Building Scenarios of Multiple Time Series that Take ...

Journal of Forecasting, J. Forecast. 33, 32–46 (2014)

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2271

Building Scenarios of Multiple Time Series that Take into Account the Effects of an Expected Intervention† VÍCTOR M. GUERRERO,1 ELIUD SILVA2* AND NICOLÁS GÓMEZ3 1 2 3

Department of Statistics, Instituto Tecnológico Autónomo de México (ITAM), México, DF, México Actuarial School, Universidad Anáhuac del Norte, Estado de México, México Banco de México, DF, México

ABSTRACT We consider a forecasting problem that arises when an intervention is expected to occur on an economic system during the forecast horizon. The time series model employed is seen as a statistical device that serves to capture the empirical regularities of the observed data on the variables of the system without relying on a particular theoretical structure. Either the deterministic or the stochastic structure of a vector autoregressive error correction model of the system is assumed to be affected by the intervention. The information about the intervention effect is just provided by some linear restrictions imposed on the future values of the variables involved. Formulas for restricted forecasts with intervention effects and their mean squared errors are derived as a particular case of Catlin’s static updating theorem. An empirical illustration uses Mexican macroeconomic data on five variables and the restricted forecasts consider targets for years 2011–2014. Copyright © 2013 John Wiley & Sons, Ltd. key words cointegration; compatibility testing; deterministic change; multiple disturbance; stochastic

change; VEC models

INTRODUCTION Intervention effects are commonly encountered when analyzing time series data. The presence of those effects in the forecast horizon could easily mislead a time series model and its forecasts, leading the analyst to reach erroneous conclusions. For instance, a forecast of the Mexican economy must take into account the structural reform agenda, because nowadays in Mexico economic as well as political efforts focus on achieving the necessary consensus to advance in the fiscal, energetic, labor and pensions reforms. These reforms are considered necessary to consolidate macroeconomic stability. If any of those reforms were approved, the forecasts based solely on the historical record could mislead economic expectations. In fact, predictions based only on historical data would be invalid if a policy change affects the economy, since the economic agents adapt their expectations and behavior to the new policy stance. In this work, we consider that an economic reform is likely to occur during the forecasting horizon, causing a change in the behavior of the system. We consider specifically the case in which a system of variables is to be predicted with the aid of a vector autoregressive error correction (VEC) model without relying on a specific economic theory. In fact, we consider this model as a statistical device composed of three elements: deterministic structure, stochastic structure and parameters. The statistical model serves to capture the empirical regularities in the historical data and should not be considered a reduced form pertaining to a specific theoretical economic structure. We assume here that an expected economic reform is an intervention that will affect either the deterministic or the stochastic part of the VEC model during the forecast horizon. Thus the probability of achieving a given target with a conventional forecast is negligible, unless additional information is taken into account. Further, we assume that all the information available on the effects of intervention is provided by means of economic expectations established by experts. Those expectations will be considered here as tentative targets and expressed as linear restrictions on the future values of the series in order to obtain restricted forecasts. The results generalize those of the univariate case obtained by Guerrero (1991). Furthermore, the univariate time series types of change presented by Tsay (1988) are considered here in a multivariate setting; five of them serve to model deterministic changes and two more capture stochastic changes. Tsay et al. (2000) generalized the four most commonly used types of deterministic disturbance effects to the multivariate case. The problem of incorporating external model information in a multivariate time series forecast setting was first considered by Doan et al. (1984), who provided a computational algorithm to deal with this problem. Then, Greene et al. (1986), Van der Knoop (1987) and Pankratz (1989) dealt with the combination of historical and additional information in the form of linear restrictions and provided formulas for vector autoregressive and moving average (VARMA) models and optimum restricted forecast in a mean squared error (MSE) sense, while Guerrero et al. (2008) also considered VEC models. Within the state-space framework, Pandher (2002) attacked the problem of modeling and forecasting multivariate time series with linear restrictions that apply during the sample period. Such *Correspondence to: Eliud Silva, Actuarial School, Universidad Anáhuac del Norte, Estado de México, Mexico. E-mail: [email protected] †The views and conclusions presented in this work are responsibility of the authors and do not necessarily reflect those of the Banco de México.

Copyright © 2013 John Wiley & Sons, Ltd.

Building Scenarios of Multiple Time Series with Structural Change 33 an approach differs from ours since we consider restrictions that apply during the forecast horizon. Our main contribution lies in incorporating in the forecasts information attributable to an expected intervention that is supposed to occur during the forecast horizon. This article is organized as follows. The next section presents the statistical methodology to obtain restricted forecasts with VEC models. The third section shows some typical disturbance functions that are used to model intervention effects. In the fourth section we derive the restricted forecasting formulas of a VEC model affected in either its deterministic or stochastic components. The fifth section illustrates the methodology with an empirical application that uses a VEC representation of a Mexican economic system. This application assumes that the economic targets proposed for years 2011–2014 will be reached with certainty and that an economic reform will take place in year 2013. By so doing we consider the resulting paths of the restricted forecasts as scenarios. Hence the restricted forecasting methodology should be considered as a tool to produce scenarios that agree with the proposed targets.

METHODOLOGY Let yt = ( y1t, . . .,ykt) 0 be a k  1 vector of variables at time t and let us assume that {yt} follows a finite pth-order Gaussian vector autoregressive (VAR) model: ΠðBÞyt ¼ Λdt þ «t

(1)

where Π(B) = I  Π1B  . . .  ΠpBp is a k  k matrix polynomial and B is the backshift operator such that Byt = yt–1. dt = (dlt,. . ., dnt)0 is an n  1 vector that includes deterministic variables to account for seasonality as well as other 0 deterministic effects that are captured by the k  n parameter P «t = (elt,. . ., ekt) is a k  1 vector of indepenP matrix Λ. dent and identically distributed (i.i.d.) random errors N(0, e) with e a positive-definite covariance matrix whose i, jth element is sij = cov(eit,ejt), for i, j = 1, 2, . . ., k and t = 1, 2, . . ., T. The elements of «t are assumed to be serially uncorrelated but may be contemporaneously correlated and cov(yt,«s) = 0 for t < s. We also assume that all the zeros of the determinant |Π(z)| are on or outside the unit circle. Subtracting yt  1 from both sides of equation (1) and rearranging terms we have (2) Π ðBÞryt ¼ Λdt  Πð1ÞByt þ «t Xp1 where r is the first difference operator and Π ðBÞ ¼ I þ Π Bi is a matrix polynomial of order p  1 with Πi ¼ i¼1 i Xp1 Π ; thus Π(B) = Π(1)B + Π*(B) r. If Π(1) 6¼ 0 there is cointegration in { yt} and we write Π(1) =  gb, where g j¼iþ1 j

and b are k  r and r  k matrices with r the rank of Π(1). Moreover, we assume that the elements of { yt} are at most integrated of order 1, which we write I(1). When r > 0 the variables are cointegrated, in the sense that there exists at least one I(0) linear combination b yt, with b = (b1, . . .,bk) 6¼ 0. Then, equation (2) becomes Π ðBÞryt ¼ Λdt þ g zt1 þ «t

(3)

where zt  1 = b yt  1 is stationary. The rows of b are cointegration vectors of the system and equation (3) is its VEC representation. It should be stressed that we will use this model as a tool to capture the empirical regularities of the data and will not consider any specific structure that may arise from economic theory. Furthermore, the VAR and VEC models are equivalent in terms of the information they capture, but the appropriate way to estimate the model is in VEC form to avoid omitted-variable bias. Once the model is estimated we will employ the VAR representation for forecasting purposes. Thus, to derive the forecasting formulas, we consider the model parameters known, although in fact they will be estimated. To get the forecast and its MSE let us start by defining Y = (y 0  p + 1, . . ., y 0 T) 0 , a k(T + p)  1 vector containing all the past information of the multiple time series, and let YF = (y 0 T + 1, . . ., y 0 T + H) 0 denote a kH  1 vector that contains the H ≥ 1 values to be forecast for each series. The optimal (in MSE sense) linear forecast of yT + h, for h = 1, . . ., H, is its conditional expectation EðyTþh j YÞ ¼ ΛdTþh þ Π1 EðyTþh1 j YÞ þ . . . þ Πp EðyTþhp j YÞ with E(yT + h  i| Y) = yT + h  i for i ≥ h. Such a forecast produces the forecast error vector yTþh  EðyTþh j YÞ ¼

h1 X

Ψj «Tþhj for h ¼ 1; . . . ; H

j¼0

where the matrices Ψj are calculated as in Wei (1990, p. 364) from the recursion Ψ0 = I andΨi ¼ k > p or k < 0) for i > 0. Stacking all the forecast errors in (4) we have Copyright © 2013 John Wiley & Sons, Ltd.

(4) Xi k¼1

Ψik Πk (with Πk = 0 for

J. Forecast. 33, 32–46 (2014)

34 V. M. Guerrero, E. Silva and N. Gómez

P

YF  EðYF j YÞ ¼ Ψ«F

where «F = (« 0 T + 1, . . ., « 0 T + H) 0 ~ N(0, I  e) is a kH  1 random vector, with  the Kronecker product and the kH  kH matrix Ψ is lower triangular with Ψ0 in its main diagonal, Ψ1 in its first subdiagonal, Ψ2 in the second subdiagonal and so on. Finally, the MSE of E(YF| Y) is given by  X MSE½EðYF j YÞ ¼ Ψ I e Ψ0

VAR WITH INTERVENTION EFFECTS When interventions occur, the original series {yt} is disturbed and becomes unobservable. In such a case, obtaining forecasts without taking into account the future effects of those interventions is a worthless task. Therefore, we consider some specific ways in which the model may be modified due to an intervention on the system. Following Tsay (1988), let us assume that the observed series fe yt g follows the model e y t ¼ y t þ f ðt Þ

(5)

where f(t) is the vector disturbance function representing the intervention effects that modify {yt}. This function may be deterministic or stochastic, depending on the type of disturbances considered. As a generalization of the univariate case, we present some typical disturbances that may affect a VAR model. It should be clear that the list of proposed effects is by no means exhaustive and excludes, for example, the case where the cointegrating space of the process may change, or the dynamic short-run structure of the model could be similarly altered. Instead, we present cases that have appeared previously in the literature and that we consider simpler to work with. Deterministic changes Suppose that the intervention produces a deterministic effect (D) that is expected to occur at time t = T + h with 1 ≤ h ≤ H (an expected effect in the forecast horizon). Using the terminology and notation of Tsay et al. (2000) we know that this effect can be represented by means of the following k  1 vector disturbance function: f ðt Þ ¼ aðBÞvxðt TþhÞ where a(B) is a matrix polynomial in B to be defined below, v = (o1, . . .,ok) 0 is the initial impact vector of a disturbance ðTþhÞ on the series {yt} and xðt TþhÞ is an indicator time index of T + h, that is xTþh ¼ 1 and xðt TþhÞ ¼ 0 if t 6¼ T + h. The function f(t) belongs to the class of intervention functions proposed by Box and Tiao (1975) and can be used to describe many dynamic disturbance behaviors. Multiple disturbances can be treated in the same manner by considering a vector disturbance function for each period. To analyze the effect of the model dynamics on the intervention dynamics it is convenient to compute the filtered series {at} given by yt  Λdt  at ¼ e

p X

yti Πi e

(6)

i¼1

where e yt ¼ yt and at = «t for t < T + h. In the presence of a disturbance, at 6¼ «t for some t points, otherwise at = «t for all t. From (1) and (6) we have at ¼ «t þ ΠðBÞaðBÞvxðt TþhÞ

(7)

so that the effect of an intervention on the filter depends on the interaction between Π(B)a(B) and v. The first four of the following specifications appear in Tsay et al. (2000), who generalized to the k-dimensional case the most commonly used deterministic disturbances and considered the effects of cointegration explicitly. The fifth is a generalization relevant for the present work. (a) An innovational outlier occurs when a(B) = Ψ(B) = I  Ψ1B  Ψ2B  . . ., where the sequence {Ψi} is not absolutely summable, but the Ψi matrices can be obtained recursively as indicated with model (4). This case represents a change in the innovational series {«t} and its   dynamic effect on {yt} propagates through the weights yt ¼ ΨðBÞ vxðt TþhÞ þ Λdt þ «t : Ψi, in such a way that e (b) The additive outlier is obtained when a(B) = I, so that the observed series is given by e yt ¼ vxðt TþhÞ þ ΨðBÞðΛdt þ «t Þ and the disturbance affects only yT + h. Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

Building Scenarios of Multiple Time Series with Structural Change 35 (c) A temporary change is produced when a(B) = (1  rB) 1I with 0 < r < 1, i.e. e yt ¼ ð1  rBÞ1 vxðt TþhÞ þ ΨðBÞðΛdt þ «t Þ: This model describes a disturbance v that occurs at time t = T + h and decays exponentially to the zero vector at rate r; its effect is transient, but affects {yt} for t ≥ T + h. (d) A level shift will occur when r a(B) = I, so that re yt ¼ vxðt TþhÞ þ rΨðBÞðΛdt þ «t Þ: Thus a level shift of size v occurs at time t = T + h and the change is permanent, so that the disturbance will affect {yt} for t ≥ T + h. yt ¼ ð1  rBÞ1 vxðt TþhÞ þ (e) A gradual change is produced when r a(B) = (1  rB) 1I with 0 < r < 1, i.e. re rΨðBÞðΛdt þ «t Þ: This is a model for a disturbance v that occurs at t = T + h and grows to (1  r) 1v, so that the gradual change will affect {yt} for t ≥ T + h. Only cases (d) and (e) are considered appropriate to represent permanent effects. Moreover, case (e) contains as a particular case the level shift when we allow the rate of growth to be r = 0, which is why we prefer to work with a gradual change to represent permanent deterministic effects in the empirical illustration. Stochastic changes Suppose that a stochastic change (V) is expected to occur at time t = T + h and let ðTþhÞ

f ðt Þ ¼ aðBÞ§t St

where {§t} is a sequence of k  1 i.i.d. random vectors N(0,ΣB) uncorrelated with {«t}, where ΣB is the contaminating ðTþhÞ covariance matrix. The step variable St takes on the value 1 for t ≥ T + h and is zero otherwise. This is a very simple but practical approach to contaminate the original process and the variance of {yt} will change for t ≥ T + h.   ðTþhÞ (f) A variance innovational change is produced when a(B) = Ψ(B), in which case we get e y t ¼ Ψ ðB Þ § t S t þ Λdt þ «t : Its univariate counterpart can be found in Tsay (1988), where it is called variance change. Then, equation (7) ðTþhÞ becomes at ¼ «t þ §t St , so that the variance innovational change implies at ~ N(0, Σe + ΣB) for t ≥ T + h. ðTþhÞ (g) The variance additive change is produced when a(B) = I, in such a way that e yt ¼ §t St þ ΨðBÞðΛdt þ «t Þ. The ðTþhÞ filtered series is given by at ¼ «t þ ΠðBÞ§t St , so that the variance additive change affects {at} for t ≥ T + h. In practice, to model the effect of an intervention in a multiple time series setting with variance change, we prefer to use specification (f) since the variables of the system are then allowed to incorporate the dynamics of the VAR model. Otherwise the disturbance effects could be unrealistic by affecting just one particular variable of the system. Moreover, we should note that by assuming stochastic changes the corresponding effects may be persistent but not permanent. The forecast An expression that allows for a deterministic intervention effect during the forecast horizon can be obtained directly from equation (5) as YF;D ¼ YF þ DF  0 0 0 where YF;D ¼ e y Tþ1 ; . . . ; e y TþH Þ is a kH  1 vector containing the future observed values of the multiple time series  0 0 0 and DF ¼ d Tþ1 ; . . . ; d TþH Þ is a kH  1 vector that accounts for the deterministic effects; here dt ¼ aðBÞvxðt TþhÞ is a k  1 vector. For example, an innovational outlier corresponds to dt ¼ ΨðBÞvxðt TþhÞ and can be written as ΠðBÞdt ¼ vxðt TþhÞ ; for t≥T þ hand dt ¼ 0 otherwise: Similarly, an expression that allows for a stochastic change can be written as YF;V ¼ YF þ VF 0

0

0

where YF,V is a kH  1 vector containing the future observed values and VF ¼ ðv Tþ1 ; . . . ; v TþH Þ is a kH  1 ðTþhÞ stochastic vector that accounts for a stochastic change; here vt ¼ aðBÞ§t St is a k  1 random vector. For instance, ðTþhÞ a variance innovational change is obtained with vt ¼ ΨðBÞ§t St , which can also be written as ðTþhÞ

ΠðBÞvt ¼ §t St

; for t≥T þ h

and vt = 0 otherwise. An expression that includes both deterministic and stochastic changes is written as YF,D,V = YF + DF + VF and the forecast, given the historical record of the series, becomes  E YF;D;V jYÞ ¼ E ðYF jYÞ þ DF whose forecast error vector is given by Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

36 V. M. Guerrero, E. Silva and N. Gómez  YF;D;V  E YF;D;V jYÞ ¼ Ψ«F þ DF So, when a variance innovational change occurs, the MSE of E(YF,D,V|Y) becomes    0    MSE E YF;D;V jY ¼ Ψ ðIΣe Þ þ eIΣB Ψ where eI ¼ diagð00 11Þ is an H  H diagonal matrix whose h  1 first elements are 0 and the rest are 1. The corresponding expression for a variance additive change is given by      0 MSE E YF;D;V jY ¼ ΨðIΣe ÞΨ þ eIΣB

RESTRICTED FORECASTS OF A VAR MODEL WITH INTERVENTION EFFECTS We are concerned with obtaining the vector of forecasts when additional information about the future values of the series is given in the form of linear restrictions, i.e. when R ¼ CYF;D;V

(8)

where C is an M  kH matrix, useful to represent some linear combinations of YF,D,V, i.e. 0 1 ⋯ C1;kH C1;1 ⋯ C1;k ⋯ C1;kðH1Þþ1 A C¼@ ⋯ CM;1 ⋯ CM;k ⋯ CM;kðH1Þþ1 ⋯ CM;kH where the first k columns are linked to yT + 1 and the last k columns to yT + H. The rows are linearly independent, so that the rank of C is M and R = (r1, . . ., rM)0 is the M  1 vector of values that these linear combinations take on. The problem of finding the optimal restricted linear forecast vector (in MSE sense) of YF,D,V can be solved with the aid of the following result provided by Catlin (1989, p. 128). Static updating theorem. Suppose the random vectors X and R2 are related by R2 = HX + W, where H is a known ^ m  n matrix and W is a random vector such that U = E(WW0 ) is known. Further suppose the minimum h that X 1is  0 i ^1 ^ ^ are MSE linear predictor of X based on a random vector R1 and that both X 1 and P1 ¼ E X  X 1 X  X   0 0 known. Finally, suppose that E XW ¼ 0 and E R1 W ¼ 0. ^ 2 of X based on R1 and R2 is given by Then, the minimum MSE linear predictor X   ^2 ¼ X ^ 1 þ P1 H 0 ðHP1 H 0 þ U Þþ R2  H X ^1 X and the new MSE matrix is given by h  0 i ^ 2 ¼ P1  P1 H 0 ðHP1 H 0 þ U Þþ HP1 ^2 X  X P2 ¼ E X  X where ()+ denotes the Moore–Penrose inverse.  ^ 1 ¼ E YF;D;V jYÞ, P1 = MSE[E(YF,D,V|Y)] = ΣF,D,V, H = C, W = 0, U = 0, R2 = R In the present situation we have X ^ F;D;V . Thus the restricted forecast of a VAR process with intervention effects in the forecast horizon is ^2 ¼ Y and X given by     ^ F;D;V ¼ E YF;D;V jYÞ þ A R  CE YF;D;V jYÞ Y where A = ΣF,D,VC 0 (CΣF,D,VC 0 ) 1. The MSE of the forecast vector is   ^ F;D;V ¼ ðI  AC ÞΣF;D;V MSE Y Note that there is no need to use the Moore–Penrose inverse since CΣF,D,VC 0 is nonsingular because C is a full rank matrix. Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

Building Scenarios of Multiple Time Series with Structural Change 37 EMPIRICAL ILLUSTRATION As already mentioned, several economic reforms (fiscal, energetic, pensions, etc.) are currently considered necessary to achieve faster growth of the Mexican economy. Since an economic reform in the short run seems likely to occur, we consider the Mexican case appropriate to illustrate our methodology. To that end we note that 2012 is a Presidential election year (the new administration will come to power on 1 December 2012) so we suppose that a reform will occur in year 2013 that will come into effect in 2014 and will produce a fast-growing effect on the economy. We consider a five-dimensional VAR model with Mexican data to obtain restricted forecasts that fulfill some proposed targets for years 2011, 2012 and 2013, and take into account the 2014 expected intervention effects. Two scenarios are considered: the first one of deterministic type and the other where the change is supposed to be stochastic. Program routines for estimation and forecasting of the model were done in Matlab 7.0 (MathWorks, Inc. software). The data The dataset consist of 60 quarterly observations spanning the period 1996:Q1–2010:Q4 for the following variables: • Consumer price index (LCPI). Monthly consumer price index base 2003 = 100. The quarterly consumer price index (ipcmex) is the value at the end of each quarter of the monthly series and log-transformed, LCPIt = ln(ipcmex t). • Gross domestic product (LGDP). Gross domestic product measured in thousands of Mexican pesos at constant prices of 2003 and log-transformed, LGDPt = ln(GDP t). • Real demand of money (LMONB). Currency held by the public plus domestic currency and checking accounts in resident banks is a monthly series expressed in nominal terms in thousand of Mexican pesos (MONB). The quarterly series is the average of the three monthly values of each quarter deflated with ipcmex and log-transformed, LMONBt = ln(MONBt/ipcmex t). • Trade balance deficit (TRDB). This is defined as income minus expenditure of the foreign sector (DEF). This is a quarterly series expressed in millions of dollars. The series is divided by 10,000 to homogenize the data scales of all variables, LTRDBt = DEFt/10,000. Unemployment rate (LUNMP). This is the unemployed population relative to the economically active population per 100. The unemployment data (UNMP) are log-transformed, LUNMPt = ln(UNMPt). The data source of ipcmex, GDP and UNMP is Instituto Nacional de Estadística y Geografía: Price Index System, National Accounts System and National Employment Survey, respectively, while the source of MONB and DEF is the Bank of México. We consider LCPI, LGDP, LMONB, TRDB and LUNMP as the observed variables. The log transformations were applied basically to stabilize the variance of each series, so that the vector of variables to be used is yt = F(xt), where xt is the k  1 vector of actual observed variables at time t and F() is the transformation applied. Order of integration The order of integration of the transformed series was decided by applying augmented Dickey–Fuller (ADF) tests. The regression model used included a constant term, a deterministic trend centered seasonal dummies and a dummy variable to take into account a relevant deterministic effect, as needed. The general equation was rzt ¼ a þ b0 t þ

4 X

bi dit þ c0 zt1 þ

i¼1

p X

cj rztj þ et

j¼1

Table I shows the results of the ADF tests, where tnc is the statistic applicable to the regression without constant, tc applies to a model with constant and tct when there are both constant and trend. Each statistic in Table I allows us to test H0: c0 = 0. The asterisk indicates rejection of the null hypothesis at the 5% significance level when comparing the statistic against its critical values (Table 20.1 in Davidson and MacKinnon, 1993). The order of the model, p, was selected so as to guarantee no residual autocorrelation. As a result of these tests, we declared all variables to be I(1). VEC estimation The model includes yt = (LCPIt, LGDPt, LMONBt, TRDBt, LUNMPt)0 plus a constant term, centered seasonal dummy variables and dummy variables to account for the Y2K effect (I2000:Q1) on the demand for money and the 2008 worldwide financial crisis (I2008:Q3 and I2009:Q1), i.e. Dt = (const, S1,t, S2,t, S3,t, I2000 : I,t, I2008 : III,t, I2009 : I,t)0 . The system turned out to be CI(1, 1) with an integrated VAR(2) model providing a reasonable representation for the variables in levels, although the dummy variable I2008:Q3,t was not significant at the 10% level and was excluded from the model. Therefore, the economic system was represented by the following VEC model: ryt ¼ ΛDt þ gbyt1 þ Π1 ryt1 þ Π2 ryt2 þ «t with estimation results shown below (t-values in parenthesis): Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

38 V. M. Guerrero, E. Silva and N. Gómez Table I. ADF unit root test results H0: I(1)

Variable

LCPI LGDP LMONB TRDB LUNMP

H0: I(2)

p

tct

p

tnc

tc

0 0a 2b 2c 0

8.36** 2.61 4.07** 2.68 2.55

2 0a 0 1c 0

2.78* — — 8.89** 6.54**

— 8.15** 8.59** — —

Asterisks indicate rejection of H0 at the *5% significance level and at the **1% level. a Including a dummy for 2009:Q1; b including a dummy for 2000:Q1; c including a dummy for 2008:Q2.

0

 -0:019

ð3:620Þ

ð2:394Þ B B B  -0:122 B ð7:151Þ B B  -0:130 ^ ¼B Λ ð4:515Þ B B  B B @  

 ð5:499Þ



B B B B ^ ¼ B Π B 1 B B B @ 0

^ ¼ Π 2

B B B B B B B @



0:424

 0

0:331 -0:137

ð2:721Þ



-0:981



0:088 - 0:235

ð2:124Þ

ð3:484Þ

ð2:719Þ



0:070 







- 0:307 









0:045 



-0:377

C C C C C C C C C C C A

18:972 -7:901 -1:519 3:610Þ

1 C C C C C C and C C C A

ð1:765Þ

1



-4:176



ð1:839Þ

 





ð1:739Þ



 

^ ¼ ð1 ; b









ð1:737Þ



ð3:050Þ

ð3:521Þ



ð1:966Þ



0:612

2:111

-0:464

ð2:029Þ

ð2:850Þ

3:509 

ð2:496Þ

0:026 - 0:044

 

ð2:007Þ

ð1:786Þ



ð1:739Þ



ð3:071Þ

-0:045

ð2:639Þ



0:762

ð2:670Þ

-0:082

ð3:184Þ

ð4:741Þ



0:294

-0:120

1

-0:019



ð4:661Þ

-0:135

ð1:588Þ ð3:912Þ ð0:956Þ

ð1:963Þ



-0:104

 ^ ¼ 0:008 -0:043 0:018 g 0



-0:024

 -0:344

C C C C C C C A

ð2:285Þ

the adjusted R2 values were equal to 0.727, 0.943, 0.905, 0.660 and 0.839 for the series in first differences DLCPI, ^ Π ^  and Π ^  show only the numerical values DLGDP, DLMONB, DTRDB and DLUNMP, respectively. The matrices Λ, 1 2 of the entries found significant at the 10% level. The following (symmetric) matrix shows the contemporaneous residual correlations and Table II reports the results of Johansen tests. LCPI LGDP LMONB TRDB LUNMP 0 1 1:000 LCPI C 1:000 LGDP B B 0:186 C B C 0:024 1:000 LMONB B 0:091 C @ A 0:031  0:124  0:330 1:000 TRDB 0:057  0:550 0:103 0:126 1:000 LUNMP At the 10% significance level, there may be two cointegrating relationships and only one at the 5% level. Thus we decided to use only one cointegration relationship, whose graph is shown in Figure 1, and we do not attempt to Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

Building Scenarios of Multiple Time Series with Structural Change 39 Table II. Johansen cointegration analysis Null

Trace statistic

Crit. 90%

Crit. 95%

Eigen statistic

Crit. 90%

Crit. 95%

r≤0 r≤1 r≤2 r≤3 r 0 and Q ¼ diag r12 ; r22 ; 0; 0; 0 is a 5  5 matrix. We should note that the contaminating effect appears only in DLCPI and DLGDP and is proportional to r12 and to r22 respectively. The problem now lies in selecting the contaminating parameter a appropriately. To that end, let us define the following distance random vector:  h ¼ R2  C2 E YF;V jYÞ ¼ C2 Ψ«F þ C2 VF

then R2 and E(YF,V|Y) are said to be compatible if the distance vector is close to zero. From the normality assumption of  P    «F and zTþh we have that h ~ N(0, Ω(a)), where ΩðaÞ ¼ C2 Ψ I e þ a eIQ Ψ0 C02 is an M2  M2 symmetric post itive semi-definite covariance matrix. So, if Ω(a) is non-singular, a statistic for testing compatibility between restrictions and unrestricted forecasts can be defined as K ðaÞ h’Ω1 ðaÞh w2M2 e which is of the type derived in Guerrero et al. (2008). We say that h lies in the compatibility region at level a if K ðaÞ≤w2M2 ðaÞ, where w2M2 ðaÞ denotes the upper a percentage point of the w2M2 ðaÞ distribution. Since, for a given realization of h, the observed test statistic K(a) decreases as the contaminating parameter a grows (see Appendix B) we should choose a large enough to obtain compatibility, but keeping in mind that the larger its value the greater the uncertainty in the forecasts. We should also realize that in practice K(a) will only have an approximate chi-square distribution for large samples, since the parameters involved are estimated. In the present case, the compatibility test statistic without contamination took on the value K(0) = 3.82 which, when compared to w22 ð0:1Þ ¼ 4:61, does not lead to rejecting the compatibility hypothesis at the 10% significance level. Nonetheless, just for illustrative purposes, we decided to use the contaminating parameter a = 0.015 in what Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

Building Scenarios of Multiple Time Series with Structural Change 43

Figure 6. Unrestricted and restricted forecasts with stochastic change and 80% prediction intervals (forecast origin at 2010:Q3). Upper left: DLCPI; upper right: DLGDP; middle left: DLMONB: middle right: DTRDB; and lower: DLUNMP

Table IV. Restricted forecasts with stochastic change for the Mexican system Variable

2011

2012

2013

2014

DLCPI DLGDP

3.78% 4.30%

3.81% 4.00%

3.26% 3.00%

2.50% 8.00%

follows. With this choice of a the calculated test statistic became K(0.015) = 3.78, so that compatibility is even less significant. Figure 6 shows the unrestricted and restricted forecasts of the original variables with stochastic change, together with 80% prediction intervals.   ^ F;V ¼ F 1 Y ^ F;V are ^ F and the restricted ones X As can be seen in Figure 6, the paths of the unrestricted forecasts X very similar. Nevertheless, the restricted paths with stochastic change are closer to those of the unrestricted forecasts than the paths of the restricted forecasts with deterministic change, even for the variables directly affected by the disturbance (DLCPI and DLGDP). In Table IV we show the forecasts of the original variables with stochastic change and clearly see that the targets are attained exactly, as expected. The amplitude of the prediction intervals is a bit larger in this case than with the deterministic change. It is clear now that the DLUNMP restricted and unrestricted paths behave as they should, contrary to the previous situation.

CONCLUSIONS The suggested methodology allows the analyst to take into account an intervention effect that is expected to occur in the forecast horizon. The types of changes due to the intervention are presented in a multivariate setting as a generalization of those in the univariate case. Since the forecasts must take into account the dynamics of the model, we prefer to use either the deterministic gradual change or the variance innovational change separately. A combination of these changes can be entertained, but the limited amount of information provided by the restrictions, as well as the confusion between deterministic and stochastic effects, prevented us from carrying out that exercise. The restricted forecast formulas of a VAR model with intervention effects are derived as a particular case of Catlin’s static updating theorem. The methodology was illustrated with a five-dimensional system with Mexican macroeconomic data, since an economic reform has been pushed through by economic and political sectors of the country. The application assumed a scenario where the economic targets for 2011–2013 as well as the expected effect of an economic reform in 2014 will be reached with certainty, not because we believe that it will actually occur, but to be able to appreciate the future paths of the variables that agree with that scenario. It is assumed that the economic reform will modify either the deterministic or the stochastic part of a VAR model and that its effect will initially impact GDP and prices; then the effect will propagate to the other variables in the system according to their own dynamics. Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

44 V. M. Guerrero, E. Silva and N. Gómez In the empirical illustration we obtained different paths with the two types of changes for the variables included, although the targets were attained exactly in both cases. The restricted path of the unemployment rate is similar in both the deterministic and the stochastic change situations up to 2013 and it is easier to understand it in the latter one from 2013 onwards. Thus, because of the simplicity of application of the variance innovational change formulation and its clarity of interpretation, we prefer to use this approach in this empirical illustration. Moreover, the combination of extra model information with VAR forecasts resulted in more realistic predictions since expectations from experts were taken into account. Further, even though we did not consider using a Bayesian approach to derive the optimal restricted forecasts, we can loosely interpret the restricted and unrestricted forecasts as a rough estimation of the limits within which the proper Bayesian forecasts might vary (depending on the prior employed).

ACKNOWLEDGEMENTS We acknowledge the comments and suggestions provided by two anonymous referees and an Associate Editor. V. M. Guerrero thankfully acknowledges the support provided by Asociación Mexicana de Cultura AC through a professorship on Time Series Analysis and Forecasting in Econometrics.

APPENDIX A: SENSITIVITY ANALYSIS OF THE DETERMINISTIC RATE OF DECAY

Figure A.1. Deterministic gradual change of the system with two choices of the rate of decay r = 0.5 (solid line ) and r = 0.7 (dotted line). Upper left: DLCPI  103; upper right: DLGDP; middle left: DLMONB; middle right: DTRDB; and lower: DLUNMP

APPENDIX B: BEHAVIOR OF K(A)

We show that K(a) is a decreasing function of a, for h fixed. We start by computing dKðaÞ ¼ h’Ω1 ðaÞ½dΩðaÞΩ1 ðaÞh where   0 dΩðaÞ ¼ C2 Ψ eIQðaÞ Ψ C’2 da see Magnus and Neudecker (2002). Thus the derivative of K(a) is given by n o   0 dKðaÞ 0 ¼ h Ω1 ðaÞ C2 Ψ eIQðaÞ Ψ C’2 Ω1 ðaÞh da   which is negative since C2 Ψ eIQðaÞ Ψ’C’2 is positive definite. The conclusion follows. Copyright © 2013 John Wiley & Sons, Ltd.

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Building Scenarios of Multiple Time Series with Structural Change 45

Figure A.2. Unrestricted and restricted forecasts with a deterministic change and 80% prediction intervals (forecast origin at 2010:Q3) with rates of decay r = 0.5 (left) and r = 0.7 (right). From top: DLCPI, DLGDP, DLMONB, DTRDB and DLUNMP

REFERENCES Box GEP and Tiao GC. 1975. Intervention analysis with applications to environmental and economic problems. Journal of the American Statistical Association 70: 70–79. Catlin DE. 1989. Estimation, Control, and the Discrete Kalman Filter. Springer: New York. Davidson R and MacKinnon JG. 1993. Estimation and Inference in Econometrics. Oxford University Press: Oxford. Doan T, Litterman R, Sims Ch. 1984. Forecasting and conditional projection using realistic prior distributions. Econometric Reviews 3: 1–100. Greene MN, Howrey EP, Hymans SW. 1986. The use of outside information in econometric forecasting. In Model Reliability, Kuh E, Belsley DA (eds) MIT Press: Cambridge, MA; 90–116. Guerrero VM. 1991. ARIMA forecast with restrictions derived from a structural change. International Journal of Forecasting 7: 339–347. Guerrero VM, Pena B, Senra E, Alegría A. 2008. Restricted forecasting with a VEC model: validating the feasibility of economic targets. ESTADISTICA 60: 83–98. Magnus JR and Neudecker H. 2002. Matrix Differential Calculus. Wiley: New York. Pandher GS. 2002. Forecasting multivariate time series with linear restrictions using constrained structural state-space models. Journal of Forecasting 21: 281–300. Pankratz AE. 1989. Time series forecasts and extra-model information. Journal of Forecasting 8: 75–83. Tsay RS. 1988. Outliers, level shifts, and variance changes in time series. Journal of Forecasting 7: 1–20. Tsay RS, Peña D and Pankratz AE. 2000. Outliers in multivariate time series. Biometrika 87: 789–804. Van der Knoop HS. 1987. Conditional forecasting with a multivariate time series model. Economics Letters 22: 233–236. Wei WWS. 1990. Time Series Analysis. Addison-Wesley: Menlo Park, CA. Authors’ biographies: Víctor M. Guerrero is Full Professor of Statistics at ITAM. He has published several papers in such journals as the Journal of Forecasting and the International Journal of Forecasting, among others. He has also written some textbooks in Spanish. His main academic interests are time series analysis and forecasting. Eliud Silva is Full Professor of Applied Statistics in the Actuarial School of Universidad Anahuac del Norte. His main research interests are time series analysis and mathematical demography. Nicolás Gómez is in charge of Cash Analysis and Research Studies Division in Banco de México.His main research interest lies in time series analysis and forecasting.

Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)

46 V. M. Guerrero, E. Silva and N. Gómez Authors’ addresses: Víctor M. Guerrero, Department of Statistics, Instituto Tecnológico Autónomo de México (ITAM), México, DF, México. Eliud Silva, Actuarial School, Universidad Anáhuac del Norte, Estado de México, México. Nicolás Gómez, Cash Analysis and Research Studies Division, Banco de México, DF, México.

Copyright © 2013 John Wiley & Sons, Ltd.

J. Forecast. 33, 32–46 (2014)