How Interest Rate, Loans, and Deposits affect Qatar National Bank Profit - Statistics Project Example

Group Project: Stat 222 How Interest Rate, Loans, and Deposits affect Qatar National Bank Profit 0 Variables Dependent Variable The dependent variable in this study is Corporate profit (QNB). Corporate profit was selected as the dependent variable in this study because naturally, a bank’s corporate profit is measured on the basis of many other variables on which it depends. Corporate profit is defined as a statistic that is reported quarterly by a bank that summarizes the net income of the banking corporation (Ross, 2001). It is also an economic indicator that is applied in calculation of net income on the basis of a number of different measures including book profits, profits from current production, and after-tax profits. Banking corporations report their corporate profits after every quarter of their financial year. Independent Variables This study included three independent variables; Interest Rates, loans, and deposits. According to Ross (2001), various factors directly affect the corporate profits of a company. These include loans given out to the customers, the interest rates charged on the loans, deposits, and other factors. Loans and the interests charged on them directly generate profits for banks and therefore the level of rate of interest also affects the amount of profits that a banking corporation gets. On the other hand, deposits by customers enable a bank to have capital to transact business including giving out loans to other customers alongside other profitable ventures (Edmister, 2002). Overall, these factors affect corporate profit and therefore have a relationship with it. Why the Variables Exhibit Relationship The four variables in this study (Corporate Profit, Interest Rates, Loans, and Deposits) were selected because they exhibit a relationship with each other. Firstly, Corporate Profit (dependent variable) is directly and indirectly affected by the other three variables (independent variables). According to Metawa and Almossawi (2008), a company’s corporate profit has a direct relationship with the volume of transactions the business carries out in terms of deposits and giving out loans. Therefore loans to customers and the deposits made by customers to their accounts affect a bank’s profit directly and have a relationship with it. On the other hand Corporate Profit directly and indirectly depends on interests charged on loans. 2.0 Sample Regression Equation For this study data was collected and organized in an excel spreadsheet for analysis. The data was entered with the first column representing the year of an entry, ranging from 2003 to 2012. The regression was calculated with an intercept and three regressors (The independent variables) including Interest rates , Loans, and Deposits over a period of ten years (The table is provided in Appendix 1). The regression model for Corporate Profit can be expressed as: y = β1 + β2 x2 + β3 x3 + u It is assumed that the error u is homoscedastic, meaning it is independent with constant variance. The regression line equation can thus be estimated as: y = b1 + b2 x2 + b3 x3 + b4b4 + b5b5 Running the regression test using excel, the output consists of three main components; the regression statistics table, the ANOVA table, and the regression coefficients table. The following findings are obtained: Regression Statistics Table Regression Statistics Multiple R 0.779524782 R Square 0.607658886 Adjusted R Square 0.215317773 Standard Error 6.115428082 Observations 9 Multiple R= 0.779523: It refers to the square toot of R2 R2= 0.607659 Adjusted R2 = 0.215318 and it is used if there is more than one x variable like in the case of this analysis. The standard error refers to the sample estimate of the standard deviation of error u. The results therefore give an overall goodness of fit measure of R2= 0.6077. Thus the correlation between y and y-hat is 0.779523 (0.6077 when it is squared). Thus, Adjusted R2= R2 – (1- R2)*(k-1)/(n-k) R2= 0.6077 implies that 60.77% of the variation of ybar (y’s mean) is explained by the three regressors (Interest Rates, Loans, and Deposits). ANOVA Table ANOVA   df SS MS F Significance F Regression 4 231.6913131 57.92282826 1.548802472 0.341007 Residual 4 149.5938425 37.39846062 Total 8 381.2851556       The ANOVA (Analysis of Variance) table usually splits the sum of squares into its components. The total sum of squares = Residual (error) sum of squares + Regression (or explained) sum of squares. The value of F is generally interpreted as shown below: The column labeled significance F has the associated P-value. Therefore since 0.3410 > 0.05, H0 is not rejected at significance level 0.05. This means that the findings of the regression model are significant at the 0.05 significance level (99% level of confidence). Regression Coefficients Table   Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -17.916 29.165 -0.614 0.572 -98.893 63.061 -98.893 63.061 Loans (% of Capital) 0.265 0.385 0.690 0.527 -0.803 1.334 -0.803 1.334 Deposits (% of Capital) -3.868 7.515 -0.514 0.633 -24.734 16.997 -24.734 16.997 Interest Rate (%) 1.126 0.552 2.039 0.111 -0.407 2.659 -0.407 2.659 (All values reduced to 3 decimal places) Letting βn denote the population coefficient of the nth regressor (exports, imports, population growth), then The column "Coefficient" gives the least squares estimates of βn. The column "Standard error" gives the standard errors (i.e. the estimated standard deviation) of the least squares estimates bn of βn. The column "t Stat" gives the computed t-statistic for H0: βn = 0 against Ha: βn ≠ 0. The column "P-value" gives the p-value for test of H0: βn = 0 against Ha: βn ≠ 0. The columns "Lower 95%" and "Upper 95%" values define a 95% confidence interval for βn. The output above can be simply summarized by indicating that the fitted line is: y = b1 + b2 x2 + b3 x3 + b4b4 y = -17.916 + 0.265x2 - 0.368x3 + 0.492x4 Where; y = Corporate Profits x2 = Loans x3 = Deposits x4 = Interest Rates Coefficient of Determination The coefficient of determination is denoted by R. Multiple R= 0.779523: It refers to the square toot of R2. In this case, R2= 0.607659. This implies that 60.77% of the variation of the dependent variable (Corporate Profits) is explained by the four independent variables (Loans, Deposits, and Interest Rates). Difference between R2 and Adjusted R2 The coefficient of determination can be adjusted for degrees of freedom, in which case it is referred to as adjusted coefficient of determination (Adjusted R2). Generally the coefficient of determination R2 illustrates how well data points fit into the regression equation. The main difference between R2 and adjusted R2 is that the former assumed that every single variable in the sample explains the variation in the dependent variable. On the other hand, adjusted R2 actually indicates the percentage of variation that is explained by only the independent variables that really affect the dependent variable, thus it leaves out any independent variables that do not explain the variation in the dependent. Adjusted R2 mainly penalizes the researcher for adding independent variables into the model that do not fit it. For example, by adding variables that do not actually relate to the dependent variable. In most cases adjusted R2 is applied when dealing with samples because in this case variables are added into the model at the discretion of the researcher. However, when dealing with a whole population it is not necessary. Adjusted R2 is applicable only if there is more than one x variable like in the case of this analysis. In the case of this study, the results give an overall goodness of fit measure of R2= 0.6077. Thus the correlation between y and y-hat is 0.779523 (0.6077 when it is squared). Thus, Adjusted R2= R2 – (1- R2)*(k-1)/(n-k) Adjusted R2 = 0.215318 This implies that one or more of the independent variables does not fit into the correlation model. 3.0 Model Validity According to Draper and Smith (2008), before application of a regression model, the model should be tested for validity in order to ensure that it produces the right outcomes. Several procedures have been proposed for model validation, these include comparison of the model predictions and coefficients with the physical theory used in developing the model, collecting new data to check the model predictions for consistency, comparing the results of a model with theoretical models and simulated data, and reserving a portion of the available data to get an independent measure of the model prediction accuracy. However, when using Excel to carry out regression analysis, the easiest way to test the validity of a model is by examining the outcome of significance F (Savage, 2002). In this study Significance F= 0.341007. This can be interpreted to mean that the probability that the Regression output could have been obtained by chance (Thus making the model invalid) is 34%. This means that there is a 34% chance that the model and the regression outputs were obtained by chance. 4.0 Test of Linearity of Relationship between Dependent Variable and Independent Variables Linearity is described by the algebraic equation of a straight line defined by y = mx + b. the equation generally describes a set of data with one independent variable (x), y being the dependent variable, and m being the gradient of the line. When the line represents more than one independent variable, the equation of the line assumes the form y = m1x1 + m2x2 … + mnxn + b. The relationship between the dependent and independent variables to be regarded as linear, the regression findings have to fit into the equation accurately. In this case, x1 through xn represent the independent variables. y = -17.916 + 0.265x2 - 0.368x3 + 0.492x4 + b Where; y = Corporate Profits (dependent variable) x2 = Loans x3 = Deposits x4 = Interest Rates One of the conditions for linearity of the relationship between the dependent variable and the independent variables is that the contributions of the various different independent variables to the predictions should be additive. From the equation above, it is evident that the contribution of (x3: % of deposits) is negative. This implies that the relationship between corporate profits and the independent variables is not linear. 5.0 Residuals and Predicted Values The residuals and predicted values for the regression model were obtained automatically upon running the data and are shown in the table below. The residuals illustrate how far the actual data points are scattered from the points predicted by the linear regression equation. Observation Corporate Profits (% of Capital) Residuals 1 7.024375742 -3.534375742 2 12.84646473 6.37353527 3 22.18438272 3.985617284 4 18.97760662 -0.987606622 5 22.83758778 -5.177587778 6 14.67299927 -2.712999271 7 12.47485359 4.255146411 8 11.69087121 2.429128785 9 10.83085834 -4.630858336 6.0 Distribution of Error Error distribution is considered to be normal when it is not skewed by the presence of large outliers. Normality is usually violated by a few large outliers. In a regression analysis, parameters are usually estimated on the basis of minimization of the squared error. A few large outliers can therefore cause a disproportionate influence on the estimates of the parameters thus bringing about a skewed error of distribution. Lack of non-normality may significantly affect confidence intervals because they are computed on the basis of normally distributed errors. The normal probability plot for the residuals represents the best test for normally distributed errors. This plot presents the fractiles of error distribution versus the fractiles of normal distributions with the same mean and variance (Ezekiel & Fox, 2009). The normal probability plot for this regression analysis is illustrated above and it indicates a slightly s-shaped pattern implying that the residuals have excessive kurtosis, meaning that there are too many large or small errors in both directions. The variance of the error is said to be constant when plotting the error against the expected value produces a constant variance of the error predicted value. The distribution of the dots representing residuals is supposed to be approximately equal on both sides of the 0 residual line for the variance error to be constant. In all the four independent variables, this criterion was met as shown in the figure below. 7.0 Autocorrelation The autocorrelation function in Excel regression is computed as; Where: yt = value of time series at a particular time t. h = the lag order. T = the number of missing values in the time series data. ybar = is the sample mean of the time series. 8.0 Plot of Residuals versus Time Periods The observations are already arranged in time-series with the first observation referring to the year 2003 progressively up to 2012. Observation Residuals 1 -3.534375742 2 6.37353527 3 3.985617284 4 -0.987606622 5 -5.177587778 6 -2.712999271 7 4.255146411 8 2.429128785 9 -4.630858336 The residual table is produced by the Excel regression once included in the command and it is shown below. The plot of residuals versus time is then obtained by developing a scatter plot as indicated above. The plot reveals that there appears to be autocorrelation in the data. 9.0 Dublin-Watson Test In the Dublin-Watson test, the d statistic is calculated using the statistic below: Where: ei = yi – ŷi are the residuals, and n = the number elements in the sample. d assumes values between 0 and 4. A value of 2, which is exactly the middle of the range, means that there is absence of autocorrelation. Any value significantly below 2 (particularly values below 1) indicates the presence of positive autocorrelation in the data. On the other hand a value substantially above 2 indicates negative autocorrelation in the data. The Dublin-Watson value (d statistic) to measure autocorrelation was calculated in Excel using the following function; d =SUM(SUMXMY2(D41:D48,D40:D47))/SUMSQ(F40:F48) The following findings were obtained: Observation Predicted Corporate Profits(% Capital) Residuals Percentile Corporate Profits(% Capital) 1 7.024375742 -3.534375742 5.555555556 3.49 2 12.84646473 6.37353527 16.66666667 6.2 3 22.18438272 3.985617284 27.77777778 11.96 4 18.97760662 -0.987606622 38.88888889 14.12 5 22.83758778 -5.177587778 50 16.73 6 14.67299927 -2.712999271 61.11111111 17.66 7 12.47485359 4.255146411 72.22222222 17.99 8 11.69087121 2.429128785 83.33333333 19.22 9 10.83085834 -4.630858336 94.44444444 26.17 d=SUM(SUMXMY2(D41:D48,D40:D47))/SUMSQ(F40:F48) 0.008491701 From the outcome d= 0.0085; this is substantially below 2, indicating that the data in this regression displays a strong positive autocorrelation. References Draper, N. & Smith, H. (2008). Applied Regression Analysis. New York: John Wiley & Sons. Edmister, R. (2002). Margin Analysis for Consumer Deposit Interest Rate Policy. Journal of Bank Research, 9, 179-84. Ezekiel, M. & Fox, K. A. (2009). Methods of Correlation and Regression Analysis, Third Edition. New York: John Wiley & Sons. Metawa, S.A., and Almossawi, M. (2008). Banking Behaviour of Bank Customers: Perspectives and Implications. International Journal of Bank Marketing, 16(7), 299-313. Ross, P. S. (2001). Commercial Bank Management. New York: Richard Irwin Inc. Savage, L. J. (2002). The Foundations of Statistica1 Inference. London: Methuen. London. Read More