Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol66.Issue3.Mar2000:

Economics Letters 66 (2000) 261–264

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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_{hX , X , . . .j and Y(t)}5_{hY , Y , . . .j if X}

t t21 t t21 t11

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: triacca@istat.it (U. Triacca)

262 U. Triacca / Economics Letters 66 (2000) 261 –264

2. Definitions

Let hY , t[Ij be an n3_{1 multivariate stochastic process on the integers I, with finite second}

t

9

9

9

moments; let Y5_{(Y , Y , Y )}9_{, where Y} 5_{(Y , . . . ,Y )}9_{, i}5_{1,2,3. Further, let H (t) be the}

t 1,t 2,t 3,t i,t i 1,t in ,ti y

Hilbert space generated by the components of Y for_t t#_{t, let H (t) be the closed subspace of H (t)}

y \y₃ v

9

9

generated by the components of (Y , Y )9_fort#_{t, similarly define H (t) and let H (t) be the}

1,t 2,t y \y₂ y \y y_{2 3}

closed subspace of H (t) generated by the components of Y fort#_{t. For any closed subspace, M,}

y 1,t

of H (t) and for 1#_i#_{n , we denote (Y uM ) the orthogonal projection of Y on M and}

y 1 1i,t1k 1i,t1k

(Y uM )5_{[(Y uM ), . . . ,(Y uM )]}9_{, similarly define (Y uM ).}

1,t1k 11,t1k 1n ,t1 1k 2,t1k Now, we consider the following definitions of non-causality.

Definition 1. The vector Y does not Granger cause Y , with respect to H (t), iff (Y uH (t))5

causal link from Y to Y . This condition may be satisfied and the information in the past and present_{3 1}

Y may still be helpful in predicting Y more than one period ahead. Intuitively, this may happen,_{3 1}

because Y may have an impact on Y which in turn may affect Y (‘indirect causality’)._{3 2 1}

Definition 2. The vector Y does not Hsiao cause Y , with respect to H (t), when either_{3 1 y}

(i) (Y uH (t))5_{(Y uH (t))} ;_t[I

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.

p q

Lemma 1. Let V , V be two subvectors of Y with V [R , V [R , p1_q5_{n. Let H (t) be}

1,t 2,t t 1,t 2,t y \v₂

the closed subspace of H (t) generated by the component of V fort#_{t. V does not Granger cause}

U. Triacca / Economics Letters 66 (2000) 261 –264 263

On the other hand, by applying the law of iterated projections, we have that

(F uH (t))5_{((V uH (t}1_{h))uH (t))}5_{(V uH (t))};_{t[I. (4)}

We remember that Lemma 1 is a suitable version of a theorem firstly proved by Kohn (1981).

Theorem 1. If Y does not Hsiao cause Y , with respect to H (t), then_{3 1 y}

264 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 , on the variables of interest, Y , when a third3 1

vector of variables, Y (environment variables), is considered in the analysis. This theoretical result is₂ 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.