Thursday, July 12, 2012

Links between GDP, Consumption, Investment and Government Expenditure

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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.

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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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