Relationship between Consumption and Income - Essay Example

Introduction Economic theory exemplifies that there is direct relationship between consumption and income. The theory further s that consumption is on the other hand negatively or indirectly associated to real interest rate and unemployment. It is hence the general objective of the study to express the relationships of these variables using models. Specifically, this study aims to: a. conduct misspecification tests for the two models provided; b. estimate parameters of the model using the Ordinary Least Squares method; c. determine which of the two models provided is valid to be used for inference; d. interpret the results of the statistical analysis; and e. enhance the skills in using E-views to analyze time series data and to perform the specific objectives stated above. Hence, for this particular undertaking, time series analysis was conducted. The variables considered for the study include the natural logarithm of real per-capita consumption (LCJ), the natural logarithm of real per-capita income (LYJ), the real interest rate (RJ) and the rate of unemployment (UJ). In addition to these four variables, there included in the model is a dummy variable (D83). This particular variable takes the value of one if the year being considered is 1983 and zero, otherwise. Moreover, annual data for Belgian aggregates for forty years, which is from 1955 to 1994, was used in the study. Statistical analysis of the time series data considered in this study is facilitated by the use of E-views. According to Judge (2003), E-views is a "modern, powerful but intuitive econometrics software". It operates by manipulating objects such as data series and equations. With the computer package, the author mentions that the "data series can also be viewed as a table of values or as points plotted on a graph". For the 40 time series observations provided, the ordinary least squares estimation procedure was used to estimate the parameters of the model with real per-capita consumption as the dependent variable. The models considered in the study depicting the relationship of the variables consumption, interest rate, per capita income and unemployment are: (a) LCJ = LYJ + RIJ + UJ + u (Model 1) (b) LCJ = C + LYJ + RIJ + UJ + LCJ(-1) + LYJ(-1) + RIJ(-1) + UJ(-1). (Model 2) Results and Discussion Before the final model (model 3) was arrived at, models 1 and 2 were both tested for various assumptions. To determine if there is presence of first order autocorrelation, the Durbin-Watson test and the Breusch-Godfrey test were used. Ramsey's RESET test, on the one hand was utilized to determine if non-linearity of the functional form. Normality of residuals was checked by way of graphs as well as statistics provided by the Jarque-Bera test. The White's test was used to examine the presence of heteroscedasticity in the data set. Finally, first-order ARCH effects were also investigated. All tests of hypotheses were conducted using the 5% level of significance. A summary of the results of the verification of the assumptions (misspecification tests), estimated coefficients for the parameters of the model with the corresponding t-ratios and the adjusted coefficient of determination are provided in table 1 below for the three models. A discussion of the results obtained for the analysis of the time series data on consumption is also provided below. Table 1. Summary of the Results for the Three Models Model 1 Model 2 Model 3 Intercept -0.101 -0.033 -0.020 t-statistic value -1.235 -0.174 -0.580 LYJt 1.019 0.756 0.822 t-statistic value 95.857 5.957 8.477 RIJt 98.734 35.773 30.969 t-statistic value 5.759 3.042 2.918 UJt 0.310 0.065 t-statistic value 0.320 0.050 LCJt-1 0.904 0.767 t-statistic value 7.968 9.299 LYJt-1 -0.663 -0.585 t-statistic value -5.368 -6.012 RIJt-1 -23.756 t-statistic value -1.679 UJt-1 -0.084 t-statistic value -0.070 Fit Measures Adjusted R2 0.998 0.999 0.999 Standard Error F(23) 0.023 0.012 0.012 1.061 0.380 Misspecification Test DW 0.752 1.809 1.694 FSC1 16.655 0.465 1.070 0.000 0.495 0.301 FF1 2.421 2.605 2.889 0.120 0.107 0.089 2N2 0.125 0.600 0.088 0.940 0.741 0.957 FH 12.017 21.595 14.200 0.062 0.087 0.077 FARCH1 3.282 0.227 0.617 0.070 0.634 0.432 Misspecification Test Results for Model 1 The Durbin-Watson test for 1st order autocorrelation provided a test statistic value of 0.752. To determine if there is evidence or not of positive first-order correlation, the critical values considered were 1.328 and 1.658 which are the lower and upper bounds, respectively of the Durbin-Watson distribution. Since the computed d value is less than 1.328, there is sufficient evidence to conclude that there exists a positive first-order autocorrelation. Model 1 was also tested for the presence of higher order autocorrelation using the Breusch-Godfrey Serial Correlation test in addition to the Durbin-Watson test. The computer generated output for this test reveals a probability value of almost zero. To be able to determine if there is autocorrelation, the probability value should not exceed the level of significance. For the data, since the probability value is almost zero and it is very much smaller than 5%, then there is evidence to say that there is indeed significant autocorrelation. The third assumption tested for model 1 is the assumption of non-linearity. The Ramsey's RESET test was used for this purpose. Again, from the generated computer output, a probability value of 0.12. This value however exceeded the level of significance which is 5%. Hence, it can be concluded with 95% confidence that non-linearity of the functional form of the model is not significant. To determine if the assumption on the normality of residuals is satisfied, the Jarque-Bera test was conducted. The computed test statistic value is 0.1246 which translates to a probability value of 0.9396. The decision as to the rejection or acceptance of the null hypothesis of normality of residuals depends upon the relationship that exists between the probability value and the level of significance. If the probability value is less than the set level of significance, then it can be inferred that normality of residuals is not satisfied. Fortunately for the data, normality of residuals is satisfied since 0.9396 is much greater than 5%. This was also shown by the graph as it depicts a nearly symmetrical bell-shaped curve. The fifth misspecification test if the White's test (without cross terms). This test is conducted to determine if the residuals are heteroscedastic or not. A probability of 0.62 indicates heteroscedasticity is insignificant at the 5% level because 0.62 is greater than the set level of significance. The last misspecification test is the ARCH (1) test. The output generated reveals P[ARCH (1)] equal to 0.0701. This probability value is greater than 5% which means that the first-order ARCH effect for the residuals is not significant at 5%. The overall assessment for model 1 indicates that misspecification is evident in terms of the first-order autocorrelation which implies that the OLS estimates cannot be used to attain valid inference. The model then needs to be re-specified. The re-specified model is now terms of the lag 1 of the variables considered in the study and this is the 2nd model. The first model consisted of 5 components including the dependent variable which the logarithm of consumption. The second model which is the re-specified one is now composed of 9 coefficients including the dependent variable. Misspecification Test Results for Model 2 The generated Durbin-Watson test statistic value for model 2 is 1.809. Using the same critical values of model 1 for the same misspecification test, 1.809 is greater than the upper bound of the distribution which is 1.658. This suggests no positive first-order autocorrelation. To test for higher order autocorrelation, again the Breusch-Godfrey Serial Correlation test was used. The generated probability value of 0.50 states that there is no autocorrelation and it is not significant at 5% level. To test model 2 for the satisfaction of the assumption of non-linearity, Ramsey's RESET test was also used. A probability value of 0.110 which is greater than 5% indicates that the assumption of the linearity of the model is now satisfied with model 2 at 5%. Testing the normality of the residuals, the Jarque-Bera test was used. By closely examining the graph, the normal curve projects a slightly negatively-skewed distribution. However, the deviation from symmetry was not that significant as evidenced by the probability value obtained using the test itself. A probability value of 0.74 exceeds 0.05. This reveals that indeed the assumption of normality of the residuals is satisfied. The fifth misspecification test White's Test, without cross terms to determine if the residuals are homoscedastic or heteroscedastic was conducted subjecting model 2 for evaluation. A probability value of 0.09 was obtained when the data was analysed using E-views. Again, the rule is to reject the null hypothesis of homoscedasticity of the errors when the probability value is greater than 5%. For the data, it seems that this is the case. Since the probability value of 0.09 exceeds 5%, then heteroscedasticity of errors is not significant. Finally, model 2 is also subjected to the ARCH (1) test which will determine if ARCH(2) effects is evident in the residuals. The computer output which is listed in the appendix section showed Pr[ARCH (2)] equal to 0.63. The probability value of 0.63 suggests that 2nd order ARCH effect is not evident. This is because 0.63 is very much larger than the set level of significance which is 0.05. The overall assessment for model 2 indicates that misspecification is not evident for the second model. The assumptions on the absence of first-order and higher order autocorrelation, normality and homoscedasticity of the residuals, linearity of the model and the absence of the second-order ARCH effect on the residuals were all satisfied. The principle of parsimony in model building is also very important. Milton Friedman (as cited by Gujarati, 1995) simply describes the significance of this principle by saying, "a hypothesis [model] is important if 'explains' much by little". Gujarati (1995) expounds on this by saying that in model building, what one should do is to introduce in the model only a few major variables which already capture the essence of the event under study. All other minor and random effects or influences should be captured by the error term of the stochastic model. Hence, for the data at hand, model 3 was obtained. Results of the misspecification tests for the 3rd model were also given by table 1. First and foremost, the Durbin-Watson test statistic value is 1.694. This value is still beyond the upper bound of the distribution which is 1.658. Hence, this is indicative of the absence of positive first-order autocorrelation in the data set. The Breusch-Godfrey Serial Correlation test moreover suggests that there is no higher order autocorrelation. A probability value of 0.301 which is greater than 5% was obtained. Hence, at that particular level, higher order autocorrelation is not significant. The third test is on the assumption of non-linearity of the model for which the Ramsey's RESET test can be of use. A probability value of 0.089 also indicates that model 3 satisfied the assumption of linearity because the probability value exceeded that of the level of significance. In terms of the normality of the residuals, the Jarque Bera test proved that indeed model 3 was able to satisfy this assumption since the residuals proved to be normal as evidenced by a 0.957 probability value. The last two assumptions are on the heteroscedasticity of the residuals and the presence of ARCH effect in the residuals. For these two assumptions, the White's test and the ARCH test were used respectively. The residuals are homoscedastic. This is evident because of the probability value of 0.077 which does not exceed 5%. There is also absence of the ARCH effect among the residuals because the probability value of 0.432 is also greater than the set level of significance. As such, it can be deduced that for model 3, all assumptions were satisfied. The decision now is choosing between models 2 and 3. Both of these models were able to satisfy all the assumptions for the inference generated from them to be valid. To be able to decide which among the two models will be the final one, the adjusted coefficients of determination of the two were also evaluated as well as their respective standard errors. In terms of the adjusted coefficient of determination, both of the models were able to show that 99.9% of the total variation in the logarithm of consumption is explained by the regressors of each of the two models. Hence, the adjusted R2 may not be enough to be able to decide which model to use. The standard error, on the other hand, also showed a value of 0.012. Hence, in terms of the precision of the estimates models 2 and 3 are equally preferred. The principle of parsimony however can help in the decision-making. Model 3 is only composed of 5 components. This translates to one dependent variable and 4 regressors. Model 2, on the one hand consists of 9 components. Hence, by following the principle of parsimony model 3 is the most logical model to choose. The final model therefore for the data on Belgian aggregates from 1955 to 1994 is given by: LCJ = -0.020 + 0.822LYJt + 30.969RIJt + 0.767LCJt-1 - 0.585LYJt-1. Hence, the natural logarithm of real per-capita consumption is a function of natural logarithm of real per-capita income at time t, the real interest rate at time t, the natural logarithm of real per-capita consumption at time t - 1 and the natural logarithm of real per-capita income at time t - 1. If all the values of the independent variables are equal to zero, the natural logarithm of real per-capita consumption is -0.020 which is equivalent to 0.9802 real per-capita consumption. The natural logarithm of real per-capita consumption has direct association with natural logarithm of real per-capita income at time t, the real interest rate at time t and the natural logarithm of real per-capita consumption at time t - 1 but is indirectly related to the natural logarithm of real per-capita income at time t - 1. Conclusion As an ending remark, analysis of a time series data was done for this study. Three models were fitted involving the variables consumption, interest rate, income and unemployment rate. The three models were subjected each to the six misspecification tests. Models 2 and 3 both satisfied the assumptions however by the principle of parsimony of parameters, model 3 was selected. The results of the study showed that real per-capita consumption at time t is significantly affected by real per-capita income and real interest rate at time t, real per-capita income at time t - 1 and real per-capita consumption at time t - 1. Given the data on Belgian aggregates, unemployment rate was not found to affect real per-capita consumption. References Gujarati, D. N. (1995). Basic Econometrics Third Edition. Singapore: McGraw-Hill, Inc., 1995. Judge, G. (2003). Introduction to Eviews. April 2003. University of Portsmouth Business School. 13 August 2006 Read More