The Demand for Drink and Tobacco - Assignment Example

Part The Demand for Drink and Tobacco Descriptive Statistics Table Descriptive Statistics of the variables of interest P Q X G  Mean  0.164142  4442.111  5854.911  0.165827  Median  0.171100  4501.000  5771.000  0.164500  Maximum  0.206900  5598.000  8737.000  0.210700  Minimum  0.127100  3454.000  4000.000  0.137400  Std. Dev.  0.024228  505.4463  1221.354  0.020988  Skewness  0.121218 -0.019092  0.424303  0.454036  Kurtosis  1.802041  2.653931  2.260744  2.099150  Jarque-Bera  2.801025  0.227291  2.374938  3.067732  Probability  0.246471  0.892574  0.304992  0.215700  Sum  7.386400  199895.0  263471.0  7.462200  Sum Sq. Dev.  0.025827  11240942  65635008  0.019383  Observations  45  45  45  45 Table 1 above presents the descriptive statistics for our variables of interest. The only point of concern that may arise in this situation is that all the variables reflect some degree of skewness which violates the normality assumption. Additionally, the fact that the number of observations is only 45 may also be a point of concern since this can lead to small sample bias. 2. Time plots Figure 1: Time plot of P There are no seasonal patterns evident in the time plot of P. Figure 2: Time plot of Q The time plot of Q exhibits strong seasonal variations. Figure 3: Time plot of G As is evident from figure 3 above, similar to the time plot of P, the time plot of G also does not exhibit seasonal fluctuations. Figure 4: Time plot of X Figure 4 shows that X also follows a seasonally fluctuating pattern 3. Thus, there is strong evidence of seasonal fluctuations among the Q and X series. This is visible in the oscillatory patterns that these series seem to follow. The series P and G exhibit no seasonal patterns. Additionally, all the series reflect a steady upward trend. Therefore inclusion of seasonal dummies is important since our dependent variable Q does exhibit seasonal fluctuations. Additionally, the dummies would be important for the explanatory variable X. 4. Estimation Table 2: Simple OLS estimation Dependent Variable: Q Method: Least Squares Date: 09/01/11 Time: 00:57 Sample: 1980Q1 1991Q1 Included observations: 45 Coefficient Std. Error t-Statistic Prob.   C 8780.852 479.3384 18.31869 0.0000 P -7530.197 6092.988 -1.235879 0.2235 G -84559.50 9770.479 -8.654591 0.0000 X 1.865016 0.111494 16.72746 0.0000 R-squared 0.915605     Mean dependent var 4442.111 Adjusted R-squared 0.909430     S.D. dependent var 505.4463 S.E. of regression 152.1132     Akaike info criterion 12.97181 Sum squared resid 948675.2     Schwarz criterion 13.13241 Log likelihood -287.8658     Hannan-Quinn criter. 13.03168 F-statistic 148.2710     Durbin-Watson stat 1.390217 Prob(F-statistic) 0.000000 From table 2 above we find that the estimated coefficients for both P and X are significantly different from zero (evident from the t-statistic). G however is not a significant determinant of Q. The coefficients reflect that the demand for drink and tobacco is negatively influenced by the price of the items and positively influenced by the total consumer expenditure. The coefficient on G is also negative but since it is not significantly different from 0 at the 5% level, we conclude that it does not have an influence on drink and tobacco demand. Thus, our results imply that an increase in the prices of drinks and tobacco will lead to a reduction in its demand while an increase in overall consumer expenditure leads to an increase in the demand. 5. Attempting to include all four dummies leads to perfect multicollinearity. Thus we modify the equation and include dummies for the 1st 3 quarters only. Table 3 presents the results. Table 3: OLS estimation with quarterly dummies Dependent Variable: Q Method: Least Squares Date: 09/01/11 Time: 01:01 Sample: 1980Q1 1991Q1 Included observations: 45 Coefficient Std. Error t-Statistic Prob.   C 5127.935 356.9563 14.36572 0.0000 P -8713.964 2700.994 -3.226206 0.0026 X 0.805451 0.096091 8.382187 0.0000 G -23150.70 6406.847 -3.613431 0.0009 Q1 137.2931 31.47450 4.362044 0.0001 Q2 -537.9634 43.07804 -12.48811 0.0000 Q3 -153.1908 29.49682 -5.193468 0.0000 R-squared 0.984746     Mean dependent var 4442.111 Adjusted R-squared 0.982338     S.D. dependent var 505.4463 S.E. of regression 67.17371     Akaike info criterion 11.39448 Sum squared resid 171467.7     Schwarz criterion 11.67551 Log likelihood -249.3757     Hannan-Quinn criter. 11.49924 F-statistic 408.8624     Durbin-Watson stat 1.293568 Prob(F-statistic) 0.000000 Observe from the table above that inclusion of the seasonal dummies leads to all the independent variables becoming significant. As we found in the previous section, P and G have negative impacts on the demand while X has a positive impact here. Thus, we find that the general price index also has a negative impact on the demand for drinks and tobacco. This in all probability reflects a substitution effect. Additionally, due to the inclusion of the time dummies we find that while there is an increase in the demand in the 1st quarter, there is a strong decline in the 2nd quarter and there is a weaker decline in the 3rd quarter. 6. Table 4 presents the results of running the model in logs. Table 4: OLS log specification with seasonal dummies Dependent Variable: LNQ Method: Least Squares Date: 09/01/11 Time: 01:03 Sample: 1980Q1 1991Q1 Included observations: 45 Coefficient Std. Error t-Statistic Prob.   C -1.269855 1.428850 -0.888725 0.3797 LNP -0.571294 0.105096 -5.435911 0.0000 LNG -0.354533 0.199891 -1.773632 0.0841 LNX 0.926634 0.124145 7.464141 0.0000 Q1 0.030868 0.007208 4.282297 0.0001 Q2 -0.139478 0.009838 -14.17764 0.0000 Q3 -0.037755 0.006783 -5.565883 0.0000 R-squared 0.984152     Mean dependent var 8.392438 Adjusted R-squared 0.981649     S.D. dependent var 0.115502 S.E. of regression 0.015646     Akaike info criterion -5.335112 Sum squared resid 0.009303     Schwarz criterion -5.054076 Log likelihood 127.0400     Hannan-Quinn criter. -5.230344 F-statistic 393.2921     Durbin-Watson stat 1.354088 Prob(F-statistic) 0.000000 Comparing the coefficients from the table above (4) to those obtained in the previous section (table 3) we find that the values of the coefficients are substantially smaller. However, the important fact is that the coefficients have the same signs. Thus, our conclusions regarding the direction of the impacts remain valid. Comparing the significances of the coefficients we find that the constant is no longer significant. Also, the natural log of G is significant at the 10% level. All other variables are significant at the 5% level. 7. Figure 5 below presents the scatter of the natural log of X with the residual to identify whether any functional patterns are evident. If we obtain evidence of any dependence of the residual on LNX, we would conclude that there is heteroscedasticity. However, from the figure, we find no evidence of any such dependence. Figure 5: Scatter plot of residuals and the natural log of X 8. We use White’s test for Heteroscedasticity to check the validity of assuming homoscedastic errors. Table 5 presents the results of the test. Table 5: Whites test for heteroscedasticity Heteroskedasticity Test: White F-statistic 0.714657     Prob. F(20,24) 0.7755 Obs*R-squared 16.79651     Prob. Chi-Square(20) 0.6661 Scaled explained SS 10.79137     Prob. Chi-Square(20) 0.9515 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 09/01/11 Time: 01:04 Sample: 1980Q1 1991Q1 Included observations: 45 Collinear test regressors dropped from specification Coefficient Std. Error t-Statistic Prob.   C 0.214673 1.072952 0.200077 0.8431 LNP -2.874355 2.425792 -1.184914 0.2477 LNP^2 0.090338 0.095195 0.948977 0.3521 LNP*LNG -0.621933 0.310947 -2.000126 0.0569 LNP*LNX 0.238333 0.210318 1.133203 0.2683 LNP*Q1 -0.001076 0.007545 -0.142562 0.8878 LNP*Q2 0.016294 0.013487 1.208137 0.2388 LNP*Q3 0.009115 0.007184 1.268784 0.2167 LNG 3.003677 3.035055 0.989662 0.3322 LNG^2 0.535635 0.366080 1.463163 0.1564 LNG*LNX -0.251387 0.253628 -0.991166 0.3315 LNG*Q1 0.007284 0.013189 0.552277 0.5859 LNG*Q2 -0.001574 0.020502 -0.076758 0.9395 LNG*Q3 -0.000877 0.012470 -0.070347 0.9445 LNX -0.022887 0.094015 -0.243439 0.8097 LNX*Q1 -0.004625 0.007645 -0.605019 0.5508 LNX*Q2 -0.009566 0.007564 -1.264606 0.2182 LNX*Q3 -0.006465 0.008807 -0.734003 0.4701 Q1 0.051414 0.088429 0.581409 0.5664 Q2 0.109141 0.089224 1.223228 0.2331 Q3 0.070871 0.100460 0.705466 0.4873 R-squared 0.373256     Mean dependent var 0.000207 Adjusted R-squared -0.149031     S.D. dependent var 0.000281 S.E. of regression 0.000301     Akaike info criterion -13.07533 Sum squared resid 2.17E-06     Schwarz criterion -12.23222 Log likelihood 315.1949     Hannan-Quinn criter. -12.76103 F-statistic 0.714657     Durbin-Watson stat 2.097690 Prob(F-statistic) 0.775479 The critical value of the Chi square distribution is 9.49. Since the computed value is 16.79 we reject the null of no-heteroscedasticity at the 5% level. Thus, we conclude that there is some heteroscedasticity in the data. 8. We utilize the Breusch-Godfrey Serial Correlation LM Test to test for autocorrelation. The results are presented in table 6 below. Table 6: Testing for auto-correlation Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.061711     Prob. F(4,34) 0.3904 Obs*R-squared 4.996699     Prob. Chi-Square(4) 0.2876 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 09/01/11 Time: 01:06 Sample: 1980Q1 1991Q1 Included observations: 45 Presample missing value lagged residuals set to zero. Coefficient Std. Error t-Statistic Prob.   C -0.417310 1.498929 -0.278405 0.7824 LNP -0.015758 0.113124 -0.139298 0.8900 LNG -0.040602 0.202744 -0.200260 0.8425 LNX 0.036424 0.130508 0.279091 0.7819 Q1 -0.001025 0.007291 -0.140521 0.8891 Q2 0.001758 0.010113 0.173827 0.8630 Q3 0.000151 0.006777 0.022272 0.9824 RESID(-1) 0.331429 0.173825 1.906688 0.0650 RESID(-2) -0.169628 0.186243 -0.910789 0.3688 RESID(-3) 0.064283 0.186100 0.345421 0.7319 RESID(-4) 0.078190 0.179761 0.434964 0.6663 R-squared 0.111038     Mean dependent var 9.71E-16 Adjusted R-squared -0.150422     S.D. dependent var 0.014541 S.E. of regression 0.015596     Akaike info criterion -5.275035 Sum squared resid 0.008270     Schwarz criterion -4.833406 Log likelihood 129.6883     Hannan-Quinn criter. -5.110400 F-statistic 0.424684     Durbin-Watson stat 1.841927 Prob(F-statistic) 0.924465 We fail to reject the null of autocorrelation. Thus, we conclude there is some serial correlation in the data. Heteroscedasticity does not bias the estimates although the standard errors tend to be very large which leads to the possibility of incorrect inferences. Thus, we use White’s heteroscedasticity consistent standard errors and covariances in estimation process. Further, we also include lagged values of all the variables to account correct for the autocorrelation. The estimates are presented in table 7 below. Table 7: Modified specification to correct for autocorrelation and heteroscedasticity Dependent Variable: LNQ Method: Least Squares Date: 09/01/11 Time: 01:28 Sample (adjusted): 1980Q2 1991Q1 Included observations: 44 after adjustments White Heteroskedasticity-Consistent Standard Errors & Covariance Coefficient Std. Error t-Statistic Prob.   C -0.071919 1.270056 -0.056626 0.9552 LNP -0.714942 0.193215 -3.700232 0.0008 LNP(-1) 0.163323 0.190163 0.858861 0.3964 LNG 0.077874 0.505817 0.153956 0.8786 LNG(-1) -0.302861 0.459058 -0.659744 0.5139 LNX 0.990473 0.134646 7.356141 0.0000 LNX(-1) -0.171285 0.130877 -1.308750 0.1994 Q1 0.028864 0.009415 3.065842 0.0042 Q2 -0.131394 0.011553 -11.37270 0.0000 Q3 -0.050149 0.012451 -4.027864 0.0003 R-squared 0.986676     Mean dependent var 8.392789 Adjusted R-squared 0.983150     S.D. dependent var 0.116814 S.E. of regression 0.015163     Akaike info criterion -5.343144 Sum squared resid 0.007818     Schwarz criterion -4.937646 Log likelihood 127.5492     Hannan-Quinn criter. -5.192766 F-statistic 279.7645     Durbin-Watson stat 1.571933 Prob(F-statistic) 0.000000 Note from table 7 above, none of the lagged values of the variables turn out to be significant. The estimated coefficients on the levels of the variables have the same signs as in the previous estimation. However, the natural log of G turns out to be insignificant under this specification when we account for heteroscedasticity and autocorrelation. 11. We test for redundancy of including the lagged values of the independent variables in the section below. The results are shown in table 8. We fail to reject the null that the lagged values are redundant. Therefore, we conclude that the model estimated in section 6 is good enough. However it should be noted that given the non-stationarity of the variables we found in the time plots, it is possible the results of these specifications are spurious. Although it is beyond the scope of this assignment, running unit root tests and then checking for co-integration in the variables and running an error corrected model is highly recommended to obtain better estimates of the true underlying relationship. Table 8: Testing for redundant variables Redundant Variables: LNP(-1) LNG(-1) LNX(-1)  F-statistic 0.547251     Prob. F(3,34) 0.6534 Log likelihood ratio 2.074918     Prob. Chi-Square(3) 0.5570 Test Equation: Dependent Variable: LNQ Method: Least Squares Date: 09/01/11 Time: 01:32 Sample: 1980Q2 1991Q1 Included observations: 44 White Heteroskedasticity-Consistent Standard Errors & Covariance Coefficient Std. Error t-Statistic Prob.   C -0.840518 1.246657 -0.674217 0.5044 LNP -0.608472 0.099133 -6.137966 0.0000 LNG -0.259304 0.182411 -1.421534 0.1635 LNX 0.889138 0.108763 8.175029 0.0000 Q1 0.034550 0.006148 5.619464 0.0000 Q2 -0.141984 0.009402 -15.10189 0.0000 Q3 -0.038296 0.005845 -6.551380 0.0000 R-squared 0.986033     Mean dependent var 8.392789 Adjusted R-squared 0.983768     S.D. dependent var 0.116814 S.E. of regression 0.014883     Akaike info criterion -5.432350 Sum squared resid 0.008195     Schwarz criterion -5.148502 Log likelihood 126.5117     Hannan-Quinn criter. -5.327086 F-statistic 435.3547     Durbin-Watson stat 1.507312 Prob(F-statistic) 0.000000 Part 2: Supply and Demand a. Estimation in reduced form: 2SLS methodology Structural form: (1) (2) Reduced form: Step 1: OLS on reduced form [equation (3) and (5)] In pursuit of obtaining our instruments which are the fitted values of the two dependent variables we run simple OLS regressions of the reduced forms. Table 9: Results of OLS estimation of equation (3) Dependent Variable: P Method: Least Squares Date: 09/01/11 Time: 15:21 Sample: 1960 1986 Included observations: 27 White Heteroskedasticity-Consistent Standard Errors & Covariance Coefficient Std. Error t-Statistic Prob.   C -19.51515 12.19712 -1.599979 0.1239 L -0.715854 0.220681 -3.243834 0.0037 NPCOST 3.031424 0.509235 5.952898 0.0000 CPI -0.841652 0.166766 -5.046912 0.0000 INCOME 0.004240 0.001438 2.948255 0.0074 R-squared 0.971087     Mean dependent var 85.40741 Adjusted R-squared 0.965830     S.D. dependent var 32.78515 S.E. of regression 6.060362     Akaike info criterion 6.606992 Sum squared resid 808.0158     Schwarz criterion 6.846962 Log likelihood -84.19439     Hannan-Quinn criter. 6.678348 F-statistic 184.7263     Durbin-Watson stat 1.838978 Prob(F-statistic) 0.000000 Table 9 shows that all variables apart from the constant are significant at the 5% level. We find that land and the consumer price index exerts a negative influence on the equilibrium price while input costs exert a positive influence. Income also has a positive albeit very small influence on the price. Table 10 below presents the reduced form estimates for the equilibrium quantity. Only the constant and the income turn out to be significant. Income seems to have a very small but positive influence on the equilibrium quantity. Table 10: OLS on equation (5) Dependent Variable: Q Method: Least Squares Date: 09/01/11 Time: 15:32 Sample: 1960 1986 Included observations: 27 White Heteroskedasticity-Consistent Standard Errors & Covariance Coefficient Std. Error t-Statistic Prob.   C 35.01470 10.22509 3.424390 0.0024 L 0.183183 0.150576 1.216553 0.2367 NPCOST -0.052441 0.302667 -0.173263 0.8640 CPI 0.017963 0.095046 0.188997 0.8518 INCOME 0.005004 0.001750 2.859242 0.0091 R-squared 0.870387     Mean dependent var 89.96296 Adjusted R-squared 0.846821     S.D. dependent var 15.97710 S.E. of regression 6.253125     Akaike info criterion 6.669616 Sum squared resid 860.2346     Schwarz criterion 6.909586 Log likelihood -85.03981     Hannan-Quinn criter. 6.740971 F-statistic 36.93412     Durbin-Watson stat 2.504654 Prob(F-statistic) 0.000000 asdh Step 2: Estimate structural equations (1) and (2) using the fitted values from the reduced form estimates as instruments. In this stage we run OLS regressions on equations (1) and (2) with the modification that we include the fitted values as instruments. Table 11: The Estimated equation for Demand Dependent Variable: Q Method: Least Squares Date: 09/01/11 Time: 15:44 Sample: 1960 1986 Included observations: 27 Coefficient Std. Error t-Statistic Prob.   C 29.31849 11.90463 2.462780 0.0217 PHAT 0.128447 0.098352 1.305992 0.2045 CPI 0.049126 0.038838 1.264873 0.2186 INCOME 0.004852 0.002113 2.296906 0.0311 R-squared 0.855845     Mean dependent var 89.96296 Adjusted R-squared 0.837042     S.D. dependent var 15.97710 S.E. of regression 6.449642     Akaike info criterion 6.701880 Sum squared resid 956.7512     Schwarz criterion 6.893856 Log likelihood -86.47538     Hannan-Quinn criter. 6.758964 F-statistic 45.51684     Durbin-Watson stat 2.386946 Prob(F-statistic) 0.000000 Table 11 shows that in the estimated structural equation for demand, only income has a significant and positive influence on the quantity demanded. Table 12: Estimated equation for supply Dependent Variable: QHAT Method: Least Squares Date: 09/01/11 Time: 15:49 Sample: 1960 1986 Included observations: 27 Coefficient Std. Error t-Statistic Prob.   C 62.23651 2.772080 22.45120 0.0000 P 0.039882 0.062145 0.641748 0.5274 NPCOST 0.156261 0.086654 1.803268 0.0845 L 0.139570 0.073523 1.898325 0.0703 R-squared 0.954256     Mean dependent var 89.96296 Adjusted R-squared 0.948289     S.D. dependent var 14.90577 S.E. of regression 3.389582     Akaike info criterion 5.415244 Sum squared resid 264.2532     Schwarz criterion 5.607220 Log likelihood -69.10580     Hannan-Quinn criter. 5.472329 F-statistic 159.9311     Durbin-Watson stat 0.183435 Prob(F-statistic) 0.000000 Table 12 above shows that the constant is significant at the 5% level of significance. Land and input costs turn out to be significant determinants of the supply at the 10% level only. Notably, both these factors have a positive impact on the quantity supplied. b. Estimation of demand and supply as a system of equations This section presents the results of estimating the model as a system of equations. Table 13 presents the results. Table 13: Estimation of demand and supply as a system of equation System: UNTITLED Estimation Method: Least Squares Date: 09/01/11 Time: 02:20 Sample: 1960 1986 Included observations: 27 Total system (balanced) observations 54 Coefficient Std. Error t-Statistic Prob.   C(1) 28.13234 12.06687 2.331371 0.0242 C(2) 0.084849 0.091836 0.923911 0.3604 C(3) 0.058059 0.038675 1.501195 0.1401 C(4) 0.005250 0.002119 2.477754 0.0170 C(5) 63.31042 5.730398 11.04817 0.0000 C(6) -9.56E-05 0.128466 -0.000744 0.9994 C(7) 0.149289 0.151985 0.982258 0.3311 C(8) 0.174503 0.179130 0.974169 0.3351 Determinant residual covariance 562.4070 Equation: Q=C(1) +C(2)*P+C(3)*CPI+C(4)*INCOME Observations: 27 R-squared 0.850696     Mean dependent var 89.96296 Adjusted R-squared 0.831222     S.D. dependent var 15.97710 S.E. of regression 6.563816     Sum squared resid 990.9245 Prob(F-statistic) 2.288646 Equation: Q=C(5) +C(6)*P+C(7)*L+C(8)*NPCOST Observations: 27 R-squared 0.829859     Mean dependent var 89.96296 Adjusted R-squared 0.807667     S.D. dependent var 15.97710 S.E. of regression 7.006889     Sum squared resid 1129.219 Prob(F-statistic) 2.027203 The lower half of the table details the exact specifications. C(1), C(4) and C(5) are the only statistically significant coefficients. While C(1) and C(5) are the constants, C(5) is the coefficient of income for the demand equation. Further, in support to the results obtained in the previous section, the coefficient on income is positive and almost equal to 0.005 which is what we obtained in the previous section. No other coefficients are found to be significant. Therefore, the results from either type of estimation yield the same result. The implication is that positive income changes induce small but positive changes on the demand. However, we fail to obtain any evidence that suggests any of the other variables included have any statistically significant impact on either demand or supply. Read More