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In this paper we discuss sensitivity of forecasts with respect to the information set considered in prediction; a sensitivity measure called impact factor, IF, is defined. This notion is specialized to the case of VAR processes integrated of order 0, 1 and 2. For stationary VARs this measure corresponds to the sum of the impulse response coefficients. For integrated VAR systems, the IF has a direct interpretation in terms of long-run forecasts. Various applications of this concept are reviewed; they include questions of policy effectiveness and of forecast uncertainty due to data revisions. A unified approach to inference on the IF is given, showing under what circumstances standard asymptotic inference can be conducted also in systems integrated of order 1 and 2. It is shown how the results reported here can be used to calculate similar sensitivity measures for models with a simultaneity structure.

Forecasting is one of the major enterprises in time-series econometrics, see Clements and Hendry (2002) and reference therein. In this paper we consider model-based long-run forecasts and their sensitivity with respect to information variables. We define a sensitivity indicator, called impact factor, IF. It is shown how this indicator allows to formulate questions on policy effectiveness and on the forecast uncertainty due to data revisions. Sensitivity indicators have long been advocated in econometrics; see Banerjee and Magnus, 1999 and Banerjee and Magnus, 2000 for recent references. By definition, they describe the sensitivity of a given procedure with respect e.g. to some possible source of model mis-specification. In the present case we apply this concept to mis-measurement of the information variables that are used in long-run forecasts. Variations in the information variables can be caused by data revisions. Data revisions may hence alter the long-run forecasts of key macroeconomic indicators. Given that many economic decisions are based on forecasts made using preliminary data, it would be of interest to measure forecast uncertainty due to this source of data errors. Improving the quality of preliminary figures for variables to which forecasts are most sensitive would greatly improve the quality of the associated economic decisions. Conversely if data revisions on some variables do not have any impact on long run forecast, then there would be no need to obtain more timely or precise data. Variations in the information variables may also be associated with the effects of policy interventions. In this perspective, it is of interest to find how long-run forecasts of key indicators are affected by possible policy actions. Absence of sensitivity would indicate long-run ineffectiveness of the policy measure. Although policy analysis and data revisions are the main economic areas of applications of this concept, the notion of IF can be defined and discussed in general for any dynamic system and forecast function. The IF is not calculated on actual forecasts, but it is defined as a function of the model parameters and possibly of sample data. It measures long-run properties of the system; it is hence suggested as a tool of model interpretation, rather than of forecast performance. Quite obviously, the notion of IF does not account for the possible occurrence of model breaks between the past and the future. The concept of IF is related to many standard econometric notions, like dynamic multipliers and impulse responses. Like a dynamic multiplier, the IF measures the sensitivity of a function. However, a dynamic multiplier is defined only between some endogenous variable yy and some exogenous variable xx; impact factors, instead, are well defined for any dynamic systems, including VARs. Moreover long-run multipliers are usually defined in terms of the static relation implied by a dynamic model for yy and xx, see e.g. Hendry (1995, p. 339), Gourieroux and Monfort (1995, p. 34, 35), whereas the IF measures the accumulated effects on forecasts of perturbations in past information. Impact factors turn out to be the limit of cumulated impulse responses (IR) in case of VARs. The definition of (economically meaningful) shocks is the subject of a vast debate in the VAR literature, to which the present paper does not contribute. We note here instead that, while IFs are defined in terms of input variables, they are part of all limit cumulated IRs of current use. Hence the analysis of the IF can be coupled with many possible definitions of structural shocks to obtain long-run sensitivity measures of forecast with respect to any particular shock definition. Moreover, the explicit expressions of the IF we derive in this paper may be used to impose long-run restrictions on cumulated IR. While the definition of IF is based on stationary processes, the concept is motivated by and applied to non-stationary integrated systems. We consider I(1) and I(2) processes and compute IFs for these processes. For I(1) systems, the present paper builds on ideas introduced in Bedini and Mosconi (2000). They defined the concept of ‘long-run adjustment coefficient’ with respect to the disequilibrium associated with an error correction term. We here offer different insights on the I(1) case and extend the concept to I(2) systems. For the I(1) case we show how the long-run adjustment coefficient is related to the forecast function, and more in general to the concept of IF. This concept is linked to the choice of state vector and the timing of variables, and we discuss the relation among different choices. Explicit expressions of the IF for the I(1) and I(2) cases are given. These formulae do not involve infinite summations, and reveal the prominent role of the moving average impact matrices both in the I(1) and I(2) cases. Inference on these matrices has been considered in Paruolo, 1997a, Paruolo, 1997b and Paruolo, 2002. These matrices and other parameters enter the expressions of the IF; this observation motivates the present extensions. The explicit expressions of the IF allow to simplify and reduce the amount of computations needed to evaluate long-run effects. More importantly, the explicit expressions reveal the different contribution of various VAR parameters to the long-run effect. Several parameters are indeed shown to have no effect in the long run. Finally one can ascertain if there are any zero long run effects by analyzing the rank of specific blocks of the explicit form of the IF. This paper also analyzes the influence of timing in variable definitions on the long-run effects. It is found that some IF are invariant to timing, while others are not. Inference on the IF is presented in a compact and unified form for stationary, I(1) and I(2) systems. Wald tests on the IF are presented. Standard arguments imply that Wald tests on any smooth function of the IF are easily derived from the ones in this paper through the delta method. We show how this analysis can be coupled with any definition of simultaneity structure to define sensitivity measures with respect to ‘structural’ perturbations. The rest of the paper is organized as follows. Section 2 reports relevant definitions and Section 3 collects basic properties. Section 4 presents impact factors in I(1) and I(2) processes. Section 5 discusses applications of this concept to policy analysis and data revisions. Section 6 treats the estimation of IF, while Section 7 contains an empirical analysis of a price system for Australia. Finally Section 8 concludes. All proofs are placed in three final appendices. In the following a≔ba≔b and b≕ab≕a indicate that aa is defined by View the MathML sourceb;(a:b) indicates the matrix obtained by horizontally concatenating aa and bb. For any full column rank matrices View the MathML sourceH,A,B,sp(H) is the linear span of the columns of View the MathML sourceH,H¯ indicates H(H′H)-1H(H′H)-1 and H⊥H⊥ indicates a basis of sp(H)⊥sp(H)⊥, the orthogonal complement of sp(H)sp(H). ∥·∥∥·∥ indicates a matrix norm and its associated vector norm. Moreover View the MathML sourcePH≔HH¯′,HAB≔A¯′HB¯, View the MathML sourceHAB.C≔HAB-HACHCC-1HCB while HA≔H(A′H)-1HA≔H(A′H)-1. Finally (·)ij(·)ij indicates the ij-th element of the argument matrix, vec is the column stacking operator, ⊗⊗ is the Kronecker product (i.e. A⊗BA⊗B is the matrix with generic block aijBaijB, where A≔[aij])A≔[aij]) and View the MathML source→d indicates convergence in distribution.

In this paper we have defined impact factors as a sensitivity measure of long-run forecasts, and discussed their properties. We have applied the definition to vector autoregressive processes, in the stationary, I(1) and I(2) cases. Not surprisingly, the impact factors are functions of the moving average total impact matrix of the stationary representation of the systems, which is singular in cointegrated processes. An application to price mark-up in Australia shows, among other things, how perturbations to labor cost can have a permanent positive effect on inflation and a permanent negative effect on the mark-up. This is in line with imperfect competition models.