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On the Hsiao definition of non-causality
Economics Letters 66 (2000) 261–264 www.elsevier.com / locate / econbase
On the Hsiao definition of non-causality Umberto Triacca Istituto Nazionale di Statistica, Via Cesare Balbo 16, 00184 Roma, Italy
Abstract Granger (Granger, C.W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.) defined causality between two variables X and Y in terms of predictability. A difficulty with this definition is that it is restricted to one-step ahead prediction. In the presence of a third environment variable Z the non-causality properties depend on the horizon of the involved prediction. Hsiao (Hsaio, C., 1982. Autoregressive modelling and causal ordering of economic variables. Journal of Economic Dynamic and Control 4, 243–259.) proposed a generalization of the Granger notion of causality. The main purpose of this paper is to show that the Hsiao non-causality properties do not depend on the horizon of involved prediction. 2000 Elsevier Science S.A. All rights reserved. Keywords: Environment variable; Granger causality; Hsiao non-causality; Prediction; Time series JEL classification: C32
1. Introduction Granger (1969) defined causality between two variables X and Y in terms of predictability. According to Granger’s definition a variable Y is said to cause another variable X with respect to a given universe or information set that includes X(t) 5 hXt , Xt21 , . . . j and Y(t) 5 hYt , Yt 21 , . . . j if Xt 11 can be better predicted by using the information in Y(t) than by not doing so, all other relevant information (including the present and the past of X) being used in either case. In presence of a third environment variable Z the non-causality properties depend on the horizon of the involved prediction. Hsiao (1982) proposed a generalization of the Granger notion of causality. The main purpose of this paper is to show that the Hsiao non-causality properties do not depend on the horizon of involved prediction. The paper is organized as follows. In Section 2 we introduce the definitions of non-causality utilized in the sequel. Section 3 is devoted to the proof of the main results. Conclusions are in Section 4. E-mail address: [email protected] (U. Triacca) 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 99 )00217-7
2000 Elsevier Science S.A. All rights reserved.
U. Triacca / Economics Letters 66 (2000) 261 – 264
262
2. Definitions Let hY t , t [ Ij be an n 3 1 multivariate stochastic process on the integers I, with finite second 9 , Y 2,t 9 , Y 3,t 9 )9, where Yi,t 5(Yi 1,t , . . . ,Yin i ,t )9, i 5 1,2,3. Further, let Hy (t) be the moments; let Y t 5(Y 1,t Hilbert space generated by the components of Yt for t # t, let Hy\y 3 (t) be the closed subspace of Hv (t) 9 t , Y 2,9 t )9 for t # t, similarly define Hy\y 2 (t) and let Hy\y 2 y 3 (t) be the generated by the components of (Y 1, closed subspace of Hy (t) generated by the components of Y 1,t for t # t. For any closed subspace, M, of Hy (t) and for 1 # i # n 1 , we denote (Y1i,t 1k uM) the orthogonal projection of Y1i,t1k on M and (Y 1,t1k uM)5[(Y11,t 1k uM), . . . ,(Y1n 1 ,t1k uM)]9, similarly define (Y 2,t1k uM). Now, we consider the following definitions of non-causality. Definition 1. The vector Y 3 does not Granger cause Y 1 , with respect to Hy (t), iff (Y 1,t11 uHy (t))5 (Y 1,t11 uHy \y 3 (t)) ;t [ I. We observe that the condition (Y 1,t 11 uHy (t))5(Y 1,t11 uHy \y 3 (t)) ;t [ I is not sufficient to exclude any causal link from Y 3 to Y 1 . This condition may be satisfied and the information in the past and present Y 3 may still be helpful in predicting Y 1 more than one period ahead. Intuitively, this may happen, because Y 3 may have an impact on Y 2 which in turn may affect Y 1 (‘indirect causality’). Definition 2. The vector Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), when either
(i) (Y 1,t11 uHy (t))5(Y 1,t11 uHy\y 2 y 3 (t)) ;t [ I or (ii) (Y 1,t11 uHy (t))5(Y 1,t 11 uHy \y 3 (t)) and (Y 2,t11 uHy (t))5(Y 2,t11 uHy\y 3 (t)) ;t [ I. In other words, we have that Y 3 does not Hsiao cause Y 1 , with respect to Hy (t) if and only if (Y 92 , Y 93 )9 does not Granger cause Y 1 , or Y 3 does not Granger cause Y 1 and Y 3 does not Granger cause Y 2 .
3. Results In this section we shall show that the Hsiao non-causality does not depend on the horizon of involved prediction. In order to do that, it is useful to prove first the following lemma. Lemma 1. Let V1,t , V2,t be two subvectors of Y t with V1,t [ R p , V2,t [ R q , p 1 q 5 n. Let Hy \v 2 (t) be the closed subspace of Hy (t) generated by the component of V1,t for t # t. V2 does not Granger cause V1 , with respect to Hy (t), if and only if (V1,t 1k uHy (t))5(V1,t 1k uHy\v 2 (t)) ;k $ 1, ;t [ I. Proof. Clearly if (V1,t1k uHy (t))5(V1,t1k uHy \v 2 (t)) ;k $ 1, ;t [ I, then V2 does not Granger cause V1 , with respect to Hy (t). If V2 does not Granger cause V1 , with respect to Hy (t), then (V1,t 11 uHy (t)) 5 (V1,t11 uHy \v 2 (t)) ;t [ I.
(1)
U. Triacca / Economics Letters 66 (2000) 261 – 264
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For h $ 1, suppose that for 1 # k # h (V1,t 1k uHy (t)) 5 (V1,t1k uHy \v 2 (t)) ;t [ I.
(2)
Then putting Ft1h 11 5 (V1,t1h 11 uHy (t 1 h)) ;t [ I it follows that Ft1h 11 5 (V1,t1h 11 uHy \v 2 (t 1 h))5(F1,t 1h11 , . . . ,Fp,t1h 11 )9 from (1). Thus Fi,t1h 11 [ Hy \v 2 (t 1 h) i 5 1, . . . , p Hence by induction hypothesis (2) (Ft 1h11 uHy (t)) 5 (Ft1h 11 uHy\v 2 (t)) ;t [ I.
(3)
On the other hand, by applying the law of iterated projections, we have that (Ft 1h11 uHy (t)) 5 ((V1,t1h 11 uHy (t 1 h))uHy (t)) 5 (V1,t 1h11 uHy (t)) ;t [ I.
(4)
(Ft 1h11 uHy \v 2 (t)) 5 ((V1,t1h 11 uHy\v 2 (t 1 h))uHy \v 2 (t)) 5 (V1,t 1h 11 uHy \v 2 (t)) ;t [ I.
(5)
and
By (3), the right-hand sides of (4) and (5), are equal, so (2) holds for k 5 h 1 1. But (2) is true for h 5 1. Hence (2) is true for all h. We remember that Lemma 1 is a suitable version of a theorem firstly proved by Kohn (1981). Theorem 1. If Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), then
( i) (Y 1,t1k uHy (t)) 5 (Y 1,t 1k uHy\y 2 y 3 (t)) ;k $ 1, ;t [ I or ( ii) (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 3 ) and (Y 2,t 1k uHy (t)) 5 (Y 2,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I. Proof. If Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), then (i) (Y t 11 uHy (t)) 5 (Y t11 uHy\y 2 y 3 (t)) ;t [ 1 or (ii) (Y t11 uHy (t)) 5 (Y t 11 uHy \y 3 ) and (Y 2t11 uHy (t)) 5 (Y 2t11 uHy \y 3 (t)) ;t [ I. If (i) then, posing V1 5Y 1 and V2 5(Y 92 , Y 39 )9, by Lemma 1, it follows that (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 2 y 3 (t)) ;k $ 1, ;t [ I. If (ii) then, posing V1 5(Y 91 , Y 29 )9 and V2 5Y 3 , by Lemma 1, it follows that (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I and (Y 2,t1k uHy (t)) 5 (Y 2,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I. The converse is trivial. This theorem shows that Hsiao non-causality properties do not depend on the horizon of the involved predictions.
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4. Conclusions In this paper we have shown that the restriction of the Hsiao’s definition to one-step ahead prediction causes no problem. Therefore, it seems that Hsiao non-causality is an appropriate notion for detecting unidirectional effects of a set of variables, Y 3 , on the variables of interest, Y 1 , when a third vector of variables, Y 2 (environment variables), is considered in the analysis. This theoretical result is relevant since causality inference in the presence of environment variables is a common practice in empirical economic research.
References Granger, C.W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438. Hsiao, C., 1982. Autoregressive modelling and causal ordering of economic variables. Journal of Economic Dynamic and Control 4, 243–259. Kohn, R., 1981. A characterization of Granger-Sims exogeneity. Economics Letters 8, 129–133.
On the Hsiao definition of non-causality Umberto Triacca Istituto Nazionale di Statistica, Via Cesare Balbo 16, 00184 Roma, Italy
Abstract Granger (Granger, C.W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.) defined causality between two variables X and Y in terms of predictability. A difficulty with this definition is that it is restricted to one-step ahead prediction. In the presence of a third environment variable Z the non-causality properties depend on the horizon of the involved prediction. Hsiao (Hsaio, C., 1982. Autoregressive modelling and causal ordering of economic variables. Journal of Economic Dynamic and Control 4, 243–259.) proposed a generalization of the Granger notion of causality. The main purpose of this paper is to show that the Hsiao non-causality properties do not depend on the horizon of involved prediction. 2000 Elsevier Science S.A. All rights reserved. Keywords: Environment variable; Granger causality; Hsiao non-causality; Prediction; Time series JEL classification: C32
1. Introduction Granger (1969) defined causality between two variables X and Y in terms of predictability. According to Granger’s definition a variable Y is said to cause another variable X with respect to a given universe or information set that includes X(t) 5 hXt , Xt21 , . . . j and Y(t) 5 hYt , Yt 21 , . . . j if Xt 11 can be better predicted by using the information in Y(t) than by not doing so, all other relevant information (including the present and the past of X) being used in either case. In presence of a third environment variable Z the non-causality properties depend on the horizon of the involved prediction. Hsiao (1982) proposed a generalization of the Granger notion of causality. The main purpose of this paper is to show that the Hsiao non-causality properties do not depend on the horizon of involved prediction. The paper is organized as follows. In Section 2 we introduce the definitions of non-causality utilized in the sequel. Section 3 is devoted to the proof of the main results. Conclusions are in Section 4. E-mail address: [email protected] (U. Triacca) 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 99 )00217-7
2000 Elsevier Science S.A. All rights reserved.
U. Triacca / Economics Letters 66 (2000) 261 – 264
262
2. Definitions Let hY t , t [ Ij be an n 3 1 multivariate stochastic process on the integers I, with finite second 9 , Y 2,t 9 , Y 3,t 9 )9, where Yi,t 5(Yi 1,t , . . . ,Yin i ,t )9, i 5 1,2,3. Further, let Hy (t) be the moments; let Y t 5(Y 1,t Hilbert space generated by the components of Yt for t # t, let Hy\y 3 (t) be the closed subspace of Hv (t) 9 t , Y 2,9 t )9 for t # t, similarly define Hy\y 2 (t) and let Hy\y 2 y 3 (t) be the generated by the components of (Y 1, closed subspace of Hy (t) generated by the components of Y 1,t for t # t. For any closed subspace, M, of Hy (t) and for 1 # i # n 1 , we denote (Y1i,t 1k uM) the orthogonal projection of Y1i,t1k on M and (Y 1,t1k uM)5[(Y11,t 1k uM), . . . ,(Y1n 1 ,t1k uM)]9, similarly define (Y 2,t1k uM). Now, we consider the following definitions of non-causality. Definition 1. The vector Y 3 does not Granger cause Y 1 , with respect to Hy (t), iff (Y 1,t11 uHy (t))5 (Y 1,t11 uHy \y 3 (t)) ;t [ I. We observe that the condition (Y 1,t 11 uHy (t))5(Y 1,t11 uHy \y 3 (t)) ;t [ I is not sufficient to exclude any causal link from Y 3 to Y 1 . This condition may be satisfied and the information in the past and present Y 3 may still be helpful in predicting Y 1 more than one period ahead. Intuitively, this may happen, because Y 3 may have an impact on Y 2 which in turn may affect Y 1 (‘indirect causality’). Definition 2. The vector Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), when either
(i) (Y 1,t11 uHy (t))5(Y 1,t11 uHy\y 2 y 3 (t)) ;t [ I or (ii) (Y 1,t11 uHy (t))5(Y 1,t 11 uHy \y 3 (t)) and (Y 2,t11 uHy (t))5(Y 2,t11 uHy\y 3 (t)) ;t [ I. In other words, we have that Y 3 does not Hsiao cause Y 1 , with respect to Hy (t) if and only if (Y 92 , Y 93 )9 does not Granger cause Y 1 , or Y 3 does not Granger cause Y 1 and Y 3 does not Granger cause Y 2 .
3. Results In this section we shall show that the Hsiao non-causality does not depend on the horizon of involved prediction. In order to do that, it is useful to prove first the following lemma. Lemma 1. Let V1,t , V2,t be two subvectors of Y t with V1,t [ R p , V2,t [ R q , p 1 q 5 n. Let Hy \v 2 (t) be the closed subspace of Hy (t) generated by the component of V1,t for t # t. V2 does not Granger cause V1 , with respect to Hy (t), if and only if (V1,t 1k uHy (t))5(V1,t 1k uHy\v 2 (t)) ;k $ 1, ;t [ I. Proof. Clearly if (V1,t1k uHy (t))5(V1,t1k uHy \v 2 (t)) ;k $ 1, ;t [ I, then V2 does not Granger cause V1 , with respect to Hy (t). If V2 does not Granger cause V1 , with respect to Hy (t), then (V1,t 11 uHy (t)) 5 (V1,t11 uHy \v 2 (t)) ;t [ I.
(1)
U. Triacca / Economics Letters 66 (2000) 261 – 264
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For h $ 1, suppose that for 1 # k # h (V1,t 1k uHy (t)) 5 (V1,t1k uHy \v 2 (t)) ;t [ I.
(2)
Then putting Ft1h 11 5 (V1,t1h 11 uHy (t 1 h)) ;t [ I it follows that Ft1h 11 5 (V1,t1h 11 uHy \v 2 (t 1 h))5(F1,t 1h11 , . . . ,Fp,t1h 11 )9 from (1). Thus Fi,t1h 11 [ Hy \v 2 (t 1 h) i 5 1, . . . , p Hence by induction hypothesis (2) (Ft 1h11 uHy (t)) 5 (Ft1h 11 uHy\v 2 (t)) ;t [ I.
(3)
On the other hand, by applying the law of iterated projections, we have that (Ft 1h11 uHy (t)) 5 ((V1,t1h 11 uHy (t 1 h))uHy (t)) 5 (V1,t 1h11 uHy (t)) ;t [ I.
(4)
(Ft 1h11 uHy \v 2 (t)) 5 ((V1,t1h 11 uHy\v 2 (t 1 h))uHy \v 2 (t)) 5 (V1,t 1h 11 uHy \v 2 (t)) ;t [ I.
(5)
and
By (3), the right-hand sides of (4) and (5), are equal, so (2) holds for k 5 h 1 1. But (2) is true for h 5 1. Hence (2) is true for all h. We remember that Lemma 1 is a suitable version of a theorem firstly proved by Kohn (1981). Theorem 1. If Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), then
( i) (Y 1,t1k uHy (t)) 5 (Y 1,t 1k uHy\y 2 y 3 (t)) ;k $ 1, ;t [ I or ( ii) (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 3 ) and (Y 2,t 1k uHy (t)) 5 (Y 2,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I. Proof. If Y 3 does not Hsiao cause Y 1 , with respect to Hy (t), then (i) (Y t 11 uHy (t)) 5 (Y t11 uHy\y 2 y 3 (t)) ;t [ 1 or (ii) (Y t11 uHy (t)) 5 (Y t 11 uHy \y 3 ) and (Y 2t11 uHy (t)) 5 (Y 2t11 uHy \y 3 (t)) ;t [ I. If (i) then, posing V1 5Y 1 and V2 5(Y 92 , Y 39 )9, by Lemma 1, it follows that (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 2 y 3 (t)) ;k $ 1, ;t [ I. If (ii) then, posing V1 5(Y 91 , Y 29 )9 and V2 5Y 3 , by Lemma 1, it follows that (Y 1,t 1k uHy (t)) 5 (Y 1,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I and (Y 2,t1k uHy (t)) 5 (Y 2,t1k uHy \y 3 (t)) ;k $ 1, ;t [ I. The converse is trivial. This theorem shows that Hsiao non-causality properties do not depend on the horizon of the involved predictions.
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U. Triacca / Economics Letters 66 (2000) 261 – 264
4. Conclusions In this paper we have shown that the restriction of the Hsiao’s definition to one-step ahead prediction causes no problem. Therefore, it seems that Hsiao non-causality is an appropriate notion for detecting unidirectional effects of a set of variables, Y 3 , on the variables of interest, Y 1 , when a third vector of variables, Y 2 (environment variables), is considered in the analysis. This theoretical result is relevant since causality inference in the presence of environment variables is a common practice in empirical economic research.
References Granger, C.W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438. Hsiao, C., 1982. Autoregressive modelling and causal ordering of economic variables. Journal of Economic Dynamic and Control 4, 243–259. Kohn, R., 1981. A characterization of Granger-Sims exogeneity. Economics Letters 8, 129–133.