T R E N D S A N D C Y C L E S : H O W I M P O R T A N T A R E L O N G - A N D S H O R T - R U N R E S T R I C T I O N S ?

T H E C A S E O F M E X I C O *

J o r g e H e r r e r a Hernández Banco de México

Ramón A . C a s t i l l o P o n c e Banco de México

E l Colegio de la Frontera Norte

Resumen: Se presenta una prueba para identificar la existencia de restricciones válidas de corto y largo plazo (tendencia común-ciclo común) en la dinámica de un conjunto de variables macroeconómicas de México. Estas restricciones se imponen en un V A R para descarticular las se­ries tanto en su componente permanente como transitorio. E l análisis muestra que la magnitud de choques transitorios (nominales) es subes­t imada cuando dichas restricciones no se consideran. Adicionalmente, encontramos que las fechas y la duración de los periodos de recesión y expansión son más acertadamente estimadas cuando la descomposición tendencia-ciclo se realiza incorporando restricciones de cointegración (largo-plazo) y de componente común (corto-plazo).

Abstract: The document presents a test for the existence of binding long- and short -run (common trend-common cycle) restrictions in the dynamics of a set of Mexican macroeconomic variables. These restrictions are imposed i n a V A R to decompose the series into their permanent and transitory components. The analysis shows that the magnitude of t ran­sitory (nominal) shocks is underestimated when such restrictions are not considered. I n addit ion, we find that the t i m i n g and duration of re­cession and expansion periods are more accurately estimated when the trend-cycle decomposition is conducted w i t h the imposit ion of cointe-grat ing ( long-run) and common feature (short-run) restrictions.

J E L Classifications: C32, C53, E32

Fecha de recepción: 09 IV 2002 Fecha de aceptación: 03 IX 2002

* We thank Angel Palerm, Raúl Razo, Mario Reyna and Abraham Vela for helpful comments and conversations. The comments of an anonymous referee are also acknowledged. The usual disclaimer applies. jherrera@banxico.org.mx, rcastill@banxico.org.mx

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134 ESTUDIOS ECONÓMICOS

1. I n t r o d u c t i o n

T h e decomposit ion of t ime series into the ir permanent and t rans i t o ry components is an exercise t h a t has occupied economists for decades. However, most of the work has been performed using data f r om devel­oped economies, f rom the US i n part icular . I n the case of developing countries, empir ical analysis of macroeconomie variables t h a t i d e n t i ­fies the importance of permanent and t rans i tory components i n t i m e series is scarce.

Aside f rom the bu lk of applications on the subject, a certa in number of papers f o rm the core of methodologies developed to ana­lyze this topic. I n their 1981 seminal paper, Beveridge and Nelson introduced a general procedure to decompose non-stat ionary t ime se­ries into their permanent and t rans i tory components and applied i t to the measuring and dat ing of business cycles. As Blanchard and Quah (1989) indicate, however, the earlier a t tempt to measure t r end and cyclical components d i d not consider the dynamic effects of per­manent and t rans i tory shocks. To address this shortcoming, B l a n ­chard and Quah proposed a methodology t h a t restricted the impact of transitory , or nomina l shocks, on the permanent component of the series. T h a t is, they appended to the neutra l i ty of nomina l shocks theory a further restr ic t ion : t h a t nominal shocks have zero impact on real variables i n the long r u n . The authors f ind t h a t demand (nom­inal) shocks have a short - l ived effect on output whi le supply (real) shocks affect o u t p u t permanently. A l t h o u g h the methodology em­ployed in that paper provided valuable tools to further evaluate the importance of t rans i tory shocks, cointegration and the shor t - run re­lationships among the variables examined were not considered. I n 1991, K i n g et al. performed an analysis that expl i c i t ly imposed coin-tegrat ing restrictions i n the decomposition of t ime series. Using a Vector Error Correction Model, V E G M , to include such restrict ions,

the authors showed t h a t nominal shocks were, i n general, more i m ­p o r t a n t (of greater magnitude) t h a n previously thought . T h i s f ind ing indicated that nominal shocks, which include the ins t rumentat i on of monetary and fiscal policy, significantly influence the magnitude and d u r a t i o n of business cycles.

More recently, Vah id and Engle (1993, V E hereafter) and Issler and Vah id (2001) highl ighted the importance of considering common cyclical components when per forming trend-cycle decomposition of cointegrated t ime series. The authors show that when common cycle (short -run) restrict ions are imposed on a system of variables, nomina l shocks are even greater in magnitude compared to t h a t found i n K i n g et al. They conclude that there are efficiency gains when a V E C M is

TRENDS AND CYCLES 135

estimated imposing s ho r t - run restrictions relative to the case when the V E G M is est imated w i t h o u t such restr i c t ions . 1

I n th i s document we follow V E and Issler and Vahid (2001) i n an­alyzing the importance of t rend and cyclical components of M e x i c a n maeroeconomic series. We choose Mexican Gross Domestic Produc t , G D P , aggregate consumption, and real credit granted by commercial banks to conduct the analysis. Recently, these three series have been the subject of s tudy i n various papers, generating a quite interest ing debate. O n the one hand, contrary to w h a t most economists w o u l d expect, some studies f ind t h a t pr ivate consumption and the M e x i c a n G D P do not cointegrate. I n response, other studies have shown t h a t i f the conditions of the credit market are considered, then i t is possible to find co integrat ion between consumption and the G D P . 2 U s i n g the V E methodology, we show t h a t the series not only comtegrate, but t h a t they also share a common cycle. I n addi t ion , we f ind t h a t the magni tude of t rans i to ry shocks when considering short and l ong r u n restrict ions is larger t h a n when such restrictions are not accounted for. Th i s result is par t i cu lar ly interest ing for purposes of monetary and fiscal policy, since the ins t rumenta t i on of b o t h might be perceived as nomina l shocks.

A l t h o u g h showing that the series share short and long r u n com­ponents, and t h a t t rans i tory shocks explain a significant p o r t i o n of their var iat ion , are interesting i n and of themselves, the analysis we conduct i n this document may be used for more pract ical purposes, such as forecasting, for instance. I n part i cular , consider the case when two series are shown to share a common cycle. I f the release o f one anticipates the release of the other, then in format ion contained i n the former may help i n forecasting the behavior of the latter , most of all the upcoming t u r n i n g point . Also, the decomposition methodology we employ here can be used i n any model t h a t includes i n its analysis a trended series or a gap between a series and its t rend . For example, one could employ this methodology to estimate the ou tput gap i n an in f la t i on model. Obviously, this methodology can be employed i n the t r a d i t i o n a l analysis of determining the magnitude and d u r a t i o n of business cycles. As w i l l become evident, the results of the t r e n d -cycle decomposition obtained when short- and l ong -run restrictions

Parallel to the advances i n the study of decomposition methods for non-stationary series, Engle and Kozicki (1993) developed common feature analysis of stationary variables based upon an instrumental variables technique.

2 For the former argument see for instance Gonzalez Garcia (2000), for the latter Garces (2001) and Castillo (2001).

136 E S T U D I O S ECONÓMICOS

are imposed are more efficient t h a n other decomposition methods. The paper is organized as follows. Section 2 outlines the m e t h o d ­

ology employed i n the analysis. I n section 3 we present the d a t a along w i t h a series of tests to identi fy the ir stochastic properties. Also, we conduct the common trend-common cycle tests and the permanent -t rans i tory decomposit ion of the series. I n section 4 we per form some estimations to measure the importance of nominal shocks on the v a r i ­ations of real series. Section 5 concludes.

2. M e t h o d o l o g y

2 .1 . Fundamentals

The methodology used to identify the permanent and t rans i to ry com­ponents of a series is, i n general, as follows: consider a vector yt of n variables integrated of order 1 (1(1)) and whose first difference, A y t , follows an autoregressive process. I t is said t h a t the variables i n yt are cointegrated i f there exists r ( < n) l inear combinations of t h e m t h a t are integrated of order 0 (1(0)). Such combinations can be grouped i n a m a t r i x ¡3 of dimension n x r . The rank of this m a t r i x is r and i t defines the cointegration space.

Accord ing to Engle and Koz ick i (1993), the elements of Ayt share serial correlation common features i f there exists a linear combinat ion of t h e m that represents an innovat ion w i t h respect to the i n f o r m a t i o n available pr ior to period t (i.e. t h a t removes the serial corre lat ion p a t t e r n in Ayt)' These linear combinations are denoted cofeature combinations and Engle and Kozicki demonstrates t h a t , condi t iona l upon the presence of cointegration among the variables included i n yt, there can only exist s(< n) such combinations and r -f- s < n. These cofeature vectors can be grouped i n a m a t r i x a w i t h dimension n x s where a Ayt represents an innovation. T h e rank of the m a t r i x a defines the subspace of al l possible cofeature vectors.

Given that b o t h , the cointegration and the cofeature subspaces may project the f u l l space spanned by the complete set of variables i n yu and given t h a t , by def init ion, the cointegrating (r) and the cofeature (s) relationships are l inearly independent and unique when they exist, there can exist a m a x i m u m of n — r ( > s) cofeature vec­tors. Therefore i t is possible to decompose the elements i n yt in to their permanent and trans i tory components (a la Beveridge - Nelson) t a k i n g into account these s t ruc tura l restrictions.

T h e most widely applied technique i n this k i n d of analysis is the one proposed by K i n g et al (1991), where the restrictions included

TRENDS AND CYCLES 137

i n the model consider the cointegration space leaving the short r u n movements unconstrained. Ignor ing short r u n comovements between t ime series when they exist, however, induces efficiency losses (Vahid and Engle, 1993; and Issler and Vahid , 2001). The losses arise because of the use of l i m i t e d in f o rmat i on methods ( short - run parameters u n ­constrained when they should be restricted) i n the est imat ion of such models.

Issler and Vah id show, by considering the same set of variables as i n K i n g et. al, t h a t the out-of-sample forecast error is lower i n a fu l l i n f o rmat i on environment (considering short r u n restrictions) t h a n i n a l i m i t e d i n f o r m a t i o n set t ing ( tak ing into account only cointegrat ion restrict ions) .

Given the efficiency gains obtained by inc lud ing common t r e n d -common cycle restrictions, i t is possible to assess accurately the i m ­portance of t rans i tory shocks to real variables.

2.2. Common Cycles Test

Fol lowing V E , the purpose of the present exercise is to f ind , condit ional on the existence of cointegration, a linear combinat ion of the first differences of the variables i n such that the serial correlation p a t t e r n i n the variables is e l iminated. Once the cointegration and short r u n restrictions are identif ied, we can proceed to construct the common trend and common cycle components of the series.

We briefly describe the methodology to identi fy the existence of a common cycle. Let alAyt be any linear combinat ion of first dif ­ferenced variables. A stat ist ica l measure of the correlation between this combinat ion, and the relevant past is T x R2 of the regression of a'Ayt on the variables t h a t represent the history of past observa­tions. T h e n , a n a t u r a l candidate to be a cofeature vector is the l inear combinat ion t h a t minimizes T x R2.

Fol lowing the notat i on i n V E , let Xbe a m a t r i x of transposed first differences and Z the corresponding m a t r i x of error correction terms. Define the past h istory w i t h the m a t r i x W = {X\,..., Xp, Z-i} and let Pw represent the orthogonal pro ject ion to the history space. Hence, the cofeature vector a minimizes the fol lowing expression

, M a'X'PwXa a'X'Xa

V E show t h a t the solution to the previous expression is the Lim­ited Maximum Likelihood, L I M L , estimator of the regression of one of

138 ESTUDIOS ECONÓMICOS

the elements i n Ayt w i t h the rest of them taken as instruments of t h e past history [W). I n add i t i on , the authors show t h a t the L I M L est i ­mator of a is the canonical covariate corresponding to the smallest canonical correlat ion between X and W.

By defining the cofeature vectors as the canonical correlations t h a t are not significant, V E derive a test for the number of l inear ly independent canonical correlations that are stat ist ical ly equal to zero. W h e n there are more t h a n two variables i n the system, the number of not-significant canonical correlations between X and W defines t h e dimension of the cofeature space.

The statist ic for the test of the n u l l hypothesis t h a t the cofeature space is at least s ( i n other words, t h a t there are no more t h a n n — s common cycles) is:

s

c ( p , 5 ) = - ( r - p - i ) ^ i o g ( i - A 22 )

i=l

where A^Vz = 1 , s are the s smallest canonical correlations between X and W. Under the n u l l hypothesis, this statist ic is d i s t r ibuted x1

w i t h s2 + snp + sr — sn degrees of freedom; where n is the dimension of the system, p is the number of lags i n the differences (which is one less t h a n the number i n levels) and r is the number of co integrat ion vectors i n the system.

3. E m p i r i c a l E x e r c i s e

I n this section, we apply the previous methodology to the seasonally adjusted series corresponding to the Mexican GDP, private aggregate consumption, and the credit granted by commercial banks to the non-banking private sector f rom the first quarter of 1982 to the t h i r d quarter of 2001. The source of the data is the system of economic in format ion of Banco de Mexico (SIE-Banxico) .

3.1. Stochastic Properties of the Series

Firs t , we examine the stochastic nature of the series. The logar i thms of the G D P , consumption and credit series are presented i n graphs 1, 2, and 3 respectively.

The series present a trend and hence, i t is l ikely t h a t a u n i t root characterizes their Data Generating Process, DGP. To test for

TRENDS AND CYCLES 139

this possibi l i ty we implement the Augmented Dickey Fuller, A D F , and Phi l l ips -Perron , PP tests. The results are presented i n table 1.

G r a p h 1 GDP, 1982 - 2001

o

G r a p h 2 Consumption, 1982 - 2001

~ï • ' M • ' • : 1 1 > I • • » I ' ' ' ! • 1 " l • • " l • « • ! I - I J « i - Ï « 1 » ; « > » i l * « ; I I < i . I . : . » . i . I i I I . I J < l » ^ I

82 84 86 88 90 92 94 96 98 00

140 E S T U D I O S ECONÓMICOS

T a b l e 1 Unit Root Tests: GDP, Consumption and Credit

Series ADF Critical Value*

PP Critical Value *

G D P -2.73 -3.46 -72.76 -3.46

D G D P -7 .61 -2.90 — —

Consumpt ion -2 .41 -3.46 -57.69 -3.46

D C o n s u m p t i o n -7.00 -2.90 — —

Credit -1.04 -2.90 -0.83 -2.90

D C r e d i t - 3 . 5 0 - 1 . 9 4 -4 .43 - 1 . 9 4 * A t 5%.

The A D F and PP results indicate unambiguously t h a t the credit series is integrated of order 1. I n the case of G D P and consumption,

TRENDS AND CYCLES 141

however, the results are contradict ing . T h e A D F test suggests that the series is integrated of order 1, whi le the PP test indicates an integrat ion order of 0. Nonetheless, re ly ing on the results of the A D F test results and the results of various documents t h a t test for s ta t i onar i ty on consumption and G D P , inc lud ing Casti l lo Diaz (2002) and Garces (2001), we w i l l assume t h a t b o t h series are integrated of order 1.

3.2. Trend-Cycle Decomposition

Since the methodology for analyzing trends and cycles is condit ional on the number of lags chosen, we first determine th is number by evaluating the series in an uncondit ional Vector Aut or egression, VAR, fo l lowing the mul t ivar ia te Akaike Information Criterion, A I C . 3 The results of the test are presented i n table 2, and suggest t h a t the o p t i ­mal number of lags for the variables i n levels is 2. Hence the o p t i m a l number of lags for the variables i n first differences is 1, t h a t is, p = 1.

T a b l e 2 Multivariate Akaike Information Criterion from

an Unrestricted VAR in Levels GDP, Private Consumption and Credit (seasonally adjusted series)

1982:1 - 2001:3

No. of bags AIC

1 -14.00 2 -14.10 3 -13.94 4 -13.93 5 -13.91 6 -13.67

The A I C is defined in this case as

In ( Ù ) + ^ + n ( l + ZO0 (2 T T ) )

142 E S T U D I O S ECONÓMICOS

N e x t , we per form the cointegration and common feature tests. We apply the methodology suggested by Johansen (1991) and the test suggested by V E , respectively. 4 The results, presented in table 3, i n ­dicate that the G D P , consumption and credit series share one cointe-g r a t i n g re lat ion and one cofeature vector at conventional significance levels. Based on Johansen's trace stat ist ic we reject the n u l l h y p o t h ­esis of no cointegration at 5 percent significance level, b u t could not reject the n u l l of one cointegrat ing vector (table 3). T h e evidence found by apply ing the cofeature test developed by V E , constrained to the existence of one cointegrating relationship, shows t h a t the n u l l of no common feature between these variables is rejected at 5 per­cent significance level, bu t we could not reject the presence of one cofeature vector at the same level (five last rows i n table 3 ) .

T a b l e 3 Common Trends and Common Cycles Tests: GDP,

Consumption and Credit, 1982:1 - 2001:3

Cointegration Test* Number of Cointegrating Vectors (r)

N u l l hypothesis r = 0 R<1 r < 2

Trace test 29.85 8.31 0.11

9 5 % Critical value c 29.68 15.41 3.76

Cofeature Test Number of Cofeature Vectors (s)

Nul l hypothesis s > 0 S > 1 s > 2

Squared correlations 0.049 0.133 0.423

4.075 15.660 60.194

Degrees of freedom 2 6 12

p- value 0.130 0.0157 « 0 . 0 0

*Johansen's (1991) trace statistic test, ^ Taken from the tables contained in Osterwald-Lenum (1992), f Statistic represented in equation (1) w i t h V — 1.

The coefficients estimated for the cointegrating and the cofeature vectors are presented i n table 4. Given that they are m a x i m u m l ike ­l ihood estimates, the normal izat ion procedure does not involve any d i s tor t i on i n the presentation of results. For the cointegrating re lat ion

We thank Joao Victor Issler for providing us w i t h the Gauss code for per­forming these estimations.

T R E N D S A N D C Y C L E S 143

we normalize on consumption. We find t h a t the long r u n elast ic i ty of consumption w i t h respect to G D P is 0.52, and w i t h respect to the credit variable is 0.14.

The cofeature vector (i.e. the l inear combinat ion of the variables i n the system which eliminates the serial correlation p a t t e r n present i n the first differences of these series) normalized on the credit v a r i ­able indicates t h a t its short r u n f luctuations are a combinat ion of 0.74 times the one per iod changes i n G D P and 0.65 times those of consumption. T h i s is only a representation of the short r u n dynamics of the system normalized on one of its variables (credit) .

T a b l e 4 Co integration and Cofeature Vectors: GDP, Private

Consumption and Credit, 1982:1 - 2001:3

GDP Consumption Credit Cointegrating Vector -0.52 1.00 -0.14 Cofeature Vector -0.74 -0.65 1.00

Trends

GDP Consumption Credit

G D P 1.00 -0.93 0.50 Consumption 1.00 -0.93 0.50 Credit 2.61 0.36 1.00

V E show t h a t , when the number of cointegration and cofeature relations are equal to the number of variables i n the system (r-hs = n ) , i t is possible to represent the decomposition of yt i n the Beveridge-Nelson form, t h a t is, as the sum of a permanent and a t rans i tory component (Vahid and Engle, 1993; and Issler and Vahid , 2001) . 5 I n the current analysis, however, there exists one cointegrating vector (r = 1) and one cofeature vector (s = 1) (presented in table 4). Since they are orthogonal , we cannot construct a base to project the

I n fact, Issler and Vahid (2001) show that when r + S — n the trends and cycles of a series can be recovered as a linear combination of the cointegrating and cofeature vectors. For our own purposes, i t would had been quite interesting to have found a group of series such that r -\- S — n. I n that case, we could have also used the Issler and Vahid decomposition technique. This is an exercise that we leave for future research.

144 E S T U D I O S ECONÓMICOS

space t h a t spans the variables i n the system (n — 3). Thus, we cannot per form the decomposition as suggested by V E . Nonetheless, i n the case when r + s < n there exist alternative methodologies to per form the trend-cycle decomposition, inc luding K i n g et al (1991) or Gonzalo and N g (2001).

We choose to implement the methodology suggested by K i n g et al I n part i cu lar , we consider the long r u n restrictions imposed by t h e co integrat ing vector and estimate a V E C M from where we obta in t h e permanent and t rans i tory components of the series. 6 Fol lowing t h i s methodology, we f ind two common trends. I n other words, we f ind t h a t two random walks guide the stochastic common trends for t h e variables considered i n the system. 7

Graphs 4-6 show the stochastic trends t h a t best fit each of t h e analyzed series. I n part i cu lar , i t is evident that one of the trends closely follows the G D P and consumption series, whereas the second t r e n d better fits the credit series. Note that the common trend series we obtained is not smooth. A t first, this result may appear somewhat puzzl ing. A f ter a l l , one wou ld expect t h a t a trended series would no t present pronounced peaks and valleys. However, we believe t h a t p a r t of the gains i n est imat ing the trended series using the present m e t h o d ­ology is that the series obtained more accurately represent the mag­n i tude of t rans i tory shocks, which, since we use seasonally adjusted series, are more l ikely to be the factors t h a t cause the f luctuations i n the estimated series.

T h e actual level of G D P for the second part of the period ex­amined (1999:01 - 2000:03) is above the estimated trend, and i t is below for the rest of the period of analysis (graph 4). This last result captures well the recession observed i n Mexico and worldwide d u r i n g 2001.

As is well known, the V E C M has two representations: autoregressive and moving average (Engle and Granger, 1987). By imposing the restrictions pre­sented i n the upper part of table 4 i n the autoregressive form of the V E C M , one can recover the moving average representation of the system including those restrictions. Af ter an algebraic manipulat ion of the moving average form i t is possible to find the trend-cycle decomposition of the variables included in the V E C M . For details we refer the interested reader to the original paper ( K i n g et al. 1991).

7 Since the cointegration relationship eliminates one stochastic trend from the system.

T R E N D S A N D C Y C L E S

G r a p h 4 GDP and its Trend

14.4

" M I | I M | . l l | l l l | M I . I I ' | l l l | M I | N I | . l l ; l l l | l l l | l l ' | l . l | . l l | l , l | . l i : i . l | l l l |

8 2 8 4 8 6 8 8 9 0 9 2 9 4 9 6 9 8 0 0

G D P C O M M O N T R E N D 1

G r a p h 5 Consumption and its Trend

1 4 . 0

| ' I I | I I I | I I I . I I I | I I I | I I I | I I I ; I I ' | I I I | I I ' | I I I | I I I | I . I | I I I | I , I | . I I | I , I | . H | I I I |

82 8.4 86 88 90 92 94 96 98 00

' — - Consumption Common Trend 1

146 E S T U D I O S ECONÓMICOS

Similarly , the observed level of consumption for the p e r i o d 1995: 03 - 2000:03, w i t h the exception of four quarters, was also located above the first common trend (graph 5). Beginning i n the f o u r t h quarter of 2000, however, i t has remained below the c o m m o n t r e n d .

G r a p h 6 Credit and its Trend

8 2 8 4 8 6 81

Credit

| i i i | M i ; i i ' | i : i | : i i | i i i | i i i | i i i | i :

9 0 9 2 9 4 9 6 9 8 0 0

— Common Trend 2

I n the case of the credit series, the observed level remained above the t rend f rom the first quarter of 1989 to the second quarter of 1995, a fact that is consistent w i t h the "credit boom" experienced d u r i n g this period (graph 6). From the t h i r d quarter of 1995 to the second quarter of 2000, the level remained below the t rend consistently. O n l y i n recent periods does there appear to be a rebound i n the level of the credit series.

3.3. Methodology Comparison

I n graphs 7-9 we compare the cycles obtained using the K i n g et al. methodology w i t h those obtained using the commonly used Hodr i ck -Prescott, H P filter.8 Note that the magnitude of the cycles obtained

8 First , we filtered the seasonally adjusted series and then obtain the logarith­mic difference between the original and the filtered series.

T R E N D S A N D C Y C L E S 147

w i t h the methodology used i n this document is greater t h a n t h a t of the cycles obtained w i t h the H P f i l t e r . 9 I n add i t i on , we f ind t h a t the expansion and recession 1 0 dates suggested by the two cyclical series do not precisely coincide.

W i t h the H P methodology, we find t h a t G D P was above its t r e n d between the t h i r d quarter of 1999 and the f o u r t h quarter of 2000 ("posit ive" cycle), whi le f rom the first quarter of 2001 to the end of the sample period, we find t h a t i t was located below its t rend (graph 7). O n the other hand, the t r e n d we obtained applying the K i n g et al. methodology shows t h a t the previously mentioned first cycle on the G D P series, appears before i t does when using the H P filter.

G r a p h 7 Cycle in GDP: Common Cycle 1 vs. HP Filter

1111111111111111 II 1111111111111 u1111111

8 2 8 4 8 6 8 8 9 0 9 2 94 9 6 9 8 0 0

Common Cycle 1 HP Cycle

9 I t is not surprising to find this result, since the trended series we estimated presents pronounced peaks and valleys, in contrast w i t h the trended series es­t imated w i t h the H P methodology. As we argued in the text , we believe that the common trend series is so volatile is because i t estimates the magnitude of transitory shocks more accurately, and not because of seasonal factors, since we are using seasonally adjusted series.

1 0 The term "expansion" ("recession") refers to the case when the level of the observed variable is located above (below) the level of the trended series

148 E S T U D I O S E C O N Ó M I C O S

I n the case of the consumption series, the "expansion" periods found w i t h the HP f i l ter appear at a later date compared w i t h those identif ied w i t h the methodology we employed (graph 8). T h e H P f i l ter shows that the last expansion period began by the end of 1999 and finished i n the t h i r d quarter of 2001. I n contrast, our results show a sustained expansion ( w i t h two or three outliers) f rom the second hal f of 1995 u n t i l the t h i r d quarter of 2000.

I n the case of credit (graph 9), we also obtained contrast ing ev­idence for the "recession" and "expansion" dates and the magni tude of the cycles when using b o t h methodologies. For the credit expan­sion per iod (1988-1994), the HP methodology finds that the level is below the trend (negative gap), a result t h a t is clearly c o u n t e r i n t u ­i t ive . So i n this case the advantage of inc lud ing s t ruc tura l restrict ions instead of using numerical methods to estimate the business cycles in macroeconomic variables is clear.

G r a p h 8 Cycle in Consumption: Common Cycle 1

vs. HP Filter

.04 h

A ! ! i

! a |

! ,V -< ¡

8 2 8 4 8 6 8 8 9 0 9 2 9 4 9 6 9 8 0 0

Common Cycle 1 HP Cycle

Once we have shown the significant influence of t rans i tory shocks on the t ime series we consider, the relevant issue becomes ident i f y ing the factors t h a t cause the shocks. Evidently , this is not a t r i v i a l task,

TRENDS AND CYCLES 149

since i t is di f f icult for anyone to properly identi fy all the variables t h a t may influence the behavior of any t ime series. As an exploratory exer­cise, in the next section we evaluate the importance of some nominal variables i n expla in ing the var iat ion of G D P , consumption and credit . I n part icular , we estimate several b ivariate s t ruc tura l vector autore gressions t h a t include a nomina l variable and one of the real variables previously m e n t i o n e d . 1 1

G r a p h 9 Cycle in Credit: Common Cycle 2

vs. HP Filter

.5

82 84 86 88 90 92 94 96 98 00

Common Cycle2 HP Cycle

4. N o m i n a l S h o c k s

T h e relative importance of nominal shocks can be evaluated imple ­ment ing a V A R . As i t is well known, to solve a V A R i t is necessary to impose ident i fy ing restrictions.

I t would also be possible to estimate a mult ivariate V A R in order to deter­mine the importance of the transitory shocks. However, we are pr imar i ly inter­ested in showing how each indiv idual nominal factor affects each ind iv idua l real variable.

150 E S T U D I O S ECONÓMICOS

There exist various procedures for identi f icat ion. B lanchard and D i a m o n d (1989, 1990), for instance, used a priori assumptions of s t r u c t u r a l parameters. The commonly used Cholesky factor izat ion can also serve as an ident i fy ing methodology. I n this document , nonetheless, we follow the ident i f i cat ion procedure suggested by B l a n ­chard and Quah (1989). This methodology has been extensively used i n studies t h a t analyze nominal and real shocks, inc lud ing t h a t of exchange rates between the US and some of its major t r a d i n g p a r t -

12 ners.

T h e Blanchard and Quah ident i f i cat ion procedure requires a long - r u n restr i c t ion of the nominal shock. T h a t is, assuming l ong - run neutra l i ty , the effect of the nominal shock on the real variable is con­strained to be zero i n the long r u n . Th is condit ion is imposed by res tr i c t ing the mov ing average representation of the V A R . 1 3

T h e long-run neutra l i ty restr ic t ion is imposed in a V A R consid­ering the real and nominal variables and a s t ruc tura l VAR, ( S V A R ) , is estimated. The real variables we consider are those examined i n the previous section. The nominal variables are interest rates of treasury bills at 28 days, CETES28, the Consumer Price Index, I N P C , and the money aggregate M l . For the interest-rate variable and the I N P C , we obtained quarter ly data by averaging m o n t h l y observations. I n the case of M l we used the end of the per iod observation. The results of the SVAR's are presented i n tables 5-7.

T a b l e 5 Variance Decomposition of GDP

Period Cetes28 INPC Ml

1 6.1 5.4 7.4

2 8.3 9.5 8.5

3 9.4 9.0 8.3

4 9.6 9.0 8.4

5 9.6 9.1 8.4

6 9.7 9.1 8.4

7 9.7 9.1 8.4

Including Enders and Lee (1997), Clarida and Gali (1994) and Lastrapes (1992).

1 3 For a detailed description of the methodology see Lastrapes (1992).

TRENDS AND CYCLES 151

T a b l e 5 (continued)

Period Cetes28 INPC Ml

8 9.7 9.1 8.4 9 9.7 9.1 8.4 10 9.7 9.1 8.4

Table 5 reports the variance decomposition of GDP. The change i n the interest rate explains nearly 10 percent of the variance i n the g r o w t h rate of G D P after 10 periods, the largest p o r t i o n explained by any of the three nominal variables considered.

Table 6 presents the variance decomposition for consumption. A s i n the case of G D P , the interest rate explains a significant p o r t i o n of the var ia t i on i n the dependent variable.

T a b l e 6 Variance Decomposition of Consumption

Period Cetes28 INPC Ml

1 7.8 2.8 2.3 2 9.0 5.6 2.2 3 11.6 5.5 2.2 4 11.7 5.5 2.2 5 11.8 5.6 2.2 6 11.8 5.6 2.2 7 11.8 5.6 2.2 8 11.8 5.6 2.2 9 11.8 5.6 2.2 10 11.8 5.6 2.2

The evidence found for the credit variable, table 7, is qua l i ta ­t ive ly the same as i n the previous cases. However, as could be ex­pected, the p o r t i o n of the var iat ion i n the credit variable explained by the interest rate is much higher t h a n for G D P and consumption, up to 19 per cent after 10 periods.

152 E S T U D I O S E C O N Ó M I C O S

T a b l e 7 Variance Decomposition of Credit

Period Cetes28 INPC Ml 1 0.0 8.8 4.6

2 15.1 8.6 6.7

3 18.8 7.3 7.0

4 19.2 7.1 7.1

5 18.9 7.0 7.1

6 19.0 6.9 7.1

7 19.2 6.9 7.0

8 19.2 6.9 7.0

9 19.2 6.8 7.0

10 19.2 6.8 7.0

T h e results of the variance decompositions previously est imated show t h a t the nomina l variables explain a significant p o r t i o n of t h e var ia t i on in the real variables. T h e nomina l interest rate is the v a r i ­able t h a t provides the highest coefficients in all cases. I n the case of G D P and consumption, M l is the variable t h a t explains the least. I n comparison w i t h studies on the US, the propor t i on of the var ia t i on t h a t the nominal variables explain i n the cases presented above is relatively high, Enders and Lee (1997), for example, f ind t h a t M l ex­plains up to 4 percent of the var ia t i on of the G D P i n the US, while we f ind t h a t for the case of Mexican G D P , i t explains around 8 percent. These results are clearly consistent w i t h those found i n the t r e n d -cycle decomposition analysis conducted i n the previous section, t h a t is, nomina l shocks appear to explain a significant p o r t i o n i n the v a r i ­at ion of real variables.

5. C o n c l u s i o n

T h e evidence found i n this paper sheds some l ight on the behav­ior of three Mexican macroeconomic series: G D P , aggregate pr ivate consumption, and bank credit granted to the private sector. T h e analysis shows t h a t the data generating process guiding these v a r i ­ables consists of two common trends (one cointegrating relation) and

TRENDS A N D CYCLES 153

two common cycles (one cofeature vector) . These characteristics a l ­lows for the impos i t i on of restrictions on the permanent - t rans i tory component decomposit ion of the series. As shown i n V a h i d a n d E n -gle (1993), and Issler and V a h i d (2001), imposing such restrict ions leads to a more efficient ident i f i cat ion of the common trends and cy­cles. This efficiency is ma in ly reflected i n the estimated effects of the t rans i tory shocks on the variables in the system. I n our par t i cu lar exercise, we f ind t h a t the restricted decomposition provides in forma­t i o n that improves the ident i f i cat ion of cyclical periods. I n add i t i on , we f ind t h a t the influence of nomina l shocks on the previously men­t ioned series is i m p o r t a n t , and greater relative to the case w h e n no restrictions are imposed on the decomposition of the series.

A l t h o u g h i n th is paper we focus main ly on presenting the t r e n d -cycle decomposit ion methodology using short and long-run restr ic ­tions, i t is evident t h a t this methodology can be applied to more pract ical enterprises. For example, given the efficiency gains obtained when using the present methodology, i t would be desirable for fore­casting purposes. T h a t is, i f one conducts an exercise to ident i fy series t h a t share common cycles, and the series are characterized by having one of t h e m lead the other, then one could provide an efficient forecast of the lagged variable using the in format ion provided b y the leading variable. Also, the methodology can be employed i n exer­cises t h a t require the der ivat ion of gaps between variables and their trends, such as in f la t i on models. Tradi t ional ly , t h e der ivat ion of out ­p u t gaps i n in f la t i on models follows less efficient methods inc lud ing filters and estimations of VAR's or VEC's . As shown i n this art ic le , however, one can ob ta in more accurate estimates of trends when using the methodology presented i n this paper. These two applications, of course, are jus t a few of the possible scenarios where the trend-cycle decomposit ion t h a t imposes short- and long-run restrictions can be applied.

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154 E S T U D I O S ECONÓMICOS

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