A Multivariate Linear Regression on the Effect of Foreign Trade on Foreign Excha

Abstract： This paperfits a multivariate linear Regression model for the relationship between four response variables； foreign trades （cost of oil imported， cost of non-oil imported， cost of oil exported and cost of non-oil exported） and two predictor variables； foreign exchange rate of US Dollar and Pounds Sterling between 19602010. The null hypothesis that no linear relationship exists between foreign trades and foreign exchange rate were proposed. The model indicates an adequate relationship between the foreign trade and exchange rate. We observed from model 1 to model 4 of the multivariate linear regression that the multivariate bootstrap method when compared with the analytic method yielded a reduced standard error. This shows that multivariate bootstrap is good methods in controlling variation among observations. We further observed that the exchange rate of US Dollar and Pounds Sterling contributed 68% effect on the Nigerian foreign trades while 32% may be attributed to other unexplained factors. This indicates that multivariate bootstrap method offer considerable potentials in the estimation of multivariate linear regression parameters. Thus， the results reveal that bootstrap resampling allows empirical assessment of the analytical model and it is a good approximation to the analytical model.

Key words： Multivariate； Bootstrap； Foreign trade； Foreign exchange rate； Linear regression

1. INTRODUCTION

The evolution of the foreign exchange in Nigeria was influenced by a number of factors such as the changing pattern of international trade， institutional changes in the economy and structural shifts in production. The increased export of crude oil in the early 1970’s， following the sharp rise in its prices， enhanced official foreign exchange receipts. The market witness a boom during this period and the management of foreign exchange resourcesfinds it imperative to ensure that shortage did not arise [7]. Adubi and Okunmadewa [1] carried out a study on price exchange role volatility and Nigeria’s Agricultural tradeflows. They argued that changes in income earning of export crop producer income as a result of either increase/decrease in international world price of the exports of devaluation of the currency and the subsequent increase in producer prices. Adubi and Okunmadewa [1] said that price/exchange rate changes may lead to major decline in future output if they are unpredictable and erratic.

Johnson and Wichern [4] define multivariate linear regression as modeling the relationship between m responses and a single set of predictor variables， where each response variable is assumed to follow its own linear regression model. Bootstrapping is a resampling technique in which sample of size n are obtained with replacement from the original sample of size n. The basic idea behind bootstrapping is to analyze the population by replacing the unknown distribution function F by the empirical distribution function？F obtained from the sample （see [2，3，5]）.

2. MATERIAL AND METHOD

General Form of Multivariate Linear Regression Model

According to Johnson and Wichern [4]， multivariate linear regression model defines the relationship between m responses Y1，Y2，...，Ymand a single set of r predictors， Z1，Z2，...，Zr.

whereβis the （（r + 1）×m） matrix of parameters. Y is the （n×m） matrix of the response variables.εis the （n×m） matrix of the errors or the residuals. Then， the

multivariate linear regression model is

3. THE PROPOSED ALGORITHMS FOR ESTIMATING MULTIVARIATE LINEAR REGRESSION PARAMETERS

3.1. The Multivariate Bootstrap Algorithm on the Resampling Observations for Estimating the Multivariate Linear Regression Parameters

Obtain the probability distributions F%？β（b）&of multivariate bootstrap estimates？β（b1），？β（b2），...，？β（bB）and use F%？β（b）&to estimate the multivariate regression parameters， standard error and confidence interval. The multivariate bootstrap estimates are the means （see [6]）

The data obtained from Central Bank of Nigeria Bulletin 2010 edition on exchange of US Dollar and Pounds to Nigeria Naira currency and Nigeria foreign trade from 1960-2010 and shown in the Table 1.

4. RESULTS AND DISCUSSION

The data obtained from Central Bank of Nigeria Bulletin 2011 edition is on the exchange of US Dollar and Pounds to Nigeria Naira currency and Nigeria foreign trade （Oil Import and Export， Non-Oil Import and Export） from 1960-2010.

We intend to obtain the multivariate linear model that describes the relationship between foreign trade and foreign exchange rate of Naira per US Dollar and Pounds Sterling. Let

For the models 1-4， the values of multiple R-square are 0.6257， 0.7126， 0.7535 and 0.6431 respectively. Adjusted R-square are 0.6101， 0.7006， 0.7432， and 0.6282 respectively at various P values of 5.736×10？11， 1.0101×10？13， 2.541×10？15and 1.823×10？11respectively. Averagely the R-square is 0.683725； Adjusted R-square is 0.670525 and P-value 2.777×10？13. Since the P-value is less than 0.05， there is enough evidence to reject the null hypothesis and conclude that the models are adequate. We observed that the exchange rates of US Dollar and Pounds Sterling contributed 68% effect on the Nigerian Foreign Trades.

Furthermore， implementing the proposed multivariate bootstrap （see Appendix） to estimation of the parameters of models 1-4 yields Table 3.

Table 4 reveals great reduction in standard error for the multivariate Bootstrap and delete-5 algorithms when compared with the analytic method. This shows that both algorithms can be used in controlling variation among observations.