Quantitive_Methold

|Subject Code: ECON20003 |Subject Name: Quantitative Methods 2 | |Tutorial Day/Time: Wednesday/ |Tutor Name: Emma Seyoum | |(4.15pm – 5.15pm) | | |Assignment Name or Number: Second Assignment | | |Student ID Number |Student Name | |1. |321388 |YINGYU LUO (Rosaline) | |2. |346473 |CHEN YUN YUN (ANITA) | |3. |355616 |GOON THIM LOONG, STANLEY | |4. |395180 |BUWANEKA JAYANETTI (BUWA) | 1. The relationships between average income levels (GDP per capita and the two measures – fertility and infant mortality The scatter plot shows the relationship between Birth Rate (dependent, Y variable) and GDP per capita (Independent, X variable) in selected countries hence demonstrates a rough negative relationship between the two. This means that as the income of women increases, the birth rates would decline. Moreover, this graph shows non-linear relationship between birth rate and the GDP per capita. The outliers are circled; however we believe that this is not due to any errors. Dependent variable (Y) is Birth Rate (per 1,000 people). Independent variable (X) is GDP per capita (current US$). The scatter plot shows the relationship between Mortality Rate (dependent, Y variable) and GDP per capita (Independent, X variable) in selected countries hence demonstrates a rough negative relationship between the two. This means that as the income levels increase, the number of deaths would decline. Moreover, this graph shows non-linear relationship between birth rate and the GDP per capita. The outliers are circled; however we believe that this is not due to any errors. Dependent variable (Y) is infant mortality rate (per 1,000 live births). Independent variable (X) is GDP per capita (current US$). 2. Transforming into linear form To convert the non-linear relationship between birth rate and GDP in order to estimate a linear relationship using Ordinary Least Squares (OLS), we need to use logarithm. This could be done by taking natural logarithms (ln) of both sides of the equation as shown below ln(Y) = ln (A)+b. ln(X) This is also called log-log (double log) transformation as we could use the above equation and transform this model to estimate lnA and b using Ordinary Least Squares. By using EViews, we ‘generate’ a new variable which is similar to the given variable. In order to form the logarithm of the given variable using EViews, the following was done. • lbirth = log(birth) • lmortality = log(mortality) • lgdp = log(gdp) Once the transformation is taken place, we got to see a clear negative relationship between the variables; GDP and Birth Rate. |Dependent Variable: LBIRTH_RATE | | |Method: Least Squares | | | |Date: 09/30/10 Time: 13:31 | | | |Sample (adjusted): 1 236 | | | |Included observations: 201 after adjustments | | | | | | | | | | | | | | | |Coefficient |Std. Error |t-Statistic |Prob.   | | | | | | | | | | | | | |LGDP |-0.239349 |0.012285 |-19.48308 |0.0000 | |C |5.015253 |0.105488 |47.54330 |0.0000 | | | | | | | | | | | | | |R-squared |0.656061 |    Mean dependent var |2.996030 | |Adjusted R-squared |0.654332 |    S.D. dependent var |0.474082 | |S.E. of regression |0.278730 |    Akaike info criterion |0.292751 | |Sum squared resid |15.46035 |    Schwarz criterion |0.325620 | |Log likelihood |-27.42150 |    Hannan-Quinn criter. |0.306051 | |F-statistic |379.5903 |    Durbin-Watson stat |1.964366 | |Prob(F-statistic) |0.000000 | | | | | | | | | | | | | | | | 3. Y=b0 +b1x + ε (Y= LBIRTH_RATE, X =LGDP) Y= 5.016-0.239X + ε Interpretation - On average, an increase in birth rate by 1 unit would result in a a decrease in 0.239units of GDP Goodness of Fit - R2= 0.656, thus 65.6% of variation of GDP can be explained by one explanatory variable, Birth Rate. ……………………………………………………………………………………………………… 4. Required conditions for using OLS a) Distribution of ε is normal b) Mean of the distribution of ε is 0 c) Standard deviation of ε (σ ε) is same for all values of x d) The ε’s are independent of each other e) The ε’s are independent of the x’s Of these conditions we can check only for a, c and d. The remaining two required conditions cannot be checked using the estimated model’s residual. However they are important as they ensure that our OLS estimates are unbiased. As we know, the sum of residuals equals zero by construction (∑ei = 0). Therefore, this is an outcome of OLS but not an assumption. Another outcome of OLS is that the residuals are constructed to be orthogonal. So, OLS residuals (ei’s) and the x’s are independent by construction. (a) Distribution of ε is normal - HISTOGRAM H0: ε = normal H1: ε ≠ normal As the p-value (0.056), 0.05 < p-value < 0.10 there is weak evidence to reject the null hypothesis. Therefore we can conclude that the distribution of residual is normal. (c) Standard deviation of ε (σ ε) is same for all values of x – SCATTER PLOT Since there is no apparent change in variation in residual value across ‘YHAT’, we assume it is homoscedesticity. Hence, the second condition of OLS is satisfied. (d) The ε’s are independent of each other – LINE GRAPH Since data used is a cross-section data, we do not need to check for autocorrelation in ε. Autocorrelation is only important if data used is time series. ……………………………………………………………………………………………………………….. 5. Test whether there is a negative relationship between fertility and average income levels across countries Step 1: H0: β1 = 0 H1: β1 < 0 Step 2: Since ε is normally distributed and σε is the same for all values of x, the required conditions for OLS holds and test-statistic is Step 3: α = 0.05 Step 4: Decision Rule: Reject H0 if t < t 0.05, 199 = (-1.645) Step 5: Calculate statistic = -0.239- 0 0.0122 = -19.48 Step 6: Since our computed t-statistic (-19.48) < (-1.645), we strongly reject the null hypothesis and conclude that at 5% level of significance, there is a negative relationship between fertility and average income levels across countries. 6. The relationship between fertility and average income levels, now including the measure of infant mortality as a second regressor. |Dependent Variable: LBIRTH_RATE | | |Method: Least Squares | | | |Date: 09/30/10 Time: 13:38 | | | |Sample (adjusted): 1 236 | | | |Included observations: 196 after adjustments | | | | | | | | | | | | | | | |Coefficient |Std. Error |t-Statistic |Prob.   | | | | | | | | | | | | | |LMORTALITY_RATE |0.313740 |0.030426 |10.31158 |0.0000 | |LGDP |-0.041486 |0.021908 |-1.893646 |0.0598 | |C |2.416367 |0.268183 |9.010131 |0.0000 | | | | | | | | | | | | | |R-squared |0.776514 |    Mean dependent var |3.004650 | |Adjusted R-squared |0.774198 |    S.D. dependent var |0.476135 | |S.E. of regression |0.226253 |    Akaike info criterion |-0.119141 | |Sum squared resid |9.879718 |    Schwarz criterion |-0.068966 | |Log likelihood |14.67585 |    Hannan-Quinn criter. |-0.098828 | |F-statistic |335.2946 |    Durbin-Watson stat |1.857063 | |Prob(F-statistic) |0.000000 | | | | | | | | | | | | | | | | Y=b0 +b1x1 + b2x2 + ε (Y= LBIRTH_RATE, X1 =LGDP, X2= LMORTALITY_RATE) Y=2.416-0.041X1+ 0.314X2 + ε • Holding GDP constant, an increase of Mortality rate by 1 unit will increase the birth rate on average by 0.314 units • Holding Mortality rate constant, an increase in GDP rate by 1 unit will decrease the birth rate on average by 0/041 units 7. Test whether there is a negative relationship between average income levels and fertility Step 1: H0: β1 = β2=0 H1: At least one βj is not equal to zero Step 2: Since ε is normally distributed and σε is the same for all values of x, the required conditions for OLS holds and test-statistic is F= SSR/ k SSE/ (n-k-1) Step 3: α = 0.05 Step 4: Decision Rule: Reject H0 if f > f 0.05, 198 = 3.04 Step 5: Calculate statistic F= SSR/k = 335.2946 SSE/ (n-k-1) Step 6: Since our computed f (335.2946) is greater than 3.04, we strongly reject the null hypothesis and conclude that at 5% level of significance, there is a negative relationship between fertility and average income levels across countries. Provide an explanation for any difference in the outcome of this test as compared to the test in part (v) above Based on the two results, we can conclude that both show a negative relationship between fertility and average income levels across countries.