Dealing with an Error Correction Model When Trade Balances Are Trend-Stationary

Dealing with an error correction model when trade balances are trend-stationary

2014 / 04

Dealing with an error correction model when trade balances are trend-stationary

Manuel Cantavella-Jordá Universitat Jaume I Department of Economics & IEI [email protected]

2014 / 04

Abstract

The present research shows how one can deal with stationary plus trend trade balance variables in a trade model where the rest of the variables contain a unit root. Data are used in a monthly and a quarterly basis from January 1980 to June 2011 and applied to four countries (Germany, France, Italy and United Kingdom). It is proved that an error correction mechanism suits better in terms of both econometrics and economics when detrending trade balances once they have been verified to have a deterministic trend.

Keywords: Trade model, difference stationary process, trend stationary process, error correction mechanism

JEL classification: C32, F10, F31

Dealing with an error correction model when trade balances are trend-stationary

Manuel Cantavella-Jordá

Departamento de Economia and Instituto de Economia Internacional

Universitat Jaume I

12071 Castellon, Spain

Abstract

JEL Classification: C32, F10, F31

Keywords: Trade model, difference stationary process, trend stationary process, error correction mechanism

1.Introduction

It is commonly accepted that most macroeconomic time series have a stochastic trend, in other words, they contain a unit root. Short sample periods and type of data frequency may affect the hypotheses testing, especially when low power unit root tests are employed as Enders states (2010). However, there are cases where the order of integration is not that clear since the unit root is in the margin of being rejected. Including trend-stationary variables in a cointegration system may become a fuss. Thus, there is a prevailing tendency to finally work with I(1) variables in the model. This research tries to explore an appropriate strategy taking into consideration a period of 31 years from 1980 to 2011, in a monthly (366 observations) and also in a quarterly (122 observations) basis along with three unit root tests, Dickey and Fuller (1979, 1981), Phillips and Perron (1988) and Durbin-Hausman (Choi 1992). The aim is the estimation of a suitable error correction model that can be valid for economic policy purposes once the kind of trends (deterministic or stochastic) involved in each variable has been determined. This is illustrated with the application of a classical trade model to four European Union countries such as Germany, France, Italy and UK.

2.Model and data

The trade model of this analysis is a variant of the imperfect substitutes model as Goldstein and Khan (1985) named. It essentially concentrates on a reduced-form equation where the trade balance ( TB ) depends on real effective exchange rates ( XR ), domestic income ( Y) and foreign income ( Y* ). In econometric terms: ∗ =∝+∝ +∝ +∝ + (1) t = 1980M 1………2011M 6 or t = 1980Q1…….2011Q2, where M stands for month, Q represents quarter and is the error term that captures omitted factors.

This approach allows us to directly examine the impact of exchange rates and different incomes on trade balances. The latter are expressed as value of exports over value of imports ratio. Domestic and foreign income are proxied by an industrial production index when dealing with monthly data and by gross domestic product when dealing with quarterly data. All the variables are real (2005 base year) and transformed in natural logarithms, so, estimates can be interpreted as elasticities. All variables are collected from International Financial Statistics (InternationaL Monetary Fund e- Library).

3.Methodology and results

Knowing that non-stationary variables in levels can turn out in spurious regressions, then, it is important that we first test not only for the order of integration but even more important for the type of trends involved (whether stochastic or deterministic) before a final error correction mechanism can be modeled. Table 1 reports the unit root results (monthly and quarterly data) of Dickey-Fuller (DF), Phillips-Perron (PP) and also Durbin-Hausman (DH) tests whose Monte Carlo simulation results in Choi (1992) turned out to have more power in testing the presence of either a stochastic or a deterministic trend. In Table 1 (Panel A, monthly data) ADF and PP tests indicate that trade balances for France and Italy are I(0)+trend while DH indicates that all trade balances in monthly data are I(0) + trend. In principle we give some confidence to the test that has more power (DH). Moreover, if this is the right decision we will check it as to the suitability of the final error correction model once the trend (stochastic or deterministic) has been eliminated by trying both methods such as differencing and detrending. The rest of the variables are I(1). When using quarterly data all variables contain a unit root even for trade balances. It is likely that the amount of variability according to the data frequency might have affected the hypotheses testing.

Table 1

Panel A: Tests for Unit Roots, Stochastic and Deterministic Trends, monthly data

Variables ADF stat ADF stat ADF stat PP stat PPstat PPstat DH stat I(1)vs I(2)vs DSPvsTSP I(1)vs I(2)vs I(1) DSP vs DSP I(0) I(1) I(0) TSP vsTSP

GETB -1.65 (4) -5.87(6) -2.45(4) -1.87(3) -6.34(3) -2.83(3) 72.50 GEXR -0.39(2) -10.88(1) -2.02(2) -0.59(4) -9.81(2) -1.75(2) 11.09 GEY -1.55(1) -22.96(1) -1.99(1) -1.78(2) -24.20(2) -1.94(3) 38.77 FRTB -2.66(6) -7.86(5) -3.48(6) -2.52(7) -7.58(5) -3.72(4) 207.25 FRXR -2.13(2) -9.61(1) -1.95(2) -2.46(4) -8.43(3) -1.64(2) 22.57 FR Y -1.98(1) -23.50(1) -2.47(1) -1.83(2) -18.25(2) -2.10(1) 48.55 ITTB -2.75(5) -10.77(3) -3.59(4) -2.64(5) -11.22(4) -3.62(5) 184.12 ITXR -0.46(1) -8.65(1) -3.14(4) -1.22(2) -9.58(1) -1.56(4) 6.53 ITY -2.27(1) -16.21(1) -2.19(3) -2.09(1) -15.85(2) -2.32(2) 23.47 UKTB -2.61(5) -6.77(3) -2.43(6) -2.34(4) -4.82(3) -2.77(3) 249.16 UKXR -2.06(3) -5.75(2) -1.85(3) -2.25(2) -5.69(2) -1.38(1) 14.69 UKY -0.84(2) -7.21(1) -1.98(2) -0.56(1) -8.34(3) -2.15(2) 42.23 Y* -1.04(3) -10.44(2) -1.24(3) -1.54(3) -9.36(2) -1.57(3) 26.83 Critical values(5%):-2.89 -2.89 -3.46 -2.89 -2.89 -3.46 47.37

Panel B: Tests for Unit Roots, Stochastic and Deterministic Trends, quarterly data

Variables ADF stat ADF stat ADF stat PP stat PPstat PPstat DH stat I(1)vs I(2) vs DSP vs I(1) vs I(2) vs DSP vs DSP I(0) I(1) TSP I(0) I(1) TSP vsTSP

GETB -1.34 (1) -10.27(0) -1.18(1) -1.64(1) -11.24(1) -1.57(0) 14.22 GEXR -0.07(0) -7.52 (0) -1.84(0) -0.25(1) -6.32(0) -1.78(1) 892 GEY -0.78(4) -3.47(3) -1.98(4) -1.23(0) -3.81(3) -2.07(3) 36.58 FRTB -2.15(4) -12.65(0) -2.78(4) -2.38(3) -10.54(3) -2.35(4) 34.80 FRXR -2.06(0) -8.90(0) -3.25(1) -1.84(2) -9.65(2) -2.74(1) 29.58 FR Y -2.14(0) -6.21(0) -1.14(0 -2.05(1) -6.35(1) -1.42(1) 8.11 ITTB -2.80(3) -9.37(2) -3.42(3) -2.67(4) -8.26(3) -2.53(3) 45.42 ITXR -2.55(0) -8.42(0) -2.81(0) -2.32(3) -9.42(1) -2.26(2) 6.15 ITY -2.18(0) -5.84(0) -1.46(1) -1.96(2) -4.87(1) -1.19(3) 10.42 UKTB -1.46(1) -8.59(0) -3.25(3) -1.79(2) -7.59(0) -3.34(2) 48.26 UKXR -0.87(0) -6.4(0) -3.14(0) -0.83(0) -6.88(1) -2.93(2) 8.14 UKY -2.27(0) -12.21(0) -2.19(0) -2.07(1) -11.43(0) -1.92(1) 16.08 Y* -1.43(1) -5.64(0) -1.24(1) -1.55(1) -5.93(1) -0.95(1) 6.23 Critical values (5%):-2.89 -2.89 -3.46 -2.89 -2.89 -3.46 55.35

Notes: aGETB stands for Germany trade balance,GEXR for Germany real effective exchange rate,GEY for Germany income, analogously for the rest of the countries;Y* stands for foreign income. b In parentheses appear the augmented lags. c DSP is difference stationary process and TSP trend stationary process. dMicrofit 4.0, E-Views 5.0 and TSP (Time Series Processor) were the programs used.

Thus, the modeling strategy that we apply is the following: quarterly data are used in a cointegration framework whereas monthly data are used in the error correction representation. The reason is that cointegration provides appropriate tools to work with non-stationary variables and particularly with I(1) variables. Monthly data conveys more information about the dynamics of the model. But at this stage, how different data frequency can relate to each other? The answer to that question is that imposing the long-run equilibrium into a vector autoregression (VAR) model should be independent of the applied data frequency (monthly, quarter, annual) since a cointegrating vector expresses a long-run relationship. The general VAR based on Johansen methodology (1988) is:

= + ∑ + ∑ + (2)

Where either and include thelogarithms of the four trade variables (TB , XR , Y, Y* ); is the intercept; is the deterministic seasonal (quarterly) dummies and is a disturbance term independently and identically distributed with zero mean and constant variance. The results for cointegration tests are reported in Table 2. The number of lags for each equation has been included following the Akaike (1974) information criterion. Two cointegrating vector exists for each country except for UK. From the econometric point of view, the first one has a stronger long-run relationship. From the economic point of view according to signs and magnitude of elasticities it seems that Vector 1 is more plausible than Vector 2. Taking into account both criteria it is the first vector which is finally included in the error correction model.

Table 2

Cointegration results

Null Germany France Italy UK hypothesis λMax λMax Trace λMax Trace λMax Trace Trace r=0 35.65* 78.54* 46.25* 83.74* 41.23* 74.56* 48.13* 84.56* r≤1 28.34* 42.57* 25.77* 42.85* 25.00* 33.03* 24.63* 35.76* r≤2 9.24 14.45 11.24 16.06 7.26 7.04 7.87 11.17 r≤3 2.01 2.01 3.91 3.91 0.75 0.75 3.26 3.26 Note: * indicates significance at 5 percent level.

Parameter estimates (normalised)

Germany France Italy UK Vector 1 Vector 2 Vector 1 Vector 2 Vector 1 Vector 2 Vector TB -1 -1 -1 -1 -1 -1 -1 XR (-) -0.09 2.18 -1.37 -5.84 -0.81 1.65 -0.21 Y (-) -1.83 -3.25 -1.42 -10.47 -3.16 -8.43 -1.54 Y*(+) 1.54 4.82 1.34 8.56 3.12 5.92 1.05 Note: Expected signs are in brackets.

The final step, then, is the representation of the error correction model for trade balances by incorporating the long-run information derived from cointegrating relationships (Table 2) and the short-run adjustment. As said before, monthly data are used in order to provide a better framework of dynamic responses. Knowing that the four trade balances are finally I (0) plus trend the suitable approach is to take the stationary transform of the trade balance in equation (3):

= + + (3)

The residuals of that regression ( TB* ) should be stationary as the rest of the terms in equation (4) so that one can apply standard OLS and interpret t-ratios:

∗ ∗ ∗ = + ∑ + ∑ ∆ + ∑ ∆ + ∑ ∆ + ∑ + + (4)

Only significant variables at 5 and 10 percent significance level are reported in Table 3. Seasonal dummies ( M) are included but not shown. AIC is applied for the number of lags. As it is seen, when trade balances are treated as difference stationary process, they display significant negative autocorrelation for each of the lagged dependent (trade balance) variables used as regressors. Serial correlation diagnostic tests detect that problem. Whenever trade balances are treated as trend stationary process the model validates all the diagnostic tests. The error correction term (ECT) is also significant. Any long-run disequilibrium is adjusted by about a 10 percent every six months. The rest of variables behave as expected in sign and amount.

Table 3

Error correction models for trade balances

Germany France Italy UK Germany France Italy (Trade balances treated as DSP) UK (Trade balances treated as TSP) C 0.14 0.47 0.04 0.51 0.11 0.33 0.10 0.51 (2.60) (2.78) (0.46) (2.24) (2.41) (2.15) (1.41) (2.25) TB -1 -0.58 -0.84 -0.84 -0.58 0.38 0.14 0.16 0.42 (-7.98) (-11.76) (-11.91) (-8.17) (5.70) (2.09) (2.53) (6.63) TB -2 -0.24 -0.56 -0.51 -0.38 0.30 0.26 0.32 0.19 (-2.96) (-6.10) (-5.64) (-4.50) (4.52) (3.69) (5.41) (2.82) TB -3 -0.36 -0.32 -0.37 0.17 0.21 (-4.44) (-3.32) (-4.08) (2.39) (3.01) TB -4 -0.33 -0.17 -0.22 -0.16 (-4.19) (-2.14) (-2.99) (-2.04) TB -5 0.15 0.26 (2.35) (4.17) TB -6 0.13 0.13 0.14 (1.94) (2.00) (2.17) ∆XR -1 0.17 0.36 0.21 0.48 0.39 (1.76) (1.67) (1.82) (1.95) (2.22) ∆Y-1 -0.41 (-1.98) ∆Y-2 -0.61 (-2.31) ∆Y-4 -0.42 -0.27 -0.32 (-2.27) (-1.73) (2.10) ∆Y-5 -0.41 (-1.89) ∆Y-6 -0.41 -0.56 (-2.53) (-2.50) ∆Y* -1 0.53 (1.94) ∆Y* -2 -0.74 (-1.67) ∆Y* -3 0.91 0.81 0.18 0.23 0.27 (3.03) (1.83) (3.39) (1.85) (2.04) ∆Y* -4 0.87 0.64 (3.22) (3.20) ∆Y* -6 0.44 (1.87) ECT -5 -0.073 -0.089 -0.048 -0.084 (-2.01) (-1.98) (-1.65) (-1.88) ECT -6 -0.081 -0.072 -0.064 -0.095 (-3.85) (-3.77) (-3.43) (-3.68)

R2 0.51 0.63 0.60 0.66 0.83 0.67 0.74 0.63 F 7.04 11.5 10.19 13.15 47.98 11.42 28.64 14.66 s.e. 0.037 0.041 0.060 0.077 0.036 0.039 0.054 0.074

s.c. (0.053) (0.038) (0.027) (0.048) (0.270) (0.272) (0.082) (0.118) f.f. (0.340) (0.285) (0.148) (0.343) (0.590) (0.302) (0.176) (0.543) n. (0.064) (0.580) (0.126) 0.254) (0.086) (0.907) (0.237) (0.470) h. (0.306) (0.420) (0.342) (0.090) (0.706) (0.825) (0.779) (0.102) Notes: a TB is in differences when treated as DSP. b A sub-index represents the corresponding variable in lags. c R2 is the coefficient of determination. d F is the joint test. e s.e. is the standard error of the regression. fs.c. stands for serial correlation. g f.f. means functional form. h n. denotes normality. i h. expresses heteroskedasticity. j The numbers between parentheses are the percentages at which the null hypothesis is rejected.

4.Conclusions

It has been shown how to manage a time series strategy according to the nature of variables. This is illustrated with the application of a classical trade model to four EU countries (Germany, France, Italy and UK). When dealing with monthly data trade balance variables are proved to be trend-stationary meanwhile exchange rates, domestic and foreign income are difference-stationary. Final ECMs are tested in two different ways, first where all variables are treated as DSP and secondly detrending the variables (trade balance) that have a deterministic trend. In the first model negative autocorrelation dominates the dynamics confirming the fact that trade balances have a deterministic trend. In the second one a better dynamics structure is exhibited which means that the trend has been properly modeled. Shocks on trade balances are temporary and adjust in the long term at a reasonable speed.

References

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Choi, I., 1992. Durbin-Hausman test for a unit root. Oxford Bulletin of Economics and Statistics 53, 289-304.

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Enders, W., 2010. Applied Econometric Time Series 3e. Wiley, 181-271. Goldstein, M., Khan, M.S., 1985. Income and price effects in foreign trade, in: Jones,R. W. and Kenen,P.B. (Eds.), Handbook of International Economics, Amsterdam, North Holland, 1041-1105.

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Appendix

Durbin-Hausman Tests

Choi (1992) proposes Durbin-Hausman (1954, 1978) tests for unit root based on the traditional parameterization from which Dickey and Fuller (1979, 1981) derive their own tests. For this purpose, the OLS estimator and an instrumental variable are used. Unlike ARMA models which usually work with lagged variables as instruments Choi

(1992) employs yt to instrument yt-1. Thus, yt is not a real instrument but a pseudo- instrument.

The maintained model is the same as for DF tests,

Yt=α+βyt-1+ϒt+ ϵt (A.1)

The null hypothesis is that of β=1 against the alternative of β<1 , that is I(1) against I(0)+trend. The test statistic is,

2 DH=(b iv -b) /est V(b) where b denotes the OLS estimate of β and biv indicates the pseudo-instrumental variables estimate of β using ytto instrument yt-1.