Calculating YearOnYear Growth of GDP
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Introduction
The model which is to be developed is real GDP in the UK. From such a series of real values, it is straightforward to calculate yearonyear growth of GDP.
Selection of variables
To model GDP, key factors identified by Easton (2004) include labour costs, savings ratio, taxation issues, inflation and terms of trade. However, many of these variables are not available for the required 40 year time span.
The variables eventually chosen and the justification were as follows:
GDP: the dependent variable, measured at 1950 prices. As GDP deflator figures were not available back to 1960, the eventual starting point of the analysis, the RPI inflation measure was used to convert the series into real prices.
Exim: this variable is the sum of imports and exports, at constant 1950 prices. As a measure of trade volumes, EXIM would be expected to increase as GDP also increases. The RPI deflator was also used for this series. Total trade was plasced into one variable was to abide by the constraint of no more than four independent variables.
Energy: energy consumption was calculated as production plus imports minus exports in tonnes of oil equivalent. As energy use increases, we would expect to see an increase in the proportion of GDP attributable to manufacturing.^{[1]}
Labour: this variable is the total number of days lost through disputes. We would expect this variable to have a negative coefficient, since an increase in the number of days lost will lead to a reduction of GDP.
Scatter diagrammes showing the relationship between the dependent variable GDP and each of the independent variables is sown in Appendix 1. These diagrammes support each of the hypotheses outlined above.
Main results
The regression equation produced by EViews, once the energy variable is excluded, is as follows:
GDP = 73223.22384 + 1.062678514*EXIM  0.1391051564*LABOUR + 1.565374397*POPN
The adjusted R^{2} is equal to 0.978; or, 97.8% of the variation in GDP is accounted for by the variation in EXIM, LABOUR and POPN.
Each of the coefficients of the three independent variables, EXIM, LABOUR and POPN, have tstatistics sufficiently high to reject the null hypothesis that any of the coefficients is equal to zero; in other words, each variable makes a significant contribution to the overall equation.
To test the overall fit of the equation, the F value of 703 allows us similarly to reject the hypothesis that the coefficients are simultaneously all equal to zero.
Dependent Variable: GDP 

Method: Least Squares 

Date: 04/15/08 Time: 09:10 

Sample: 1960 2006 

Included observations: 47 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
73223.22 
23204.60 
3.155548 
0.0029 
EXIM 
1.062679 
0.117445 
9.048297 
0.0000 
LABOUR 
0.139105 
0.036951 
3.764585 
0.0005 
POPN 
1.565374 
0.443541 
3.529270 
0.0010 
Rsquared 
0.980046 
Mean dependent var 
32813.25 

Adjusted Rsquared 
0.978654 
S.D. dependent var 
10905.60 

S.E. of regression 
1593.331 
Akaike info criterion 
17.66631 

Sum squared resid 
1.09E+08 
Schwarz criterion 
17.82377 

Log likelihood 
411.1582 
Fstatistic 
703.9962 

DurbinWatson stat 
0.746519 
Prob(Fstatistic) 
0.000000 
The Akaike and Schwartz criteria are used principally to compare two or more models (a model with a lower value of either of these statistics is preferred). As we are analysing only one model here, we will not discuss these two further.
Using tables provided by Gujarati (2004), the upper and lower limits for the DW test are:
D_{L} = 1.383 D_{U} = 1.666
The DW statistic calculated by EViews is 0.746, which is below DL. This results leads us to infer that there is no positive autocorrelation in the model. This is an unlikely result, given that we are dealing with increasing variables over time, but we shall examine the issue of autocorrelation in detail later on.
Multicollinearity
Ideally, there should be little or no significant correlation between the dependent variables; if two dependent variables are perfectly correlated, then one variable is redundant and the OLS equations could not be solved.
The correlation of variables table below shows that EXIM and POPN have a particularly high level of correlation (the removal of the ENERGY variable early on solved two other cases of multicollinearity).
It is important, however, to point out that multicollinearity does not violate any assumptions of the OLS process and Gujarati points out the multicollinearity is a consequence of the data being observed (indeed, section 10.4 of his book is entitled “Multicollinearity; much ado about nothing?”).
Correlations of Variables
GDP 
EXIM 
POPN 
ENERGY 

GDP 
1.000000 

EXIM 
0.984644 

POPN 
0.960960 
0.957558 

ENERGY 
0.835053 
0.836279 
0.914026 

LABOUR 
0.380830 
0.320518 
0.259193 
0.166407 
Analysis of Residuals
Overview
The following graph shows the relationship between actual, fitted and residual values. At first glance, the residuals appear to be reasonably well behaved; the values are not increasing over time and there several points at which the residual switches from positive to negative. A more detailed tabular version of this graph may be found at Appendix 2.
Heteroscedascicity
To examine the issue of heteroscedascicity more closely, we will employ White’s test. As we are using a model with only three independent variables, we may use the version of the test which uses the crossterms between the independent variables.
White Heteroskedasticity Test: 

Fstatistic 
1.174056 
Probability 
0.339611 

Obs*Rsquared 
10.44066 
Probability 
0.316002 

Test Equation: 

Dependent Variable: RESID^2 

Method: Least Squares 

Date: 04/16/08 Time: 08:24 

Sample: 1960 2006 

Included observations: 47 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
2.99E+09 
4.06E+09 
0.735744 
0.4665 
EXIM 
49439.98 
45383.77 
1.089376 
0.2830 
EXIM^2 
0.175428 
0.128496 
1.365249 
0.1804 
EXIM*LABOUR 
0.049223 
0.047215 
1.042532 
0.3039 
EXIM*POPN 
0.982165 
0.879151 
1.117174 
0.2711 
LABOUR 
18039.83 
18496.29 
0.975322 
0.3357 
LABOUR^2 
0.018423 
0.009986 
1.844849 
0.0731 
LABOUR*POPN 
0.344698 
0.336446 
1.024526 
0.3122 
POPN 
120773.0 
157305.5 
0.767761 
0.4475 
POPN^2 
1.217523 
1.523271 
0.799282 
0.4292 
Rsquared 
0.222142 
Mean dependent var 
2322644. 

Adjusted Rsquared 
0.032933 
S.D. dependent var 
3306810. 

S.E. of regression 
3251902. 
Akaike info criterion 
33.01368 

Sum squared resid 
3.91E+14 
Schwarz criterion 
33.40733 

Log likelihood 
765.8215 
Fstatistic 
1.174056 

DurbinWatson stat 
1.306019 
Prob(Fstatistic) 
0.339611 
The 5% critical value for chisquared with nine degrees of freedom is 16.919, whilst the computed value of White’s statistic is 10.44. We may therefore conclude that, on the basis of the White test, there is no evidence of heteroscedascicity.
Autocorrelation
The existence of autocorrelation exists in the model if there exists correlation between residuals. In the context of a time series, we are particularly interested to see if successive residual values are related to prior values.
To determine autocorrelation, Gujarati’s rule of thumb of using between a third and a quarter of the length of the time series was used. In this particular case, a lag of 15 was selected.
Date: 04/16/08 Time: 08:05 

Sample: 1960 2006 

Included observations: 47 

Autocorrelation 
Partial Correlation 
AC 
PAC 
QStat 
Prob 

. ****  
. ****  
1 
0.494 
0.494 
12.234 
0.000 
. ***  
. **  
2 
0.423 
0.237 
21.409 
0.000 
. *.  
.* .  
3 
0.155 
0.171 
22.669 
0.000 
.  .  
.* .  
4 
0.007 
0.145 
22.672 
0.000 
.* .  
.* .  
5 
0.109 
0.069 
23.319 
0.000 
** .  
.* .  
6 
0.244 
0.160 
26.674 
0.000 
** .  
.  .  
7 
0.194 
0.037 
28.845 
0.000 
** .  
.  .  
8 
0.202 
0.004 
31.247 
0.000 
** .  
.* .  
9 
0.226 
0.162 
34.344 
0.000 
** .  
.* .  
10 
0.269 
0.186 
38.859 
0.000 
.* .  
. *.  
11 
0.134 
0.122 
40.013 
0.000 
.* .  
.  .  
12 
0.079 
0.047 
40.428 
0.000 
.* .  
.* .  
13 
0.078 
0.151 
40.837 
0.000 
.  .  
.  .  
14 
0.013 
0.029 
40.849 
0.000 
.  .  
.  .  
15 
0.041 
0.018 
40.970 
0.000 
The results of the Q statistic indicate that the data is nonstationary; in other words, the mean and standard deviation of the data do indeed vary over time. This is not a surprising result, given growth in the UK’s economy and population since 1960.
A further test available to test for autocorrelation is the BreuschGodfrey test. The results of this test on the model are detailed below.
BreuschGodfrey Serial Correlation LM Test: 

Fstatistic 
15.53618 
Probability 
0.000010 

Obs*Rsquared 
20.26299 
Probability 
0.000040 

Test Equation: 

Dependent Variable: RESID 

Method: Least Squares 

Date: 04/16/08 Time: 09:23 

Presample missing value lagged residuals set to zero. 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
9294.879 
18204.51 
0.510581 
0.6124 
EXIM 
0.047292 
0.092176 
0.513065 
0.6107 
LABOUR 
0.039181 
0.031072 
1.260967 
0.2144 
POPN 
0.182287 
0.348222 
0.523479 
0.6035 
RESID(1) 
0.788084 
0.154144 
5.112655 
0.0000 
RESID(2) 
0.180226 
0.160485 
1.123009 
0.2680 
Rsquared 
0.431127 
Mean dependent var 
0.000100 

Adjusted Rsquared 
0.361753 
S.D. dependent var 
1540.499 

S.E. of regression 
1230.710 
Akaike info criterion 
17.18731 

Sum squared resid 
62100572 
Schwarz criterion 
17.42350 

Log likelihood 
397.9019 
Fstatistic 
6.214475 

DurbinWatson stat 
1.734584 
Prob(Fstatistic) 
0.000225 
We can observe from the results above that RESID(1) has a high t value. In other words, we would reject the hypothesis of no first order autocorrelation. By contrast, second order autocorrelation does not appear to be present in the model.
Overcoming serial correlation
A method to overcome the problem of nonstationarity is to undertake a difference of the dependent variable (ie GDP_{year1} – GDP_{year0}) An initial attempt to improve the equation by using this differencing method produced a very poor result, as can be seen below.
Dependent Variable: GDPDIFF 

Method: Least Squares 

Date: 04/16/08 Time: 08:17 

Sample: 1961 2006 

Included observations: 46 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
14037.58 
12694.29 
1.105818 
0.2753 
EXIM 
0.084287 
0.052601 
1.602398 
0.1167 
ENERGY 
0.011470 
0.011710 
0.979487 
0.3331 
LABOUR 
0.004251 
0.014304 
0.297230 
0.7678 
POPN 
0.300942 
0.265082 
1.135279 
0.2629 
Rsquared 
0.207408 
Mean dependent var 
816.6959 

Adjusted Rsquared 
0.130082 
S.D. dependent var 
657.1886 

S.E. of regression 
612.9557 
Akaike info criterion 
15.77678 

Sum squared resid 
15404304 
Schwarz criterion 
15.97555 

Log likelihood 
357.8660 
Fstatistic 
2.682255 

DurbinWatson stat 
1.401626 
Prob(Fstatistic) 
0.044754 
Forecasting
The forecasts for the dependent variables are based on Kirby (2008) and are presented below.
The calculation of EXIM for future years was based upon growth rates for exports (47% of the 2006 total) and imports (53%) separately. The two streams were added together to produce the 1950 level GDP figure, from which yearonyear increases in GDP could be calculated. The results of the forecast are shown below.
The 2008 figure was felt to be particularly unrealistic, so a sensitivity test was applied to EXIM (population growth is relatively certain in the short term and calculating a forecast of labour days lost is a particularly difficult challenge).
Instead of EXIM growing by an average of 1.7% per annum during the forecast period, its growth was constrained to 0.7%. As we can see from the “GDP2” column, GDP forecast growth is significantly lower in 2008 and 2009 as a result.
Critical evaluation of the econometric approach to model building and forecasting
GDP is dependent on many factors, many of which were excluded from this analysis due to the unavailability of data covering forty years. Although the main regression results appear highly significant, there are many activities which should be trialled to try to improve the approach:
a shorter time series with more available variables: using a short time series would enable a more intuitive set of variables to be trialled. For example, labour days lost is effectively a surrogate for productivity and cost per labour hour, but this is unavailable over 40 years;
transformation of variables: a logarithmic or other transformation should be trialled to ascertain if some of the problems observed, such as autocorrelation, could be mitigated to any extent. The other, more relevant transformation is to undertake differencing of the data to remove autocorrelation; the one attempt made in this paper was particularly unsuccessful!
Approximate word count, excluding all tables, charts and appendices: 1,400 Appendix 1 – Scatter diagrammes of GDP against dependent variables
Appendix 2
obs 
Actual 
Fitted 
Residual 
Residual Plot 
1960 
17460.5 
15933.8 
1526.78 
 .  *  
1961 
17816.1 
16494.5 
1321.57 
 .  *.  
1962 
17883.8 
16714.1 
1169.67 
 .  * .  
1963 
18556.7 
18153.6 
403.108 
 . * .  
1964 
19618.0 
19117.8 
500.191 
 .  * .  
1965 
20209.7 
19558.9 
650.773 
 .  * .  
1966 
20699.1 
20272.1 
426.905 
 . * .  
1967 
21303.1 
20973.3 
329.754 
 . * .  
1968 
22037.1 
22395.3 
358.204 
 . * .  
1969 
22518.6 
22824.6 
305.982 
 . * .  
1970 
23272.7 
23147.8 
124.912 
 . * .  
1971 
23729.9 
23395.8 
334.070 
 . * .  
1972 
24806.3 
22418.6 
2387.67 
 .  . *  
1973 
26134.9 
27249.5 
1114.60 
 . *  .  
1974 
25506.2 
28880.9 
3374.64 
 * .  .  
1975 
25944.6 
28401.8 
2457.14 
 * .  .  
1976 
26343.7 
30306.2 
3962.47 
* .  .  
1977 
26468.8 
29829.1 
3360.31 
 * .  .  
1978 
28174.4 
29922.0 
1747.61 
 *  .  
1979 
29232.7 
27846.9 
1385.71 
 .  *.  
1980 
28957.2 
29271.0 
313.855 
 . * .  
1981 
28384.0 
29590.8 
1206.86 
 .*  .  
1982 
28626.2 
29526.2 
899.933 
 . *  .  
1983 
29915.3 
30883.9 
968.627 
 . *  .  
1984 
30531.7 
29677.7 
853.960 
 .  * .  
1985 
31494.3 
33289.4 
1795.09 
 *  .  
1986 
32748.5 
33293.0 
544.520 
 . *  .  
1987 
34609.2 
34223.2 
385.976 
 . * .  
1988 
36842.2 
34669.4 
2172.76 
 .  . *  
1989 
37539.8 
35938.6 
1601.20 
 .  *  
1990 
37187.7 
35988.5 
1199.22 
 .  *.  
1991 
36922.2 
35080.4 
1841.84 
 .  .*  
1992 
37116.4 
35793.7 
1322.74 
 .  *.  
1993 
38357.7 
38051.2 
306.418 
 . * .  
1994 
39696.7 
39790.8 
94.0617 
 . * .  
1995 
40541.7 
41908.5 
1366.77 
 .*  .  
1996 
42100.6 
43803.1 
1702.49 
 *  .  
1997 
43329.5 
44112.0 
782.512 
 . *  .  
1998 
44464.0 
43725.4 
738.594 
 .  * .  
1999 
46146.5 
44906.9 
1239.65 
 .  *.  
2000 
47110.8 
47458.2 
347.410 
 . * .  
2001 
48418.9 
48530.2 
111.276 
 . * .  
2002 
50100.6 
48901.4 
1199.23 
 .  *.  
2003 
51568.7 
49351.6 
2217.02 
 .  . *  
2004 
53023.9 
50419.2 
2604.71 
 .  . *  
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