Regression Models Using Cross Section Data And Time Series Analysis In Essay.

Q1.There is a growing interest among economists and policy makers with regard to the effect socioeconomic and political environment has on country specific income and development. consists data on log of real gdp per capita in dollars (lgdpc), median age in population measured in years (medage), urbanisation rate with an index of one or less (ur) and political right index that ranges from zero to one (plr) of 125 countries of the world in 2000.    

(a)Provide a descriptive statistics of the variables real gdp per capita and medage. Comment on their mean, median, standard deviation, skewness and kurtosis.

(b)Plot lgdpc against medage and ur in two separate scatter diagrams. What do you observe?

(c)Estimate the following regression model,

(d)Interpret the estimates ofand  in terms of percentage changes.

(e)Based on your estimated model, what are the predicted lgdpc and actual GDP per capita for Australia and Japan in 2000?

(f)Does the variable plr significantly explain the dependent variable lgdpc? Explain.

(g)Comment on the overall significance of the model.

Q2.Use the data containing data on Thailand’s gross domestic product (gdp), consumer price index (cpi), exchange rate (exrate) and interest rate (ir).

(a)Estimate the following model and comment on the significance of the independent variables,  

(b)Re estimate the model in (a) by adding a single lag of all the independent variables and report the results in the usual form.

(c)Compare the estimated long run impact of the effect of interest rate and exchange rates on cpi.  

(d)Are the three lagged variables included in the model in (b) above jointly significant at the 5% level?

Mean – In accordance with the highlighted table, average real GDP per capital for the year 2000 stood at $ 8,454.97. Also, it is evident that this average or mean value is in excess of the given variable’s median value which may be inferred to the distortion caused on account of outliers on the higher side primarily in the form of developed nations.

Median – The median value of the given variable for the year 2000 has been derived as $ 5,141.26.  It is indicative that 50% of the nations taken for the computation have a real per capita GDP lower than the above median value.

Standard Deviation –  The standard deviation needs to be interpret in the context of mean and evaluating the same, it is imperative that standard deviation is large which is on expected lines taking into consideration high disparity amongst the developed and underdeveloped nations.

Skewness – The calculated value of skew has come out to be positive which is indicative of the presence of rightward tail which may be attributed to the presence of developed nations having exceptionally high per capita real GDP. Further, the presence of skew confirms that the distribution would not be normal and non-symmetric.

Kurtosis – The kurtosis value needs to be viewed in the context of normal distribution kurtosis of 3. It is apparent that kurtosis of per capita real GDP is lesser and leads to peaks that tend to short and tails that tend to be thinner when compared with a corresponding normal curve.

Mean – The mean age of the various national population in the year 2000 was 26.27 years. Also, it is evident that this average or mean value is in excess of the given variable’s median value which may be inferred to the distortion caused on account of outliers on the higher side primarily due to aging experienced in the developed countries.

Median – The median of the given national populations in the year 2000 was 24 years. The figure implies that 50% of nations had a national age of 24 years or less in the year 2000.

Standard Deviation –  The extent of dispersion measured by standard deviation is low when the corresponding figure of 8.3 years is compared with the mean national age. This is expected since the average age for nations would not vary too much and is likely to lie in a narrow range.

Scatter Plots

Skewness – The presence of positive skew in the meian age signifies that a skew on the right would be present leading to a long right tail and  this may be caused by the concentration of certain developed countries which are experiencing an aging population.

Kurtosis – The negative value of kurtosis is indicative of lighter tails and represents that extreme values for age are rather limited in comparison to the other variables.

The scatter-pot between median age (independent variable) and log of per capita real GDP (Dependent variable) is shown below.The scatter-pot between urbanization index (independent variable) and log of per capita real GDP (Dependent variable) is shown below.

For both the above scatter-plots, the respective figures indicate the presence of a linear relationship with a positive slope. As a result, improvement in the independent variables represented by higher urbanization index and higher median age would lead to higher per capita real GDP as observed for developed nations.

The multiple regression output for the given case is indicated above based on which the regression equation can be highlighted as indicated below.

lgdpc = 5.339 + 0.366plr +0.066medage +2.105ur + 0.502

β₂ tends to represent the slope of the medage variable which has a value of 0.066 and thus implies that a unit change in the median change of a given country would have caused a change in log of per capital real GDP (lgdpc) by 0.066 and the direction of the change would have been as the independent variable. Taking antilog, the corresponding change in per capita real GDP in the year 2000 would have been USD 1.07.

Β₃ tends to represent the urban index which on change of 0.1 would be expected to yield a corresponding lgdpc change of 2.105 directed in the same direction thus leading to an absolute change of USD 0.82.

The multiple regression equation as derived from the model above is summarized below.

Lgdpc = 5.339 + 0.366plr +0.066medage +2.105ur + 0.502

The corresponding inputs for Australia are indicated below.

plr = 1, medage =35.2 and ur=0.8469

Substituting the given inputs in the regression equation, we get the following.

lgdpc (Australia)  = 5.339 +0.366*1+ 0.066*35.2 + 2.105*0.8469 + 0.502 = 10.325

The computed value would lead to a corresponding real GDP per capita value of USD 30,466.3in 2000.

The corresponding inputs for Japan are indicated below.

plr = 1, medage =41.3 and ur=0.788

lgdpc (Japan) = 5.339 +0.366*1+ 0.066*41.3+ 2.105*788 + 0.502 = 10.606

Multiple Regression Analysis

The computed value would lead to a corresponding real GDP per capita value of USD 40,370 in 2000.

Null Hypothesis: β_plr = 0 i.e. there is no statistical significance of the variable plr slope and therefore it can be assumed as zero.

Alternative Hypothesis: β_plr ≠ 0 i.e. there is statistical significance of the variable plr slope and therefore it cannot be assumed as zero.

The regression output derived above indicates that the t statistic computed value comes at 2.2 which transforms to a p value of 0.03.

Since, p value (0.03) < Significance Level (0.05), hence rejection of null hypothesis would take place thus ensuring acceptance of the alternative hypothesis. This, it may be concluded that for lgdpc determination,  one of the significant independent variables is plr.

In order to ascertain the significance of the regression model, a F test needs to be deployed. In this regards, the applicable hypotheses are as follows.

Null hypothesis: β₁ = β₂ = β₃ which implies that all the slope coefficient associated with the model are insignificant and can be assumed as zero.

Alternative hypothesis: One of the slopes at a minimum of the given three slopes is significant an therefore cannot be assumed as zero.

The output of the excel indicates that F value for the regression is 166.21 and the corresponding p value is 0.00

Since, p value (0.00) < Significance Level (0.05), hence rejection of null hypothesis would take place thus ensuring acceptance of the alternative hypothesis. This, it may be concluded that the given regression model is indeed significant.

Cpi = 95.183 + 0.00004gdp -1.05997exrate – 0.227ir + 3.5775

To determine the significance of the independent variables, hypothesis testing of the slope needs to be performed which has been already performed in the output obtained above.

The corresponding t values of coefficients of the variables gdp, exrate and ir are 14.558, -7.76 and -1.79 respectively.

For, two variables namely gdp and exrate, the p value derived from the t statistics stated above is lesser than the significance level (5%), which implies that these variables are significant unlike ur while has a p value greater than the significance level (5%) and hence it is not a significant independent variable.

Cpi = 94.14 + 0.00004gdp -1.0424exrate -0.13296ir +3.478

• The output highlighting multiple regression indicates that out of interest rate and exchange rate, the latter would be more significant. This becomes apparent from determining the slope of the respective slope coefficients where it is found that exchange rate is able to establish higher significance.  This is on account of a higher t value and a lower p value observed for exchange rate. Further, on account of slope being larger in magnitude for exchange rate, it can also be derived that only the significance of exchange rate is higher but essentially the role of the same would be higher in consumer price index determination.
• The independent variables joint significance may be ascertained through the application of the F Test for the multiple regression model derived above. The first step in this process is to state the relevant hypotheses which is carried out below.

Null hypothesis or H_o: β₁ = β₂ = β₃ which implies that all the slope coefficient associated with the model are insignificant and can be assumed as zero.

Alternative hypothesis or H₁: One of the slopes at a minimum of the given three slopes is significant an therefore cannot be assumed as zero.

The output of the excel indicates that F value for the regression is 141.26 and the corresponding p value is 0.00

Since, p value (0.00) < Significance Level (0.05), hence rejection of null hypothesis would take place thus ensuring acceptance of the alternative hypothesis. This, it may be concluded that the given regression model is indeed significant.