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Links between GDP, Consumption, Investment and Government Expenditure
EC010 ECONOMETRICS PROJECT
(1) Collect and analyse data sets of his/her own choosing in order to illustrate a postulated economic relationship over time.
For my project I have decided to look at the relationship between Gross Domestic Product (GDP), Household Consumption (C), Investment (I) and Government Expenditure (G). The source from which my data was derived was 'Economic Trends Aug-Dec 1 Inc Suppl UK/CES E600'. This was used as my source of data as it is compiled by the government and therefore hopefully is relatively accurate. Using just the one source should remove any error, as the values should be consistent with the market price at the time so trends should be easily identified.
Year     GDP     C     G     I     NX     GDP GR     C     GDP
155     1107     18     5     06     110          18.     11.07
156     0541     144     41     15     87     6.81%     14.4     05.41
157     1671     147     650     600     4     5.14%     14.7     16.71
158     585     1501     7     74     80     4.047%     150.1     5.85
15     877     1580     70     80     14     5.411%     158.0     8.77
160     544     1657     405     468     65     6.4%     165.7     54.4
161     671     174     454     488     856     5.476%     174.     6.71
16     870     1848     4868     504     76     4.55%     184.8     8.7
16     0     1565     417     55     117     6.46%     15.65     0.
164     06     0868     5458     67     14     8.515%     08.68     0.6
165     5574     151     5     6804     107     7.14%     1.51     55.74
166     785     1     651     761     168     6.06%     .1     78.5
167     88     457     75     745     1816     5.075%     45.7     8.8
168     48     6451     7685     878     1     7.78%     64.51     4.8
16     46541     8054     8048     066     6     7.118%     80.54     465.41
170     51168     0547     07     1006     4564     .04%     05.47     511.68
171     57080     450     1046     114     5675     10.57%     4.5     570.8
17     6     8780     1171     147     577     10.77%     87.8     6.
17     755     4460     1455     157     768     1.078%     44.6     75.5
174     817     5116     1681     1814     145     11.565%     511.6     81.7
175     1047     6881     70     1856     15751     0.780%     68.81     104.7
176     14     7060     770     5516          15.551%     70.6     14.
177     144840     8504     60     801     55     14.15%     85.04     1448.4
178     1670     668     68     08     85     1.1%     6.68     1670.
17     1658     114458     086     811     777     15.001%     1144.58     165.8
180     58     166     41     48     840     14.71%     16.6     5.8
181     544     14710     55667     41     08     8.84%     1471.     5.44
18     75851     1607     60600     474     017     8.558%     160.7     758.51
18     0154     176881     654     5140     4517     8.514%     1768.81     015.4
184     08     1844     681     5868     744     6.677%     18.44     0.8
185     54     06600     77     6446     44477     8.788%     066     54.
186     8057     8848     76     68718     758     6.8%     88.48     805.7
187     4181     5114     85077     784     8806     8.6%     511.4     418.1
188     46650     845     1658     6076     0006     10.5%     84.5     4665.
18     51188     104     88     1110     06     8.86%     104.     5118.8
By simply looking at my explanatory variables it can be seen that in all cases there is a general increase over time. This is also true of the dependant variable GDP, this strongly demonstrates that there is a strong link between these variables. This is not unexpected as the macroeconomic model for GDP shows that household consumption, government spending and investment are three major components in calculating GDP. The general formula for calculating GDP is as follows
GDP = C + I + G + NX, where NX represents Net Exports. In this project I have decided to plot each of these variables against GDP in order to obtain a relationship, if any, between them. This means that I shall be using GDP as my dependant variable and C, I and G will be explanatory variables. I have added data for NX, GDP GR (GDP growth rate), C, and GDP, these will come into play later in question .
It is clear to see just by looking at the diagram for consumption against GDP that there is a strong positive relationship between the two variables, implying that it is essential in calculating GDP and has a significant impact on it if it changes.
In the above graph plotting GDP against investment there is again a strong positive relationship between the two variables suggesting it is also essential in calculating GDP, it will also have a proportional effect on the GDP if it changes.
The above diagram plotting GDP against government spending also displays a strong positive relationship where government spending is needed to calculate the GDP, again inducing a proportional change on GDP if it changes.
The above diagram plotting GDP against net exports displays a fairly weak positive relationship, however there is still a clear relationship. Again it is essential in calculating GDP and is likely to induce a proportional change in GDP if the value for NX changes.
The mathematical model I shall be using in this project is
GDP = ß1 + ßC + ßI + ß4G
.a) Conduct significance tests on the individual coefficients of your model and on the overall regression. State your chosen level of significance and give the critical values of your test statistics to which your results are compared. Interpret your estimated coefficients.
As mentioned previously I shall be using the mathematical model; 
GDP = ß1 + ßC + ßI + ß4G, and shall be using linear regression models where GDP is the dependant variable and and the rest will be explanatory variables. The software used to calculate the linear regression estimates is MICROFIT using the method of Ordinary Least Squares. The following are my results
Ordinary Least Squares Estimation
Dependent variable is GDP
5 observations used for estimation from 155 to 18
Regressor              Coefficient       Standard Error        T-Ratio     [Prob]
INT                       85.68         46.15                     1.717     [.05]
C                           .868             .1056                     .64     [.000]
I                            .4554             .168                     .5876     [.015]
G                          1.5785              .161                     .00     [.000]
R-Squared                          .85        R-Bar-Squared                               .8
S.E. of Regression                 18.        F-stat.    F(  ,  1)             6808.8[.000]
Mean of Dependent Variable  14868.5        S.D. of Dependent Variable      147467.1 Residual Sum of Squares          1.1E+08        Equation Log-likelihood          -11.76
Akaike Info. Criterion          -15.76        Schwarz Bayesian Criterion     -18.87
DW-statistic                       1.5
In my calculations H0 will represent the null hypothesis and H1 will represent the alterative hypothisis.
Significance test for Consumption expenditure
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.868  0           T table=1.67
0.1056
= .6076
= .64 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Investment
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.4554 - 0          T table=1.67
0.168
= .587571
=.5876 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Government Spending
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 1.5785 - 0          t table=1.67
0.161
= .74761
= .7 (4d.p)
Reject H0 as tcalc  1.67
I shall use the analysis-of-variables (ANOVA) to test the overall regression, in which the GDP is the dependant variable and the rest are explanatory variables. After obtaining the relevant information I conducted the following tests using the model
F =  R / (k-1)
(1-R)/(n-k)
Significance test for overall regression.
H0 ß = ß = ß4 = 0          Ha ß ¹ ß ¹ ß4 ¹ 0               wo tail test
Ftable=8.6
F =     0.85/(4-1)         
(1-0.85)/(5-4) 
F = 0.8
0.00004887
= 6887.864
Reject H0 as Fcalc  8.6
The size of the coefficients alters the value of the explanatory variables. If the coefficient is positive or negative dictates whether it has a positive or negative effect on the dependant variable. Also if it is less than or greater than 1 dictates how great the impact is. In my results the coefficients for Consumption and Investment are less than 1 meaning that a 1% increase in their value would have a less than 1% increase in GDP. However Government Spending has a value greater than 1, meaning a 1% rise in its value would lead to a greater than 1% rise in the value of GDP. This runs alongside the concept of the Multiplier effect.
(b) Add a further variable to your model, compare and comment on the values of R and R you obtain compared to your results in (a)
I have decided to add Net Exports (NX) as my extra variable to my model. These are the results obtained from the regression. This changes the mathematical model to
GDP = ß1 + ßC + ßI + ß4G + ß5NX.
Ordinary Least Squares Estimation
Dependent variable is GDP
5 observations used for estimation from 155 to 18
Regressor              Coefficient       Standard Error         T-Ratio     [Prob]
INT                        41.1840         6.81            1.188        [.64]
C                         1.178             .0745              14.1     [.000]
I                          .676             .140                 .167     [.04]                         G                         .7457             .174                .857                 [.001]
NX                        .4576             .085              5.557      [.000]
R-Squared                          .        R-Bar-Squared                               .
S.E. of Regression                 155.        F-stat.    F(  4,  0)            10065.1[.000]
Mean of Dependent Variable  14868.5       S.D. of Dependent Variable      147467.1
Residual Sum of Squares          5.51E+07        Equation Log-likelihood          -.774
Akaike Info. Criterion          -04.774        Schwarz Bayesian Criterion     -08.658
DW-statistic                       .4717
The R² value in the second model with the added explanatory variable of NX is higher in value to that in the first model. (0.85 and 0. respectively). This is expected as the R² value explains the variance, as more variables are introduced, more of the variance is explained and therefore a higher value is obtained. It can also be said that the error term (or stochastic term) has been reduced. The value represents the percentage of the variance that is explained in this case, the value has in creased from .85% to .% by adding the extra variable.
(c) 'Change the units' of one of your explanatory variables (e.g. divide all the data on one of your explanatory variables by 100) and repeat exercise (a). Comment. Then change your data on your dependant variable in a similar way and repeat (a). Comment, concentrating on the interpretation of the estimated coefficients in your model
I decided to change the units of my consumption variable by dividing it by 100. It will be referred to as Consumption (C). These are the results of the regression             
Ordinary Least Squares Estimation
Dependent variable is GDP
5 observations used for estimation from 155 to 18
Regressor              Coefficient       Standard Error         T-Ratio[Prob]
INT                      85.68           46.15             1.717[.05]
C                        8.68            10.56             .64[.000]
I                          .4554             .168             .5876[.015]
G                          1.5785             .161             .00[.000]
R-Squared                          .85        R-Bar-Squared                               .8
S.E. of Regression                18.        F-stat.    F(  ,  1)               6808.8[.000]
Mean of Dependent Variable  14868.5        S.D. of Dependent Variable      147467.1
Residual Sum of Squares          1.1E+08        Equation Log-likelihood          -11.76
Akaike Info. Criterion          -15.76        Schwarz Bayesian Criterion     -18.87
DW-statistic                       1.5
Significance test for Consumption expenditure
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 8.68  0           T table=1.67
10.56
= .64506
= .64 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Investment
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.4554 - 0          T table=1.67
0.168
= .587571
=.5876 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Government Spending
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 1.5785 - 0          t table=1.67
0.161
= .74761
= .7 (4d.p)
Reject H0 as tcalc  1.67
Significance test for overall regression.
Model  F =  R / (k-1)
(1-R)/(n-k)
H0 ß = ß = ß4 = 0          Ha ß ¹ ß ¹ ß4 ¹ 0               wo tail test
F =     0.85/(4-1)                                  Ftable=8.6
(1-0.85)/(5-4) 
F = 0.8
0.00004887
= 6887.864
Reject H0 as Fcalc  8.6
Even though the figures for consumption have been altered, the results for all significance testing remain the same. Again I have rejected all of the hypotheses.
ii) I decided to change the units of my GDP variable by dividing it by 100. It will be referred to as GDP. These are the results of the regression
Ordinary Least Squares Estimation
Dependent variable is GDP
5 observations used for estimation from 155 to 18
Regressor              Coefficient       Standard Error         T-Ratio     [Prob]
INT                       8.5             4.6                    1.717          [.05]
C                          .00868         .001056                 .64          [.000]
I                          .004554          .00168                .5876          [.015]
G                         .015785            .00161                 .00          [.000]
R-Squared                            .85        R-Bar-Squared                               .8
S.E. of Regression                  18.4        F-stat.    F(  ,  1)               6808.8[.000]
Mean of Dependent Variable    148.7        S.D. of Dependent Variable          1474.7
Residual Sum of Squares             1118.        Equation Log-likelihood          -150.58
Akaike Info. Criterion            -154.58        Schwarz Bayesian Criterion     -157.60
DW-statistic                         1.5
Significance test for Consumption expenditure
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.00868  0           T table=1.67
0.001056
= .6076
= .64 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Investment
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.004554 - 0          T table=1.67
0.00168
= .587571
=.5876 (4d.p)
Reject H0 as tcalc  1.67
Significance test for Government Spending
H0 ß = 0          
H1 ß ¹ 0               wo tail test to a 5% significance level
Tcalc = 0.015785 - 0          t table=1.67
0.00161
= .74761
= .7 (4d.p)
Reject H0 as tcalc  1.67
Significance test for overall regression.
Model  F =  R / (k-1)
(1-R)/(n-k)
H0 ß = ß = ß4 = 0          Ha ß ¹ ß ¹ ß4 ¹ 0               wo tail test
F =     0.0085/(4-1)                                  Ftable=8.6
(1-0.0085)/(5-4) 
F = 0.008
0.0155
= 0.104617
Reject H0 as Fcalc  8.6
The end result remains the same as I still reject all my hypotheses
(d) Using your model estimated in (b), look for possible multicollinearity in your results. If there is multicollinearity, indicate how you could proceed. (You do not need to do any new calculations here).
Perfect collinearity is when two variables (eg. Price and Income) have a perfect linear relationship between each other. Multicollinearity refers to more than one such relationship; multicollinearity is often used to refer to both cases however.
The first test to see if there are signs of multicollinearity are if there are high values of R²  but few significant t ratios. It is evident from the data obtained in part b) that there is a high value for R² however there is no evidence in the individual t tests show that none or very few partial slope coefficients are statistically different from zero. This does not indicate multicollinearity in the data set. 
The second test is to see if there is high pairwise correlations among explanatory variables, this is done by regressing individual explanatory variables against each other. In testing for any high correlations ( R² above 0.8) it emerged that there were high correlations between Consumption and Government Spending, Consumption and Investment, Consumption and Net Exports. This suggests possible collinearity between them. For instance consumption and government spending are directly related through taxation.
Another test is to do Auxiliary regressions, where each explanatory variable is regressed against all the other explanatory variables to compute the corresponding R², it can identify if there is a high R² but few of the individual coefficients are significant so you can identify the variable or variables which have perfect or near perfect linear combination of the other variables. These are my results
Regression of C on I, G, NX      R² = 0.80
Regression of I on G, NX, C      R² = 0.578
Regression of G on NX, C, I      R² = 0.844
These high values of R² suggest some form of collinearity exists.
How I could proceed
It is evident from the tests for collinearity that it does exist in some form in my model.
In order to eliminate or reduce the effects of collinearity, a larger or completely new sample can be taken as collinearity is defined as a sample specific problem.
It may even be that another key variable has been overlooked on my part and it would be a inducing the effects of collinearity in their abcence. Another solution is to drop a variable, this causes problems though as in this model the variables are based on economic theory it will have a detrimental effect on the whole model. However this does not mean that you cannot alter the variables, in this instance we could change government spending to something like government debt/surplus, as it would remove the direct link of taxation between consumption and government spending.
Even though collinearity is evident in my model it isn't necessarily a serious problem due to the high R² value as the model will still be accurate to forcast future GDP values.
(e) Test your model in (b) for possible structural breaks halfway through your sample period. The possible breaks to be investigated are (i) a possible parallel shift in your relationship at he halfway point (change in intercept only). (ii) a possible change in the overall regression (a change in both intercept and slope coefficients)
Ordinary Least Squares Estimation
Dependent variable is GDP
5 observations used for estimation from 155 to 18
Regressor              Coefficient       Standard Error         T-Ratio     [Prob]
INT                      -848.11           51.878            -.8175          [.81]
C                         1.105                 .06074              11.664          [.000]
I                           1.51                1.0788                1.54          [.]
G                          -.1761              1.5668               -.08788          [.1]
NX                       .4601                .6150                .74          [.460]
DINT                   67.7100            16.0                .6860          [.70]
DC                       -.01865             .01174             -1.656          [.10]
DI                        -1.058               1.0805                -.8000          [.6]
DG                       .74                1.555                .680          [.5]
DNX                    -.01400             .6186              -.0664          [.8]
R-Squared                     .5             R-Bar-Squared                   .
S.E. of Regression            1.1             F-stat.    F(  ,  5)               540.5[.000]
Mean of Dependent Variable  14868.5        S.D. of Dependent Variable      147467.1
Residual Sum of Squares     .80E+07        Equation Log-likelihood             -.880
Akaike Info. Criterion     -0.880             Schwarz Bayesian Criterion     -10.6577
DW-statistic                  1.765
In order to test for structural breaks, the data has to be split into two sections and 'Dummy variables' have to be introduced in order to do so, these will be known as the same corresponding variable but with a 'D' preceding it. The dummy variables will split the data into two by only holding their true values in one of the sections, the rest will have values of zero. The first of which will cover all of my data from 155 to 17 inclusive. This will have the mathematical formula of
GDP = .1 + .C + .I + .4G + .5NX
= (-848.11) + 1.105 + 1.51 + (-0.1761) + (+0.1761)
The second section covers 17 to 18 inclusive, it has the following mathematical formula
GDP     = (.1 + D.1) + (.C + DC) + (.I + DI) + (.4G + DG) + (.5NX + DNX).
= (-848.11 + 67.7100) + (1.105 - 0.01865) + (1.51 - 1.058)                                                                        + (-0.1761 + 0.74) + (.4601 - 0.01400)
Just by looking at the data obtained you can see that there has been a structural change in the data sample where the overall regression has changed, as both the intercept and slope coefficients are different which suggests that their impact on GDP have also changed.
f) For annual data calculate the (instantaneous) annual growth rate of the dependant variable.
The data for the annual growth rate of my dependant variable (GDP) is listed in the column labelled 'GDP GR'. There is no clear, strong correlation between the results as they change constantly although from 167 to 180 there was a general rise in the GDP growth rate, with a high in 175 with a growth rate of 0.780%. The growth rate was fairly consistent towards the end of my data selection from the period of 181 to to 18. The lowest rate of growth was in 158 with a growth rate of 4.047%.
These are the average growth rates of the first second and third sections of the data
1st     156-167      6.01%
nd     168-178     11.54%
rd     17-18       .6%
This suggests there was an increase in growth over the period sampled, with a slight slowdown in the last period.
g) Test for the presence of serially correlated errors in your model. If there is evidence of such correlation, indicate how you could proceed.
The presence of serially correlated errors can be detected through a number of tests. I shall be using a test known as the Durbin-Watson Test.
The Durbin Watson Test
The Durbin-Watson Test is so commonly used and is based on the Ordinary Least Squares residual, so is often seen alongside other summary statistics such as R², t and F ratios. The Durbin Watson Test uses the following formula
d =   
There are some assumptions for the d statistic to work, these are
·     The regression model includes an intercept term as does not work for models where the line goes through the origin.
·     The X variables are nonstochastic. This means that the values are fixed in repeated sampling.
·     The disturbances Ut are generated by the Makov first-order autoregressive scheme.
·     The regressiondoes not contain lagged values of the dependant variable as one of the explanatory variables. These models are known as autoregressive models.
From my model used in .b) with four explanatory variables the Durbin-Watson d Statistic has a value of 0.4717, by using the tables, the upper and lower limits can be found. For my model the upper limit is 1.76 and the lower limit is 1. at 5% significance level.
H0 No positive autocorrelation          H0 No negative autocorrelation
I have to reject the null hypothesis that there is no positive autocorrelation as 00.47171.. This says that there is a positive autocorrelation in my model.
By my model having positive autocorrelation occurring means that the stochastic shock term is increasing in my model. This is not wanted in my model as the consequences can be very serious. A suitable remedy for this would be to introduce either use generalized least squares (GLS) or the 'Prais-Winstein Transformation' method of transformation.
h) On the basis of your results, and on any other tests that you care to undertake, indicate whether you feel your model could be improved, and if so, how. Attempt to improve your model, including handling and serial correlation problems if appropriate. Explain why you think your results do or do not show a model improvement.
I feel that my model has proved itself to be quite accurate in explaining the GDP as the R² value obtained was 0. which means that .% of the variance is explained by the explanatory variables. My model could undergo improvement as shown in part g), my model has experienced autocorrelation meaning that the stochastic shock to my model is ever increasing. Also there was some evidence of multicollinearity, which could be eliminated with a larger sample size.
To eliminate the effects of autocorrelation the autocorrelation parameter must be found, to do this I shall use the Durbin-Watson d statistic to estimate it value
The relationship between d and p 
This formula can be rearranged with p as the subject to form
Previously the value for d has been calculated
This value could then be used in the generalised difference equation to eliminate the effects of autocorrelation which would improve my models accuracy.
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