARIABLES 2 where wy (t =w y (t w y (r dr and w x (t =w x (t w x (rdr are two mutually independent, de-meaned Wiener processes. Analogous to the limits above, it is also easy to show that 2 (y t ȳ(x t x t ( σ y σ x w y (rw x (r dr =: σ y σ x m yx. w y (r dr w x (r dr he following results on ˆα, ˆβ the limits above. and their t-ratios now can be easily derived from heorem 7.9 Let y t = y t + u t and x t = x t + v t,where{u t } and {v t } are two mutually independent processes satisfying [C2]. hen for the specification y t = α + βx t + e t, we have: (i ˆβ σ y m yx σ x m x, (ii /2 ˆα σ y ( (iii /2 t β w y (rdr m yx m x m yx (m y m x m 2 yx /2, w x (rdr, (iv /2 t α m x w y(rdr m yx w x(rdr [ (my m x m 2 yx w x (r2 dr ] /2, where w x and w y are two mutually independent, standard Wiener processes. When y t and x t are mutually independent, the true parameters of this regression should be α o = β o =. he first two assertions of heorem 7.9 show, however, that the OLS estimators do not converge in probability to zero. Instead, ˆβ has a non-degenerate limiting distribution, whereas ˆα diverges at the rate /2. heorem 7.9(iii and (iv further indicate that t α and t β both diverge at the rate /2. hus, one would easily infer that these coefficients are significantly different from zero if the critical values of the t-ratio were taken from the standard normal distribution. hese results together suggest that, when the variables are I( series, one should be extremely careful in drawing statistical inferences from the t tests, for the t tests do not have the standard normal distribution in the limit. c Chung-Ming Kuan, 27

16 22 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II Nelson and Kang (984 also showed that, given the time trend specification: y t = a + bt+ e t, it is likely to falsely infer that the time trend is significant in explaining the behavior of y t when {y t } is a random walk. his is known as the problem of spurious trend. Phillips and Durlauf (986 analyzed this problem as in heorem 7.9 and demonstrated that the F test of b o = diverges at the rate. he divergence of the F test (and hence the t-ratio explains why an incorrect inference would result Cointegration he results in the preceding sub-section indicate that the relation between I( variables found using standard asymptotics may be a spurious one. hey do not mean that there can be no relation between I( variables. In this section, we formally characterize the relations between I( variables. Consider two variables y and x that obey an equilibrium relationship ay bx =. With real data (y t,x t, z t := ay t bx t are understood as equilibrium errors because they need not be zero all the time. When y t and x t are both I(, a linear combination of them is, in general, also an I( series. hus, {z t } would be an I( series that wanders away from zero and has growing variances over time. If that is the case, {z t } rarely crosses zero (the horizontal axis, so that the equilibrium condition entails little empirical restriction on z t. On the other hand, when y t and x t are both I( but involve the same random walk q t such that y t = q t + u t and x t = cq t + v t,where{u t } and {v t } are two I( series. It is then easily seen that z t = cy t x t = cu t v t, which is a linear combination of I( series and hence is also I(. his example shows that when two I( series share the same trending (random walk component, it is possible to find a linear combination of these series that annihilates the common trend and becomes an I( series. In this case, the equilibrium condition is empirically relevant because z t is I( and hence must cross zero often. Formally, two I( series are said to be cointegrated if a linear combination of them is I(. he concept of cointegration was originally proposed by Granger (98 and Granger and Weiss (983 and subsequently formalized in Engle and Granger (987. c Chung-Ming Kuan, 27

17 7.4. REGRESSIONS OF I( VARIABLES 23 his concept is readily generalized to characterize the relationships among di( time series. Let y t be a d-dimensional vector I( series such that each element is an I( series. hese elements are said to be cointegrated if there exists a d vector,α, such that z t = α y t is I(. We say that the elements of y t are CI(, for simplicity, indicating that a linear combination of the elements of y t is capable of reducing the integrated order by one. he vector α is referred to as a cointegrating vector. When d > 2, there may be more than one cointegrating vector. Clearly, if α and α 2 are two cointegrating vectors, so are their linear combinations. Hence, we are primarily interested in the cointegrating vectors that are not linearly dependent. he space spanned by linearly independent cointegating vectors is the cointegrating space; the number of linearly independent cointegrating vectors is known as the cointegrating rank which is the dimension of the cointegrating space. If the cointegrating rank is r, we can put these r linearly independent cointegrating vectors together and form the d r matrix A such that z t = A y is a vector I(series. Notethatforad-dimensional vector I( series y t, the cointegrating rank is at most d. For if the cointegrating rank is d, A would be a d d nonsingular matrix, so that A z t = y t must be a vector I( series as well. his contradicts the assumption that y t is a vector of I( series. A simple way to find a cointegration relationship among the elements of y t is to regress one element, say, y,t on all other elements, y 2,t, as suggested by Engle and Granger (987. A cointegrating regression is y,t = α y 2,t + z t, where the vector ( α is the cointegrating vector with the first element normalized to one, and z t are the regression errors and also the equilibrium errors. he estimated cointegrating regression is y,t = ˆα y 2,t + ẑ t, where ˆα is the OLS estimate, and ẑ t are the OLS residuals which approximate the equilibrium errors. It should not be surprising to find that the estimator ˆα is - consistent, as in the case of autoregression with a unit root. When the elements of y t are cointegrated, they must be determined jointly. he equilibrium errors z t are thus also correlated with y 2,t. As far as the consistency of the OLS estimator is concerned, the correlations between z t and y 2,t do not matter asymptotically, but they would result in finite-sample bias and efficiency loss. o correct these correlations and obtain more efficient estimates, Saikkonen (99 proposed c Chung-Ming Kuan, 27

18 24 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II estimating a modified co-integrating regression that includes additional k leads and lags of Δy 2,t = y 2,t y 2,t : y,t = α y 2,t + k j= k Δy 2,t j b j + e t. It has been shown that the OLS estimator of α is asymptotically efficient in the sense of Saikkonen (99, Definition 2.2; we omit the details. Phillips and Hansen (99 proposed a different way to compute efficient estimates. When cointegration exists, the true equilibrium errors z t should be an I( series; otherwise, they should be I(. One can then verify the cointegration relationship by applying unit-root tests, such as the augmented Dickey-Fuller test and the Phillips- Perron test, to ẑ t. he null hypothesis that a unit root is present is equivalent to the hypothesis of no cointegration. Failing to reject the null hypothesis of no cointegration suggests that the regression is in fact a spurious one, in the sense of Granger and Newbold (974. o implement a unit-root test on cointegration residuals ẑ, a difficulty is that ẑ is not a raw series but a result of OLS fitting. hus, even when z t may be I(, the residuals ẑ t may not have much variation and hence behave like a stationary series. Consequently, the null hypothesis would be rejected too often if the original Dickey- Fuller critical values were used. Engle and Granger (987, Engle and Yoo (987, and Davidson and MacKinnon (993 simulated proper critical values for the unit-root tests on cointegrating residuals. Similar to the unit-root tests discussed earlier, these critical values are all model dependent. In particular, the critical values vary with d, the number of variables (dependent variables and regressors in the cointegrating regression. Let τ c denote the t-ratio of an auxiliary autoregression on ẑ t with a constant term. able 7.3 summarizes some critical values of the τ c test of no cointegration based on Davidson and MacKinnon (993. he cointegrating regression approach has some drawbacks in practice. First, the estimation result is somewhat arbitrary because it is determined by the dependent variable in the regression. As far as cointegration is conerned, any variable in the vector series could serve as a dependent variable. Although the choice of the dependent variable does not matter asymptotically, it does affect the estimated cointegration relationships in finite samples. Second, this approach is more suitable for finding only one cointegrating relationship, despite that Engle and Granger (987 proposed estimating multiple cointegration relationships by a vector regression. It is now typical to adopt the maximum likelihood approach of Johansen (988 to estimate the cointegrating space directly. c Chung-Ming Kuan, 27

19 7.4. REGRESSIONS OF I( VARIABLES 25 able 7.3: Some percentiles of the distributions of the cointegration τ c test. d % 2.5% 5% % Cointegration has an important implication. When the elements of y t are cointegrated such that A y t = z t, then there must exist an error correction model (ECM in the sense that Δy t = Bz t + C Δy t + + C k Δy t k + ν t, where B is d r matrix of coefficients associated with the vector of equilibrium errors and C j, j =,...,k, are the coefficient matrices associated with lagged differences. It must be emphasized that cointegration characterizes the long-run equilibrium relationship among the variables because it deals with the levels of I( variables. On the other hand, the corresponding ECM describes short-run dynamics of these variables, in the sense that it is a dynamic vector regression on the differences of these variables. hus, the long-run equilibrium relationships are useful in explaining the short-run adjustment when cointegration exists. he result here also indicates that, when cointegration exists, a vector AR model of Δy t is misspecified because it omits the important variable z t, the lagged equilibrium errors. Omitting this variable would render the estimates in the AR model of Δy t inconsistent. herefore, it is important to identify the cointegrating relationship before estimating an ECM. On the other hand, the Johansen approach mentioned above permits joint estimation of the cointegrating space and ECM. In practice, an ECM can be estimated by replacing z t with the residuals of a cointegrating regression ẑ t and then regressing Δy t on ẑ t and lagged Δy t. Note that standard asymptotic theory applies here because ECM involves only stationary variables when cointegration exists. c Chung-Ming Kuan, 27

20 26 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II Appendix Proof of Lemma 7.: By invoking a suitable FCL, the proofs of the assertions (i and (ii are the same as those in Example o prove (iii, we apply the formula of summation by parts. For two sequences {a t } and {b t }, put A n = n a t for n and A =. Summation by parts is such that, for p q, q q a t b t = A t (b t b t+ +A q b q A p b p. t=p t=p Now, setting a t = ɛ t and b t = y t we get A t = y t and Hence, y t ɛ t = y y y t ɛ t = y 2 ɛ 2 t y t ɛ t. y t ɛ t = 2 ( y2 ɛ 2 t 2 [σ2 w(2 σɛ 2 ]. Note that the stochastic integral w(r dw(r =[w(2 ]/2; see e.g., Davidson (994, p. 57. An alternative expression of the weak limit of y t ɛ t / is thus σ 2 w(r dw(r+ 2 (σ2 σ2 ɛ. Proof of heorem 7.2: he first assertion follows directly from Lemma 7. and the continuous mapping theorem. he second assertion follows from the first by noting that σ 2 = σ2 ɛ when ɛ t are i.i.d. Proof of Lemma 7.3: By Lemma 7.(i and (ii, we have 2 (y t ȳ 2 = 2 ( yt 2 3/2 2 y t ( 2 σ 2 w(r 2 dr σ 2 w(r dr. It is easy to show that the limit on the right-hand side is just σ 2 w (r 2 dr, asasserted in (i. o prove (ii, note that ( ( ( ȳ ɛ t = 3/2 y t y σ 2 w(r dr w(. c Chung-Ming Kuan, 27

21 7.4. REGRESSIONS OF I( VARIABLES 27 It follows from Lemma 7.(iii that (y t ȳ ɛ t σ 2 = σ 2 w(r dw(r+ 2 (σ2 σ 2 ɛ σ 2 w (r dw(r+ 2 (σ2 σ2 ɛ, ( w(r dr w( where the last equality is due to the fact that dw(r =w(. Proof of heorem 7.4: he assertions on (ˆα follow directly from Lemma 7.3. For ĉ,wehave ĉ = ( ˆα ȳ + ɛ t t=2 A (σ w(r dr + σ w(. Proof of heorem 7.5: First note that ˆσ 2 = 2 t=2 ɛ 2 t ˆα 2 a.s. y t ɛ t σɛ 2, t=2 by Kolmogorov s SLLN. When ɛ t are i.i.d., σɛ 2 is also the long-run variance σ2. he t statistic τ is thus τ = ( t=2 y2 t / 2 /2 (ˆα ˆσ 2 [ w( 2 ] [ w(r2 dr ] /2, by Lemma 7.(ii and the second assertion of heorem 7.2. Similarly, the weak limit of τ c follows from Lemma 7.3(i and the second assertion of heorem 7.4. Proof of heorem 7.6: When {y t } is a general I( process, it follows from Lemma 7.(ii and the first assertion of heorem 7.2 that τ = ( t=2 y2 t / ( 2 /2 (ˆα σ [ 2 w( 2 σɛ 2 ] /σ2 ˆσ σ ɛ (. w(r2 dr /2 c Chung-Ming Kuan, 27

22 28 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II he second assertion can be derived similarly. Proof of Corollary 7.7: he limits follow straightforwardly from heorem 7.6. Proof of heorem 7.8: For ê t = y t ȳ, /2 [r] i= êi σ w (r,asshowninthe text. As s n is consistent for σ, the first assertion follows from the continuous mapping theorem. he proof for the second assertion is left to Exercise 7.4. Proof of heorem 7.9: he assertion (i follows easily from the expression ˆβ = (x t x(y t ȳ/ 2 (x t x2 / 2. oprove(ii,wenotethat /2 ˆα = 3/2 y t 3/2 ˆβ ( σy m yx σ y w y (r dr (σ σ x m x x x t w x (r dr. In Exercise 7.5, it is shown that the OLS variance estimator ˆσ 2 ( ˆσ 2 / σ2 y m y m2 yx. m x diverges such that It follows that t β = ˆβ [ (x t x 2 / 2] /2 ˆσ / m yx /m /2 x (m y m 2 yx /m x /2. his proves (iii. For the assertion (iv, we have t α = [ ˆα ] /2 (x t x 2 ˆσ x2 t [ w y(rdr (m yx /m x w x(r dr ] [ (my m 2 yx /m x w x (r2 dr ]. /2 m/2 x c Chung-Ming Kuan, 27

23 7.4. REGRESSIONS OF I( VARIABLES 29 Exercises 7. For the specification (7.3, derive the weak limit of the t-ratio for c =. 7.2 Suppose that y t = c o + y t + ɛ t with c o. Find the orders of y t and y2 t and compare these ordier with those in Lemma Given the specification y t = c + αy t + e t, suppose that y t = c o + y t + ɛ t with c o andɛ t i.i.d. Find the weak limit of (ˆα and compare with the result in heorem Given y t = a + b t + ɛ t,letê t = y t â ˆb t denote the OLS residuals. Show that σ [r] ê t w(r+(2r 3r 2 w( (6r 6r 2 w(sds, r (, ]. 7.5 As in Section 7.4., consider the specification y t = α + βx t + e t,wherey t and x t are two independent random walks. Let ˆσ 2 = (y t ˆα ˆβ x t 2 /( 2 denote the standard OLS variance estimator. Show that ( ˆσ 2 / σy 2 m y m2 yx. m x 7.6 As in Section 7.4., consider the specification y t = α + βx t + e t,wherey t and x t are two independent random walks. Let d denote the Durbin-Watson statistic. Granger and Newbold (974 also observed that it is typical to have a small value of d. Provethat d (σ2 u /σ2 y +(m yx /m x 2 (σv 2/σ2 x m y m 2 yx /m, x and explain how this result is related to Granger and Newbold s observation. References Davidson, James (994. Stochastic Limit heory, New York, NY: Oxford University Press. Davidson, R. and James G. MacKinnon (993. Estimation and Inference in Econometrics, New York, NY: Oxford University Press. c Chung-Ming Kuan, 27

24 2 CHAPER 7. ASYMPOIC LEAS SQUARES HEORY: PAR II Dickey, David A. and Wayne A. Fuller (979. Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, 74, Dickey, David A. and Wayne A. Fuller (98. Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, Engle, Robert F. and Clive W. J. Granger (987. Co-integration and error correction: Representation, estimation and testing, Econometrica, 55, Engle, Robert F. and Byung S. Yoo (987. Forecasting and testing co-integrated systems, Journal of Econometrics, 35, Fuller, Wayne A. (996. Introduction to Statistical ime Series, second edition, New York, NY: John Wiley and Sons. Granger, Clive W. J. (98. Some properties of time series data and their use in econometric model specification, Journal of Econometrics, 6, 2 3. Granger, Clive W. J. and Paul Newbold (974. Spurious regressions in econometrics, Journal of Econometrics, 2, 2. Granger, Clive W. J. and Andrew A. Weiss (983. ime series analysis of errorcorrection models, in Samuel Karlin, akeshi Amemiya, & Leo A. Goodman ed., Studies in Econometrics, ime Series, and Multivariate Statistics, , New York: Academic Press. Hamilton, James D. (994. ime Series Analysis, Priceton, NJ: Princeton University Press. Johansen, Søren (988. Statistical analysis of cointegration vectors, Journal of Economic Dynamics and Control, 2, Kwiawtkowski, Denis, Peter C. B. Phillips, Peter Schmidt, and Yongcheol Shin (992. esting the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics, 54, Nabeya, Seiji and Katsuto anaka (988. Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, Annals of Statistics, 6, Nelson, Charles R. and Heejoon Kang (984. Pitfalls in the use of time as an exc Chung-Ming Kuan, 27

25 7.4. REGRESSIONS OF I( VARIABLES 2 planatory variable in regression, Journal of Business and Economic Statistics, 2, Nelson, Charles R. and Charles I. Plosser (982. rends and random walks in macroeconomic time series: Some evidence and implications, Journal of Monetary Economics,, Phillips, Peter C. B. (986. Understanding spurious regressions in econometrics, Journal of Econometrics, 33, Phillips, Peter C. B. (987. ime series regression with a unit root, Econometrica, 55, Phillips, Peter C. B. and Steven N. Durlauf (986. Multiple time series regression with integrated processes, Review of Economic Studies, 53, Phillips, Peter C. B. and Bruce E. Hansen (99. Statistical inference in instrumental variables regression with I( processes, Review of Economic Studies, 57, Phillips, Peter C. B. and Pierre Perron (988. esting for a unit root in time series regression, Biometrika, 75, Said, Said E. and David A. Dickey (984. esting for unit roots in autoregressivemoving average models of unknown order, Biometrika, 7, Saikkonen, Pentti (99. Asymptotically efficient estimation of cointegration regressions, Econometric heory, 7, 2. c Chung-Ming Kuan, 27