Introduction

The topic selected shall be Gross Domestic Product. However, due to the complexity of such data and its review the research shall base on the GDP trend for United States. This way aim to establish whether GDP has been instrumental in enabling the US to achieve equitable economic development. In the analysis part the focus is to develop a single-equation linear regression model and eventually test for GLS and non-linear regression models. In econometrics there are a number of ways to estimate a model within a single equation. For instance, the commonly used one is the ordinary least squares approach adopted in estimating linear regressions. In the case of non-linear models it applies mostly where the dependent variable is discrete and such may include logit, probit or tobit. As such a linear regression is simple to analyze, interpret and be scientifically acceptable. However, its limitation is that in most of the real-world phenomena such does not correspond to the assumptions deriving from a linear model. However, in order to make a well substantiated analysis the model estimate was developed as follows:

RGDPt = ?0 + ?1GS + ?2MS + ?3FDI + ?t

Where in the model, the parameters are:

RGDP = GDP

GS = Government Spending

MS = Money Supply

FDI = Foreign Direct Investment (Net Outflows)

Thus, the focus shall be to test the model above using a number of regression models to as to ascertain whether GDP in the United States has been of any significance to other development initiatives in the country such as government spending, money supply and foreign direct investment where the latter three are the indicator variables for sustainable development. The following NULL hypotheses are proposed:

H1: GDP has been a significant predictor of government spending in the U.S.

H2: GDP has been a significant predictor of money supply in the U.S.

H3: GDP has been a significant predictor of foreign direct investment in the U.S.

The model shall be estimated and tested based on both linear and non-linear regression parameters. E-Views shall be used to generate all statistical analysis.

Data analysis and evaluation

To start with are the descriptive statistics for the main variables.

Table 1: Descriptive statistics results
FDI ($, billions) GDP ($ Trillions) GS ($ Trillions) MS ($ Trillions)
Mean 157 111 137 114
Median 109 108 118 118
Maximum 448 158 254 246
Minimum 607 653 455 465
Std. Dev. 142 308 678 443
Skewness 0.69 0.03 0.46 0.78
Kurtosis 2.10 1.58 1.91 4.20

Jarque-Bera 3.85 2.88 2.88 5.49
Probability 0.15 0.24 0.24 0.06

Sum 5350 3760 4660 3890
Sum Sq. Dev. 6660 31300 15200 64700

Observations 34 34 34 34

The key descriptive statistics indicate the mean for FDI to be at 157 billion; then for the GDP it shows a mean of 111 in terms of trillions while GS has an average score of 137. Lastly, is the mean score for money supply resting at 114 trillion: Across the board it can be seen that these development indicators report varying mean differences which could be because they all have an effect on the other upon any changes to them.

The trend for GDP is as shown below.

Figure 1: GDP trend performance quarterly in the United States

The trend has been generated using E-Views and indicates an upward performance across the period.

Prior to conducting and presenting the results for the regressions it would be worthwhile to check the dataset for its normal distribution. It is imperative that a dataset to have a normal distribution before it can be subjected to regression analysis. In normal distribution the hypotheses would be as follows:

H0: The dataset is a normal distribution

H1: The dataset is not a normal distribution

Figure 2: Normality model from E-Views

The indication is that with a Jarque-Bera score of 1.462090 and Probability value at 0.48 then the null hypothesis cannot be rejected. It means the dataset used in the study was a normal distribution paving way for regression analysis on it.

Table 1: Regression model estimate using Least Squares (NLS and ARMA)
Dependent Variable: LOG(GDP)
Method: Least Squares
Date: 05/20/16 Time: 10:45
Sample: 1 34
Included observations: 34

Variable Coefficient Std. Error t-Statistic Prob.

C 29.39719 0.058676 501.0124 0.0000
FDI -6.94E-14 3.59E-13 -0.193354 0.8480
GS 3.78E-13 9.18E-14 4.114965 0.0003
MS 7.93E-14 7.82E-14 1.014486 0.3185

R-squared 0.914828 Mean dependent var 29.99469
Adjusted R-squared 0.906311 S.D. dependent var 0.291169
S.E. of regression 0.089123 Akaike info criterion -1.887462
Sum squared resid 0.238289 Schwarz criterion -1.707891
Log likelihood 36.08686 Hannan-Quinn criter. -1.826223
F-statistic 107.4092 Durbin-Watson stat 0.144342
Prob(F-statistic) 0.000000

In the regression model it can be seen that The R squared is at .915 which means 91.5% of the cases for FDI (Outflows), Money Supply (MS) and Government Spending (GS) are explained by GDP. From the adjusted R squared it can be said 90.6% are the cases of MS, GS, and FDI that are explained by GDP in the United States for the selected quarterly period. The F statistic is at .000 indicating there is statistical significant difference between GDP and the other variables. So, a change in GDP would affect GS, MS, or FDI (Outflows). In the p values it can be seen that FDI (outflows) at .848 meaning it is not significantly predicted by GDP. On the other hand, GS (.0003) shows high significance meaning it is significantly predicted by GDP. Then MS (.3185) meaning it is not significantly predicted by GDP.

The same model estimate was assessed based on a non-linear regression model (i.e. probit); the results are as shown below.

Table 2: Regression model estimate using Probit
Dependent Variable: LOG(GDP)
Method: Probit
Date: 05/20/16 Time: 10:35
Sample: 1 34
Included observations: 34
Number of always included regressors: 4
No search regressors
Selection method: Probit
Stopping criterion: p-value forwards/backwards = 0.5/0.5

Variable Coefficient Std. Error t-Statistic Prob.

C 29.39719 0.058676 501.0124 0.0000
MS 7.93E-14 7.82E-14 1.014486 0.3185
GS 3.78E-13 9.18E-14 4.114965 0.0003
FDI -6.94E-14 3.59E-13 -0.193354 0.8480

R-squared 0.914828 Mean dependent var 29.99469
Adjusted R-squared 0.906311 S.D. dependent var 0.291169
S.E. of regression 0.089123 Akaike info criterion -1.887462
Sum squared resid 0.238289 Schwarz criterion -1.707891
Log likelihood 36.08686 Hannan-Quinn criter. -1.826223
F-statistic 107.4092 Durbin-Watson stat 0.144342
Prob(F-statistic) 0.000000

Selection Summary

Overall the model outputs for probit indicate same trends compared to OLS regression in table 1; therefore, explanations would still be the same in terms of R square, Anova and P values. Thus, the hypotheses stated indicate that only the first one (H1) can be confirmed but the rest two rejected. It means as follows:

GDP has been a significant predictor of government spending in the U.S.
GDP has NOT been a significant predictor of money supply in the U.S.
GDP NOT has been a significant predictor of foreign direct investment in the U.S.

It would be worthwhile to run more diagnostic tests on the data. See the results below for heteroscedasticity and specification error test (Ramsey REST test).

Table 3: Test for heteroscedasticity
Heteroskedasticity Test: Breusch-Pagan-Godfrey

F-statistic 3.957368 Prob. F(3,30) 0.0172
Obs*R-squared 9.640106 Prob. Chi-Square(3) 0.0219
Scaled explained SS 3.784364 Prob. Chi-Square(3) 0.2857

The hypothesis would be as follows:

H0: There is homoscedasticity

H1: There is heteroscedasticity

The results (F-statistics 3.957, Prob 0.0172) means the null hypothesis must be rejected; it follows that the model has heteroscedasticity problems. In the regression where heteroscedasticity is a problem it means the coefficients are inefficient but then unbiased. However, the biggest problem would be the bias projection in the standard errors.

Table 4: Results for Ramsey RESET test
Ramsey RESET Test
Equation: UNTITLED
Specification: LOG(GDP) C FDI GS MS
Omitted Variables: Squares of fitted values

Value df Probability
t-statistic 17.09345 29 0.0000
F-statistic 292.1860 (1, 29) 0.0000
Likelihood ratio 81.76063 1 0.0000

F-test summary:
Sum of Sq. df Mean Squares
Test SSR 0.216773 1 0.216773
Restricted SSR 0.238289 30 0.007943
Unrestricted SSR 0.021515 29 0.000742
Unrestricted SSR 0.021515 29 0.000742

LR test summary:
Value df
Restricted LogL 36.08686 30
Unrestricted LogL 76.96718 29

H0: The model is correctly specified

H1: There is a functional form of misspecification

The Ramsey Regression Equation Specification Error test shows if the non-linear combinations in relation to the fitted values enable to explain the response variable. The test results indicate t-statistic of 17.09345 and probability value of 0.0000 hence the null hypothesis must be rejected. The model is, therefore, mis-specified meaning the non-linear combinations of the regressors have capacity to explain the response variable.

The other test shall be multicollinearity. It exists in a dataset that has two or more of its predictors as moderately or highly correlated. The results are as generated below using correlation coefficient.

Table 5: Multicollinearity test results
GDP FDI GS MS
GDP 1 0.93 0.98 0.87
FDI 0.93 1 0.95 0.84
GS 0.98 0.95 1 0.89
MS 0.87 0.84 0.89 1

The most important column is the one marked red because it depicts the multicollinearity results for between GDP and the predictor variables namely FDI, GS, and MS. Essentially, any value above +6 is a show of high multicollinearity; as can be seen FDI (0.93), GS (0.98) and MS (0.87) are all above 0.6. That way it means there is high multicollinearity between each of these predictor variables to GDP. Unfortunately, it can be seen that there is severe multicollinearity in the model the risk of increasing variance of the coefficient estimates is high. Thus, the coefficient estimates could be said to be unstable and complex to interpret. In fact, basing on the R square in the regression model it can be see it is at .92 which is a sign of high multicollinearity.

Chow’s forecast test is an estimation of multiple models where the coefficients in two linear regressions are tested for equality.

Table 6: Results for Chow’s forecast test
Value df Probability
t-statistic 0.499907 29 0.6209
F-statistic 0.249907 (1, 29) 0.6209
Likelihood ratio 0.291739 1 0.5891

F-test summary:
Sum of Sq. df Mean Squares
Test SSR 0.002036 1 0.002036
Restricted SSR 0.238289 30 0.007943
Unrestricted SSR 0.236253 29 0.008147
Unrestricted SSR 0.236253 29 0.008147

LR test summary:
Value df
Restricted LogL 36.08686 30
Unrestricted LogL 36.23273 29

Unrestricted log likelihood adjusts test equation results to account for
observations in forecast sample

It can be seen that none of the forecast test statistics refute the null hypothesis of lack of structural change.

Conclusion

The study has focused on the U.S. economy under the topic gross domestic product. The regressor variables were FDI (out flow), government spending and money supply where all these are the cornerstone of economic development not only in the United States but also in other parts of the world. The data results indicated the presence of high multi collinearity in these variables meaning they all explain each other across the period. Other observations were that GDP has positively influenced the trend in government spending but not in terms of money supply neither foreign direct investment (out flow). Other observation was the misspecification with the model used to determine U.S. economy in lieu of the variables meaning FDI, Money Supply and GS have capacity to explain the response variable i.e. GDP.
Reference

McKisney, G. 2013. Econometrics. 3rd edition, Cambridge University Press
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Posted on May 26, 2016Author TutorCategories Question, Questions