Logarithmic spurious regressions

Logarithmic spurious regressions Robert M. e Jong Michigan State University February 5, 22 Abstract Spurious regressions, i.e. regressions in which an integrate process is regresse on another integrate process while there is no cointegration, are well unerstoo in contemporary time series econometrics. In this paper, I investigate the properties of regressions in which the logarithm of an integrate process is regresse on the logarithm of another integrate process. It is shown that, exactly as in spurious regressions, in this setup too, the estimate slope coefficient is asymptotically ranom an the t-value for the slope coefficient is of a stochastic orer equal to the square root of sample size. Therefore, spurious regressions results can alternatively be explaine as an artifact of regressing the logarithm of an integrate process on the logarithm of another integrate process. 1 Introuction In this paper, an alternative type of spurious regressions is consiere. The issue of spurious regressions is well-ocumente; it was consiere by Granger an Newbol (1974) an a first full mathematical analysis appeare in Phillips (1986). For an accessible explanation of the mathematics of spurious regressions the reaer is referre to Hamilton (1994), paragraph 18.3. Spurious regressions occur when two (possibly inepenent) integrate processes x t an y t are regresse on each other. In that case, we fin that (i) the estimate slope coefficient is asymptotically ranom; (ii) the t-value for the slope coefficient ˆt 2 of the regression of y t on x t Corresponing author: Robert M. e Jong, Department of Economics, Michigan State University, 217 Marshall Hall, East Lansing, MI 48824, USA, Email ejongr@msu.eu, fax (517) 432-168. 1

with intercept satisfies /2ˆt 2 T for a nonegenerate ranom variable T. Therefore, we may spuriously reject the null hypothesis of a zero slope coefficient in a regression of y t on x t. Here, I consier the analysis of regressions in which log y t is regresse on log x t, where x t an y t are integrate processes. Given the tenency of applie time series econometricians to apply logarithmic transformations in many settings, this analysis may quite well approximate econometric practice in some instances. An important tool that is use is a result of e Jong (21), where it is establishe that for general integrate processes x t, an functions T (.) that are explicitly allowe to have a pole, we have uner some regularity conitions T (/2 x t ) T (σ x W x (r))r (1) where σx 2 = lim n E(/2 x n ) 2 an W x (.) enotes the Brownian motion process associate with the x t, as long as K T (x) x < for all K >. Note that this result is not K as straightforwar as it may seem at first sight; the possible pole in T (.) makes that the continuous mapping theorem cannot be applie to obtain the above result. Examples of the above result are T (x) = log x an T (x) = (log x ) 2 ; clearly K log x x < an K K log K x 2 x < for all K >, an therefore it follows that an log /2 x t (log /2 x t ) 2 log σ x W x (r) r (2) (log σ x W x (r) ) 2 r. (3) Therefore, the usual type of result that is typically obtaine through an application of the functional central limit theorem an the continuous mapping theorem will remain vali for the logarithmic function an its square, even though the functional central limit theorem cannot be applie irectly. Note that for a function such as T (x) = x 2, this type of reasoning is incorrect, because for such a function T (.), W (r) 2 r = almost surely. Similar results extening those of e Jong (21) have recently been obtaine by Pötscher (21). 2

2 Main results Similarly to Park an Phillips (1999), it is assume that x t = x t 1 + w t an y t = y t 1 + v t, (4) where w t an v t are generate accoring to w t = φ x kε t k an v t = k= φ y k η t k k= where (ε t, η t ) is assume to be a sequence of i.i.. ranom vectors with mean zero an a nonsingular covariance matrix Σ, an where it is assume that k= φx k an k= φy k. In aition, I will assume that (x, y ) is an arbitrary ranom vector that is inepenent of all (w t, v t ), t 1. The main assumption in this paper is similar to Assumption 2.2 from Park an Phillips (1999): Assumption 1 (a) k= k( φx k + φy k ) < an E ε t p + E η t p < for some p > 2. (b) The istributions of ε t an η t are absolutely continuous with respect to the Lebesgue measure an have characteristic functions ψ x (s) an ψ y (s) for which lim s s η (ψ x (s)+ ψ y (s)) = for some η >. (5) (6) Assumption 1 implies that for n M for some fixe value of M, the ensities of /2 x n an /2 y n will both exist an be boune. This assumption is a multivariate version of the assumption use in e Jong (21) an Park an Phillips (1999). Assumption 1 also implies that (/2 x [nr1 ], /2 y [nr2 ]) (σ x W x (r 1 ), σ y W y (r 2 )), where enotes weak convergence an W x (.) an W y (.) enote the Brownian motions associate with the x t an the y t respectively, an σ 2 y is efine analogously to σ 2 x. Consier a regression where log y t is regresse on log x t an an intercept. Let ˆβ 1 an ˆβ 2 enote the estimate intercept an slope coefficient, an efine ˆβ = ( ˆβ 1, ˆβ 2 ). The asymptotic behavior of ˆβ is as follows: 3

Theorem 1 Uner Assumption 1, ˆβ 2 (log σ xw x (s) log σ xw x (r) r)(log σ y W y (s) log σ xw x (r) r)s (log σ xw x (s) log σ xw x (r) r) 2 s an enoting the above limit ranom variable by B 2, we have (7) (log(n)) 1 ˆβ1 (1/2) (1/2)B 2. (8) In orer to etermine the limit behavior of the t-statistic for ˆβ 2, the following lemma is neee. Let s 2 enote the usual regression error variance estimator. Lemma 1 Uner Assumption 1, s 2 S, (9) for some nonegenerate ranom variable S. Finally, I will establish the limit behavior of the t-statistics. Let ˆt 1 an ˆt 2 enote the usual t-statistics for ˆβ 1 an ˆβ 2. Theorem 2 Uner Assumption 1, /2ˆt 1 T 1 an /2ˆt 2 T 2, (1) for nonegenerate ranom variables T 1 an T 2. 3 Conclusions Theorem 1 an Theorem 2 provie results similar to the results that can be obtaine for spurious regressions; the estimate slope coefficient is asymptotically ranom, an Theorem 2 implies that the t-value will asymptotically ten to infinity as sample size increases at the rate n 1/2. This result is similar to the situation for spurious regressions, where the t-value for the slope coefficient is also of stochastic orer n 1/2. Therefore, the conclusion has to be that the spurious regression effect that is observe in applie time series scenarios can equally well be explaine as an artifact of having regresse the logarithm of an integrate process on the logarithm of another integrate process. 4

Appenix: Mathematical Proofs Proof of Theorem 1: Note that ˆβ 2 = n (log x t log x t )(log y t log y t ) n (log x t log x t ) 2 = n 1 n (log n 1/2 x t log /2 x t )(log /2 y t log /2 y t ) n 1 n (log, (11) n 1/2 x t log /2 x t ) 2 where a t enotes n 1 n a t. By applying the results of e Jong (21) as quote in Equations (2) an (3) an noting that the convergence in istribution of the various summations is joint, the only result that remains to be proven is log /2 x t log /2 y t log σ y W y (r) log σ x W x (r) r. (12) Note that max 1 t n /2 x t + max 1 t n /2 y t = O P (1), an therefore with arbitrarily large probability, the above statistic equals n log /2 x t log /2 y t I( /2 x t K)I( /2 y t K), (13) an therefore it suffices to show that log /2 x t log /2 y t I( /2 x t K)I( /2 y t K) log σ y W y (r) log σ x W x (r) I( σ x W x (r) K)I( σ y W y (r) K)r. (14) Note that the combination of the functional central limit theorem an the continuous mapping theorem fails to obtain this result, because of the poles of the logarithmic functions at. The strategy of the proof will be to apply the functional central limit theorem an the continuous mapping theorem to conclue that f δ (/2 x t )f δ (/2 y t )I( /2 x t K)I( /2 y t K) 5

f δ (σ x W x (r))f δ (σ y W y (r))i( σ x W x (r) K)I( σ y W y (r) K)r (15) for a boune an continuous approximation f δ (.) of log x, an show that ifferences are negligible asymptotically. Furthermore note that, for M as efine in the text following Assumption 1, M For any δ >, let log /2 x t log /2 y t I( /2 x t K)I( /2 y t K) = o P (1). (16) f δ (x) = log x I( x > δ) + log(δ)i( x δ). (17) I will now show that E log /2 x t log /2 y t I( /2 x t K)I( /2 y t K) f δ (/2 x t )f δ (/2 y t )I( /2 x t K)I( /2 y t K) h(δ) (18) for some function h(.) that oes not epen on n such that h(δ) as δ, an also that f δ (σ x W x (r))f δ (σ y W y (r))r as which suffices to prove the result. To show the result of Equation (18), note that = + log σ y W y (r) log σ x W x (r) r as δ, (19) log /2 x t log /2 y t I( /2 y t K)I( /2 x t K) (log /2 x t f δ (/2 x t )) log /2 y t I( /2 x t K)I( /2 y t K) f δ (/2 x t )(log /2 y t f δ (/2 y t ))I( /2 x t K)I( /2 y t K) 6

+ f δ (/2 x t )f δ (/2 y t )I( /2 x t K)I( /2 y t K). (2) I will only emonstrate the asserte result for the secon term, but the proof for the thir term is analogous. To eal with the first term, note that by the Cauchy-Schwartz inequality, (E (log /2 x t f δ (/2 x t )) log /2 y t I( /2 x t K)I( /2 y t K)) 2 E (log /2 x t f δ (/2 x t )) 2 E (log /2 y t ) 2 I( /2 y t K). (21) Noting that the ensities of t 1/2 x t an t 1/2 y t are both uniformly boune over t M + 1, it follows that E (log /2 y t ) 2 I( /2 y t K) (log /2 t 1/2 r ) 2 I( /2 t 1/2 r K)r K n 1/2 t 1/2 (log s ) 2 s < C (22) K for some constant C. Therefore, the expression of Equation (21) can be boune by C (log /2 t 1/2 r f δ (/2 t 1/2 r)) 2 r δ C t 1/2 n 1/2 (log s log δ ) 2 s δ δ C (log s log δ ) 2 s as δ. (23) δ Now in orer to show the result of Equation (19) an complete the proof, note that f δ (σ x W x (r))f δ (σ y W y (r))i( σ y W y (r) K)I( σ x W x (r) K)r 7

= + + (f δ (σ x W x (r)) log σ x W x (r) )f δ (σ y W y (r))i( σ x W x (r) K)I( σ y W y (r) K)r log σ x W x (r) (f δ (σ y W y (r)) log σ y W y (r) )I( σ x W x (r) K)I( σ y W y (r) K)r log σ x W x (r) log σ y W y (r) I( σ x W x (r) K)I( σ y W y (r) K)r, (24) an the first an secon term converge to zero almost surely as δ. I will show this for the first term only. By the Cauchy-Schwartz inequality, the square of the first term can be boune by (f δ (σ x W x (r)) log σ x W x (r) ) 2 I( σ x W x (r) K)r (log σ y W y (r) ) 2 I( σ y W y (r) K)r, (25) an by the occupation times formula (see for example Park an Phillips (1999)), letting L y (.,.) enote the Brownian local time of W y (.), = (log σ y W y (r) ) 2 I( σ y W y (r) K)r K K while similarly, = L(1, r)(log σ y r ) 2 r sup L(1, r) r K K K (f δ (σ x W x (r)) log σ x W x (r) ) 2 I( σ x W x (r) K)r L x (1, r)(f δ (σ x r) log σ x r ) 2 I( σ x r K)r sup L x (1, r) r K/σ x δ δ (log σ y r ) 2 r = O P (1), (26) (log δ log r ) 2 r as δ, (27) 8

which completes the proof of the asserte result for ˆβ 2. Now note that ˆβ 1 = log y t ˆβ 2 log x t = log /2 y t ˆβ 2 log /2 x t log(/2 ) + ˆβ 2 log(/2 ), (28) an therefore (log(n)) 1 ˆβ1 = o P (1) + (1/2) (1/2) ˆβ 2 (1/2) (1/2)B 2, (29) which completes the proof. Proof of Lemma 1: This result follows from noting that s 2 = (n 2) 1 ((log /2 y t log /2 y t ) ˆβ 2 (log /2 y t log /2 y t )) 2, (3) an when working out the square in the above expression, all terms can be ealt with by Theorem 1 in combination with the results of Equations (2) an (3) to show convergence in istribution to a ranom limit S. Proof of Theorem 2: Note that ˆt 2 = n 1/2 ˆβ2 s 1 ( (log /2 x t log /2 x t ) 2 ) 1/2, (31) an because ˆβ 2, s, an n (log n 1/2 x t log /2 x t ) 2 all converge jointly in istribution, the result now follows. For showing the result for ˆt 1, note that ˆt 1 = n 1/2 (log(n)) 1 ˆβ1 )( n (log n 1/2 x t log /2 x t ) 2 ) 1/2 (s 2 (log(n)) 2 n (log x t ) 2 ) 1/2, (32) 9

an because (log(n)) 2 (log y t ) 2 p (1/4) (33) an because all other terms converge in istribution, the result now follows. References e Jong, R.M. (21), A continuous mapping theorem-type result without continuity, mimeo, Department of Economics, Michigan State University. Granger, C.W.J. an P. Newbol (1974), Spurious regressions in econometrics, Journal of Econometrics 2, 111-12. Hamilton, J.D. (1995), Time Series Analysis. Princeton: Princeton University Press. Park, J.Y. an P.C.B. Phillips (1999), Asymptotics for nonlinear transformations of integrate time series, Econometric Theory 15, 269-298. Phillips, P.C.B. (1986), Unerstaning spurious regressions in econometrics, Journal of Econometrics 33, 311-34. Pötscher, B.M. (21), Nonlinear functions an convergence to Brownian motion: beyon the continuous mapping theorem, mimeo, University of Vienna. 1