Question

2. (40 marks) The following equations have been estimated by OLS using annual data for a Canadian city for 1991-2018. QW per capita consumption of wine in litres RPW = relative price of wine RPB relative price of beer RPS relative price of spirits RY real per capita disposable income (1) QW- 10.50-7.30RPW + 0.003RY 4.96/(2.00) 0.0002) R2= 0.90 a2= 3.50 2) QW.00H6.9ORPW 0.0050RY 1.10RPB 1.50RPS 10.40 2.30 (0.0009) (7.65) (0.55) (a) (10 marks) Perform a test of the significance of the regression on equation (1) using 1 % level of significance. (b) (10 marks) Using equation (2) determine whether the cross price elasticity of demand for wine is elastic or inelastic with respect to the price of beer. Briefly explain the meaning of this cross elasticity. (Note that QW-6.00 and RPB-0.55) (c) (10 marks) Perform a test on the null hypothesis that the coefficients on RPB and RPS are jointly equal to zero using 5% level of significance (d) (10 marks) In 1997 RY-4200, RPW-1.50, RPB-0.5, RPS 5.3. As part of its 2019 budget the local government proposes to impose a 5% consurnption tax on wine. Assuming that the beer and spirits producers are not planning to change prices and that real income is expected to grow by 5% what is your forecast of the wine consumption in 2019 (Use equation 2).
Please answer all four subquestions with specific steps

Answer 1

Consider the following notations: QW per capita consumption of wine in liters RPW relative price of wine RPB relative price of beer RPS relative price of spirits RY real per capita disposable income - - The equations estimated by OLS using annual data for a Canadian city for 1991-2018 are given as follows: OW 10.50- 7.30 RPW+0.003 RY (2.00) (1) (4.96 R2 0.9 (0.0002) 2 3.5 = (2) QW -4 -6.90 RPW+ 0.005 RY +1.1RPB+1.5 RPS (0.0009) = (7.65) (10.4) (2.30) (0.55) R2 0.94 G2 2.50 = (a) To test the significance of the regression hypothesis as follows: Null Hypothesis, Ho B 0 Alternative Hypothesis, H:B B, 0 (two-tailed alternative) equation (1), frame the null and alternative on is the co-efficient of RPW and B, is the co-efficient of RY Here, The test statistic is given as follows: MSR F = MSE

Here, as follows MSR is the regression mean square and is given SSR MSR k-1 And MSE is the error mean square which is given as follows: SSE MSE n-k Here, k denotes the number of parameters in the regression model and represents the total number of data points Therefore, the test statistic F follows the F -distribution with k-1 and n-k degrees of freedom The test is to reject the null hypothesis if F is greater than the tabulated value at F -ln-k* Where a is the level of significance. Since the data is given from 1991-2018, therefore, n = 28 Also, in equation (1), there are 3 regression coefficients, therefore, k = 3 MSR The test statistic F = can also be written as follows: MSE MSR F MSE R2 n-k k -11-R2 Here, R2 is the coefficient of determination

The value of the test statistic is computed as follows 28-3 F = 3 -11-0.90 0.90 12.5 (9) =112.5 Now the tabulated value of F with 2 and 25 degrees of freedom at 1% level of significance is 5.57 Since the calculated value of F is greater than the tabulated value of F, that is, 112.5 is greater than 5.57, as a result of which we B , that is, the relative price of wine and real per capita disposable income has a significant effect on the per capita consumption of wine in liters. reject the null hypothesis and conclude that (b) The formula for cross-price elasticity is given as follows %change inquantity demanded in product A Cross price elasticity %change in price of product B Thus, -6.9 Cross price elasticity 1.1 =-6.272 Since the cross-price elasticity is less than 1, thus, the cross-price elasticity is inelastic Here, the cross-price elasticity indicates that as the price of beer falls, the quantity demanded for wine will increase

(c) as follows The hypothesis for testing that the coefficients on RPB and RPS is given Null Hypothesis, Ho B B4 =0 Alternative Hypothesis, H B B4 #0 is the co-efficient of RPB and B, is the co-efficient of RPS Here, 4 The test statistic for testing Ho H is given as follows: versus (SSR F - SSR restricted orestricted /(п-k-1) SSR urestricted The test statistic F follows the F -distribution with q and n-k degrees of freedom. Here, SSRicted is the sum of squares of residuals from the restricted regression and SSR is the sum of squares of residuals from the unrestricted regression, q is the ures tricted number of restrictions under the null hypothesis and k is the number of regressors in the unrestricted regression Here, the restricted model is represented by equation (1) and the unrestricted model is represented by equation (2) To obtain the value of the test statistic, proceed as follows: For equation (1), o = 3.5 and c = MSE. Also SSE MSE n-k SSE (n-k) MSE = Therefore (28-3)3.5 SSEresticted = SSE 46.7707 'resticted

For equation (2), 6 =2.50 and c = MSE Therefore (28 -5)V2.5 = toresti cted SSE 36.3662 1oresticted Thus, the test statistic is calculated as follows: (46.7707-36.3661)/2 F 36.3662/(28-5-1) - 3.4333 The tabulated value of F with 2 and 24 degrees of freedom at 5% level of significance is 3.40 Since the calculated value of F is greater than the tabulated value of F, that is, 3.4333 is greater than 3.40, as a result of which we reject the null hypothesis and conclude that RPB and RPS differ from zero (d) It is given that in 1997, RY = 4200,RPW = 1.50,RPB = 0.5 and RPS = 5.3 In 2019, the local government proposes to impose 5% consumption tax on wine, therefore, RPW for the year 2019 is computed as follows: RPW 1.55%(1.5) = 1.5 0.075 =1.575 Also, the real income is expected to grow by 5% in 2019, therefore, the real income for the year 2019 is calculated as follows: RY 4200 5%(4200) = 4200 210

= 4410 The forecast for wine consumption using equation (2) is obtained as follows: QW -4.00 6.90(1.575)+0.005 (4410)+1.1(0.5)+1.5 (5.3) QW 15.6825 Therefore, wine consumption in 2019 is 15.6825