1. (a) i. As the sample size n, gets larger, the sample variance of the observed variables will tend to decrease.
ii. The covariance of the two regressors will tend to decrease if the sample size increases. This is because the sample size n, is in the denominator in the formula of covariance.
iii. In the regression model, the OLS coefficients will tend to more accurate estimates, if the sample size increases. They will tend to population parameter. It will yield more accurate results.
iv. The R squared will tend to be less biased as the sample size increases. This will further lead to less difference between R squared and adjusted R squared.
v. Here, the dependent and independent variables are related due to the presence of unobserved variable v3i. As n gets larger, the estimation tends to be more accurate. Therefore, there are chances that the coefficient of x2i will tend to zero.
vi. In this case, when x2i is regressed on x1i then, the coefficient will tend to zero as the sample size increases.
vii. In this model, the independent variables are not related to each other. Therefore, as n gets larger, the estimates will be more accurate and will show the exact change in Y due to individual changes in x1i and x2i.
viii. As n gets larger, R square will tend to 1.
ix. As the n gets larger, the OLS estimates will tend to normal distribution with mean 0 and variance 1.
2. (a) Residual sum of squares of regression A can be calculated as:
RSS = ∑ui2 = TSS – ESS= ∑Yi2 – β2҇ ∑YiX1i – β3҇ ∑YiX2i
RSS = 11400-(0.4*13000) – (0.7*6000)
RSS = 114000-5200-4200 = 2000
R squared can be calculated as:
R2 = ESS/TSS = 1 – (RSS/TSS) = 1 – (2000/11400) = 0.82456
R squared also known as coefficient of distribution shows the proportion of changes in Y explained by the regressors in the regression. This highlights that 82.45% of the variations in Y is explained by the regressors X1i and X2i in regression A.
(b) Residual sum of squares of regression B can be calculated as:
RSS = ∑ui2 = TSS – ESS= ∑Yi2 – β2҇ ∑YiX1i – β3҇ ∑YiX2i – β4҇ ∑YiX3i
RSS = 11400-(0.3*13000) – (0.6*6000) – (-0.9*500)
RSS = 114000 – 3900 – 3600 + 450 = 4350
R squared can be calculated as:
R2 = ESS/TSS = 1 – (RSS/TSS) = 1 – (4350/11400) = 0.61842
R squared also known as coefficient of distribution shows the proportion of changes in Y explained by the regressors in the regression. This highlights that 61.82% of the variations in Y is explained by the regressors X1i and X2i and X3i in regression B.
(c) No, the inclusion of X3i did not reduce the residual sum of squares as seen with the calculations above. In fact, the RSS increased from 2000 to 4350. By adding annual savings, the variation in Y due to the regressors is reduced. Moreover, the proportional change in the unexplained variation in Y in regression A is increased in regression B by 0.21 approximately. The coefficient of annual savings shows that on an average, an increase in one unit in the annual savings reduces the family spending by 0.9 units.
(d) i. Null Hypothesis: H0: β3 = 0; This means that the coefficient of annual savings is zero and annual savings do not affect the family spending.
Alternative Hypothesis: H1: β3 not equal to zero; This means that annual savings affect the family spending.
ii. Test statistic can be taken as:
t = (Estimated Value of β3 – Hypothesised Value of β3)
Standard error of estimated β3 / √n
t = Estimated value of β3 - 0 * 10
SE(β3)
t = -0.9/ SE(β3) * 10
SE (β3) = √ (∑ui2 / (N-K)) = 4350 / (100-4) = √45.31= 6.73
t = -0.9/6.73 * 10 = -1.337
iii. For two tailed test, we will use level of significance α/2 = 0.01/2 = 0.005.
The critical region can be found by looking at t table. The critical value is calculated as 1 – α/2 = 0.995.
t table is used as the population variance is unknown.
iv.The decision rule is that, if the absolute value of the calculated test statistic is greater than the critical value, then the null hypothesis H0 is rejected. However, if the value of calculated test statistic is less than the critical value, null hypothesis is not rejected. In the above case, absolute calculated t statistic is more than the critical value. Therefore, the null hypothesis is rejected. The coefficient of annual saving is statistically significant.
3. To calculate the probability, we need to calculate the Z value. From the question, mean is given as 30 grams and population standard deviation is 1.5 grams of a booklet. We need to calculate the probability that the 100 booklets weigh more than 3.3 kg. This can be written as the probability that 1 booklet weigh more than 33 grams.
Z = (X – population mean) / population std. Deviation
Z = (33 – 30) / 1.5
Z = 2
Here, X is the random variable showing the weight of one booklet.
P ( X> 33) = P ( Z> 2) = 1 – P ( Z < 2) = 1 – 0.9772 = 0.0228.
This probability can be tested by using Z table. We can take the critical value at 5% level of significance.
The calculated absolute value of Z statistic should be compared with the critical value of Z.
If the calculated Z statistic is greater than the critical value of Z, then the null hypothesis stating that the probability is not 0.0228 should be rejected and vice-versa.
Remember, at the center of any academic work, lies clarity and evidence. Should you need further assistance, do look up to our Econometrics Assignment Help
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