Bayesian Regression Analysis

BAYESIAN LINEAR REGRESSION MODEL:

ANALYSIS OF THE FINANCIAL ACTIVITIES

OF THE BANKING SECTOR OF THE NIGERIA

STOCK EXCHANGE.

AN ASSIGNMENT SUBMITTED

BY

ABDULFATAI SHAKIRUDEEN 060806002

(Mathematics Department, University of Lagos)

SUBMITTED TO

DR M. ADAMU-IRIA

STATISTICAL PACKAGES

MAT 829

MAY 2013

ABSTRACT

Bayesian statistics is an approach to statistics which formally seeks use of prior information with

the data. Bayes Theorem provides the formal basis for making use of both sources of

information in a formal manner. The Bayesian analysis is the study of

different features of posterior density. In this study, Bayesian Regression analysis using R

software (with MCMC pack) is used to explore data extracted from the Nigeria Stock Exchange

in the Capital market (case study of two banks; Access Bank PLC and United Bank of Africa

PLC). Data collected for this project is a secondary data from Nigeria stock Exchange; All-share

Index, daily price of stock, interest rate, exchange rate and daily oil price for the period of five

years 2005-2009. For this purpose, we define the response variable as the Nigeria Stock

Exchange All-share Index (NSEAI). The covariates are the Daily price of stock, Interest rate

(Lending rate), Exchange rate and Oil price. Simulation approach of Bayesian analysis was

found to be the most useful one in this study.

BAYESIAN REGRESSION METHODS

1. Introduction

Prior Probabilities and Bayes Theorem

The task of Bayesian analysis is to build a model for the relationship between parameters () and

observables (y), and then calculate the probability distribution of parameters conditional on the

data, p(|y). In addition, the Bayesian analysis may calculate the predicted distribution of

unobserved data.

Bayesian statistics begins with a model for the joint probability distribution of and y, p(,y).

may be a single parameter or a vector of many parameters, and y may be a vector of

observations of a single variable or a matrix with multiple observations of many variables. The

function p is a probability distribution. An example of a model is the familiar one for estimating

the mean and variance of a normally distributed population, in which p(,y) is a normal

distribution with mean and variance given by the parameter vector , and y is a sample of

independent measurements. Using the definition of conditional probability (Mangel and Clark

1988, Howson and Urbach 1989), p(,y) can be decomposed into two components:

p(,y) = p() p(y| ).. .1

By convention, p() is called the prior distribution of (i.e. the distribution prior to observing the

data y) and p(y| ) is called the likelihood function (i.e. the likelihood of observing the data given

a particular parameter value ). Bayes theorem provides the posterior probability distribution

p(|y) (i.e. the distribution of obtained after observing y and combining the information in the

data with the information in the prior distribution):

p(|y) = p() p(y| ) / p(y)..2

Equation (2) provides a probability distribution of given observations of the data y.

In this equation, p(y) is the sum (or integral) of p() p(y| ) over all possible values of -Mangel

and Clark (1988) or Howson and Urbach (1989).

2. Subjectivity

Bayesian probabilities are sometimes called subjective probabilities. It is important to

understand exactly what is meant by subjective in this context. Decision analyses are often

unique. The situation in which one is making the decision may occur only once. It cannot be

replicated, so there is no possibility for measuring probabilities by repeated sampling.

Nevertheless, Bayesian analysis may be used to compute the probabilities needed to make

decisions. Because these probabilities cannot be measured by repeated sampling, they are called

subjective and they represent a degree of belief in a particular outcome-Lindley (1985),

Howson and Urbach (1989) and Pratt et al. (1995).

Also, if there is no basis in observed data for estimating the prior probability distribution, then

the analyst may simply assume a particular prior distribution. The consequences of this

assumption can be tested by sensitivity analyses that compare the response of the posterior

distribution to different assumptions about the prior distribution. Most commonly, a non-

informative prior distribution is assumed. A non-informative prior distribution assigns the same

probability to each possible value of the parameters. If the number of observations is at least

moderately large, a non-informative prior distribution will have negligible impact on the

posterior distribution. If the data y is limited, however, the choice of prior distribution may have

a substantial impact on the posterior distribution. In this case, sensitivity analysis is needed to

evaluate the consequences of different assumptions about the prior distribution.

3. Linear Regression with Non-informative Prior

In linear regression, the observations consist of a response variable in a vector y and one or more

predictor variables in a matrix X. The vector y has n elements, corresponding to n observations.

The matrix X has n rows, corresponding to the observations, and k columns corresponding to the

number of predictors. If the regression includes an intercept, one of the columns of X is a column

of ones. The parameters are the regression coefficients and the error variance of the fitted

model, 2. The model that relates observations and parameters is written:

(y | , 2, X) ~ Normal(X , 2 I)..3

In words, this model states that the distribution of y given parameters and 2 and predictors X

is a normal distribution with mean X and variance 2. The identity matrix is I. A normal

distribution is completely specified by its mean and variance.

Once the model is specified, the Bayesian analysis seeks the posterior distribution for the

parameters and a predictive distribution for the models predictions. The analysis begins with a

prior distribution. A non-informative prior distribution that is commonly used for linear

regression is p(, 2) 1/2..4

In words, this expression means that the joint probability distribution of and 2 given X is a

flat surface with a constant level proportional to 1/2.

The posterior distribution of given 2 is | 2, y ~ Normal (E, V 2) (A.5)

Expression (5) states that the probability distribution of given 2 and y is normal with mean E

and variance V 2. The parameters of this normal distribution are computed from

E = (X X)-1 X y.6

V = (X X)-1.7

The apostrophe () denotes matrix transposition. The marginal posterior distribution of 2 (i.e.

the integral over all possible values of of the joint distribution of and 2) is

2 | y ~ Inverse 2 (n - k, s2).8

Expression (8) says that the probability distribution of 2 given y follows an inverse 2

distribution. The inverse 2 distribution, presented by Gelman et al. (1995), is fully defined by

two parameters, the degrees of freedom and the scale factor. In this case there are n k degrees

of freedom (where n is the number of observations of y and k is the number of parameters to be

estimated, i.e. the number of columns of X). The scale factor s2 is computed by

s2 = (y - X E) (y - X E) / (n k)..10

Note that y - X E is the vector of residuals, or deviations of observations from predictions.

The marginal posterior distribution of given y is written

| y ~ Multivariate Student t (n k, E, s2)11

The multivariate Student t distribution (presented by Gelman et al. 1995) has three parameters,

the degrees of freedom n k, the mean E, and the scale factor s2. This distribution is derived

by integrating the posterior distribution of given 2 (5) over all possible values of 2 (8).

Regressions are often fitted in order to make predictions. The predictive distribution, yp, given a

new set of predictors Xp has mean

E(yp | y) = Xp E..12

The marginal posterior distribution of the variance of this prediction is

var(yp | 2, y) = (I + Xp V Xp) 2.13

where I is the identity matrix. This variance formula has two components, I 2 for sampling

variance of the new observations and Xp V Xp 2 for uncertainty about . The marginal

posterior distribution of yp given y is

p(yp | y ) ~ Multivariate Student-t [n k, Xp E, (I + Xp V Xp) s2]..14

Bayesian analysis using the non-informative prior of equation (4). The classical estimates of

and 2 are E and s2, respectively. The classical standard error estimate for is V s2. The

classical prediction for new data is yp = Xp E with variance (I + Xp V Xp) s2.

4. Linear Regression with Informative Prior

Bayesian analysis can be used to combine two different sources of information in a single model

to estimate parameters or make predictions. The results can then be combined with a third

source of information to improve the parameter estimates or predictions. This process can be

repeated over and over again to combine information from any number of sources. Combining

multiple sources of information is one of the most important uses of Bayesian statistics (Hilborn

and Mangel 1997). In linear regression with an informative prior distribution, there are two

statistically-independent data sets that provide information about the model to be analyzed. We

assign one data set to be the prior, and use the other data set for the likelihood. Usually it is

convenient to assume that the prior distribution of the k regression parameters is multivariate

Student-t. This distribution has three parameters, a vector of mean regression parameters, a

matrix with variances along the main diagonal and covariance elsewhere, and degrees of

freedom. In this case, the mean vector contains the k prior estimates of the mean regression

parameters (B0) and the k x k parameter covariance matrix S0, model variance s02 and degrees

of freedom n0. For the second data set, we have a n1 x 1 response vector y1 and a n1 x k matrix

of predictors X1.

The posterior can be computed by treating the prior as additional data points, and then weighting

their contribution to the posterior (Gelman et al. 1995).

5. DATA ANALYSIS

ACCESS BANK PLC.

5.1 Simple Linear Regression analysis using R-package.

R version 2.15.2 (2012-10-26) -- "Trick or Treat"

Copyright (C) 2012 The R Foundation for Statistical Computing

ISBN 3-900051-07-0

Platform: x86_64-w64-mingw32/x64 (64-bit)

> data1 data1

> attach(data1)

> lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)

Call:

lm(formula = NSEASI ~ PRICE + INTEREST + EXCHANGE + OIL)

Coefficients:

(Intercept) PRICE INTEREST EXCHANGE OIL

3.3104 0.4697 2.3299 -5.3469 0.3923

Table 1. The estimated coefficients for Access Bank.

> summary(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))

Call:


Residuals:

Min 1Q Median 3Q Max

-0.41806 -0.16074 -0.00902 0.15483 0.46518

Table 2.

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 3.31044 0.52732 6.278 5.71e-08 ***

PRICE 0.46975 0.09486 4.952 7.35e-06 ***

INTEREST 2.32985 0.52988 4.397 5.07e-05 ***

EXCHANGE -5.34694 0.77386 -6.909 5.31e-09 ***

OIL 0.39228 0.09805 4.001 0.00019 ***

---Table 3. The estimated coefficients with standard error for Access Bank.

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.2258 on 55 degrees of freedom

Multiple R-squared: 0.827, Adjusted R-squared: 0.8144

F-statistic: 65.71 on 4 and 55 DF, p-value: < 2.2e-16

> anova(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))

Analysis of Variance Table

Response: NSEASI

Df Sum Sq Mean Sq F value Pr(>F)

PRICE 1 7.2169 7.2169 141.5014 < 2.2e-16 ***

INTEREST 1 0.3667 0.3667 7.1897 0.0096600 **

EXCHANGE 1 5.0055 5.0055 98.1435 7.798e-14 ***

OIL 1 0.8164 0.8164 16.0077 0.0001902 ***

Residuals 55 2.8051 0.0510

Table 4. Anova table for Access bank.

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

> plot(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)

1.0 1.5 2.0 2.5

1.0

1.5

2.0

2.5

OIL

NS

EA

SI

5.3 Bayesian Linear Regression Analysis using R-package.

> library(MCMCpack)

Loading required package: coda

Loading required package: lattice

Loading required package: MASS

##

## Markov Chain Monte Carlo Package (MCMCpack)

## Copyright (C) 2003-2013 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park

##

## Support provided by the U.S. National Science Foundation

## (Grants SES-0350646 and SES-0350613)

##

Warning messages:

1: package MCMCpack was built under R version 2.15.3 2: package coda was built under R version 2.15.3 > M6 summary(M6)

Iterations = 1001:11000

Thinning interval = 1

Number of chains = 1

Sample size per chain = 10000

1. Empirical mean and standard deviation for each variable, plus standard error of the mean:

Mean SD Naive SE Time-series SE

(Intercept) 3.31131 0.53659 0.0053659 0.0053659

PRICE 0.47069 0.09694 0.0009694 0.0009694

INTEREST 2.33433 0.53859 0.0053859 0.0056140

EXCHANGE -5.35102 0.78375 0.0078375 0.0078375

OIL 0.39044 0.09988 0.0009988 0.0009988

sigma2 0.05303 0.01061 0.0001061 0.0001168

Table 5. The estimated coefficients for Access Bank using Bayesian Regression Model.

2. Quantiles for each variable:

2.5% 25% 50% 75% 97.5%

(Intercept) 2.25156 2.95241 3.31539 3.66211 4.36502

PRICE 0.28003 0.40412 0.47118 0.53532 0.66045

INTEREST 1.28593 1.97042 2.34178 2.70126 3.38008

EXCHANGE -6.88844 -5.87923 -5.35285 -4.83245 -3.82332

OIL 0.19330 0.32454 0.39170 0.45712 0.58544

sigma2 0.03612 0.04546 0.05168 0.05898 0.07756

>plot(M6,trace=FALSE)

6.0 UBA PLC

6.1 Simple Linear Regression using R-package.

> data2 data2

> attach(data2)

> lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)

Call:


Coefficients

(Intercept) PRICE INTEREST EXCHANGE OIL

1.6753 0.4002 1.3577 -2.7630 0.2790

Table 8. The estimated coefficients for UBA.

> summary(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))

Call:


Residuals:

Min 1Q Median 3Q Max

-0.21473 -0.07511 -0.01378 0.05101 0.36808

Table 9.

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.67530 0.30128 5.561 8.15e-07 ***

PRICE 0.40024 0.02590 15.455 < 2e-16 ***

INTEREST 1.35771 0.27000 5.028 5.60e-06 ***

EXCHANGE -2.76303 0.43410 -6.365 4.12e-08 ***

OIL 0.27902 0.05058 5.516 9.59e-07 ***

---Table 10. The estimated coefficients with standard error for UBA.

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 0.1164 on 55 degrees of freedom

Multiple R-squared: 0.9532, Adjusted R-squared: 0.9497

F-statistic: 279.8 on 4 and 55 DF, p-value: < 2.2e-16

> anova(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))

Analysis of Variance Table

Response: NSEASI

Df Sum Sq Mean Sq F value Pr(>F)

PRICE 1 13.8116 13.8116 1018.7482 < 2.2e-16 ***

INTEREST 1 0.0006 0.0006 0.0418 0.8387

EXCHANGE 1 0.9472 0.9472 69.8640 2.280e-11 ***

OIL 1 0.4126 0.4126 30.4308 9.587e-07 ***

Residuals 55 0.7457 0.0136

Table 11. Anova table for UBA.

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

> plot(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)

1.0 1.5 2.0 2.5

1.0

1.5

2.0

OIL

NS

EA

SI

6.2 Bayesian Linear Regression Analysis using R-package.

> library(MCMCpack)

Loading required package: coda

Loading required package: lattice

Loading required package: MASS

##

## Markov Chain Monte Carlo Package (MCMCpack)

## Copyright (C) 2003-2013 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park

##

## Support provided by the U.S. National Science Foundation

## (Grants SES-0350646 and SES-0350613)

##

Warning messages:

1: package MCMCpack was built under R version 2.15.3 2: package coda was built under R version 2.15.3 > M6 summary(M6)

Iterations = 1001:11000

Thinning interval = 1

Number of chains = 1

Sample size per chain = 10000

1. Empirical mean and standard deviation for each variable, plus standard error of the mean:

Mean SD Naive SE Time-series SE

(Intercept) 1.67580 0.306723 3.06e-03 3.067e-03

PRICE 0.40046 0.026465 2.647e-04 2.647e-04

INTEREST 1.35941 0.275462 2.755e-03 2.755e-03

EXCHANGE -2.76445 0.440716 4.407e-03 4.407e-03

OIL 0.27802 0.051560 5.156e-04 5.156e-04

Sigma2 0.01411 0.002822 2.822e-05 3.109e-05

Table 12. The estimated coefficients with standard error for UBA using Bayesian Regression Model.

2. Quantiles for each variable:

2.5% 25% 50% 75% 97.5%

(Intercept) 1.070025 1.4706 1.67813 1.87632 2.27812

PRICE 0.348276 0.3827 0.40066 0.41820 0.45158

INTEREST 0.824582 1.1724 1.36324 1.54945 1.89541

EXCHANGE -3.623964 -3.0635 -2.76876 -2.46834 -1.90566

OIL 0.175582 0.2432 0.27886 0.31263 0.37938

sigma2 0.009611 0.0121 0.01375 0.01569 0.02064

>plot(M6,trace=FALSE)

0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.6

1.2

Density of (Intercept)

N = 10000 Bandw idth = 0.05086

0.30 0.35 0.40 0.45 0.50

05

10

15

Density of PRICE

N = 10000 Bandw idth = 0.004446

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.6

1.2

Density of INTEREST

N = 10000 Bandw idth = 0.04628

-4 -3 -2 -1

0.0

0.4

0.8

Density of EXCHANGE

N = 10000 Bandw idth = 0.07404

0.0 0.1 0.2 0.3 0.4 0.5

02

46

Density of OIL

N = 10000 Bandw idth = 0.008662

0.005 0.010 0.015 0.020 0.025 0.030

050

150

Density of sigma2

N = 10000 Bandw idth = 0.000451

7.0 Summary of finds, Conclusion and Recommendation.

7.1 Summary of finds and Conclusion

In this study, analysis of data extracted from Nigeria Stock Exchange was subjected to

investigate the relationship between the respond variable All-share Index and daily price of

stock, interest rate, exchange rate and oil price of the banking sector of the Nigeria Stock

Exchange; case study of two banks Access Bank Plc, and United Bank For Africa (UBA) Plc.

The result of the simple linear regression is very similar to the Bayesian Regression as shown in

the table below. Although more computationally intensive, the Bayesian Regression is similarly

easy to implement and automatically provides interval estimates for all parameters, including the

standard error. By the results of the analysis for the two banks the estimated coefficients are

statistically significant and the price of stock, Lending rate and oil price have positive effect on

the response variable All-share Index that is, as all the performance metrics increase All-share

Index increases except the Exchange rate; as the exchange rate goes down All-share Index

increases and vice versa.

SLR Std. Error BLR Std. Error

Intercept 3.31044 0.52732 3.31131 0.53659

Price of stock 0.46975 0.09486 0.47069 0.09694

Interest Rate 2.32985 0.52988 2.33433 0.53859

Exchange Rate -5.34694 0.77386 -5.35102 0.78375

Oil price 0.39228 0.09805 0.39044 0.09988

Table 14. The estimated coefficients with standard error (Std. Error) for Access Bank Plc. Using Simple Linear

Regression (SLR) and Bayesian Linear Regression (BLR).

SLR Std. Error BLR Std. Error

Intercept 1.67530 0.30128 1.67580 0.306723

Price of stock 0.40024 0.02590 0.40046 0.026465

Interest Rate 1.35771 0.27000 1.35941 0.275462

Exchange Rate -2.76303 0.43410 -2.76445 0.440716

Oil price 0.27902 0.05058 0.27802 0.051560

Table 15. The estimated coefficients with standard error (Std. Error) for UBA Plc. Using Simple Linear Regression

(SLR) and Bayesian Linear Regression (BLR).

Note: that All-share Index is an arbitrary number used in Stock Exchange for evaluation

purpose.

7.2 Recommendation.

Based on the information gathered and findings from the analysis carried out in this study, the

following recommendations are suggested.

1) There is need for the federal government to commence buying of shares of the banks on

the Nigeria Stock Exchange (NSE). When these shares are purchased, they will serve

twin purpose- being investment for the government which it can hold, earn returns and

later resell and increase the demand segment of the capital market following the upward

movement of both the market capitalization and the All-share Index.

2) There is need for advertisement for the banks involve being their public offer (IPO) so

that more people may participate in the purchasing of their shares which will increase the

percentage of their All-share Index.

3) The banks should not be over expose to the capital market because this significantly

increases the quantum of banks non- performing loans which invariably led to loss of

depositors fund with the banks.

4) The government fiscal policy on exchange rate, bank lending rate (interest rate) and oil

price should be look into because of the significant of its values in the profit of the banks.

REFERENCE

[1] Box, G. E. P., Hunter W. G., and Hunter J. S. (1978): Statistics for Experimenters. John

Wiley.

[2] Gelman, A., Carlin, J. B., Stern H. S. and Rubin, D. B. (1995): Bayesian Data Analysis.

Chapman and Hall.

[3] Kass, R. E. and Steffy, D. (1989): Approximate Bayesian inference in conditionally

independent hierarchical models (parametric empirical Bayes models). J. Amer. Statist. Assoc.,

84:717-726.

4] Lindley, D. V. and Smith, A. F. M. (1972): Bayes estimates for the linear model (with

discussion). J. R. Statist. Soc . Ser B 34: 1-41.

[5] R Development Core Team (2007). R: A language and environment for statistical

computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0,

URL http://www.R-project.org.

[6] Snedecor, G. W. and Cochran, W. G. (1989). Statistical Methods, 8th edition. IOWA State

University Press, Ames. IOWA.

[7] Tanner, M. A. (1996): Tools for Statistical Inference . Springer-Verlag

[8] Venables, W. N. and Replay, D. B. (2002). Modern Applied Statistics with S-PLUS .

Springer, New York.

APENDIX

ACCESS BANK PLC

S/NO NSEASI PRICE INTEREST RATE EXCHANGE RATE OIL PRICE

1 1.0000 1.0000 1.0000 1.0000 1.0000

2 0.8738 0.9503 0.9739 0.9999 1.1156

3 0.7844 0.8776 0.9077 0.9999 1.1531

4 0.7686 0.832 0.9899 0.9999 1.0375

5 0.9518 1.0103 0.9723 0.9997 1.0656

6 1.0761 1.2459 0.9947 1.0001 1.0953

7 0.9427 1.0279 1.0096 1.0001 1.1143

8 0.9008 0.8267 0.9184 0.9940 0.9589

9 0.8212 0.8136 0.9899 0.9747 1.0848

10 0.8520 0.8375 1.0096 0.9749 1.0381

11 0.8035 0.7890 1.0091 0.9736 1.0075

12 0.7818 0.7899 0.9189 0.9708 1.1192

13 0.8993 0.1564 0.9723 0.9708 1.0836

14 0.8989 0.1492 0.9675 0.9657 1.1076

15 0.8853 0.1395 0.9541 0.9593 1.2333

16 0.8752 0.1324 0.9616 0.9572 1.2461

17 0.9135 0.1337 0.9792 0.9571 1.2364

18 0.9464 0.1396 0.9685 0.9571 1.3185

19 1.0183 0.1404 0.9925 0.9566 1.3169

20 1.1920 0.1563 0.9808 0.9562 1.1357

21 1.2383 0.1653 0.9877 0.9559 1.0521

22 1.2427 0.3212 0.9979 0.9558 1.0607

23 1.2235 0.4102 0.9984 0.9558 1.1095

24 1.2300 0.3908 0.9952 0.9559 0.9721

25 1.3183 0.3987 0.9973 0.9558 1.0442

26 1.5093 0.5361 0.9941 0.9557 1.1213

27 1.5560 0.6228 1.0091 0.955 1.2163

28 1.7363 0.8112 0.9909 0.9537 1.2341

29 1.7938 0.9895 0.9157 0.9505 1.2802

30 1.9220 1.0771 0.9995 0.9496 1.3759

31 1.9379 1.0650 0.9792 0.9476 1.3150

32 1.9594 1.0548 0.9744 0.9256 1.4197

33 1.9373 1.0548 0.9744 0.9377 1.5181

34 1.9170 1.0938 0.9712 0.9269 1.5181

35 1.9876 1.0954 0.9755 0.8967 1.7003

36 2.0661 1.1952 0.9712 0.8798 1.6660

37 2.1883 1.3234 0.9883 0.8787 1.6909

38 2.3710 1.3491 0.9776 0.8786 1.7347

39 2.4384 1.3349 0.9419 0.8783 1.8953

40 2.3281 1.2094 0.9984 0.8780 2.0126

41 2.3017 1.1146 0.9552 0.8777 2.2850

42 2.1249 1.0065 0.9109 0.8775 2.4561

43 2.0158 0.9542 0.9557 0.8772 2.5114

44 1.8036 0.7896 0.9157 0.8770 2.1514

45 1.8048 0.7161 1.0251 0.8769 1.8536

46 1.5682 0.5817 1.0256 0.8768 1.3236

47 1.3137 0.4611 1.0117 0.8770 0.9523

48 1.1261 0.3596 1.1296 0.9714 0.7388

49 1.0000 0.3277 1.0789 1.0748 0.7950

50 0.8738 0.2448 1.2592 1.0968 0.7925

51 0.7844 0.2743 1.2752 1.1015 0.8762

52 0.7686 0.2795 1.2357 1.0975 0.9608

53 0.9518 0.4188 1.2192 1.1021 1.0905

54 1.0761 0.5341 1.2075 1.1086 1.3083

55 0.9427 0.3909 1.2160 1.1117 1.2362

56 0.9008 0.3475 1.2293 1.1357 1.3656

57 0.8212 0.3272 1.2251 1.1417 1.2856

58 0.8520 0.3797 1.2267 1.1173 1.3908

59 0.8035 0.3666 1.2320 1.1302 1.4601

60 0.7818 0.4034 1.2512 1.1202 1.4165

UBA PLC

S/NO NSEASI PRICE INTEREST RATE EXCHANGE RATE OIL PRICE

1 1.0000 1.0000 1.0000 1.0000 1.0000

2 0.8631 0.9331 0.9739 0.9999 1.1156

3 0.7837 0.8727 0.9077 0.9999 1.1531

4 0.7587 0.8247 0.9899 0.9999 1.0375

5 0.9272 0.9766 0.9723 0.9997 1.0656

6 1.0714 1.2379 0.9947 1.0001 1.0953

7 0.9368 1.0323 1.0096 1.0001 1.1143

8 0.8968 0.8232 0.9184 0.9940 0.9589

9 0.8157 0.8049 0.9899 0.9747 1.0848

10 0.8449 0.8283 1.0096 0.9749 1.0381

11 0.7979 0.7837 1.0091 0.9736 1.0075

12 0.7754 0.7820 0.9189 0.9708 1.1192

13 0.8913 0.7062 0.9723 0.9708 1.0836

14 0.8909 0.6640 0.9675 0.9657 1.1076

15 0.8774 0.6498 0.9541 0.9593 1.2333

16 0.8674 0.6974 0.9616 0.9572 1.2461

17 0.9054 0.7420 0.9792 0.9571 1.2364

18 0.9380 0.7697 0.9685 0.9571 1.3185

19 1.0092 0.8027 0.9925 0.9566 1.3169

20 1.1814 1.0475 0.9808 0.9562 1.1357

21 1.2273 1.2664 0.9877 0.9559 1.0521

22 1.2316 1.4314 0.9979 0.9558 1.0607

23 1.2126 1.3792 0.9984 0.9558 1.1095

24 1.2191 1.3788 0.9952 0.9559 0.9721

25 1.3066 1.6050 0.9973 0.9558 1.0442

26 1.4959 2.0831 0.9941 0.9557 1.1213

27 1.5422 2.0972 1.0091 0.9550 1.2163

28 1.7208 2.0972 0.9909 0.9537 1.2341

29 1.7778 2.1281 0.9157 0.9505 1.2801

30 1.9049 2.5679 0.9995 0.9496 1.3759

31 1.9207 2.9766 0.9792 0.9476 1.3150

32 1.942 2.9416 0.9744 0.9256 1.4197

33 1.9201 2.9345 0.9744 0.9377 1.5181

34 1.8999 2.9213 0.9712 0.9269 1.5181

35 1.9699 2.9358 0.9755 0.8967 1.7003

36 2.0477 2.7062 0.9712 0.8798 1.6660

37 2.1689 2.7634 0.9883 0.8787 1.6909

38 2.3499 2.761 0.9776 0.8786 1.7347

39 2.4167 2.7232 0.9419 0.8783 1.8953

40 2.3074 2.8921 0.9984 0.8780 2.0126

41 2.2812 3.2763 0.9552 0.8777 2.2850

42 2.106 1.8548 0.9109 0.8775 2.4561

43 1.9978 1.7688 0.9557 0.8772 2.5114

44 1.7876 1.5453 0.9157 0.8770 2.1514

45 1.7887 1.5439 1.0251 0.8769 1.8536

46 1.5543 1.2372 1.0256 0.8768 1.3236

47 1.3021 0.9342 1.0117 0.8770 0.9523

48 1.1161 0.7788 1.1296 0.9714 0.7388

49 0.9911 0.5766 1.0789 1.0748 0.7950

50 0.866 0.4917 1.2592 1.0968 0.7925

51 0.7774 0.4541 1.2752 1.1015 0.8762

52 0.7618 0.4883 1.2357 1.0975 0.9608

53 0.9433 0.7802 1.2192 1.1021 1.0905

54 1.0666 0.8142 1.2075 1.1086 1.3083

55 0.9343 0.6857 1.2160 1.1117 1.2362

56 0.8928 0.6524 1.2293 1.1357 1.3656

57 0.8139 0.6755 1.2251 1.1417 1.2856

58 0.8444s 0.7248 1.2267 1.1173 1.3908

59 0.7963 0.6494 1.2320 1.1302 1.4601

60 0.7748 0.6168 1.2512 1.1202 1.4165

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Bayesian Regression Analysis