Analyzing Income Level of UK Consumers - Assignment Example

Question Examining the Data Prior to Estimation Provide time series plots of the data (all three series) and describe the behaviour of the series.Figure 1: real income In net effect, the income level of UK consumers has risen since 1954. Between 1954 and around 1973, the level of income has maintained a positive rate of increase. However, between 1972 and 1973, and again between 1978 and 1979, the rate of increase was very drastic, implying that those years could have experienced an economic boom, when consumers receive a high level of incomes due to favorable economic conditions. Nevertheless, there are periods when the economy was experiencing a recession, which resulted in a reduction in real income, for example between 1975 and 1977 and again between 1980 and 1982. Figure 2: Logarithms of real price level The price level has maintained a positive rate of increase since 1954. Perhaps, the price level has maintained this trend because of inflation, which makes money lose value and hence the increase in the general prices of goods and services in the economy. Ideally, the rate of price level increase is very rapid during some periods such as between 1973 and 1975, perhaps because of the economic slowdown which results in a reduction in production level and consequent increase in prices of goods and services. Figure 3: logarithms of real consumption From 1955 through 1984, the consumption maintained an almost constant increase in the most predictable rate of increase being experienced between 1954 and 1973. Between 1973 and 1978, consumption reduced remarkably but picked again after 1978. Another slowdown in consumption was experienced between 1979 and 1983. Basically, the changes in consumption seem to move in a similar pattern with real income. The implication of this is that when consumers are earning low incomes, they tend to cut down their expenditure because they do not have adequate income to spend, but when their income increases, they also consume more because they have adequate income to spend. On the other hand, when there is a slowdown in consumption, the price level seems to increase at a higher rate. The implication of this scenario is that, potentially the UK economy has experienced economic recession from time to time, when price levels of goods and services increases hence reducing the consumer’s real income earned on them forcing them to cut down on their expenditure. An example of an economic recession was experienced approximately between 1973 and 1978. Question 2: Bivariate Static Consumption Function a) Estimate model 1 using EViews. Give the coefficient estimates, the estimated standard errors, and goodness of fit. Model 1 Table 1: Model 1 Equation Output Dependent Variable: CONSUMPTION Method: Least Squares Date: 01/20/13 Time: 21:58 Sample: 1955 1984 Included observations: 30 Variable Coefficient Std. Error t-Statistic Prob.   C 1.311413 0.107515 12.19746 0.0000 INCOME 0.879481 0.009190 95.70251 0.0000 R-squared 0.996952     Mean dependent var 11.59896 Adjusted R-squared 0.996843     S.D. dependent var 0.203186 S.E. of regression 0.011416     Akaike info criterion -6.043297 Sum squared resid 0.003649     Schwarz criterion -5.949884 Log likelihood 92.64945     Hannan-Quinn criter. -6.013413 F-statistic 9158.971     Durbin-Watson stat 0.899355 Prob(F-statistic) 0.000000 The column marked ‘coefficient’ shows the coefficient estimates. The coefficient measures the marginal contribution of the real income to the real consumption. Also, ‘C’ has been included in the list of regressors, which represents the constant or the intercept in the regression equation. Ideally, when income is changed by 1 unit, the level of consumption is increased by 0.879481 as shown in the coefficient column. The estimated standard errors measure the statistical reliability of the estimated coefficients. The standard error for the income is 0.009, which is relatively small and hence reassuring that the coefficient is quite reliable. Adjusted R-squared is 0.99, which is very close to 1, and which implied that the model is very good in predicting the dependent variable. b) Test the significance of (log) income in this model. The p-value measures the significance of income in this model. Ideally, this value is less that 0.05, which means that there is significant evidence to show that income can be used to predict the consumption level. c) Compute a 95% confidence interval for the coefficient on income and hence determine if 0.9 is a plausible value for the elasticity of consumption with respect to income. At 95% confidence interval, the p-value is o.ooo, meaning that income is a significant determinant of consumption level, however, 0.9 is more than 0.05 and hence it is not a plausible value of elasticity of consumption with respect to income. d) What is the size of this test? The size of this test is 5% e) Is the regression as a whole significant? (Use the F-test reported with the EViews regression results.) The probability of F-statistics is zero meaning that the relationship of the whole regression is significant. f) Perform tests of serial correlation and heteroscedasticity on the model residuals. What do you conclude? Table 2: serial correlation test Breusch-Godfrey Serial Correlation LM Test: F-statistic 7.670719     Prob. F(2,26) 0.0024 Obs*R-squared 11.13273     Prob. Chi-Square(2) 0.0038 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 01/20/13 Time: 23:01 Sample: 1955 1984 Included observations: 30 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob.   C 0.011547 0.088950 0.129813 0.8977 INCOME -0.000995 0.007605 -0.130788 0.8969 RESID(-1) 0.718177 0.183807 3.907226 0.0006 RESID(-2) -0.406051 0.198441 -2.046209 0.0510 R-squared 0.371091     Mean dependent var -2.19E-15 Adjusted R-squared 0.298525     S.D. dependent var 0.011217 S.E. of regression 0.009395     Akaike info criterion -6.373732 Sum squared resid 0.002295     Schwarz criterion -6.186906 Log likelihood 99.60599     Hannan-Quinn criter. -6.313965 F-statistic 5.113813     Durbin-Watson stat 1.981838 Prob(F-statistic) 0.006491 From Breusch-Godfrey Serial Correlation LM Test, there is no evidence of serial correlation because the p-value for income is more than 0.05 and hence OLS estimates are neither biased nor inconsistent. Table 3: Heteroskedasticity Test: Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic 1.045537     Prob. F(1,28) 0.3153 Obs*R-squared 1.079894     Prob. Chi-Square(1) 0.2987 Scaled explained SS 0.600590     Prob. Chi-Square(1) 0.4384 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 01/20/13 Time: 23:09 Sample: 1955 1984 Included observations: 30 Variable Coefficient Std. Error t-Statistic Prob.   C -0.001223 0.001316 -0.929866 0.3604 INCOME 0.000115 0.000112 1.022515 0.3153 R-squared 0.035996     Mean dependent var 0.000122 Adjusted R-squared 0.001568     S.D. dependent var 0.000140 S.E. of regression 0.000140     Akaike info criterion -14.85004 Sum squared resid 5.46E-07     Schwarz criterion -14.75663 Log likelihood 224.7506     Hannan-Quinn criter. -14.82015 F-statistic 1.045537     Durbin-Watson stat 2.157778 Prob(F-statistic) 0.315291 The results of heteroskedasticity above show that the p-value of income is more than 0.05, meaning that there is no evidence of heteroskedasticity in the model. g) In the light of the tests performed at f), how reliable do you consider the results of the earlier t-tests to be? Explain. On the right of the above tests, the earlier t-tests indicate that the adjustments that were made in later models have added no value, and hence the earlier model remains the most suitable in predicting the dependent variable. h) What is your general view of model 1? Is it an adequate representation of the data? Model 1 is a very good representation of the data because the Adjusted R-squared is very high at 0.99, which means the prediction is almost perfect, with very little margin being attributed to error. Question 3: A Preliminary Dynamic Model Model 2 adds the lags of consumption and income to develop a dynamic model a) Why might this have been thought desirable from an economic point of view? When the lags of the two variables are introduced, the model becomes more useful because it introduces the concept of rate of change, which is better in expressing changes in the economy from time to time rather than using the absolute values per se. In other words, the patterns can be more visible with the introduction of the rate of change from one year to the other. Also comparability of indicators from in different years becomes more credible. b) As with model 1, use statistical tests to determine if model 2 is an adequate representation of the data. You should use: Table 4: Equation Output: Model 2 Dependent Variable: CONSUMPTION Method: Least Squares Date: 01/20/13 Time: 23:31 Sample (adjusted): 1956 1984 Included observations: 29 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C 0.306248 0.272937 1.122049 0.2725 INCOME 0.715827 0.089360 8.010617 0.0000 CONSUMPTION(-1) 0.763998 0.202101 3.780273 0.0009 INCOME(-1) -0.507624 0.145681 -3.484477 0.0018 R-squared 0.997894     Mean dependent var 11.61109 Adjusted R-squared 0.997642     S.D. dependent var 0.195416 S.E. of regression 0.009490     Akaike info criterion -6.349795 Sum squared resid 0.002251     Schwarz criterion -6.161202 Log likelihood 96.07202     Hannan-Quinn criter. -6.290730 F-statistic 3949.518     Durbin-Watson stat 1.556424 Prob(F-statistic) 0.000000 From the equation output above, the variables of lags of consumption and lags of income are statistically significant since their p-values are less than 0.05. The Adjusted R-squared is 0.998, which means that the model is very good in predicting the dependent variable. The F-statistics p-value is zero which means, as a whole, the regression model is significant. Table 5: Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic 0.443825     Prob. F(3,25) 0.7238 Obs*R-squared 1.466411     Prob. Chi-Square(3) 0.6900 Scaled explained SS 0.890859     Prob. Chi-Square(3) 0.8276 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 01/20/13 Time: 23:52 Sample: 1956 1984 Included observations: 29 Variable Coefficient Std. Error t-Statistic Prob.   C -0.001910 0.002996 -0.637586 0.5295 INCOME -0.000942 0.000981 -0.960387 0.3461 CONSUMPTION(-1) 0.001245 0.002219 0.561335 0.5796 INCOME(-1) -0.000121 0.001599 -0.075491 0.9404 R-squared 0.050566     Mean dependent var 7.76E-05 Adjusted R-squared -0.063366     S.D. dependent var 0.000101 S.E. of regression 0.000104     Akaike info criterion -15.37362 Sum squared resid 2.71E-07     Schwarz criterion -15.18503 Log likelihood 226.9175     Hannan-Quinn criter. -15.31455 F-statistic 0.443825     Durbin-Watson stat 2.589709 Prob(F-statistic) 0.723765 Heteroskedasticity Test in the table above shows that p-values of all the variables are greater than 0.05, hence excluding the possibility of Heteroskedasticity. Table 6: serial correlation: Breusch-Godfrey Serial Correlation LM Test F-statistic 3.083345     Prob. F(2,23) 0.0651 Obs*R-squared 6.131447     Prob. Chi-Square(2) 0.0466 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 01/20/13 Time: 23:56 Sample: 1956 1984 Included observations: 29 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob.   C 0.083181 0.439137 0.189420 0.8514 INCOME 0.029662 0.089117 0.332839 0.7423 INCOME(-1) 0.023092 0.291881 0.079115 0.9376 CONSUMPTION(-1) -0.060468 0.321638 -0.188001 0.8525 RESID(-1) 0.287794 0.361293 0.796565 0.4338 RESID(-2) -0.466291 0.271434 -1.717878 0.0993 R-squared 0.211429     Mean dependent var -1.96E-15 Adjusted R-squared 0.040001     S.D. dependent var 0.008967 S.E. of regression 0.008786     Akaike info criterion -6.449397 Sum squared resid 0.001775     Schwarz criterion -6.166508 Log likelihood 99.51625     Hannan-Quinn criter. -6.360799 F-statistic 1.233338     Durbin-Watson stat 2.203832 Prob(F-statistic) 0.325758 From Breusch-Godfrey Serial Correlation LM Test performed as shown in table 6 above, there is no evidence of serial correlation because the p-value of income is more than 0.05 and hence OLS estimates are neither biased nor inconsistent. c) A test of ‘normality’ of the model errors. Figure 4: Normality test In the figure above, the histogram is bell shaped and the Jaque-Bera statistics are not significant. This implies that there is no evidence of certain errors being out of line (outliers). d) Do you prefer model 1 or 2? Give economic and econometric reasons. The second model is preferable because its Adjusted R-squared is higher, indicating that it is better in predicting the dependent variable. Economically, I prefer model 2 because models that incorporate rate of change are usually more indicative than those that use absolute values only. Question 4: An Omitted Variable? a) Plot the errors from model 2. At what points in time does the model appear to perform badly (i.e. Where are errors unusually large)? Provide this graph. Figure 5: plot of the errors From the figure 5 above, the residuals begin to deviate significantly from zero from 1970, and hence that is the moment the model begun to perform poorly. b) It is decided to add current and lagged prices to the model (to give model 3). What is the evidence from the plots of the data that this might be justified? There is still a room for improving the model because there are some errors. Perhaps adding this variable will reduce these errors. c) Estimate model 3. Are the additional price variables individually significant? Table 7: Model 3 equation output Dependent Variable: CONSUMPTION Method: Least Squares Date: 01/21/13 Time: 00:31 Sample (adjusted): 1956 1984 Included observations: 29 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C -0.047069 0.337533 -0.139449 0.8903 INCOME 0.670672 0.077161 8.691848 0.0000 INCOME(-1) -0.468378 0.124813 -3.752626 0.0010 PRICE -0.144795 0.043735 -3.310739 0.0031 PRICE(-1) 0.142857 0.042024 3.399454 0.0025 CONSUMPTION(-1) 0.801273 0.172072 4.656625 0.0001 R-squared 0.998607     Mean dependent var 11.61109 Adjusted R-squared 0.998305     S.D. dependent var 0.195416 S.E. of regression 0.008046     Akaike info criterion -6.625297 Sum squared resid 0.001489     Schwarz criterion -6.342408 Log likelihood 102.0668     Hannan-Quinn criter. -6.536699 F-statistic 3298.717     Durbin-Watson stat 1.950347 Prob(F-statistic) 0.000000 From table 7 above, the additional price variables are individually significant because their p-values are less than 0.05 d) Use an F-test to determine if the additional price variables are jointly significant (i.e. Whether they should be included together)? The value of F-statistics has reduced while its p-value has remained zero. This means that additional price variables are jointly significant and they are worth being included. e) Compare the coefficient estimates on the current and lagged price variables (pt and pt-1). The coefficient for pt is -0.144795, while that of pt-1 is 0.142857. This means that the current price is inversely related to consumption while the lagged price variable is directly related to the consumption. f) Why must the R square value for model 3 be greater than that for model 2? Is the same necessarily true for the adjusted R square values? For model 3 to be considered better than model 1, its R-squared as well as R-squared must be greater. g) Compare the estimated coefficients of yt, ct, and yt-1 and their significance between models 2 and 3. Coefficient Significance Model 2 Model 3 Model 2 Model 3 yt, 0.715827 0.670672 0.0000 0.0000 ct, 0.001245 0.801273 0.0009 0.0001 yt-1 -0.507624 -0.468378 0.0018 0.0010 h) Perform tests of heteroscedasticity, serial correlation and normality of the errors. Is the model adequate? Table 8: Heteroskedasticity Test: Breusch-Pagan-Godfrey – model 3 F-statistic 0.120354     Prob. F(5,23) 0.9865 Obs*R-squared 0.739409     Prob. Chi-Square(5) 0.9807 Scaled explained SS 0.318279     Prob. Chi-Square(5) 0.9973 Variable Coefficient Std. Error t-Statistic Prob.   C -0.000670 0.002793 -0.239721 0.8127 INCOME -0.000304 0.000639 -0.475908 0.6386 INCOME(-1) -0.000311 0.001033 -0.300880 0.7662 PRICE -0.000139 0.000362 -0.384410 0.7042 PRICE(-1) 0.000141 0.000348 0.406553 0.6881 CONSUMPTION(-1) 0.000684 0.001424 0.480190 0.6356 R-squared 0.025497     Mean dependent var 5.13E-05 Adjusted R-squared -0.186352     S.D. dependent var 6.11E-05 S.E. of regression 6.66E-05     Akaike info criterion -16.21426 Sum squared resid 1.02E-07     Schwarz criterion -15.93137 Log likelihood 241.1068     Hannan-Quinn criter. -16.12567 F-statistic 0.120354     Durbin-Watson stat 2.388786 Prob(F-statistic) 0.986503 Table 9: serial correlation: Model 3 F-statistic 1.530057     Prob. F(2,21) 0.2397 Obs*R-squared 3.688399     Prob. Chi-Square(2) 0.1582 Variable Coefficient Std. Error t-Statistic Prob.   C -0.158054 0.386751 -0.408670 0.6869 INCOME 0.049007 0.080684 0.607394 0.5501 INCOME(-1) -0.202191 0.215380 -0.938764 0.3585 PRICE 0.042392 0.049359 0.858853 0.4001 PRICE(-1) -0.041621 0.047647 -0.873515 0.3923 CONSUMPTION(-1) 0.167814 0.242257 0.692708 0.4961 RESID(-1) -0.273086 0.350040 -0.780156 0.4440 RESID(-2) -0.485333 0.277796 -1.747082 0.0952 R-squared 0.127186     Mean dependent var -1.23E-15 Adjusted R-squared -0.163752     S.D. dependent var 0.007292 S.E. of regression 0.007867     Akaike info criterion -6.623398 Sum squared resid 0.001300     Schwarz criterion -6.246213 Log likelihood 104.0393     Hannan-Quinn criter. -6.505269 F-statistic 0.437159     Durbin-Watson stat 2.216512 Prob(F-statistic) 0.867668 Heteroskedasticity Test in the table above shows that p-values of all the variables are greater than 0.05, thus excluding the possibility of Heteroskedasticity. From Breusch-Godfrey Serial Correlation LM Test performed as shown in table 9 above, there is no evidence of serial correlation because the p-value of income is more than 0.05 and hence OLS estimates are neither biased nor inconsistent. Lack of these errors proves that the model is adequate (Triola, 2001). Question 5: Interpreting Model 3 by writing it differently – from Model 3 to Model 4 Model 3 can be rewritten – called a reparameterisation – as model 4. (In model 4 is the differencing operator, meaning.) It is possible to work out what the coefficients of model 4 will be from those of model 3. No restrictions or additional variables have actually been used. a) Use the Generate function under the Quick tab of EViews to construct the differences of each variable. Model 4 b) Estimate model 4. Present the results. Table 10: output equation – Model 4 Dependent Variable: CONSUMPTION Method: Least Squares Date: 01/21/13 Time: 01:39 Sample (adjusted): 1956 1984 Included observations: 29 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C -0.047069 0.337533 -0.139449 0.8903 INCOME-INCOME(-1) 0.670672 0.077161 8.691848 0.0000 PRICE-PRICE(-1) -0.144795 0.043735 -3.310739 0.0031 CONSUMPTION(-1) 0.801273 0.172072 4.656625 0.0001 INCOME(-1) 0.202294 0.151661 1.333855 0.1953 PRICE(-1) -0.001938 0.005793 -0.334471 0.7411 R-squared 0.998607     Mean dependent var 11.61109 Adjusted R-squared 0.998305     S.D. dependent var 0.195416 S.E. of regression 0.008046     Akaike info criterion -6.625297 Sum squared resid 0.001489     Schwarz criterion -6.342408 Log likelihood 102.0668     Hannan-Quinn criter. -6.536699 F-statistic 3298.717     Durbin-Watson stat 1.950347 Prob(F-statistic) 0.000000 c) Compare the residual sum of squares of models 3 and 4 and compare their goodness of fit (R2). R-squared remains the same which means the goodness of fit of the two models is the same. d) In model 4, what do you observe about the estimated coefficient of pt? The coefficient of pt is inversely related to the dependent variable e) What is the economic interpretation of the variable ? Change in Pt shows the percentage rate of change of price from one year to the other. Question 6: Simplifying Model 4: Models 5 and 6 a) By estimating and testing model 5, explain if it is a statistically acceptable simplification of model 4. Model 5 Table 11: output of model 5 Dependent Variable: CONSUMPTION-CONSUMPTION(-1) Method: Least Squares Date: 01/21/13 Time: 01:52 Sample (adjusted): 1956 1984 Included observations: 29 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C 0.030554 0.240513 0.127036 0.9000 INCOME-INCOME(-1) 0.673213 0.075352 8.934245 0.0000 PRICE-PRICE(-1) -0.139589 0.040109 -3.480279 0.0019 CONSUMPTION(-1) -0.202844 0.168425 -1.204356 0.2402 INCOME(-1) 0.199866 0.148658 1.344470 0.1914 R-squared 0.853194     Mean dependent var 0.022461 Adjusted R-squared 0.828727     S.D. dependent var 0.019079 S.E. of regression 0.007896     Akaike info criterion -6.689410 Sum squared resid 0.001496     Schwarz criterion -6.453669 Log likelihood 101.9964     Hannan-Quinn criter. -6.615579 F-statistic 34.87032     Durbin-Watson stat 1.935101 Prob(F-statistic) 0.000000 The R-squared for model 4 is 0.9986 while that for model 5 is 0.8532. This means that model 5 is not a simplification of model 4; it is actually a poor model. b) Comment on the estimated coefficients of ct-1 and yt-1 in model 5 and explain whether you think this justifies the further simplification to model 6. Ct-1 is -0.2 while yt-1 is 0.199. This should be simplified to model 5 to remove the negative value of consumption. c) Estimate model 6 and present the results. Does it pass tests on the errors (residuals)? Model 6 Dependent Variable: CONSUMPTION-CONSUMPTION (-1) Method: Least Squares Date: 01/21/13 Time: 02:06 Sample (adjusted): 1956 1984 Included observations: 29 after adjustments Variable Coefficient Std. Error t-Statistic Prob.   C -0.220239 0.107718 -2.044584 0.0516 INCOME-INCOME(-1) 0.616403 0.062648 9.839157 0.0000 PRICE-PRICE(-1) -0.140641 0.041263 -3.408381 0.0022 CONSUMPTION-CONSUMPTION(-1)-INCOME-INCOME(-1) -0.010152 0.004687 -2.166127 0.0400 R-squared 0.841059     Mean dependent var 0.022461 Adjusted R-squared 0.821987     S.D. dependent var 0.019079 S.E. of regression 0.008050     Akaike info criterion -6.678955 Sum squared resid 0.001620     Schwarz criterion -6.490363 Log likelihood 100.8449     Hannan-Quinn criter. -6.619890 F-statistic 44.09715     Durbin-Watson stat 2.106077 Prob(F-statistic) 0.000000 Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic 0.145736     Prob. F(3,25) 0.9314 Obs*R-squared 0.498446     Prob. Chi-Square(3) 0.9192 Scaled explained SS 0.205356     Prob. Chi-Square(3) 0.9767 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 01/21/13 Time: 02:08 Sample: 1956 1984 Included observations: 29 Variable Coefficient Std. Error t-Statistic Prob.   C -0.000167 0.000840 -0.198220 0.8445 INCOME-INCOME(-1) -6.85E-05 0.000489 -0.140218 0.8896 PRICE-PRICE(-1) -0.000203 0.000322 -0.632088 0.5331 CONSUMPTION-CONSUMPTION(-1)-INCOME-INCOME(-1) -1.02E-05 3.66E-05 -0.279436 0.7822 R-squared 0.017188     Mean dependent var 5.59E-05 Adjusted R-squared -0.100750     S.D. dependent var 5.99E-05 S.E. of regression 6.28E-05     Akaike info criterion -16.38575 Sum squared resid 9.86E-08     Schwarz criterion -16.19716 Log likelihood 241.5934     Hannan-Quinn criter. -16.32669 F-statistic 0.145736     Durbin-Watson stat 2.189082 Prob(F-statistic) 0.931448 The model has passed the test of errors because there is no evidence of Heteroskedasticity as shown in the figure above. Question 7: Interpreting Model 6 Provide an economic interpretation of model 6 – what does it say about the way consumption depends on income? The model shows that when income increases, the level of consumption also tends to increase and vice versa. Question 8: Final Choice of Model Of all the models estimated and tested above which do you prefer and why? Model 4 is the best because its R squared is the highest, meaning it is very good in predicting the dependent variable compared to other models (Fisher, 1925). References Fisher, R.A., 1925. Statistical Methods for Research Workers. Edinburgh: Oliver and Boyd. Triola, M., 2001. Elementary statistics (8 ed.). Boston: Addison-Wesley. Read More