Output growth

OUTPUT GROWTH AND MONETARY POLICY INTERACTION IN A COMMON MONETARY AREA: FORECASTING WITH VEC MODELS IN NAMIBIA, LESOTHO, SOUTH AFRICA AND SWAZILAND. 1981-2004 KHAMFULA, Yohane^*

Abstract

In this study, we empirically attempt to investigate output growth forecasts as a result of dynamic interplay between money supplies and output growths of Southern African Common Monetary Area (SACMA) countries using Vector Error Correction Models (VECM). In general, the results show that the forecasts for output growths in SACMA countries are quite similar to the actual values. Generally, the money supply changes have had a positive effect on output growths in all these SACMA countries. This may suggest that symmetric monetary shocks dominate in SACMA economies, which is a good indicator for an optimal currency area that would foster development in the region.

JEL classification: C13; E31; E61; F32

Key Words: Common monetary area; Multivariate dynamic forecasts; Output growth;

Money supply; Cointegration; Vector Error correction model.

1. Introduction

In this study, we empirically attempt to investigate output growth forecasts as a result of dynamic interplay between money supplies and output growths of Southern African Common Monetary Area (SACMA) countries using Vector Error Correction Models (VECM). The formation of SACMA is an advanced stage of economic integration.

Economic integration is defined as an attempt to link together the economies of two or more countries, with some geographical proximity, through the removal of economic barriers, aimed at raising the living standards of the participating countries (Mulinde 2001). Monetary policy provides one of the tools through which sovereign a state can direct its economic path. The harmonization of monetary policies among geographically related members, through an arrangement like SACMA, could be expected to reduce price fluctuations and uncertainty. The promotion of greater price stability leads to long- term confidence, increased foreign investment, trade and consequently economic growth (Sparks 2002).

SACMA, originally known as the Rand Monetary Area (RMA), has been in existence since 1974. It is governed by the Multilateral Monetary Agreement of 1974, which was amended in 1986. SACMA comprises four countries: South Africa, Namibia, Swaziland and Lesotho. Botswana opted out of this monetary arrangement to pursue its own independent monetary and exchange rate polices. All member countries of SACMA maintain parity of their currencies with the Rand and have free access to South African capital and money markets. The countries share a common pool of foreign exchange

* Yohane Khamfula, Department of Economics, University of Stellenbosch, South Africa, e- mail:[email protected]

reserves, apply a common exchange control policy to the outside world and maintain current accounts with the South African Reserve Bank. SACMA is not a fully-fledged monetary union in that member countries still have their separate currencies. Though the rand is the dominant and generally acceptable currency in all SACMA countries, member countries have no irrevocable commitment to keep the currency at parity with the rand.

The paper’s objective is to check the extent to which money supplies within SACMA play an important role in forecasting output growth in individual member countries. In general, the forecasting results show that the forecasts for output growths in SACMA countries are quite similar to the actual values. The rest of the paper is organized as follows. In section two, we give an overview of SACMA economies. Section three presents the econometric methodology and data employed in this study. Section four presents and analyses the empirical results obtained. In section five, we conclude and draw some policy implications of the findings.

2. Overview of SACMA Economies

South Africa: South Africa is the dominant economy within the arrangement. It accounts for more than 80% of GDP of SACMA. The South African Reserve Bank is instrumental in monetary policy formulations, which affect all other SACMA countries. Despite the economic dominance of South Africa in SACMA, its major trading partners are the United Kingdoms, the European Union and the United States of America, which are not members of SACMA. Other member countries of SACMA have not been able to increase their trade with South Africa despite the favorable trade arrangements within the SACMA. This scenario brings into question the link between trade and monetary integration.

Namibia: Namibia was automatically a member of SACMA before independence, by virtue of its administration by South Africa. The new post-independence government decided to remain in SACMA and to maintain use of the South African rand until the Namibia dollar was introduced in September 1993. Unlike Botswana, which decided to leave SACMA arrangements in 1976, Namibia has, so far, chosen not to pursue its own exchange rate polic y. The Namibia dollar, like the lilangeni (Swaziland) and the loti (Lesotho), has remained at par with the rand and there is no immediate prospect of change. The rand has remained a legal tender for a transitional period in Namibia, but the governor of the Bank of Namibia (the central bank) has said the use of South African banknotes and coins will be phased out gradually. Foreign exchange regulations and monetary policy in Namibia continue to reflect the influence of the South African Reserve Bank. During the first five years of independence, Namibia economic growth rate averaged to close to five percent. Earnings led this growth from the booming offshore diamond, fishing and tourism. Inflation was in double -digit figures, but was stable at an average of 12%. This sound macro-economic stability enabled Namibia to fit into SACMA arrangement, based on the realities on the ground. However between 1995 and 1999, there was an economic slow down, in the country. Real GDP averaged to only 2%

and the exchange rate depreciated by more than 20% against the US dollar. Such macro economic fluctuations raise a number of questions: In the context of SACMA, to which Namibia is a member, how would it (Namibia) redress its economic downturn? Did the

Namibian experience affect other SACMA countries? Would the effects have been different if Namibia was not a member of the SACMA?

Lesotho: Lesotho’s monetary policy is closely linked to South Africa’s. Like in South Africa, the monetary, credit, and interest rate policies are directed at re-allocating financial resources from the public to the private sector, improving efficiency in the financial systems and capital markets. As most Lesotho’s imports of goods and services are from South Africa, Lesotho’s inflation rates closely track inflation trends in South Africa. Monetary shocks in the South Africa, economy are directly transmitted to Lesotho financial system because the loti (Lesotho currency) is directly pegged to the South Africa’s rand. Though not legal tender, the rand still circulates freely in Lesotho. The country has no exchange rate policy independent of SACMA. Thus with the sharp depreciation of the rand between 1996 and 2002, the loti experienced the same depreciation. The economy of Lesotho has been performing quiet well, at a growth rate above that of South Africa. Between 1993 and 1996, the real GDP growth averaged to 9.4%. This high positive economic growth could be partly attributed to the construction of Lesotho Highland Water Project, expanding exports and higher manufacturing production. In 1998, the country had economic upheaval; with wide spread looting and civil disorder in reaction to the election outcome. The economy was adversely affected and the real GDP growth contracted sharply. Though political sanity was restored, the subsequent economic growth has not recovered up to the previous level.

Swaziland: The ultimate goal of the Swaziland monetary policy is to attain price stability, full employment, sustained economic growth and balance of payment equilibrium (Swaziland Central Bank Order). Monetary policy in Swaziland is highly influenced by the country’s historical membership to SACMA. Under SCMA agreement, Swaziland has liberty to delink the lilangeni (Swaziland currency) from the rand, should circumstances dictate. Swaziland did, however, find it in the best interest to maintain the peg with the rand. The down side of pegging the lilangeni to the rand is that any change in international exchange rate of the rand culminates in an equivalent movement of the lilangeni. In essence, just like other smaller SACMA countries, the policies pursued by the country are to a large extent influenced by South Africa policy announcements.

Swaziland has strong trade links with South Africa; with 80% of its imports originating from South Africa and over 50% of its exports destined to the same country. Due to the currency parity, trade between South Africa and Swaziland takes place without uncertainty and costs that could arise from exchange rate disparities and speculation. One of the main advantages for Swaziland’s belonging to SACMA is that the arrangement frees the country from the need to make decisions concerning monetary and exchange rate policies, allowing it to concentrate largely on pressing fiscal tasks. The arrangement, however, severely limits Swaziland’s ability to formulate or influence monetary policy or respond to shocks that are unique to the country. Given the membership to SACMA, Swaziland has limited scope to determine interest rate different from that of other SACMA countries, led by South Africa. The country has been recording two-digit inflation figures between 1985 and 2000, for example, but it could not use interest rates, as a tool, to control its price increase.

3. Econometric Methodology

3.1 Vector Error Correction Model: The existence of long-run equilibrium (stationary) relationships among economic variables is referred to in the literature as cointegration.

The Johansen procedure examines the question of cointegration and provide not only an estimation methodology but also explicit procedures for testing for the number of cointegrating vectors as well as for restrictions suggested by economic theory in a multivariate setting. If the economic variables are cointegrated, we can proceed to utilize the Vector Auto-regression (VAR) representation in deriving the impulse response functions and the forecast-error decompositions. The basis of these impulse responses (triggered by monetary/fiscal innovations or shocks) and error decompositions here is the Johansen technique, which precisely looks at a Vector Error Correction Model (VECM).

The most common application of cointegration is to test the existence of long-run relationships. One argument sometimes made is that cointegration is about long-run economic relationships, and one needs really long time series (not in the number of observations but in time span) to use cointegration technique. Maddala and Kim (1999) stress that this is not a meaningful argument for two reasons. (i) If the variables are nonstationary, then existence of a long-run equilibrium economic relationship implies cointegration. But not all cointegrating relationships need have meaning in the sense of long-run economic relationships. Cointegrating relationships need not have any economic interpretation. (ii) How long the long run is depends on the speed of adjustment of the particular markets considered. For financial markets with rapid speeds of adjustment, the long run is indeed short. For goods markets the speeds of adjustment are perhaps slow for some commodities and fast for others. For example, Johansen and Juselius (1990) estimate long-run demand for money functions for Denmark (55 observations) and Finland (67 observations) using quarterly data.

3.2.Forecasting: VAR systems are widely used for the production of forecasts. However, the approach is atheoretical in the sense that there is no use of economic theory to specify explicit structural equations between various sets of variables. The VAR system rests on the general proposition that economic variables tend to co-move over time and also to be autocorrelated. Consider a column vector of k different variables, yt = [y1t, y2t, …, ykt]′ and model this in terms of past values of the vector. The result is a vector autoregression (VAR). The VAR(p) process is given as

yt = m + A1 yt-1 + A2 yt-2 + … + Ap yt-p + εt (1)

In equation (1) the Ai are kxk matrices of coefficients, with the properties

E(εt) = 0 for all t; E(εtε′t) = {^Ω₀ ^s_s⁼_≠^t_t (2) where the Ω covariance matrix is assumed to be positive definite. Thus the ε’s are serially uncorrelated but may be contemporaneously correlated.

Suppose that we have observed the vectors y1, y2, …, yk. Assuming a VAR(1), we will have used this data to estimate A and Ω. For now we will set these estimates aside and assume that we know the true values of these matrices. Suppose further that we now wish,

at the end of period k, of the y vector one, two, three, or more periods ahead. No matter how far ahead the forecast period lies, we assume that no information is available beyond that known at the end of period k. the optimal (minimum mean squared error) forecasts of yk+1 is the conditional expectation of yk+1, formed at time k. That is,

n n

k

k E y y y Ay

yˆ ₊₁= ( ₊₁| ,..., ₁)= (3) where yˆdenotes a forecast vector. For simplicity, we omit the usual vector of constants (i.e., m = 0) in equation (1). Then the optimal forecast two periods ahead is E(yk+2 | yk, …, y1). Thus, by forward induction, we have

yk+2 = A²yk + Aεk+1 + εk+2 (4) This gives us

ˆ_k+2

y = A²yk (5) Generally, we have

yk+s = A^syk + A^s-1εk+1 + …+ Aεk+s-1 + εk+s (6) which gives us

s

yˆk₊ = A^syk (7) We express the vector of forecast errors in the forecast for s periods ahead as follows:

es = yk+s - yˆ_{k+s = ε}k+s + Aεk+s-1 + … + A^s-1εk+1 (8) We can then obtain the variance-covariance matrix for the forecast errors, s periods ahead, denoted by

∑

^{( s)} as

∑

^{( s) =} Ω +AΩA′ + A²Ω(A′)² + … + A^s-1Ω(A′)^s-1 (9) However, when the variables in the VAR are not integrated of order zero, unrestricted estimation is subject to the hazards of regressions involving nonstationary variables. If there are cointegrating relations, we can then estimate the VAR, incorporating the cointegrating relations. This essentially entails estimating a VECM.

3.3. Co-integration tests. Two or more nonstationary variables, integrated of the same order, are cointegrated if they have a long-term, or equilibrium, relationship between them. Using augmented Dickey-Fuller (1979) tests, our univariate analyses of stationarity in the time series of interest (GDP growth and four money supply variables) for each country indicate that all the time series are integrated of order one.¹ This prompted us to proceed with testing for the presence of cointegrating relations for each set of variables in CMA countries. In this case, we used the method developed by Johansen (1988, 1991) to estimate the cointegrating vectors between these time series. While the method of Engle and Granger (1987) only applies to the single equation estimation to test for cointegration between variables, the estimation techniques by Johansen (1988, 1991) estimate the cointegrating vectors, and test for the order of cointegrating vectors and linear relationships in a multivariate model. Tables 1a to 1d are a summary of results of co- integration analyses for all four CMA countries using the Johansen maximum likelihood

1 Tests for unit roots in the five time series were conducted for each country. Each regression included an intercept and a trend.

approach, i.e., the co-integration likelihood ratio tests based on maximal eigenvalues and trace of the stochastic matrix.

4. Data and Empirical Results.

4.1.Data Used. The data set spans the period 1981.I-2004.IV. The variables for which data are sourced include: nominal GDP and nominal money stock. The sources of these data are IMF International Finance Statistics, the World Bank and the South African Reserve Bank. We define the measure of the nominal stock of money, M1, is as the currency outside banks plus demand deposits. The variables are defined as follows:

LGDPLE is the natural logarithm of nominal GDP for Lesotho, LM1LE the natural logarithm of nominal money supply as given by M1 for Lesotho, LGDPNA the natural logarithm of nominal GDP for Namibia, LM1NA the natural logarithm of nominal money supply as given by M1 for Namibia, LGDPSA the natural logarithm of nominal GDP for South Africa, LM1SA the natural logarithm of nominal money supply as given by M1 for South Africa, LGDPSW the natural logarithm of nominal GDP for Swaziland and LM1SW the natural logarithm of nominal money supply as given by M1 for Swaziland.

4.2.Tests Results: Both tests confirm that there are only two co-integrating vectors in the given set of variables for Lesotho. Thus, in the case of Lesotho, the null hypothesis of equal to two cointegrating relations is not rejected at the 5% level of significance. There is only one cointegrating relation for Namibia and Swaziland according to both tests conducted at 5% level of significance. But in case of South Africa, the maximal eigenvalue test indicates that there is only one cointegrating vector, while the trace test gives a result of two cointegrating relations at 5% level of significance.

Table 1a: Johansen Cointegration Results (Maximal Eigen Value and Trace Test) for Lesotho: LGDPLE, LM1LE, LM1NA, LM1SA, LM1SW; VAR = 4

Null Hyp Max. eigenvalue stat 5% crit.v Trace stat 5% crit.v.

r = 0 48.64 33.64 110.52 70.49

r ≤ 1 36.26 27.42 61.88 48.88

r ≤ 2 13.73 21.12 25.62 31.54

r ≤ 3 11.57 14.88 11.88 17.86

r ≤ 4 0.32 8.07 0.32 8.07

Note: r =number of cointegrating relationships. The null hypothesis of r ≤ 1, for example, is set against the alternative hypothesis of r = 2 for the maximum eigen value test and the null hypothesis of r ≤ 1 is set against the alternative hypothesis of r ≥ 2 for the trace test.

Table 1b: Johansen Cointegration Results (Maximal Eigen Value and Trace Test) for Namibia: LGDPNA,LM1LE,LM1NA,LM1SA,LM1SW VAR = 4

Null Hyp Max. eigenvalue stat 5% crit.v Trace stat 5% crit.v.

r = 0 41.89 33.64 90.50 70.49

r ≤ 1 22.02 27.42 48.61 48.88

r ≤ 2 17.19 21.12 26.60 31.54

r ≤ 3 8.37 14.88 9.41 17.86

r ≤ 4 1.04 8.07 1.04 8.07

Note: r denotes the number of cointegrating relationships. The null hypothesis of r ≤ 1, for example, is set against the alternative hypothesis of r = 2 for the maximum eigen value test and the null hypothesis of r ≤ 1 is set against the alternative hypothesis of r ≥ 2 for the trace test.

Table 1c: Johansen Cointegration Results (Maximal Eigen Value and Trace Test) for South Africa: LGDPSA, LM1LE, LM1NA, LM1SA, LM1SW; VAR = 4

Null Hyp Max. eigenvalue stat 5% crit.v Trace stat 5% crit.v.

r = 0 40.87 33.64 90.20 70.49 r ≤ 1 17.72 27.42 49.33 48.88 r ≤ 2 16.61 21.12 31.60 31.54 r ≤ 3 9.71 14.88 14.99 17.86 r ≤ 4 5.28 8.07 5.28 8.07

Note: r denotes the number of cointegrating relationships. The null hypothesis of r ≤ 1, for example, is set against the alternative hypothesis of r = 2 for the maximum eigen value test and the null hypothesis of r ≤ 1 is set against the alternative hypothesis of r ≥ 2 for the trace test.

Table 1d: Johansen Cointegration Results (Maximal Eigen Value and Trace Test) for Swaziland: LGDPSW, LM1LE, LM1NA, LM1SA, LM1SW; VAR = 4

Null Hyp Max. eigenvalue stat 5% crit.v Trace stat 5% crit.v.

r = 0 42.56 33.64 90.06 70.49

r ≤ 1 22.51 27.42 47.49 48.88

r ≤ 2 15.51 21.12 24.99 31.54

r ≤ 3 9.16 14.88 9.48 17.86

r ≤ 4 0.32 8.07 0.32 8.07

Note: r denotes the number of cointegrating relationships. The null hypothesis of r ≤ 1, for example, is set against the alternative hypothesis of r = 2 for the maximum eigen value test and the null hypothesis of r ≤ 1 is set against the alternative hypothesis of r ≥ 2 for the trace test.

4.3. Multivariate Dynamic Forecasts for Output Growth. According to Maddala and Kim (1998), one useful contribution of cointegration tests is in the modeling of the VAR systems, whether they should be in levels or first-differences or both, with some restrictions. For this use the cointegrating relationships need not have any economic interpretation. Their value suffices in determining the restrictions of the VAR system, which should be of value in forecasting. If a group of nonstationary variables satisfies a cointegrating relation, simple first differencing of all the variables can cause econometric problems. In the general VAR system with n variables, if all the variables are stationary, using an unrestricted VAR in levels is appropriate. If, in addition, there are r cointegrating relationships, then we require to model the system as a VAR in the r stationary combinations and (n-r) differences of the integrated of order 1 variables. Since our cointegration tests in Tables 1a to 1d indicate existence of cointegrating relations between variables in the sets of variables under study, we appropriately have imposed restrictions on the estimated cointegrating vectors. In each case, we have exactly one estimated cointegrating vector. Table 2 below gives the estimated cointegrating vectors with restrictions for the CMA countries. For each vector, the restriction is put on the estimated coefficient of the output growth by normalizing it to one to identify the long run structural relation.

Table 2: Maximum Likelihood Estimates of Restricted Cointegrating Relations Variable Lesotho

vector

Namibia vector

South Africa vector

Swaziland vector

LGDP 1.0000 1.0000 1.0000 1.0000

LM1LE 0.0215 8.3601* -4.4542 0.2286 LM1NA 0.1317** 1.0702 -0.3241** -0.0283**

LM1SA -0.0491 -5.4818 1.8531 -0.2545*

LM1SW -0.4107* -3.8988** 1.9937 -0.1358*

* Significant at 10% level only; ** significant at 5% level.

We next compute multivariate, multi-step ahead forecasts of output growths for the SACMA countries using the augmented VAR models (i.e., VECMs) obtained as a result of imposing restrictions on the cointegrating relations over the 1981(1) to 20024) period.

As can be see from the summary statistics in Table 3a (see the Annex), the size of the forecast errors and in-sample residuals are very similar. A similar picture also appears by plotting in-sample fitted values and out-of-sample forecasts of the change in output growth (see Figure 1a).

Similarly, forecasts of output growths for Namibia, South Africa and Swaziland can be computed. For Namibia, the root mean sum of squares of the forecasts errors over the 2003(1) to 2004(4) period is 1.3 per cent. This estimate is twice as much lower than the value of 2.6 per cent obtained for the root mean sum of squares of residuals over the estimation period (see Table 3b in the Annex). For South Africa, as shown in Table 3c, the root mean sum of squares of the forecasts errors over the estimated period is above that of residuals, 3.2 per cent and 1.7 per cent, respectively. It is worth noting that the growth forecasts for between 2003(2) and 2004(4) over predict the actual growth values by considerably big margins (see also Figure 1c).

In the case of output growth for Swaziland, the root mean sum of squares of the residuals is 0.1 per cent lower than that of the forecasts errors (table 3d). A plot of in- sample fitted values against out-of-sample forecasts of the change in output growth also reveals that these values are very similar (see Figure 1d). These forecasting results show that the forecasts for output growths in SACMA countries are quite similar to the actual values. One interesting outcome from this analysis is that the dynamic forecasts for the change in output growth are above zero. Thus, generally the money supply changes in Lesotho, Namibia, South Africa and Swaziland have had a positive effect on output growths in all these SACMA countries.

Figure 1a. Multivariate dynamic forecasts: change in Lesotho Output Growth

L G D P L E

F o r e c a s t

Q u a r t e r s 0 . 0 0 0

0 . 0 0 5 0 . 0 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 3 0

1 9 8 2 Q 11 9 8 5 Q 41 9 8 9 Q 31 9 9 3 Q 21 9 9 7 Q 12 0 0 0 Q 42 0 0 4 Q 4

Figure 1b.Multivariate dynamic forecasts: change in Namibian Output Growth

L G D P N A

F o r e c a s t

Q u a r t e r s - 0 . 0 2

- 0 . 0 4 - 0 . 0 6 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

1 9 8 2 Q 11 9 8 5 Q 41 9 8 9 Q 31 9 9 3 Q 21 9 9 7 Q 12 0 0 0 Q 42 0 0 4 Q 4

Figure 1c. Multivariate dynamic forecasts: change in South African Output Growth

L G D P S A

F o r e c a s t

Q u a r t e r s - 0 . 0 5

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5

1 9 8 2 Q 11 9 8 5 Q 41 9 8 9 Q 31 9 9 3 Q 21 9 9 7 Q 12 0 0 0 Q 42 0 0 4 Q 4

Figure 1d.Multivariate dynamic forecasts: change in Swaziland Output Growth

L G D P S W

F o r e c a s t

Q u a r t e r s 0 . 0 0 0

0 . 0 0 5 0 . 0 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0 2 5

1 9 8 2 Q 11 9 8 5 Q 41 9 8 9 Q 31 9 9 3 Q 21 9 9 7 Q 12 0 0 0 Q 42 0 0 4 Q 4

5. Summary and Conclusions

This study has empirically attempted to investigate output growth forecasts as a result of dynamic interplay between money supplies and output growths of Southern African Common Monetary Area (SACMA) countries using Vector Error Correction Models (VECM). The data obtained for these time series are quarterly and span the period 1981(1) to 2004(4). Maximum likelihood (Johansen) Cointegration tests show that there is at least one cointegrating relation in the set of time series for individual SACMA countries. After imposing restrictions on the estimated cointegrating vectors over the 1981(1)-2002(4) period, we have then computed multivariate, multi-step ahead forecasts of output growths for the SACMA countries for the remaining 8 quarters using the augmented VAR models (i.e., VECMs).

In general, the forecasting results show that the forecasts for output growths in SACMA countries are quite similar to the actual values. One interesting outcome from this analysis is that the multivariate dynamic forecasts for the change in output growth are above zero.

Thus, generally the money supply changes in Lesotho, Namibia, South Africa and Swaziland have had a positive effect on output growths in all these SACMA countries.

One policy implication here is that monetary shocks are likely to affect the economies of SACMA symmetrically. Furthermore, this suggests that there is a high degree of monetary integration between SACMA countries and the region can be qualify as an optimal currency area. Since the currencies of Lesotho, Namibia and Swaziland are already pegged to the South African rand, it may be optimal for SACMA countries to go for a fully-fledged monetary union. Future research should therefore concentrate on how exactly monetary shocks affect output growths of SACMA economies.

References

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Annex: Available in the on line edition of the article

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Journal AEID published by Euro-American Assocation of Economic Development Studies:

http://www.usc.es/economet/eaa.htm

Annex

A1: Cointegration and Vector Error Correction Model. Cointegration occurs when the variables in a model are nonstationary, but the trends of the variables are related in a way so that the error term observations are stationary. The presence of cointegration enables us to proceed as if the variables were stationary. For cointegration to take place, the variables must be nonstationary to the same extent, and this nonstationary aspect from the different variables must cancel each other out. Suppose that we have two nonstationary time series, x and y, which are precisely integrated of order 1. If x and y are cointegrated with parameter β, then we have additional variables that we can include in the following first-difference equation

∆yt = α0 + α1∆yt-1 + γ0∆xt + γ1∆xt-1 + ut (A2.1) Let st = yt - βxt, so that st is I(0) (i.e., stationary), and assume for the sake of simplicity that st

has zero mean. Now we can include lags of st in equation (A2.1). In the simplest case we include one lag of st:

∆yt = α0 + α1∆yt-1 + γ0∆xt + γ1∆xt-1 + δ st-1 + ut

= α0 + α1∆yt-1 + γ0∆xt + γ1∆xt-1 + δ(yt-1 - βxt-1)+ ut (A2.2)

where E(ut|It-1) = 0, and It-1 contains information on ∆xt and all past values of x and y. The term δ(yt-1 - βxt-1) is called the error correction term, and equation (A2.2) is an example of an error correction model. An error correction model allows us to study the short-run dynamics in the relationship between y and x. For simplicity consider the model without lags of ∆yt and

∆xt:

∆yt = α0 + γ0∆xt + δ(yt-1 - βxt-1)+ ut (A2.3) where δ < 0. If yt-1 > βxt-1, then y in the previous period has overshot the equilibrium; because δ < 0, he error correction term works to push y back towards the equilibrium. Similarly, if yt-1

< βxt-1, the error correction term induces a positive change in y back towards the equilibrium.

Consider the following model.

yt = α0 + α1yt-1 + α2yt-2 + … (A2.4)

The model in equation (A2.4) is one equation in what is known as a vector autoregressive (VAR) model. In an autoregressive model, we model a single series {∆yt}, for example, in terms of its own past. In vector autoregressive models, we model several series – which is where the word vector comes from – in terms of their own past. If we have two series, yt and zt, a vector autoregression consists of equations that look like

yt = δ0 + α1yt-1 +γ1zt-1 + α2yt-2 + γ2zt-2 … and

y_t = η0 + ϕ1y_t-1 + λ1z_t-1 + ϕ2y_t-2 + λ2z_t-2,… (A2.5)

where each equation contains an error that has zero expected value given past information on y and z. We can extend this to the n-variable model. Formally, the (nx1) vector x_t = (x_1t, x_2t,

…, x_nt)’ has an error-correction representation if it can be expressed in the form:

∆xt = π0 + πxt-1 + π1∆xt-1 + π2∆xt-2 + … + πp∆xt-p + εt (A2.6)

where π0 = (nx1) vector of intercept terms with elements πi0

π = matrix with elements πjk such that one or more of the πjk≠ 0 πi = (n x n) coefficient matrices with elements πjk(i)

εt = = (nx1) vector with elements εit

Note that the disturbance terms are such that εit may be correlated with εjt. Let all variables in xt be I(1). Now, if there is an error-correction representation of these variables as in (A2.6), there is necessarily a linear combination of the I(1) variables that is stationary. Solving (A2.6) for πxt-1 yields

πxt-1 = ∆xt + - π0 - Σπi∆xt-i - εt (A2.7) Since each expression on the right hand side of (A2.7) is stationary, πxt-1 must also be

stationary. Since π contains only constants, each row of π is a cointegrating vector of xt. For example, the first row can be written as (π11x1t-1 + π12x2t-1 + … +π1nxnt-1). Since each series xit-1

is I(1), (π11, π12 … π1n) must be a cointegrating vector for x_t. If this is true, equation (A2.6) is known as vector error correction model (VECM).

A2: Results of Variance-Covariance Matrices of Forecast Errors

Lesotho

LGDPLE LM1LE LM1NA LM1SA LM1SW LGDPLE 0.2734E-4 0.3031E-6 0.2799E-4 0.3547E-6 -0.2218E-4 LM1LE 0.3031E-6 0.0027793 -0.2636E-3 -0.6625E-4 0.0011320 LM1NA 0.2799E-4 -0.2636E-3 0.0033721 -0.3374E-3 0.2190E-3 LM1SA 0.3547E-6 -0.6625E-4 -0.3374E-3 0.0025680 -0.4645E-3 LM1SW -0.2218E-4 0.0011320 0.2190E-3 -0.4645E-3 0.0032749

Namibia

LGDPNA LM1NA LM1LE LM1SA LM1SW LGDPNA 0.9040E-3 -0.2074E-4 0.2355E-3 -0.1169E-3 0.1363E-3 LM1NA -0.2074E-4 0.0034586 -0.2897E-3 -0.4340E-3 0.4412E-3 LM1LE 0.2355E-3 -0.2897E-3 0.0030339 -0.1329E-3 0.0015119 LM1SA -0.1169E-3 -0.4340E-3 -0.1329E-3 0.0025659 -0.6658E-3 LM1SW 0.1363E-3 0.4412E-3 0.0015119 -0.6658E-3 0.0041621

South Africa

LGDPSA LM1SA LM1LE LM1NA LM1SW LGDPSA 0.3845E-3 -0.3618E-4 -0.2781E-3 -0.1141E-3 -0.1492E-4 LM1SA -0.3618E-4 0.0025755 -0.2860E-3 -0.3396E-3 -0.6017E-3 LM1LE -0.2781E-3 -0.2860E-3 0.0029036 -0.2007E-3 0.0015412 LM1NA -0.1141E-3 -0.3396E-3 -0.2007E-3 0.0036378 0.3007E-3 LM1SW -0.1492E-4 -0.6017E-3 0.0015412 0.3007E-3 0.0043081

Swaziland

LGDPSW LM1SW LM1LE LM1NA LM1SA LGDPSW 0.2385E-4 0.2308E-4 0.1528E-4 0.3460E-4 -0.2105E-6 LM1SW 0.2308E-4 0.0042656 0.0015531 0.3757E-3 -0.4785E-3 LM1LE 0.1528E-4 0.0015531 0.0030389 -0.3912E-3 -0.2368E-3 LM1NA 0.3460E-4 0.3757E-3 -0.3912E-3 0.0032298 -0.5496E-3 LM1SA -0.2105E-6 -0.4785E-3 -0.2368E-3 -0.5496E-3 0.0023625 A3.

Table 3a: Multivariate Dynamic Forecasts for the Lesotho Output Growth

Observation Actual Prediction Error 2003Q1 0.0044843 0.0086631 -0.0041788

2003Q2 0.0055773 0.010050 -0.0044731 2003Q3 0.0066519 0.010174 -0.0035222 2003Q4 0.010989 0.0095510 0.0014381 2004Q1 0.014108 0.0094831 0.0046246 2004Q2 0.0096515 0.0090864 0.5652E-3 2004Q3 0.011671 0.0092674 0.0024038 2004Q4 0.019844 0.0093697 0.010474 Summary Statistics for Residuals and Forecast Errors

Estimation Period Forecast Period 1982Q1 to 2002Q4 2003Q1 to 2004Q4 Mean -0.0000 0.9165E-3 Mean Absolute 0.0035129 0.0039600 Mean Sum Squares 0.2540E-4 0.2364E-4 Root Mean Sum Squares 0.0050396 0.0048624

Table 3b: Multivariate Dynamic Forecasts for the Namibian Output Growth

Observation Actual Prediction Error 2003Q1 0.0077394 0.030744 0.023005 2003Q2 0.0042189 0.0092960 -0.0050771 2003Q3 0.016456 0.0078894 0.0085668 2003Q4 0.014270 0.012888 0.0013821 2004Q1 0.021854 0.012395 0.0094590 2004Q2 0.030316 0.011155 0.019161 2004Q3 0.024542 0.010114 0.014429 2004Q4 0.0044385 0.011172 -0.0067333 Summary Statistics for Residuals and Forecast Errors

Estimation Period Forecast Period 1982Q1 to 2002Q4 2003Q1 to 2004Q4 Mean -0.0000 0.0022728 Mean Absolute 0.020610 0.010977 Mean Sum Squares 0.6780E-3 0.1676E-3 Root Mean Sum Squares 0.026038 0.012944

Table 3c: Multivariate Dynamic Forecasts for the South African Output Growth

Observation Actual Prediction Error 2003Q1 0.6465E-3 0.024006 -0.023360 2003Q2 0.0011559 0.027659 -0.026503 2003Q3 -0.4949E-3 0.042491 -0.042986 2003Q4 0.0014108 0.036345 -0.034934 2004Q1 0.9352E-3 0.036377 -0.035442 2004Q2 0.0020655 0.033845 -0.031780 2004Q3 0.0034208 0.033270 -0.029849 2004Q4 0.0017565 0.032393 -0.030636 Summary Statistics for Residuals and Forecast Errors

Estimation Period Forecast Period 1982Q1 to 2002Q4 2003Q1 to 2004Q4 Mean 0.0000 -0.031936 Mean Absolute 0.012920 0.031936 Mean Sum Squares 0.2884E-3 0.0010515 Root Mean Sum Squares 0.016982 0.032427

Table 3d: Multivariate Dynamic Forecasts for the Swaziland Output Growth

Observation Actual Prediction Error 2003Q1 0.010097 0.0033831 0.0067141

2003Q2 0.016859 0.0050147 0.011844 2003Q3 0.0068156 0.0056413 0.0011743

2003Q4 0.0082676 0.0078552 0.4124E-3 2004Q1 0.0059702 0.0076204 - 0.0016503

2004Q2 0.0022297 0.0076457 -0.0054161 2004Q3 0.0037051 0.0073597 -0.0036546 2004Q4 0.011765 0.0072187 0.0045461 Summary Statistics for Residuals and Forecast Errors

Estimation Period Forecast Period 1982Q1 to 2002Q4 2003Q1 to 2004Q4 Mean 0.0000 0.0017462 Mean Absolute 0.0035841 0.0044265 Mean Sum Squares 0.2042E-4 0.3162E-4 Root Mean Sum Squares 0.0045186 0.0056235