Quantitative Research Methods A Practical Approach Abdul Waheed
Quantitative Research Methods A Practical Approach Abdul Waheed Quantitative Research Methods by Abdul Waheed 1
Chapter 5: Bivariate Data Analysis Learning Outcomes �Explain the cross-tabulation. �Explain the scatter diagrams. �Perform simple correlation analysis. �Conduct simple linear regression analysis. �Measure the fit of the regression model. �Test the significance of the regression coefficient �Perform unit root test for stationarity of variables. �Conduct a bivariate causality test. �Conduct an Engle-Granger cointegration test. Quantitative Research Methods by Abdul Waheed 2
Cross Tabulation �Some researcher wants to know the detailed frequency information. �A cross-tabulation is a joint frequency distribution of two or more variables. �If the data has two or more categories, crosstabulation can be used to compare these categories. �It can address many research questions and mostly used with demographic data. �If we construct a statistical test using cross-tabulation, then such tables are called contingency tables. Quantitative Research Methods by Abdul Waheed 3
Cross Tabulation Quantitative Research Methods by Abdul Waheed 4
Correlation Analysis �The correlation is the degree of the relationship existing between two (or more) variables. �The correlation can be linear or non-linear (curvilinear). �Correlation is said to be linear if the ratio of change is constant, otherwise non-linear. �The correlation between two variables can be positive, negative, or zero. Quantitative Research Methods by Abdul Waheed 5
Scatter Diagram �In a scatter diagram, we plot the paired values of two variables on a graph paper and keep them scattered. �The scatter diagram gives an idea about the nature and extent of the relationship between two variables. �If the scatter diagram shows that the paired values of two variables are closed to a positively sloped line (or curve), the variables positively correlate. �If the scatter diagram shows that the paired values of two variables are closed to a negatively sloped line (or curve), the variables negatively correlate. �If the paired values of two variables in a scatter diagram are dispersed all over the surface of the X. Y. plane, then there is no correlation or zero correlation between two variables. Quantitative Research Methods by Abdul Waheed 6
Scatter Diagram Quantitative Research Methods by Abdul Waheed 7
Correlation Coefficient �The scatter diagram just gives an idea about the nature and extent of the relationship between two variables. �However, for a precise quantitative measurement of relationship, a parameter is used, called the Correlation Coefficient. This parameter is introduced by Karl Pearson (1857 -1936) and is therefore called Pearson's Correlation Coefficient. �The value of the correlation coefficient lies between -1 and +1. The value of -1 shows perfect negative, while +1 shows a perfect positive correlation. The value of zero shows no correlation. �A value closer to -1 or +1 shows a strong while closer to zero shows weak correlation. Quantitative Research Methods by Abdul Waheed 8
Simple Linear Regression Analysis �Simple correlation analysis gives an idea about the nature and degree of the relationship between two variables. � Sometimes we are interested in how much one variable will change due to change in another variable. This regression analysis is used for this purpose. �Sir Francis Galton (1886) first introduced the concept of linear aggression with the study on the height of parents and the height of children. �He found that there is a tendency for all tall parents to have tall children and for short parents to have short children. �He further discovered that the average height of the children tends to regress (or move) toward the average Quantitative Research Methods by Abdul Waheed 9 height in the population.
Simple Linear Regression Analysis Suppose Y is savings, and X is an income variable. The simple regression model (also called the two-variable linear regression model or bivariate linear regression model) is written as: �In the regression model, Y is called the dependent variable, the explained variable, response variable, the predicted variable, or the regressand. �The variable X is called the independent variable, the explanatory variable, the control variable, the predictor variable, or the regressor. Quantitative Research Methods by Abdul Waheed 10
Simple Linear Regression Analysis �The simple linear regression model can be estimated from the sample observations using the following equation. �Where Quantitative Research Methods by Abdul Waheed 11
Simple Linear Regression Analysis �The estimators that are found through Ordinary Least Square (OLS) methods are BLUE-the Best Linear Unbiased Estimator. �An estimator is unbiased if its expected value (or mean value) is equal to its true value. �The OLS estimator is efficient because they have smaller variance compare to other estimators. The OLS estimator is consistent because its estimated value approaches to the true value when population size becomes very large. �Thus, the Gauss-Markov theorem, states that when all the assumption of least square method holds, the OLS estimators are BLUE (Best Linear Unbiased Estimator). Quantitative Research Methods by Abdul Waheed 12
The Assumptions of Least Square Method �The regression model is linear in parameters. �The values of the explanatory variable (X) are fixed in repeated samples. �The mean or expected value of the random error term is zero. �The variance of the error term is the same for all observations. �The error terms are uncorrelated with each other. Quantitative Research Methods by Abdul Waheed 13
The Assumptions of Least Square Method �There is zero covariance between the error term and Xi. �The number of explanatory variables should be less than the number of observations. �The values of X in a given sample must not all be the same. �The specification of the regression model is correct. �The relationship between the explanatory variables is not perfect. Quantitative Research Methods by Abdul Waheed 14
Interpretation of Coefficients This shows that the minimum average value of Y (saving) is $20. 25, irrespective of income level (X). A value of 0. 15 shows that there is a positive effect of income (X) on saving (Y). Furthermore, one dollar change in income (X) will change saving (Y) by 0. 15 dollars. We can predict the value of saving(Y) by putting a specific value of income (X). Quantitative Research Methods by Abdul Waheed 15
The Goodness of Fit �To measure how good fit is the regression model, we use the coefficient of determination (or ), which is the ratio of ESS (Explained Sum of Squares) to TSS(Total Sum of Squares). �Figure 5. 3 shows �TSS (Total Sum of Squares), that measures the total variation of actual Y about their mean. �ESS (Explained Sum of Squares) that measures the variation of estimated Y about their mean. �RSS (Unexplained or Residual Sum of Squares), which measures the variation of Y about the regression line. Quantitative Research Methods by Abdul Waheed 16
Figure 5. 3: Breakdown of the Variation Quantitative Research Methods by Abdul Waheed 17
EViews Output for Simple Saving Model Estimates Dependent Variable: DSAV Method: Least Squares Date: 07/11/20 Time: 14: 53 Sample: 1982 2015 Included observations: 34 Variable Coefficient Std. Error t-Statistic Prob. C GDPRB -951. 1079 0. 268440 73. 83466 0. 011170 -12. 88159 24. 03209 0. 0000 R-squared Adjusted R-squared S. E. of regression Sum squared resid Log-likelihood F-statistic Prob. (F-statistic) 0. 947502 0. 945861 157. 5671 794476. 7 -219. 2482 577. 5414 0. 000000 Mean dependent var S. D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criterion. Durbin-Watson stat Quantitative Research Methods by Abdul Waheed 700. 1853 677. 1895 13. 01460 13. 10439 13. 04522 0. 803063 18
EViews Output for Simple Saving Model Estimates Quantitative Research Methods by Abdul Waheed 19
The Unit Root Test �A time-series variable whose moments (mainly mean and variance are time-invariant is said to be stationary. �A time-series variable whose mean and variance changes over time is called nonstationarity. �When non-stationary series are used to get regression results, it will be termed as spurious regression. �The unit-root tests are used to see if a variable is stationary or non-stationary. Quantitative Research Methods by Abdul Waheed 20
EViews Output for Unit Root Test for the Variables of the Saving Model Null Hypothesis: DSAV has a unit root Exogenous: Constant, Linear Trend Lag Length: 3 (Automatic - based on SIC, maxlag=8) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level *Mac. Kinnon (1996) one-sided p-values. t-Statistic Prob. * -0. 100700 -4. 296729 -3. 568379 -3. 218382 0. 9924 Quantitative Research Methods by Abdul Waheed 21
EViews Output for Unit Root Test for the Variables of the Saving Model Null Hypothesis: D(DSAV) has a unit root Exogenous: Constant, Linear Trend Lag Length: 2 (Automatic - based on SIC, max lag=8) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob. * -6. 858542 -4. 296729 -3. 568379 -3. 218382 0. 0000 Quantitative Research Methods by Abdul Waheed 22
Cointegration Test �Cointegration is the existence of a long-run relationship between two or more variables. �Granger (1981) introducedthe concept of cointegration, which was further extended by Engle and Granger (1987). �The idea is that, although a time series may have a nonstationary behavior, however, their linear combination may remove the common trend component and be stationary, and the relevant time series variables are then cointegrated. � If the time series are cointegrated, thenit means there exists a long run or steady-state equilibrium relationshipamong variables. Quantitative Research Methods by Abdul Waheed 23
EViews Output for Unit Root Test for the Residual of Saving Model Null Hypothesis: RESIDUAL has a unit root Exogenous: None Lag Length: 0 (Automatic - based on SIC, maxlag=8) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob. * -2. 949435 0. 0044 -2. 636901 -1. 951332 -1. 610747 *Mac. Kinnon (1996) one-sided p-values. Quantitative Research Methods by Abdul Waheed 24
Causality Analysis �If there is a positive or negative correlation between X and Y, it does not imply that cause Y or Y cause X. �Thus, from correlation, the direction of causation is not confirmed. �To test the direction of causation, we perform Granger causality tests. �We can have unidirectional causality, written as Yt => Xt) or bi-directional causality written as Xt Yt. Quantitative Research Methods by Abdul Waheed 25
EViews Output for. Granger Causality Tests for Variables of Savings Model Pairwise Granger Causality Tests Date: 07/11/20 Time: 16: 10 Sample: 1982 2015 Lags: 1 Null Hypothesis: GDPRB does not Granger Cause DSAV does not Granger Cause GDPRB Obs F-Statistic Prob. 33 7. 88814 4. 62268 0. 0087 0. 0397 Quantitative Research Methods by Abdul Waheed 26
Research Activity-1 Based on some theory, formulate a model and get the time series data of two (from times series data file) variables. Perform the following task using computer software. �Plot the line graph of the variables. �Draw a scatter diagram and interpret the results. �Get a descriptive statistics table and interpret the results. �Get the correlation coefficient and interpret the results. �Perform the Unit Root Test on each variable and interpret the results. �Bivariate Granger Causality Analysis. � Estimate the regression model using the OLS method and interpret the results. �Engle-Granger Cointegration Test. Quantitative Research Methods by Abdul Waheed 27
Research Activity-2 Pick any one model from Box-5. 3, get the data of the variables from time-series data file, and perform the following task using the software. �Estimate all models using the OLS method and interpret the results. �Get the actual, fitted values of the dependent variable for all models and plot the residual. �Perform unit root test and comment on the stationarity of the variables of all models. �Conduct a bivariate Granger causality test for all models and comment on the direction of the causality. �Conduct the Engle-Granger cointegration test for all models and comment on Methods the results. Quantitative Research by Abdul Waheed 28
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