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Exchange rate is very pivotal in its role in the economy of any nation especially as a result of globalization. This paper seeks to model the Nigerian economy proxied by the log of Gross Domestic Product (LGDP) and its relationship with other variables in the economy. The variables used are NGN/USD exchange rate (NAIRA), log of oil revenue (LOILREV), log of government expenditure. We estimated the model using the Vector Autoregressive (VAR) model. We tested the presence or otherwise of causality among the variables using the method of Granger. The result reveals that the optimal lag for the model was 1. The exchange rate was found to Granger cause the economy (LGDP), LOILREV (Oil Revenue) and LGEXP (Government expenditure). We also discovered that the dynamics of NAIRA was not fully captured by the variables used. We also pointed out the shock persistence of NAIRA in time.

Exchange rate is the price at which one country’s currency is exchanged for that of another country. The exchange rate can be considered as the most important relative price in the financial world [

Exchange rate volatility can be observed at any time in the day, week, month or year. The status of a country’s economic well being is reflected by the real exchange rate. When the domestic income of a country increases, then the country’ currency is likely to depreciate. However, an increase in foreign income would likely lead to appreciation of the currency. Also when a nation have a crave for domestic goods and services, it will likely lead to appreciation of the currency, while a crave for foreign goods and services will likely lead to depreciation of its currency [

The post Bretton Woods’ exchange rate models assume that there are close relationship between exchange rate movement and other macroeconomic variables [

When the monetary authority allows the exchange rate to be fixed, other macroeconomic variables are volatile. However, when exchange rate is allowed to float, it is known that the exchange rate becomes highly volatile in comparison with other macroeconomic variables [

“Economic growth is the increase in the inflation-adjusted market value of the goods and services produced by an economy over time. It is conventionally measured as the percent rate of increase in real gross domestic product, or real GDP, usually in per capita terms. [

Economic growth is measured by the increase in the amount of goods and services produced in a country. Growth is said to occur in a situation where a country’s productive capacity is on the increase. This is then applied to produce more goods and services [

The Nigerian government has employed various strategies such as the use of monetary and fiscal policy, encourage export of goods and services, encourage the use of home made goods, structural adjustment program, etc. All these were aimed at achieving the following; price stabilization, job creation, maintaining balance of payment, and growing the economy [

In this paper we seek to establish the type of relationship that exists between exchange rate economic growth and other macroeconomic variables in Nigeria using the Vector Autoregressive (VAR) model.

We recall that economic growth refer to the increase in the quantity of goods and services produced in a country as against those imported into the country. Thus economic growth has to do with productivity within a country. The question then arises. What is the relationship between exchange rate and economic growth. A strong exchange rate (the home currency appreciating over foreign currency) will lead to export being expensive. The demand for the country’s goods and services by other countries will decrease leading to decrease in the quantity of goods and services produced in the country. This will result in negative economic growth (depressed economy). If the exchange rate is weak (the foreign currency appreciating over home currency), it will make exported goods and services cheaper at the international market, this will lead to increase in demand for the home country’s goods and services. The increase in demand will stimulate the production of more goods and services in the home country. This will lead to a positive economic growth.

In an open world economy where goods are allowed to move freely across national boundaries without the restriction of tax or tariff, the exchange rate of a country is “determined by efficient labor and capital in producing tradable goods, as compared with producing non-tradable goods” [

When there is a rise in foreign demand for domestic currency it will lead to the strengthening of the domestic currency, when there is a fall in foreign demand for the domestic currency, it will lead to the weakening of the domestic currency. A rise in supply of the domestic currency will lead to the weakening of the domestic currency while a fall in the supply of the domestic currency will lead to the strengthening of the domestic currency [

Many researchers are of the opinion that exchange rate is critical to the performance of any economy. They hold that it is a major determinant of economic performance [

A cursory look at the performance of Nigeria’s exchange rate between 1970-2010 suggests a Granger causality between exchange rate and other macroeconomic variables [

The vector autoregressive (VAR) method is a simple and flexible tool used in modelling multivariate data. It has enjoyed wide usage both by researchers and practitioners in financial econometric. Its beauty is in the ability of the model to capture the dynamics of the variable of a time series data as well as its ability to forecast.

We can represent a VAR(p) model as

v t = c + X 1 v t − 1 + X 2 v t − 2 + X 3 v t − 3 + ⋯ + X p v t − p + ε t (1)

Such that p is the lag order and v t = ( v 1 t , ⋯ , v k t ) ′ is a ( k × 1 ) random terms, the X i is the coefficient matrix, c = ( c 1 , ⋯ , c k ) ′ is a ( k × 1 ) constant term where E ( v t ) can be non zero. ε t is a vector of white noise with E ( ε t ) = 0 , E ( ε t , ε ′ t ) = ξ ε satisfy the non singularity condition.

Equation (1) can be represented as a VAR process of order 1

V t = C + X V t − 1 + ϵ t (2)

where

V t = [ v t v y t − 1 ⋮ v t − p + 1 ] , C = [ c 0 ⋮ 0 ] , X = [ X 1 X 2 … X p − 1 X p I 0 … 0 0 0 I … 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 … I 0 ] , ϵ t = [ ε t 0 ⋮ 0 ]

We can easily estimate Equation (2) separately using the method of OLS.

Proposition: The VAR(p) is said to be stable if the roots of det ( I k − X 1 z − ⋯ − X p z p ) = 0 lie outside the unit root circle. We can also say V t is stable if all the eigenvalues of the Companion matrix X have modulus less than 1.

The concept of stability is important because it guarantees the convergence of the process V t .

The lag length p is selected in such a way as to minimize the information criteria (The commonly used criteria are: Akaike information criterion (AIC), Hannan-Quinn information criterion (HQ), Schwarz Information criterion (SC) and penalize large values of the selection criteria.

For a VAR(p) process, the information criteria is (IC) given by

I C ( p ) = I n | ξ ¯ ( p ) | + g T ⋅ φ ( k , p ) (3)

where

ξ ¯ ( p ) = T − 1 ∑ t = 1 T ε ^ 1 ε ^ ′ T is the residual covariance matrix without a degree of freedom correction from a VAR(p) model.

g T is a sequence indexed by the sample size T and

φ ( k , p ) is a penalty function which penalizes large VAR(p) models.

H Q ( p ) = I n | ξ ¯ ( p ) | + 2 ln T T p k 2 (4)

A I C ( p ) = I n | ξ ¯ ( p ) | + 2 T p k 2 (5)

S C ( p ) = I n | ξ ¯ ( p ) | + ln T T p k 2 (6)

We say that y t does not Granger-cause x t if

E [ x t − E ( x t | x t − 1 , y t − 1 , x t − 2 , y t − 2 , ⋯ ) ] 2 = E [ x t − E ( x t | x t − 1 , x t − 2 , ⋯ ) ] 2 (7)

If y t Granger-causes x t then one can predict x t better using the whole past of the x t and y t processes than using only the past of x t . According to Granger, “a cause cannot come after the effect”. Thus if a variable x t affect a variable y t , then x t should help improving the prediction of y t .

Let v t be a VAR process such that its moving Average (MA) representation in canonical form is

V t = μ = ∑ i = 0 ∞ ϕ i ε t − i = μ + ϕ ( L ) ε t , ϕ 0 = I k (8)

where

ε t is a white noise process with non-singular covariance matrix Σ ε . Assume that v t consist of M-dimensional process z t and the ( k − M ) -dimensional process x t and the MA representation is partitioned accordingly,

v t = [ z t x t ] = [ μ 1 μ 2 ] + [ ϕ 11 ( L ) ϕ 12 ( L ) ϕ 21 ( L ) ϕ 22 ( L ) ] [ ε 1 ε 2 ] (9)

A necessary and sufficient condition for x t being not Granger-caused for z t , that is z t is not Granger-caused by x t is

z t ( 1 | { x s | s ≤ t } ) = z t ( 1 | { z s | s ≤ t } ) ↔ ϕ 12 , t = 0 (10)

An impulse response function traces the effect of a one-time shock to one of the innovations on current and future values of the endogenous variables (LLC, 2009).

Recall the Wold representation,

y t = μ + ε t + η 1 ε t − 1 + η 2 ε t − 2 + ⋯ (11)

where

η i j s = ∂ y i , t + 1 ∂ ε j , t = ∂ y i , t ∂ ε j , t − s , i , j = 1 , ⋯ , T (12)

is the response of y in period t + s to shock in period s .

η s is the Impulse Response Function of s .

The k × k moving average matrices η s are determined recursively using

η 1 = ∑ j = 1 p − 1 η s − j A j , s = 1 , 2 , … ⋯ (13)

Exchange rate serves as our dependent variable. As s result of the heterogeneity of the variables’ sizes, we take the natural log of some of the variables in other to homogenize the variables as well as make the model more robust.

We specify our Vector Autoregressive model as

Y t = C + ∑ i = 1 p A i Y t − 1 + ϵ t . (14)

where

Y t is a ( 6 × 1 ) random vector of endogenous variables being considered as exchange rate, Gross Domestic Product, government expenditure, inflation rate, external reserve, and oil revenue, the A i are fixed coefficient matrix, C is a fixed ( 6 × 1 ) vector of intercept terms. ε t is a 6-dimensional white noise and p is the lag order.

The structural unrestricted VAR model for this study is specified as

N A I R A t = C i + ∑ i = 1 p X 1 i N A I R A t − 1 + ∑ i = 1 p X 2 i L G D P t − 1 + ∑ i = 1 p X 3 i L I N F t − 1 + ∑ i = 1 p X 4 i L E X R E S t − 1 + ∑ i = 1 p X 5 i L G E X P t − 1 + ∑ i = 1 p X 6 i L O I L R E V + ϵ i t (15)

W t = C i + ∑ i = 1 p X u i N A I R A t − 1 + ∑ i = 1 p X v i L G D P t − 1 + ∑ i = 1 p X w i L I N F t − 1 + ∑ i = 1 p X x i L E X R E S t − 1 + ∑ i = 1 p X y i L G E X P t − 1 + ∑ i = 1 p X z i L O I L R E V + ϵ i t (16)

where W t is a vector ( 5 × 1 ) matrix of other exogenous variables excluding NAIRA

W t = [ L G D P t I N F t L E X R E S t L G E X P t L O I L R E V t ] (17)

where:

NAIRA is the USD/NGN exchange rate

INF is the rate of Inflation

LEXRES is the natural log of external reserve.

LGEXP is the natural log of Government expenditure

LOILREV is the natural log of oil revenue.

All data used in this work were obtained from Central Bank of Nigeria (CBN) website http://www.cenbank.org. The exchange rate represented by NAIRA is the cost of a Nigerian Naira in terms of the US dollars. The LGDP is used as a proxy for the economy. Other variables are as listed in 4.0. As a result of the heterogeneous nature of the data, we took the natural log of the variables except inflation and foreign exchange, this is to homogenize the variables. We did not use the natural logarithm of inflation and foreign exchange because their series are homogenous and need no further transformation. The model was analysed using Eview 7.0 Enterprise Edition.

The lag exclusion criteria as well as the lag inclusion criteria (

The impulse response (see

The variance decomposition (see

We can summarize our findings as follows: The exchange rate seems to influence the economy as shown by Granger causality, though a shock on NAIRA has a weak impact on the economy; The variance decomposition (see

In conclusion, we are of the opinion that diversifying the economy will exploit the weakness of NAIRA and give us a competitive edge in the international market.

We acknowledge the contributions of the anonymous referee.

Okoronkwo, U.C., Ujumadu, R.N. and Osu, B.O. (2017) A VAR Approach to Exchange Rate and Economic Growth in Nigeria. Journal of Mathematical Finance, 7, 834-845. https://doi.org/10.4236/jmf.2017.74044

*indicates lag order selected by the criterion; LR: sequential modified LR test statistic (each test at 5% level); FPE: Final prediction error; AIC: Akaike information criterion; SC: Schwarz information criterion; HQ: Hannan-Quinn information criterion.